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Development of Time-History and Response Spectrum Analysis Procedures for Determining Bridge Response to Barge Impact Loading

Permanent Link: http://ufdc.ufl.edu/UFE0021594/00001

Material Information

Title: Development of Time-History and Response Spectrum Analysis Procedures for Determining Bridge Response to Barge Impact Loading
Physical Description: 1 online resource (240 p.)
Language: english
Creator: Cowan, David R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: analysis, barge, bridge, dynamic, impact, response, spectrum
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Bridge structures that span navigable waterways are inherently at risk for barge collision incidents, and as such, must be designed for impact loading. Current design procedures for barge impact loading use an equivalent static load determination technique. However, barge impacts are fundamentally dynamic in nature, and a static analysis procedure may not be adequate in designing bridge structures to resist a barge collision. Therefore, dynamic analysis methods for estimating the response of a bridge structure to barge collision are developed in this study. As part of this research, static barge bow crush analyses were conducted?using a general purpose finite element code?to define the barge bow force-deformation relationship for various bridge pier column shapes and sizes. Using the generated force-deformation relationships, in conjunction with barge mass and speed, relationships for the peak dynamic load and duration of loading are developed based upon principles of conservation of energy and linear momentum. The resulting relationships are then used to develop applied vessel impact load and impact response spectrum analysis procedures. Additionally, both of these methods are validated against the previously validated coupled vessel impact analysis technique.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by David R Cowan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Consolazio, Gary R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021594:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021594/00001

Material Information

Title: Development of Time-History and Response Spectrum Analysis Procedures for Determining Bridge Response to Barge Impact Loading
Physical Description: 1 online resource (240 p.)
Language: english
Creator: Cowan, David R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: analysis, barge, bridge, dynamic, impact, response, spectrum
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Bridge structures that span navigable waterways are inherently at risk for barge collision incidents, and as such, must be designed for impact loading. Current design procedures for barge impact loading use an equivalent static load determination technique. However, barge impacts are fundamentally dynamic in nature, and a static analysis procedure may not be adequate in designing bridge structures to resist a barge collision. Therefore, dynamic analysis methods for estimating the response of a bridge structure to barge collision are developed in this study. As part of this research, static barge bow crush analyses were conducted?using a general purpose finite element code?to define the barge bow force-deformation relationship for various bridge pier column shapes and sizes. Using the generated force-deformation relationships, in conjunction with barge mass and speed, relationships for the peak dynamic load and duration of loading are developed based upon principles of conservation of energy and linear momentum. The resulting relationships are then used to develop applied vessel impact load and impact response spectrum analysis procedures. Additionally, both of these methods are validated against the previously validated coupled vessel impact analysis technique.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by David R Cowan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Consolazio, Gary R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021594:00001


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DEVELOPMENT OF TIME-HISTORY AND RESPONSE SPECTRUM ANALYSIS
PROCEDURES FOR DETERMINING BRIDGE RESPONSE TO BARGE IMPACT
LOADING




















By

DAVID RONALD COWAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































O 2007 David Ronald Cowan
































To Zoey Elizabeth.









ACKNOWLEDGMENTS

Completion of this dissertation and the accompanying research would not have been

feasible without the support and guidance of a number of individuals. First, the author wishes

thank Dr. Gary Consolazio for his continual support in this endeavor. He has offered invaluable

knowledge and insight throughout the course of this research.

The author also wishes to thank his supervisory committee: Dr. Ronald Cook, Dr. Kurtis

Gurley, Dr. Trey Hamilton, Dr. Nam-Ho Kim, and Dr. Michael McVay, who have each

contributed valuable insight into multiple aspects of this research. Furthermore, the author

wishes to thank Mr. Henry Bollmann and Mr. Lex Collins for their continual leadership and

support.

Others deserving of thanks for their support and contributions include Alex Biggs, Long

Bui, Michael Davidson, Daniel Getter, Jessica Hendrix, Ben Lehr, Cory Salzano, and Bibo

Zhang. The author wishes to thank his friends and family for their support and encouragement.












TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ............ ...... .__ ...............8....


LIST OF FIGURES .............. ...............9.....


AB S TRAC T ............._. .......... ..............._ 15...


CHAPTER


1 INTRODUCTION .............. ...............16....


1 .1 Introducti on ................. ...............16......._ ....
1.2 Motivation............... ...............1
1.3 Objectives .............. ...............19....
1.4 S cope of W ork ................. ...............20......... ....

2 BACKGROUND .............. ...............22....


2.1 Vessel-Bridge Collision Incidents ................ ...............22...
2.2 Review of Experimental Vessel Impact Tests............_..._ ......_._ .... ........... .....2
2.3 Design of Bridges According to the AASHTO Barge Impact Provisions.............._...__.....28
2.3.1 Selection of Design Vessel ..........._..._ ...............29....._.._ ...
2.3.2 Method II: Probability Based Analysis .............. ...............30....
2.3.3 Barge Impact Force Determination .............. ...............32....

3 SUMMARY OF FINDINGS FROM ST. GEORGE ISLAND BARGE IMPACT
TES TING ................. ...............51........... ....


3 .1 Introducti on ................ .... .. ...... ...............51.....
3.2 Overview of Experimental Test Program ................ ...............51........... ..
3.3 Overview of Analytical Research ................. ...............53........... ..
3.3.1 FB-MultiPier Models ................... ...............54........... ..
3.3.1.1 FB-MultiPier Pier-1 model ................. ...............54...........
3.3.1.2 FB-MultiPier Pier-3 model ................. ...............54........... .
3.3.1.3 FB-MultiPier Bridge model .............. ...............54....
3.3.2 Finite Element Simulation of Models ................. ...............55........... .
3.3.2. 1 Impact test PIT7 ................. ...............56......___
3.3.2.2 Impact test P3 T3 ................. ...............57..............
3.3.2.3 Impact test B3T4 .............. ... ...............58...
3.4 Comparison of Dynamic and Static Pier Response .....__ ................ ................. .59
3.5 Observations .............. ...............62....


4 BARGE FORCE-DEFORMATION RELATIONSHIPS ................. .............................81











4. 1 Introducti on ........._.._.... ........._... ...... .. ...... .._ .. .. ..........8
4.2 Review of the Current AASHTO Load Determination Procedure ........._...... ................81
4.3 High-Fidelity Finite Element Barge Models .............. ...............83....
4.3.1 Jumbo Hopper Barge Finite Element Model ....._____ .........__ .............. .84
4.3.2 Tanker Barge Finite Element Model .............. ...............88....
4.4 High-Fidelity Finite Element Barge Crush Analyses ................ ................ ...._...89
4.4.1 Finite Element Barge Bow Crush Simulations............... ..... ...........8
4.4.2 Development of Barge Bow Force-Deformation Relationships ............................91
4.4.3 Summary of Barge Bow Force-Deformation Relationships ................. ...............96

5 COUPLED VESSEL IMPACT ANALYSIS AND SIMPLIFIED ONE-PIER TWO-SPAN
STRUCTURAL MODELING................. ..............12


5.1 Coupled Vessel Impact Analysis ................. ...............129.____ ..
5.1.1 Nonlinear Barge Bow Behavior ................... ...............129.
5.1.2 Time-Integration of Barge Equation of Motion ............__.....___ ..............13 1
5.1.3 Coupling Between Barge and Pier ................ .. ...............132..
5.2 One-Pier Two-Span Simplified Bridge Modeling Technique ............ .. ......._.......134
5.2. 1 Effective Linearly Independent Stiffness Approximation .........._.... ................13 4
5.2.2 Effective Lumped Mass Approximation .............. .. ..... ............ .........3
5.3 Coupled Vessel Impact Analysis of One-Pier Two-Span Bridge Models...................... 136

6 APPLIED VES SEL IMPACT LOAD HISTORY METHOD ........._._ ...... .. ..............15 1

6. 1 Introducti on ........._..... ......... ._._ .... ...............151
6.2 Development of Load Prediction Equations ........._.. ........ .. .. .. ..._. ..........5
6.2. 1 Prediction of Peak Impact Load from Conservation of Energy .........._..............15 1
6.2.2 Prediction of Load Duration from Conservation of Linear Momentum ..............156
6.2.3 Summary of Procedure for Constructing an Impact Load History ................... .... 160
6.3 Validation of the Applied Vessel Impact Load History Method ................. .................161

7 IMPACT RESPONSE SPECTRUM ANALYSIS .............. ...............177....

7. 1 Introducti on .................. .. ......... ...............177.....
7.2 Response Spectrum Analysis............... ...............17
7.2.1 M odal Analysis............... ...... ............17
7.2.2 General Response Spectrum Analysis............... ...............18
7.2.2.1 Modal Combination............... ..............18
7.2.2.2 Mass Participation Factors ................. ...............184...............
7.3 Dynamic Magnification Factor (DMF) .............. ...............186....
7.4 Impact Response Spectrum Analysis............... ........... .........9
7.5 Impact Re sponse Spectrum Analy si s for Nonlinear Sy stems............. ..._.........__ ...1 93
7.5.1 Load Determination and DMF Spectrum Construction ................... ...............19
7.5.2 Structural Linearization Procedure ............... ... .......... ......_. ..........19
7.6 Validation and Demonstration of Impact Response Spectrum Analysis ........._.............198
7.6. 1 Event-Specific Impact Response Spectrum Analysis (IRSA)Validation .............199
7.6.2 Design-Oriented Impact Response Spectrum Analysis Demonstration...............201












8 CONCLUSIONS AND RECOM1VENDATIONS .............. ...............232....


8.1 Concluding Remarks .............. ...............232....
8.2 Recommendations............... ....... .. ........23
8.2.1 Recommendations for Bridge Design .............. ...............235....
8.2.2 Recommendations for Future Research............... ...............23


LIST OF REFERENCES .........__.. ..... .___ ...............237...


BIOGRAPHICAL SKETCH .............. ...............240....











LIST OF TABLES

Table page

3-1 Summary of forces acting on the pier during test PIT7 ....._ ................ ................6

3-2 Dynamic and static analysis cases ................. ...............63...... ....

4-1 Barge material properties............... ...............9

6-1 Impact energies for AVIL validation............... ..............16

6-2 Maximum moments in all pier columns and piles ................. .....___.............. ....16

7-1 Impact energies for IRSA validation ........._. ........_. ...............204.

7-2 Maximum moments for all columns and piles for event-specific IRSA validation with
SRS S combination ........._ .......... ...............204....

7-3 Maximum moments for all columns and piles for event-specific IRSA validation with
CQC combination ................. ...............204............

7-4 Maximum moments for all columns and piles for design IRSA demonstration with SRSS
combination .............. ...............205....

7-5 Maximum moments for all columns and piles for design IRSA demonstration with CQC
combination .............. ...............205....

7-6 Mass participation by mode for design IRSA .............. ...............205....











LIST OF FIGURES


Figure page

2-1 Collapse of the Sunshine Skyway Bridge in Florida (1980) after being struck by the
cargo ship Summit Venture....... ...............35~~~~~~~~~~~~

2-2 Failure of the Big Bayou Canot railroad bridge in Alabama (1993) after being struck by
a bar ge flotilla ................. ...............36........... ....

2-3 Collapse of the Queen Isabella Causeway Bridge in Texas (2001) after being struck by a
barge flotilla............... ...............37

2-4 Collapse of an Interstate I-40 bridge in Oklahoma (2002) after being struck by a barge
fl otill a............... ...............3 8

2-5 Reduced scale ship-to-ship collision tests conducted by Woisin (1976) ................. ...............39

2-6 Instrumented full-scale barge-lock-gate collision tests .............. ...............40....

2-7 Instrumented 4-barge lock-wall collision tests ................. ...............41...............

2-8 Instrumented 15-barge lock-wall collision tests ................. ...............42........... .

2-9 Barge tow configuration ................ ...............43................

2-10 Design impact speed ................. ...............44...............

2-11 Bridge location correction factor ................. ...............45........... ..

2-12 Geometric probability of collision ................. ...............46..............

2-13 Probability of collapse distribution............... ..............4

2-14 AASHTO relationship between kinetic energy and barge crush depth ............... ... ........._...48

2-15 AASHTO relationship between barge crush depth and impact force .............. ..................49

2-16 Relationship between kinetic energy and impact load .............. ...............50....

3 -1 Overview of the layout of the bridge ................. ...............64.............

3-2 Schematic of Pier-1 .............. ...............65....

3-3 Schematic of Pier-3 .............. ...............66....

3-4 Test barge with payload impacting Pier-1 in the series Pl tests .............. .....................6

3-5 Series B3 tests ................. ...............68........... ...











3-6 Pier-3 in isolation for the series P3 tests............... ...............69.

3-7 Pier-1 FB-MultiPier model ................. ...............70................

3-8 Pier-3 FB-MultiPier model ................. ...............71........... ...


3-9 Bridge FB-MultiPier model ................. ...............72........... ...

3-10 Schematic of forces acting on Pier-1 ................ ...............73.............

3-11 Resistance forces mobilized during tests PIT7 .............. ...............74....

3-12 Schematic of forces acting on Pier-3 ................. ....__....._ ......__............75

3-13 Resistance forces mobilized during tests P3T3 .............. ...............76....

3 -14 Schematic of forces acting on Pier-3 during test B3 T4 ..........._._ ......_.._ .........._.....7

3-15 Resistance forces mobilized during tests B3T4............... ...............78..

3-16 Comparison dynamic and static analysis results for foundation of pier ..........._.... ..............79

3-17 Comparison of dynamic and static analysis results for pier structure ................. ...............80

4-1 Force-deformation results obtained by Meier-Doirnberg ....._____ .........__ ................99

4-2 Relationships developed from experimental barge impact tests conducted by Meier-
Doirnberg (1976) ................. ...............100..___ .....

4-3 AASHTO barge force-deformation relationship for hopper and tanker barges ................... .101

4-4 Hopper barge dimensions ................ ...............102...............

4-5 Tanker barge dimensions ................. ...............103...............

4-6 Hopper barge schematic .............. ...............104....

4-7 Hopper barge bow model with cut-section showing internal structure ................ ...............105

4-8 Internal rake truss model .............. ...............106....


4-9 Use of spot weld constraints to connect structural components ................. .....................107

4-10 A36 stress-strain curve .............. ...............108....


4-11 Barge bow model with a six-foot square impactor............... ...............10

4-12 Tanker barge bow model ................. ...............110..............

4-13 Crush analysis models ................ ...............111..............











4-14 Hopper barge bow force-deformation data for flat piers subj ected to centerline
crushing ................. ...............112._ ._ ......

4-15 Hopper barge bow force-deformation data for flat piers subj ected to corner-zone
crushing ................. ...............113._ ._ ......

4-16 Tanker barge bow force-deformation data for flat piers subj ected to centerline
crushing ................. ...............114._ ._ ......

4-17 Relationship of pier width to engaged trusses .....___.....__.___ .......____ ........15

4-18 Hopper barge bow force-deformation data for round piers subj ected to centerline
crushing ................. ...............116._ ._ ......

4-19 Hopper barge bow force-deformation data for flat piers subj ected to corner-zone
crushing ................. ...............117._ ._ ......

4-20 Gradual increase in number of trusses engaged with deformation in round pier
simulations ........._.__ ..... .._ ._ ...............118...

4-21 Elastic-perfectly plastic barge bow force-deformation curve .........____....... ._.............119

4-22 Peak barge contact force versus pier width ................. ....._._ ...... ... .........2

4-23 Peak barge contact force versus pier width ................. ....._._ ...... ... .........2

4-24 Comparison of truss-yield controlled peak force versus plate-yield controlled peak
force ................. ...............122....... ......

4-25 Design curve for peak impact force versus flat pier width .....__.___ .... ... .___ ..............123

4-26 Comparison of low diameter peak force versus large diameter peak force. ................... .....124

4-27 Design curve for peak impact force versus round pier diameter ................. ................ ...125

4-28 Initial barge bow stiffness as a function of pier width ................. ................ ......... 126

4-29 Barge bow deformation at yield versus pier width ................. ...............127........... .

4-30 Barge bow force-deformation flowchart .............. ...............128....

5-1 Barge and pier modeled as separate but coupled modules ................ .........................137

5-2 Permanent plastic deformation of a barge bow after an impact ................ .....................138

5-3 Stages of barge crush ................. ...............139.......... ....

5-4 Unloading curves ................. ...............140................











5-5 Generation of intermediate unloading curves by interpolation ................. ............. .......141

5-6 Flow-chart for nonlinear dynamic pier/soil control module............... ...............142

5-7 Flow-chart for nonlinear dynamic barge module .............. ...............143....

5-8 Treatment of oblique collision conditions ................. .....___ ........_...........4

5-9 OPTS model with linearly independent springs .......__................. .. ......._.........14

5-10 Full bridge model with impact pier ................. ...............146...._._._.

5-11 Peripheral models with applied loads ............... ...............147........... ..

5-12 Displacements of peripheral models ................. ........_._ ......_ ......... .........148

5-13 OPTS model with lumped mass .............. ...............149....

5-14 Tributary area of peripheral models for lumped mass calculation ........._.._.. ......._.._.....150

6-1 Barge bow force-deformation relationship ...._.._.._ ......_. ............ ..........16

6-2 Inelastic barge bow deformation energy .............. ...............165....

6-3 Two degree-of-freedom barge-pier-soil model .............. ...............166....

6-4 Peak impact force vs. initial barge kinetic energy using a rigid pier assumption .................167

6-5 Peak impact force vs. initial barge kinetic energy using an effective barge-pier-soil
stiffness ................ ...............168...............

6-6 Impact load histories............... ...............16

6-7 Construction of loading portion of impact force ................. .....___............ .......7

6-8 Construction of unloading portion of impact force ..........._ ........... .......__ .......17

6-9 AVIL procedure............... ...............17

6-10 AASHTO load curve indicating barge masses and velocities used in validating the
applied load history method............... ...............173

6-11 Barge bow force-deformation relationship for an impact on a six-foot round column......174

6-12 Impact load history comparisons .........._._....... ___ ...............175.

6-13 Moment results profile for the new St. George Island Causeway Bridge channel
pier ........... __..... ...............176....











7- 1 Time hi story analysis s of a structure ................. ...............206..__.__ .

7-2 Time-history versus modal analysis .............. ...............207....

7-3 Modal analysis............... ...............20

7-4 Dynamic magnification of single degree-of-freedom system .............. ......................0

7-5 Dynamic magnification factor for a specific impact load history .............. .....................21

7-6 Dynamic magnification factor ........._... ...... ..... ...............211...

7-7 Specific dynamic magnification factor for a low-energy impact vs. a broad-banded
design spectrum .............. ...............212....

7-8 Evolution of the dynamic magnification spectrum from short to long duration loading ......213

7-9 Definition of the short and long-period transition points ........._._._........ ...............214

7-10 Period of impact loading.. ......___..........._. .......___ ....___ ...... .........215

7-11 Short-period transition point data ........._..._. ....._... ...............216..

7-12 Long-period transition point data .............. ...............217....

7-13 Evolving design DMF spectrum ........._.__ ......._._ ...............218.

7-14 Event-specific and design DMF spectra for varying impact energies............... ...............21

7-15 Impact response spectrum analysis procedure............... ...............22

7-16 Static analysis stage of IRSA ................. ...............221.............

7-17 Transformation of static displacements into modal coordinates .............. ....................22

7-18 Dynamic magnification factor as a function of structural period .........._._.... ......_.._......223

7-19 Combination of amplified dynamic modal displacements into amplified dynamic
structural displacements............... .............22

7-20 Nonlinear impact response spectrum analysis procedure ......____ ........_ ..............225

7-21 Nonlinear impact response spectrum analysis procedure ................. ............... ...._...226

7-22 Barge bow force-deformation relationship for an impact on a six-foot round column.......227

7-23 Event-specific IRSA validation ................. ...............228...............











7-24 Moment results profile for the new St. George Island Causeway Bridge channel
pier ................. ...............229................

7-25 Design-oriented IRSA demonstration .............. ...............230....

7-26 Moment results profile for the new St. George Island Causeway Bridge channel
pier ................ ...............23. 1...............









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DEVELOPMENT OF TIME-HISTORY AND RESPONSE SPECTRUM ANALYSIS
PROCEDURES FOR DETERMINING BRIDGE RESPONSE TO BARGE IMPACT
LOADING

By

David Ronald Cowan

December 2007

Chair: Gary Consolazio
Major: Civil Engineering

Bridge structures that span navigable waterways are inherently at risk for barge collision

incidents, and as such, must be designed for impact loading. Current design procedures for barge

impact loading use an equivalent static load determination technique. However, barge impacts

are fundamentally dynamic in nature, and a static analysis procedure may not be adequate in

designing bridge structures to resist a barge collision. Therefore, dynamic analysis methods for

estimating the response of a bridge structure to barge collision are developed in this study. As

part of this research, static barge bow crush analyses were conducted--using a general purpose

finite element code--to define the barge bow force-deformation relationship for various bridge

pier column shapes and sizes. Using the generated force-deformation relationships, in

conjunction with barge mass and speed, relationships for the peak dynamic load and duration of

loading are developed based upon principles of conservation of energy and linear momentum.

The resulting relationships are then used to develop applied vessel impact load and impact

response spectrum analysis procedures. Additionally, both of these methods are validated against

the previously validated coupled vessel impact analysis technique.









CHAPTER 1
INTTRODUCTION

1.1 Introduction

Bridge structures that span navigable waterways are inherently susceptible to vessel

collision. In the United States, vessels operate on a vast network of coastal waterways and inland

rivers, over which bridges span, to transport cargo between destinations. Many of these

waterways have relatively shallow channels, and therefore are unable to accommodate large-

draft vessels. For this reason, large-draft cargo ships are restricted to operation in deep water

routes and ports. Barges, however, have a shallow draft, and are thus capable of operating on

shallow coastal and inland waterways. Thus, heavily-loaded barge traffic routinely passes

beneath highway bridge structures.

Vessel collisions on bridge structures may occur when vessels veer off-course, becoming

aberrant. Factors that affect vessel aberrancy include adverse weather conditions, mechanical

failures, and human error. It has been noted in the literature that, on average, at least one serious

vessel collision occurs per year (Larsen 1993). During severe vessel collisions, significant lateral

loads may be imparted to bridge structures. Engineers must therefore account for lateral vessel

impact loads when designing bridge structures over navigable waterways. If such bridges cannot

adequately resist impact loading, vessel collisions may result in failure and collapse of the

bridge; leading to expensive repairs, extensive traffic delays, and potentially, human casualties.

1.2 Motivation

In 1980, the Sunshine Skyway Bridge over Tampa Bay was struck by a cargo ship, the

Summit Venture, resulting in the collapse of a portion of the superstructure and thirty-five

human casualties. Later, in 1988, prompted by the collapse of the Sunshine Skyway Bridge, a

formal investigation was initiated to develop bridge design specifications for vessel collision. As









a result, the American Association of State Highway and Transportation Officials (AASHTO)

published the Guide Specifieation and Commentary for Vessel Collision Design of Highway

Bridges in 1991 (AASHTO 1991). Additionally, similar provisions were incorporated in the

AASHTO Load and Resistance Factor Design (LRFD) Specifieations (AASHTO 1994) a few

years later. Publication of these design specifications represented a maj or step in improving the

safety of bridge structures.

At the time of the inception of the AASHTO vessel collision specifications, relatively few

experimental studies had been conducted involving barge collisions. Thus, very little data was

available for the development of the AASHTO specifications. The data upon which the

AASHTO specifications were based came from a single experimental study conducted by Meier-

Diirnberg during the 1980's in Germany (Meier-Dirnberg 1983). In these experiments, Meier-

Diirnberg conducted pendulum drop-hammer impacts on reduced-scale European hopper barge

bow models. Additionally, static load tests were also conducted on a similarly scaled barge bow

model .

One of the maj or findings of this study was that no significant differences were observed

between static and dynamic tests. Thus, relationships between barge kinetic energy, barge bow

deformation, and static impact force were developed and recommended. With few modifications,

the relationships developed by Meier-Diirnberg were ultimately adopted by the AASHTO

Specifications. Importantly, this implies that static loads are assumed to be sufficient in the

prediction of structural capacity during a vessel collision.

Using reduced-scale models to describe a full-scale response may introduce an uncertainty

in the accuracy of experimental results. This uncertainty is even more pronounced when full-

scale experimental data are unavailable to validate results obtained from the reduced-scale










models, as is the case with the Meier-Diirnberg data. Therefore, the AASHTO vessel collision

specifications necessarily include the same degree of uncertainty given that the AASHTO

provisions are directly based upon the Meier-Diirnberg results.

Particularly, the nature of the dynamic tests Meier-Dirnberg conducted precludes the

presence of significant dynamic effects that are generated between the barge and bridge in a

collision event. The interaction between the barge, bridge, and soil significantly affects the loads,

displacements, and stresses generated. One setback of the Meier-Diirnberg experiments is the use

of a pendulum drop-hammer. The interaction between the barge and the drop-hammer is not

representative of barge-bridge-soil interaction. Additionally, for the Meier-Diirnberg study, the

barge bow models were Eixed in a stationary configuration, thereby simulating impacts on a rigid

pier structure. Hence, the fixed condition of the barge bow models used in the Meier-Diirnberg

study prevents dynamic interaction between the barge and drop-hammer. Use of static analysis

and design procedures fails to account for important dynamic effects--such as inertial forces,

damping forces, and rate effects--present in an impact event. Omission of dynamic effects from

analysis procedures may result in a non-uniform margin of safety against failure in a dynamic

impact event.

At the time that the AASHTO specifications were first published, typical analysis

procedures use in bridge design practice were linear and static. Nonlinear and dynamic analyses

required expensive, both with respect to cost and time, computational hardware and software. As

a result, nonlinear dynamic analysis was primarily used in research, being too time-consuming

for use in typical design practice. However, with technological advancements in both hardware

and software over the past two decades, analysis techniques that were historically in the domain

of research have now become common place in design. Nonlinear analysis techniques are now









commonly used in design of bridge foundations. Additionally, dynamic analyses--either modal

or time-history--are commonly employed in the design of structures to resist earthquake loads.

Given the limitations of the Meier-Dornberg data, full-scale barge impact tests were

initiated by the Florida Department of Transportation (FDOT) and researchers at the University

of Florida (UF) in 2000 (Consolazio et al. 2006). The old St. George Island Causeway Bridge

was scheduled for demolition and eventual replacement by a new structure. Located in the

panhandle of Florida, approximately 5 miles east of Apalachicola, the bridge spanned, north to

south, from Eastpoint to St. George Island. Due to environmental concerns, the alignment of the

new replacement bridge deviated nearly 1500 ft from the old alignment in several locations,

affording researchers the unique opportunity to conduct full-scale barge impact tests on the out-

of-commission structure without endangering the new bridge.

During March and April of 2004, after the new St. George Island Causeway Bridge was

opened to traffic, full-scale impact testing commenced. Two of the piers near the main

navigation channel were selected for a total of fifteen impact tests. Instrumentation was installed

on each pier, and during each test, barge impact forces, deformations, and decelerations; pier

displacement and accelerations; and soil information were recorded. Once all of the experimental

test data was reduced, analytical models of each pier were developed and validated using the

experimental results. Based on the data recorded and insights gained from full scale testing of

the old St. George Island Causeway Bridge, it was possible to initiate the development of

updated design provisions for bridge structures subj ected to barge impact loading.

1.3 Objectives

The two primary obj ectives of this research were to develop: 1) improved methods for

calculating maximum impact forces imparted to a bridge structure during a barge collision, and

2) improved procedures for determining barge impact load and corresponding bridge response









such loads. A maj or aspect of all of the proposed structural analysis methods is the ability to

quantify important dynamic effects inherently present in a barge collision event. Most design

provisions for impact loading use a static analysis to determine bridge response; however, these

static methods fail to capture significant dynamic effects. Therefore, semi-empirical methods that

can capture dynamic effects were developed as a part of this research, each providing a more

sophisticated--yet still design-oriented (i.e., practical enough for use in bridge design)--impact

analy si s.

1.4 Scope of Work

* Characterize barge bow force-deformation relationships: It was discovered during St.
George Island full-scale testing that in situations in which the barge bow undergoes
significant plastic deformation, the maximum impact force that can be generated is limited
by the load-carrying capacity of the barge. This implies that, as the barge bow crushes
beyond yield and fracture, the impact force generated does not increase with further bow
deformation. The relationships of Meier-Doirnberg and the AASHTO specifications,
however, prescribe that the impact force increases monotonically with increasing barge
bow deformation (crush depth). In this study, updated relationships between impact force
and barge bow deformation for various barge types have been developed independently of
the AASHTO relationship.

* Develop dynamic analysis techniques for use in design: It was determined during St.
George Island experimental testing that dynamic effects have a significant effect on pier
response during a barge impact. Early in an impact event, inertial (mass-proportional) and
damping (velocity-proportional) forces comprise more resistance to impact loading than
the stiffness (di splacement-proportional resistance). However, at later stages in an impact,
it has been shown that inertial forces actually change directions, and drive the motion of
the bridge, effectively becoming a source of loading on the structure. This complex
interaction of dynamic forces cannot adequately be captured through the use of static
analysis, and thus dynamic analysis is required. In this study, three dynamic analysis
techniques were developed and are available for use in design:

0 Coupled vessel impact analysis (CVIA): A time-history analysis that permits
interaction between a single degree-of-freedom barge model and a numerical
bridge.

o Applied vessel impact analysis (AVIL): A time-history analysis technique that
uses impact characteristics to generate an approximate time-varying impact load,
which in turn is applied to a bridge structure in a dynamic sense.










o Impact response spectrum analysis (IRSA): A response spectrum analysis
method that uses impact characteristics and dynamic structural characteristics
(modal shapes and periods of vibration) to determine maximum dynamic forces in
a bridge structure during an impact.

*Validate proposed dynamic analysis techniques: Using FB-MultiPier, the effectiveness
of the proposed dynamic analysis methods are evaluated. Construction drawings for the
new St. George Island Bridge were obtained, and used to develop numerical bridge
models. This structure was selected because the new St. George Island Bridge is
representative of current structures.









CHAPTER 2
BACKGROUND

2.1 Vessel-Bridge Collision Incidents

Designing bridge structures that span over vessel-navigable bodies of water requires that

careful consideration be given to the fact that cargo vessels may inadvertently collide with piers

that support the bridge superstructure. Causes of such collisions often involve poor weather

conditions, limited visibility, strong cross-currents, poor navigational aids, failure of mechanical

equipment, or operator error. Worldwide, vessel impacts occur frequently enough that, on

average, at least one serious collision occurs each year (Larsen 1993). Within the United States, a

succession of incidents involving ships and barges impacting bridge structures clearly

demonstrates that the potential for structural failure and loss of life exist.

One of the most catastrophic incidents of vessel-bridge collision in the United States was

the 1980 collapse of the Sunshine Skyway Bridge, which spanned over Tampa Bay in Florida.

Navigating in poor weather conditions and limited visibility, the cargo ship Sunanit Venture

collided with one of the anchor piers of the bridge causing the collapse of almost 1300 ft. of

bridge deck (Figure 2-1) and the loss of thirty-five lives. Due in large part to this incident,

comprehensive guidelines for designing bridge structures to resist ship and barge collision loads

were later published, in 1991, as the AASHTO Guide Specification and Conanentaly for Vessel

Collision Design ofHiginvay Bridges (AASHTO 1991).

While massive cargo ships clearly pose a significant threat to bridge structures, they are

also limited in operation to relatively deep waterways. Consequently, ships pose significant risks

primarily to bridges near maj or shipping ports. In contrast, multi-barge flotillas are able to

operate in much shallower waterways and thus pose a risk to a greater number of structures.









Considering that each individual barge within a flotilla might weigh as much as fifty fully-loaded

tractor-trailers, the potential damage that can be caused by barges striking bridge piers is clear.

In September 1993, a multi-barge tow navigating at night and in dense fog collided with

the Big Bayou Canot railroad bridge near Mobile, Alabama resulting in a significant lateral

displacement of the structure. Moments later, unaware that the bridge had just been struck by a

barge, a passenger train attempted to cross the structure at 70 mph, resulting in catastrophic

structural failure (Figure 2-2) and forty-seven fatalities (Knott 2000). In September 2001 a barge

tow navigating near South Padre, Texas veered off course in strong currents and collided with

piers supporting the Queen Isabella Causeway Bridge. As a result of this impact, three spans of

the structure collapsed (Figure 2-3) and several people died (Wilson 2003). In May 2002, an

errant barge tow struck a bridge on interstate I-40 near Webbers Falls, Oklahoma. On an average

day, this structure carried approximately 20,000 vehicles across the Arkansas River. As a result

of the impact, 580 ft. of superstructure collapsed (Figure 2-4), fourteen people died, and traffic

had to be rerouted for approximately two months (NTSB 2004).

2.2 Review of Experimental Vessel Impact Tests

Despite the significant number of vessel-bridge collisions that have occurred in recent

decades, only a small number of instrumented experimental tests have ever been performed to

quantify vessel impact loading characteristics. Generally, ship collision events have been studied

to a much greater extent than have barge collisions. Two key ship collision studies form the basis

for most current theories relating to ship impact loading. The first was conducted by Minorsky

(1959) to analyze collisions with reference to protection of nuclear powered ships, and focused

on predicting the extent of vessel damage sustained during a collision. A semi-analytical

approach was employed using data from twenty-six actual collisions. From this data a

relationship between the deformed steel volume and the absorbed impact energy was formulated.









A second key ship collision study was that of Woisin (1976) which also focused on deformation

of nuclear powered ships during collisions. Data were collected from twenty-four collision tests

(Figure 2-5) of reduced-scale (1:7.5 to 1:12) ship-bow models colliding with ship-side-hull

models. Relationships between impact energy, deformation, and force developed during this

study were later used in the development of the AASHTO equations for calculating equivalent

static ship impact forces (AASHTO 1991).

In terms of quantifying the characteristics of barge impact loads, as opposed to ship impact

loads, one of the most significant experimental studies conducted to date is that of Meier-

Diirnberg (1983). This research included both dynamic and static loading of reduced-scale (1:4.5

to 1:6) models of European Type IIa barges. In overall dimensions, European Type IIa barges are

similar to the jumbo hopper barges that are commonly found in the U.S. barge fleet. All tests in

the Meier-Dirnberg study were conducted on partial vessel models that consisted only of nose

sections of barges. In conducting the dynamic tests in this study, the partial barge models were

mounted in a stati onary (fixed-b oundary-conditi on) configurati on and then struck by a falling

impact pendulum hammer. The amount of impact energy imparted to the barge model during

each test was dictated by the weight of the hammer and its drop height. Due to limitations of

hammer drop height, repeated impacts were carried out on each partial barge model to

accumulate both impact energy and impact damage (crushing deformation). From the

experimental data collected, Meier-Diirnberg developed relationships between kinetic impact

energy, inelastic barge deformation, and force.

More recently, experimental studies have been conducted that overcome one of the key

limitations of the Meier-Diirnberg study--i.e., the use of reduced-scale models. In 1989, Bridge

Diagnostics, Inc. completed a series of fi~ll-scale tests for the U. S. Army Corps of Engineers that









involved a nine-barge flotilla impacting lock gates at Lock and Dam 26 on the Mississippi river

near Alton, Illinois (Goble 1990). Each of the impacts was performed at approximately 0.4 knots.

Force, acceleration, and velocity time histories for the impacting barge were recorded using

commercially available sensors such as strain gages and accelerometers. In addition, custom

manufactured and calibrated load cells, developed by Bridge Diagnostics, were used to measure

impact forces (Figure 2-6). Unfortunately, data obtained from this study are not directly

applicable to bridge pier design because the system struck by the barge was a lock gate, not a

bridge pier. Lock gates and bridge piers posses different structural characteristics which produce

dissimilar impact loads. More importantly, the energy levels used during these tests were

insufficient to cause significant inelastic barge deformation. Because inelastic barge

deformations are common in head-on barge-pier collisions, and given that such deformations

affect both barge stiffness and impact energy dissipation, data obtained from this set of tests are

not directly applicable to bridge design.

Several years later, full-scale barge impact tests on concrete lock walls were conducted by

the U. S. Army Corps of Engineers. In 1997, a 4-barge flotilla was used to ram a lock wall at Old

Lock and Dam 2, located north of Pittsburgh, Pennsylvania (Patev and Barker 2003). These

experiments (Figure 2-7) were conducted to measure the structural response of the lock wall at

the point of impact and to quantify barge-to-barge lashing forces during impact. Strain gages

were installed on the barge to record steel plate deformations at the point of impact. An

accelerometer was used to capture the overall acceleration history of the flotilla, and clevis pin

load cells quantified lashing forces generated during impact. A total of thirty-six impact tests

were successfully carried out on the lock wall.









Following the 4-barge tests, larger impact experiments involving a 15-barge flotilla were

initiated in December of 1998 at the decommissioned Gallipolis Lock at Robert C. Byrd Lock

and Dam in West Virginia (Arroyo et al. 2003). In contrast to the 4-barge tests, one of the

primary goals of the 15-barge tests was to recover time-histories of impact force generated

between the barge flotilla and the lock wall. To accomplish this goal, a load-measurement impact

beam was affixed to the impact corner of the barge flotilla using two uniaxial high-capacity

clevis-pin load cells (Figure 2-8). Additional instrumentation used during these tests included

accelerometers, strain gages, water pressure transducers, and smaller capacity clevis-pin load

cells that were installed in-line with the barge-to-barge cable lashings. In total, forty-four impact

tests were successfully carried out on the lock wall.

In bridge pier design for barge collision loading, maximum impact forces are generally

associated with head-on impact conditions, not oblique glancing blows of the type tested in the

4-barge and 15-barge tests performed by the Army Corps. Therefore, while the data collected

during these tests could be useful in developing load prediction models for oblique side impacts

on piers, the same data cannot be used to improve the AASHTO expressions for head-on

impacts. Additionally, neither of the Army Corps test series involved dynamic vessel-pier-soil

interactions or significant crushing deformation of the impacting barges. Data collected during

the experimental tests were later used to develop several analytical methods for approximating

impact load on structures have been developed based on the experimental test results (Patev

1999, Arroyo et al. 2003).

In 2004, full-scale experimental barge impact tests on piers of the old St. George Island

Causeway Bridge (Consolazio et al. 2006) were conducted by the University of Florida (UF) and

the department of Transportation (FDOT). The purpose of conducting these tests was to directly










quantify barge impact loads and resulting pier, soil, and superstructure responses. The research

revealed that the AASHTO static design forces currently employed in bridge design range

broadly from being overly conservative in some instances to being unconservative (due to

dynamic amplification effects) in other cases. This variability stems from differences in the

dynamic characteristics of varying bridge types and in variations of design impact conditions. A

detailed review of the St. George Island barge impact test program is provided in Chapter 3.

Researchers at the University of Kentucky (UK) conducted numerical finite element barge

impact simulations to estimate forces imparted to bridge piers when impacted by a barge flotilla

(Yuan 2005a, Yuan 2005b). High-resolution finite element models were developed of a single

jumbo hopper barge and of multi-barge flotillas for the purpose of conducting collision analyses,

using the finite element analysis code LS-DYNA (LSTC 2003), on bridge piers. From the

analysis results, expressions were developed for predicting average barge impact force and

duration of impact load. Additionally, the effects of pier shape and pier stiffness on barge impact

force were investigated. A response spectrum analysis technique involving a four degree-of-

freedom pier model that is subj ected to a rectangular pulse impact load was also developed.

Researchers at the University of Texas (UTx) have recently investigated the AASHTO

probabilistic analysis procedures for bridge impact design under vessel impact loading (Manuel

et al. 2006). Computer software has been developed to determine the annual frequency of

collapse, facilitating the AASHTO probabilistic design procedure. Additionally, high resolution

finite element simulations were conducted to determine impact forces imparted to a pier during

an impact event, as well as providing guidance to calculating the ultimate lateral strength of the

pier.









2.3 Design of Bridges According to the AASHTO Barge Impact Provisions

A pooled fund research program sponsored by eleven states and the Federal Highway

Administration (FHWA) was initiated in 1988 to develop methods of safeguarding bridges

against collapse when impacted by ships or barges. The findings of the research were adopted by

AASHTO and published in the Guide Specification and Conanentary for Vessel Collision Design

ofHiginvay Bridges and the Load and Resistance Factor Design (LRFD) Specifications.

Provisions included in these publications serve as a nationally adopted basis for bridge design

with respect to vessel collision loads. The provisions allow two approaches to collision resistant

bridge design. Either the structure can be designed to withstand the vessel impact loads alone, or

a secondary protection system can be designed that will absorb the vessel impact loads and

prevent the bridge structure itself from being struck. In addition to providing design guidelines,

the AASHTO provisions recommend methodologies for placement of the bridge structure

relative to the waterway as well as specifications for navigational aids. Both are intended to

reduce the potential risk of a vessel collision with the bridge. Nonetheless, for a wide ranging set

of reasons--windy high-current waterways; adverse weather conditions; narrow or curved

waterway geometry--most bridges that are accessible to barge impact will likely be struck at

some point during their lifetime (Knott and Prucz 2000). With this in mind, all bridges that span

navigable waterways need to be designed with due consideration being given to vessel impact

loading.

The AASHTO Guide Specification and Conanentazy for Vessel Collision Design of

Highway Bridges (1991) provides designers with three methods by which piers may be designed:

Methods I, II, and III. Method I uses a simple semi-deterministic procedure, calibrated to

Method II criteria, for the selection of a design vessel. This method however, is less accurate

than Method II, and as such, should not be used for complex structures. Method II is a









probability based method for selecting an impact design vessel. Method II is also more

complicated than Method I; however, Method II is required if documentation on the acceptable

annual frequency of collapse is necessary. Method III is a cost-effective vessel selection

procedure that is permitted when compliance with acceptable annual frequency of collapse

provisions are neither economically nor technically feasible. This method is typically used in

evaluation of vulnerability of existing bridges.

2.3.1 Selection of Design Vessel

The first step in designing a bridge structure for barge impact loading is to determine the

characteristics of the waterway over which the structure spans. Determination of waterway

characteristics includes the geometric layout of the channel (including the centerline) beneath the

bridge, the design water depth for each component, and the water currents parallel and

perpendicular to the motion of the barge.

Next, impact vessel character stics--such as, vessel type, size, cargo weight, typical speed,

and annual frequency--must be determined based upon actual vessels traversing the design

waterway. Furthermore, vessel characteristics must be calculated for waterway traffic in both

directions along the channel. Using the characteristics of individual barges, the overall length

(length-overall, LOA) of the barge tow must be calculated as the total length of the barge tow

plus the length of the tow vessel (Figure 2-9).

The design impact velocity for each bridge element is calculated based upon the typical

vessel transit speed, geometry of the channel, and overall length of the barge tow. The design

velocity for bridge elements within the area extending from the centerline of the channel to the

edge of the channel is the typical vessel transit speed (VT) (Figure 2-10). From the edge of the

channel (xc) to a distance equal to three times the overall barge tow length (3 LOA) from the

centerline (xL), the design speed linearly decreases to a minimum design speed (Vmin). Beyond a









distance of 3 LOA from the channel centerline, the design speed is held constant at a minimum

design impact speed that corresponds to the yearly mean current at that location.

2.3.2 Method II: Probability Based Analysis

AASHTO Method II design requires the designer to calculate the annual frequency of

collapse resulting from vessel collisions as:

AF = N- PA -PG -PC (2-1)

where AF is the annual frequency of collapse, N represents the annual vessel frequency for each

design element and vessel type and size, PA is the probability of vessel aberrancy, PG is the

geometric probability of impact, and PC is the probability of collapse due to impact. The annual

frequency of collapse must be calculated for each bridge element and each type and size of

vessel. Summing up all of the individual annual frequencies of collapse yields the annual

frequency of collapse for the total bridge. The annual frequency of collapse determined from this

process is required to be less than or equal to an acceptable annual frequency of collapse that

depends upon the importance of the bridge:

* For critical bridges AF < 0.0001 (i.e., a 1 in 10,000 probability of failure each year)

* For regular bridges AF < 0.001 (i.e., a 1 in 1000 probability of failure each year)

Probability of aberrancy (PA) represents the probability that a given vessel will deviate

from its course, possibly imperiling the bridge. Based upon the proposed bridge site, the most

accurate determination of probability of aberrancy is determined from historical data on vessel

aberrancy. However, if appropriate historical data are not available, ASSHTO permits an

estimation for the probability of aberrancy:

PA = BR RB Rc Rxc RD (2-2)









where BR is the base aberrancy rate (specified by AASHTO), RB is the bridge location

correction factor, Rc is the water current correction factor in the direction of barge travel, Rxc is

the water current correction factor transverse to barge travel, and RD is a vessel traffic density

factor. The base aberrancy rate (BR) was determined from historical accident data along several

U. S. waterways. For barge traffic, the base aberrancy rate specified by AASHTO is 1.2x10-4 (i.e.,

0.00012, or 1 aberrancy in every 8333 barge transits).

The bridge location correction factor (RB) WAS implemented to account for the added

difficulty in navigating a barge tow around a bend (Figure 2-11):

S1.0 straight region

RB- = 1.0 + 6/90 transition region (2-3)
1.0 + 6/45 bend region


where 6 is the angle of the bend in degrees.

In order to account for the effects that water currents have on the navigation of a barge

tow, correction factors for water currents both parallel (Rc) and transverse (Rxc) to the motion of

the barge tow are required:

Rc = 1+ Vc/10 (2-4)
Rxc = 1+ Vxc (2-5)

where Vc and Vxc are the water current velocities parallel and transverse to the barge motion (in

units of knots). Furthermore, depending upon the density of vessel traffic in the waterway, a

vessel traffic density factor is used to modify the base aberrancy rate:

* Low density traffic: RD = 1.0

* Average density traffic: RD = 1.3

* High density traffic: RD = 1.6









Geometric probability of impact is defined as the probability that a vessel will strike a

bridge element once it has become aberrant. Based on historical barge collision data, AASHTO

requires that a normal distribution be used to characterize the locations of aberrant vessels in

relation to the centerline of the channel (Figure 2-12). The normal distribution is then used to

determine the chance that a given bridge element will be struck by the aberrant barge.

Furthermore, the mean value of the distribution is situated on the centerline of the channel and

the standard deviation is assumed to be equal to the overall length of the barge tow (LOA). The

effective width of impact for each bridge element is calculated as the width of the structural

element plus half the width of the vessel on each side of the pier. Using this impact zone width,

the geometric probability of impact is calculated as the area under the normal distribution curve

that is bounded by the effective impact width and centered on the pier.

Assuming a bridge element is struck by an errant vessel, the probability that the element

(e.g., a pier) will collapse must also be determined. AASHTO provides the following

relationships for calculating the probability of collapse (PC) based upon the strength of the

bridge element and the static vessel impact force that is applied to the element (Figure 2-13):

(0.1I+ 9(0. 1- H/P) if H/P < 0.1
PC = (1 H/P9 if0.1< IH/P<~1.0 (2-6)
S0.0 if H/P > 1.0


In this equation, H is the ultimate strength of the pier element, and P is the design static vessel

impact force.

2.3.3 Barge Impact Force Determination

The AASHTO specifications use a kinetic energy based method to determine the design

impact load imparted to a bridge element. Using the total barge tow weight and design velocity









determined for each bridge element (e.g., pier) the kinetic energy of the barge is computed as

follows:

C WV2
KE = H(2-7)
29.2

where KE is the barge kinetic energy (kip-ft), CH is a hydrodynamic mass coefficient, W is the

weight of the vessel tow (in tones), and V is the design speed of the vessel tow (ft/sec). The

hydrodynamic mass coefficient (CH) is included to account for additional inertia forces caused by

the mass of the water surrounding and moving with the vessel. Several variables may be

accounted for in the determination of CH : walter depth, underkeel clearances, shape of the

vessel, speed, currents, direction of travel, and the cleanliness of the hull underwater. A

simplified expression has been adopted by AASHTO in the case of a vessel moving in a forward

direction at high velocity (the worst-case scenario). Under such conditions, the recommended

procedure depends only on the underkeel clearances:

* For large underk~eel clearances (2 0.5 draft) : CH = 1.05

* For small underkeel clearances (<;0.1 draft) : CH = 1.25

where the draft is the distance between the bottom of the vessel and the floor of the waterway.

For underkeel clearances between the two limits cited above, CH is estimated by linear

interpolation.

Once the kinetic energy of the barge tow has been determined, a two-part empirical load-

prediction model is used to determine the static-equivalent impact load. The first component of

the model consists of an empirical relationship that predicts barge crush deformation (inelastic

deformation) as a function of kinetic energy:










ag=11 562KE \12 10.2R
a=1+ 1 (2-8)


where aB is the depth (ft.) of barge crush deformation (depth of penetration of the bridge pier into

the bow of the barge), KE is the barge kinetic energy (kip-ft), and RB = BB/3 5; where BB is the

width of the barge (ft). Figure 2-14 graphically illustrates Eqn. 2-8.

The second component of the load prediction model consists of an empirical barge crush

model that predicts impact loads as a function of crush depth:

P 41 12aBRB if aB < 0.34
B 1(1349 +110aB)RB if aB > 0.34 29

where PB is the equivalent static barge impact load (kips) and aB is the barge crush depth (ft).

The AASHTO barge force-deformation relationship given in Eqn. 2-9 is illustrated in Figure 2-

15. Furthermore, combining Eqns. 2-8 and 2-9, a relationship between barge impact force and

initial barge kinetic energy may be defined (Figure 2-16). As will be discussed in additional

detail in Chapter 4, Eqns. 2-8 and 2-9 above were both adopted from research conducted by

Meier-Doirnberg (1983).


























Figure 2-1 Collapse of the Sunshine Skyway Bridge in Florida (1980)
after being struck by the cargo ship Summit Venture

































Figure 2-2 Failure of the Big Bayou Canot railroad bridge in Alabama (1993)
after being struck by a barge flotilla


x
~r ;r

"~-~si ~
?~K

~t
u-...
r
_~y~Zu .

























Figure 2-3 Collapse of the Queen Isabella Causeway Bridge in Texas (2001)
after being struck by a barge flotilla





Figure 2-4 Collapse of an Interstate I-40 bridge in Oklahoma (2002)
after being struck by a barge flotilla
(Source: Oklahoma DOT)


li~





























Figure 2-5 Reduced scale ship-to-ship collision tests conducted by Woisin (1976) A) Ship bow
model on inclined ramp prior to test, B) Permanent deformation of ship bow model
after test
























rr. P --B

Figure 2-6 Instrumented full-scale barge-lock-gate collision tests (Source: Bridge Diagnostics,
Inc.) A) Barge bow approaching lock gate, B) Load cells attached to barge bow
























Figure 2-7 Instrumented 4-barge lock-wall collision tests (Source: U.S. Army Corps of
Engineers) A) Push boat and 4-barge flotilla, B) Sensors at impact corner of barge

























Figure 2-8 Instrumented 15-barge lock-wall collision tests (Source: U.S. Army Corps of
Engineers) A) Push boat and 15-barge flotilla, B) Force measurement beam attached
to barge with clevis-pin load cells


J:'
-I ~ ...i
.":
~L~-l;*

ti 1~I

















PLAN~ (2x2 BARGE T;OWL


CC" ~---- ---


LOA


Figure 2-9 Barge tow configuration (Source: AASHTO Figure 3.5.1-2)


























Vmm


Xc XL

Distance from Centerline of Channel


Figure 2-10 Design impact speed












Tura Region












TRansitiOn Region


Straight Region -

Trpanition Region


Transition Region


Figure 2-11 Bridge location correction factor (Source: AASHTO Figure 4.8.3.2-1) A) Turn in
channel, B) Bend in channel






































wr r I~"If rr


Centsd uinlori~la of I
Path





Distribuslo


Centerline
Bridge












. -hip, Budge


Intersection Path
Ito Cnterline of Pier


Figure 2-12 Geometric probability of collision (Source: AASHTO Figure 4.8.3.3-1)


qI Si



























o~r r. go

Ultimaate Bridpe Element Strength He or HP
Vessel Imrpacl Force Payo a


Figure 2-13 Probability of collapse distribution (Source: AASHTO Figure 4.8.3.4-1)












10








10








0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Kinetic Energy, KE (kip-ft)


Figure 2-14 AASHTO relationship between kinetic energy and barge crush depth


















1000


2500





100



0 2 4 6 8 10 12

Crush depth, aB (ft)


Figure 2-15 AASHTO relationship between barge crush depth and impact force












3000


2500


S2000


S1500


1000


500




0 2000 4000 6000 8000 10000 12000 14000 16000 18000
Kinetic Energy, KE (kip-ft)


Figure 2-16 Relationship between kinetic energy and impact load









CHAPTER 3
SUMMARY OF FINDINGS FROM ST. GEORGE ISLAND BARGE IMPACT TESTING

3.1 Introduction

In 2004, the University of Florida (UF) and the Florida Department of Transportation

(FDOT) conducted full-scale barge impact tests on the old St. George Island Causeway Bridge

(Consolazio et al. 2006). Experimental results were obtained to quantify the loading and

response of pier and bridge structures subj ected to a barge impact. To compliment the physical

testing, numerical finite element analysis (FEA) techniques were also employed to aid in

interpretation of experimental test data. Comparisons of measured experimental data and FEA

results substantiated the validity of the experimental data and provided additional insights into

the nature of pier response to barge impact loading.

3.2 Overview of Experimental Test Program

Impact tests were conducted on Piers 1 and 3 on the south side of the St. George Island

Causeway Bridge, near the navigation channel (Figure 3-1). These piers were selected for the

different structural configurations that they represented.

Pier-1 (Figure 3-2), the more impact resistant pier, was a reinforced concrete pier

composed of two pier columns, a pier cap, a shear wall for lateral resistance, and a massive

concrete pile cap and tremie seal mud-line footing, all supported by forty HP 14x73 steel piles.

This pier was adj acent to the navigation channel, and as such, was most prone to impact from an

errant vessel. Therefore, Pier-1 was designed to be the most impact resistant pier supporting the

bridge. From the perspective of the barge impact test program, Pier-1 was expected to able to

resist the highest loads as well as produce the largest amounts of barge bow deformation.

In contrast to Pier-1, Pier-3 (Figure 3-3) was a more flexible, less impact resistant

reinforced concrete structure composed of two pier columns, a pier cap, and a shear strut for









lateral resistance. Unlike the massive pile and cap foundation system supporting Pier-1, Pier-3

was supported by two small waterline pile caps, each on top of four 20 in, square prestressed

concrete piles. Due to its flexibility, Pier-3 was not able to resist large impact forces. Pier-3,

however, was selected to investigate the effects low-energy impacts on secondary support piers.

Furthermore, it was anticipated that the barge would sustain negligible permanent barge bow

deformations during impacts on Pier-3.

The overall experimental test program was conducted in three distinct series, each with a

different structural configuration. The first test series (Figure 3-4), denoted series Pl, involved

tests conducted on Pier-1 in isolation, with the primary obj ective of generating maximal impact

loads and inelastic barge bow deformations. In order to achieve such loads and deformations,

the mass of the barge was increased by placing two 55 ft. spans of concrete deck from a

previously demolished section of the bridge on top of the barge, for a total barge-plus-payload

weight of 604 tons. Impact speeds in series Pl varied from 0.75 knots 3.45 knots.

Instrumentation used during test series Pl included load cells, accelerometers (on both the pier

and the barge), displacement transducers, optical break beams (used to trigger the data

acquisition (DAQ) system and provide precise barge velocity determination), and a pressure

transducer (used to measure pressure changes in the water). See Consolazio et al. (2006) for a

detailed discussion of the instrumentation systems used.

Upon completion of the series Pl tests, the payload spans on the test barge were removed

(Figure 3-5a), restoring the empty barge weight to 275 tons. After removal of the spans, impact

tests on Pier-3 began; the first series of which was denoted series B3. The B3 test series

involved low-energy impacts on Pier-3 with superstructure spans still intact between piers 2, 3,

4, 5, and beyond (Figure 3-5b). Due to the relatively high flexibility ofPier-3, impact velocities









were limited so as to avoid dislodging the superstructure. With the reduced mass and lower

velocities, the impact energies imparted to the pier during the B3 series of impacts were not

substantial enough to generate noticeable permanent barge bow deformations. The

instrumentation used during test series B3 included load cells, accelerometers (including the

addition of accelerometers on the superstructuree), displacement transducers, an optical break

beam, and strain gages (attached to the piles).

Following the B3 series tests, the superstructuree connecting piers 2, 3, and 4 was removed

leaving Pier-3 isolated (Figure 3-6). Impact tests were then conducted on Pier-3 in isolation

during tests series P3. The series P3 tests were very similar to the series B3 tests with respect to

barge velocities and mass, instrumentation, and barge bow deformations. Data from the B3 and

P3 series tests were used to compare the response of an isolated pier to a pier that is integrated

within a bridge structuree.

A total of fifteen impact tests were conducted: eight in series Pl, four in series B3, and

three in series P3. Maximum allowable impact speeds for each series were governed by

equipment limitations, weather conditions, and safety concerns. For additional details, see

Consolazio et al. (2006).

3.3 Overview of Analytical Research

Finite element models were developed using the pier analysis program, FB-MultiPier

(2005), which was selected for its ability to model both dynamic behavior and material

nonlinearity. A brief description of the finite element models is presented below, however, for a

more detailed discussion see Consolazio et al. (2006) and McVay et al. (2005).










3.3.1 FB-MultiPier Models

3.3.1.1 FB-MultiPier Pier-1 model

The FB-MultiPier Pier-1 model used frame elements to model the pier components, such

as the columns, pier cap, shear wall, and piles; and shell elements to model the behavior of the

cap and seal (Figure 3-7). Since shell elements were used to model the pile cap, the piles

connected to the mid-plane of the cap. However, in the actual structure, the piles were

embedded in both the 5 ft thick pile cap, and the 6 ft thick tremie seal. Within this embedment

length, the piles were restrained against flexure. To correctly model this embedment, a network

of cross bracing frame elements (Figure 3-7) was placed in the top 8.5 ft (half of the 5 ft cap plus

the 6 ft seal) of the piles. The material behavior of all structural elements was chosen to be linear

elastic because the physical tests were non-destructive in nature.

3.3.1.2 FB-MultiPier Pier-3 model

All structural components of Pier-3--pier columns, pier cap, shear strut, and piles--

excluding the pile caps, were modeled using frame elements (Figure 3-8). As with Pier-1, the

pile caps were modeled using shell elements at the mid-plane elevation of the caps. In the Pier-3

model, it was determined that additional elements were necessary to correctly model the shear

strut behavior. As such, cross bracing was provided to stiffen the shear strut. Furthermore,

though not monolithically cast together, the strut and the pile caps were cast such that they could

come in contact during impact. Therefore, a strut was placed connecting the pier column to the

pile caps to model this added stiffness. Similar to all previous models, the material model

chosen was linear elastic.

3.3.1.3 FB-MultiPier Bridge model

In addition to the Pier-3 model, a model of the bridge was also developed (Figure 3-9).

This model included Piers 2, 3, 4, and 5 with superstructure elements connecting adjacent piers.









Models of Piers 4 and 5 were developed from the Pier-3 model since these three piers have very

similar structural layouts--with variation only in the height of the pier column. Pier-2 had a

similar layout to Pier-1, in that it was composed of two pier columns, a pier cap, a shear wall,

and a massive mudline footing with a pile cap resting on twenty-one steel H-piles. All elements

of each pier were modeled using frame elements with the exception of the pile caps, which were

modeled using shell elements. The superstructure was modeled using frame elements with cross-

sectional properties calculated from the cross-sectional properties of the individual elements.

3.3.2 Finite Element Simulation of Models

In order to understand the sources of resistance that are mobilized during a barge impact

scenario, it was imperative to quantify all of the forces acting on the structure. However, sensors

installed on the piers were unable to measure all of the forces generated during the impact tests.

Thus, using experimental data, the numerical models discussed previously were calibrated to

yield responses similar (e.g. pier displacement) to those measured experimentally.

The general calibration procedure involved applying the impact loads measured from the

experimental tests directly to the numerical models of the piers. Various parameters (e.g. soil

properties) were then calibrated such that the numerical predictions of pier response--pier

displacement, pile displacement, pile shears, pressure forces on the foundation, etc.-agreed well

with the measured experimental data. With confidence in the numerical models established,

results from the analyses were then used to predict sources of resistance that could not be

measured during the experimental tests.

This procedure was applied to three characteristic impact tests from the experimental

program. Each test in the experimental study was given a four-character designation; the first

two characters indicate the test series in which the impact tests were conducted, and the second

two characters indicate the specific test within that series. The three tests analyzed are as









follows: 1.) PIT7 (i.e. test series Pl, test number 7) 2.) P3T3, and 3.) B3T4. These three tests

represent some of the more severe impacts conducted. The remainder of this section focuses on

results from dynamic analyses of these three tests. Specifically, the sources of resistances are

broken down and compared to the impact force (as well as each other) during each simulation.

3.3.2.1 Impact test P1T7

Figure 3-10 is a schematic of Pier-1 showing the forces acting upon the structure during

test PIT7. Due to the embedment of the pier cap and tremie seal in the soil, a complex

interaction--both static and dynamic--between the soil and the pier foundation occurred.

Forces presented in the schematic that were experimentally measured--either directly or

indirectly--include the impact force, inertia force, shear force in the instrumented pile, and the

cap+seal passive+active pressure force. The experimentally measured impact load for test PIT7

was applied to the finite element model as an external force (Figure 3-10). Calibration of the

model was then carried out to bring model and experimental results into agreement mainly

through refinement of soil parameters.

Figure 3-11la provides a comparison of all forces acting on the pier structure during the

PIT7 impact, and Figure 3-11b compares the soil forces acting on the pile cap and tremie seal.

Positive values indicate forces acting in the directions shown in Figure 3-10. The effective load

presented in these plots is a combination of the applied load and the negative portion of the

inertial (acceleration-dependent) force. This approach was used because, as the inertial force

changes from a positive value--representing a resistance to the motion of the pier--to a negative

value, the force becomes a source of loading, further driving the motion of the pier.

Table 3-1 summarizes the maximum values of the five maj or forces acting on the pier

presented in Figures 3-11. Examining the data presented in the table, it is evident that the inertial

resistance is mobilized more rapidly than all other sources of resistance. Note that at the time at









which the inertial resistance is fully mobilized (250 kips at 0.10 sec), the inertial resistance is

approximately 40-percent of the applied load of 600 kips at the same point in time. Furthermore,

the other three maj or sources of resistance indicated in Figure 3-11la are all less than 200 kips.

This indicates that at an early stage of an impact event, the mass properties of the pier structure

comprise an important contribution to the total resistance developed.

As the impact event continues, note that between 0.1 and 0.2 seconds, the inertial force has

reversed direction, effectively providing an additional source of loading (Figure 3-11a). When

combining this inertial load with the applied load, the effective load maximizes at approximately

970 kips at 0.25 sec. Comparing this value to the maj or sources of resistance present, it is

apparent that the sum of the pile shears (an indication of the soil forces acting on the piles)

contributes only 30-percent to the total resistance, whereas the soil forces acting on the cap and

seal contribute to approximately 70-percent of the total resistance, thus indicating that the cap

and seal forces have a maj or effect on the total resistance (Figure 3-11i).

3.3.2.2 Impact test P3T3

Figure 3-12 is a schematic of Pier-3 showing the forces acting upon the structure during test

P3T3. One of the key differences between this test and test PIT7, is that the soil forces acting on

the cap and seal during test PIT7 are not present in test P3T3, as the pile caps are above the

mudline. Thus, looking at the forces acting on the pier structure, the interaction of Pier-3 with

the surrounding soil was far less complex than that of Pier-1 with its surrounding soil. As with

test PIT7, the experimentally measured P3T3 load history was applied dynamically to the

structure (Figure 3-12). Furthermore, results from the calibrated P3T3 model agreed well with

the results from the experimental test (Consolazio et al. 2006).

As with test PIT7, the inertial force acting on Pier-3 mobilizes earlier--reaching a

maximum value of approximately 370 kips at 0.08 sec--than the sum of the pile shears--









maximizing at approximately 400 kips at 0.23 sec (Figure 3-13). Furthermore, this inertial

resistance clearly dominates the total resistance of the pier at this early stage, representing

approximately 75-percent of the total resistance at 0.08 sec, where as the pile shears (slightly

larger than 100 kips at 0.08 sec) only represent 25-percent. Clearly, inertial forces provide an

important source of resistance early in the P3T3 test.

3.3.2.3 Impact test B3T4

Impact test B3T4 differed from tests PIT7 and P3T3 in that during test B3T4, portions of

the superstructure connecting Pier-3 to adj acent piers were still intact. Comparing Figure 3-14 to

Figure 3-12, it can be seen that the sources of resistance present in test B3T4 are the same as

those in P3 T3 with the addition of load transferring through shear in the superstructure bearings

to the superstructure and adjacent piers. Using the bridge model discussed earlier, a dynamic

FB-MultiPier analysis was conducted. As with tests PIT7 and P3T3, the experimentally

measured B3T4 impact load was applied to Pier-3 in the numerical model (Figure 3-14).

Forces acting on the pier were extracted from the FB-MultiPier analysis of test B3T4, and

are presented in Figure 3-15. Like previous tests, pier inertia forces mobilized earlier than any

other forms of resistance, maximizing at 190 kips at approximately 0.08 seconds and accounting

for approximately 70-percent of the 280-kip applied load at this point in time, thus clearly

dominating the resistance early in the impact. As the pier displaces further, shear forces

(representing static and dynamic resistance from the soil) start to mobilize, maximizing at about

260 kips at 0.20 seconds. Following the shear forces, forces in the bearing pads (representing

resistance from adj acent piers) begin to mobilize, maximizing at 130 kips at approximately 0. 16

seconds.

As with the PIT7 and P3T3 analyses, the inertia resistance quickly decreases, and changes

to a form of loading when it becomes negative, further driving pier motion. When the bearing









shear force decreases and becomes negative, it constitutes a form of loading on the structure also

driving the motion of the pier. To illustrate this concept, the magnitudes of the inertial force and

the bearing shear force (when negative) are added to the applied load to form an effective load

history (Figure 3-15). Note that from 0.25 to 0.50 seconds, this effective load is approximately

equal to the shear in the piles (with the difference being attributable to structural damping), thus

indicating that the effective load drives the motion of the pier, and the resistance to this motion is

mobilized in the soil (acting through the pile shears).

3.4 Comparison of Dynamic and Static Pier Response

Currently, design of bridges to resist barge impact load involves using an equivalent static

load procedure--such as the method outlined in AASHTO--in which the calculated load is

applied to a static pier model. However, as presented in the results outlined in previous sections,

dynamic effects (such as damping and inertial forces) constitute a significant source of

resistance, and in some cases additional loading. In the present section, comparisons between

static and dynamic analyses for each of the three tests simulated (i.e. PIT7, P3T3, and B3T4)

will be presented. Three separate analyses were conducted for each test: 1.) dynamic analysis

using the experimentally measured load histories, 2.) static analysis using the peak load from the

experimentally measured load histories applied statically, and 3.) static analysis using the

AASHTO specified load based upon the initial impact energy measured for the tests. Each of

these cases is summarized in Table 3-2. Comparisons of results for each of these three analyses

will provide insight into the level of safety present in static analysis procedures, such as the

methods provided by AASHTO.

Cases A, D, and G in Table 3-2 represent dynamic analyses in which the time-varying

experimentally measured loads are applied to the piers in tests PIT7, P3T3, and B3T4

respectively. For cases B, E, and H, the peak load from experimentally measured load histories










(864 kips, 5 16 kips, and 328 kips for tests PIT7, P3T3, and B3 T4 respectively) was applied to

the structures statically. Finally, cases C, F, and I represent the cases in which the AASHTO

equivalent static load is calculated--using a hydrodynamic mass coefficient of 1.05--from the

measured impact energies (494 kip-ft, 108 kip-ft, and 75 kip-ft for experiments PIT7, P3T3, and

B3T4 respectively), and then applied to the structure in a static manner. All dynamic effects

(such as structural and soil damping, inertial effects, and cyclic degradation behavior) are absent

in the static analysis for each model.

The experiments are broken down into two categories: 1.) high-energy impacts (those in

which there is enough impact energy to produce significant plastic deformation of the barge

bow), and 2.) low-energy impacts (those in which negligible plastic deformation of the barge

bow is observed). The PIT7 experiment--along with the majority of the Pl series tests--falls

into the former category, where as the P3 and B3 series tests falls into the latter.

Figure 3-16 presents key results for the foundation of each pier from each analysis. For the

high-energy impact PIT7, a comparison between cases A and B shows that for the same peak

load, the maximum dynamic pier displacement is 32-percent larger than the static displacement.

Furthermore, as expected, the pile shears and moments are also larger for the dynamic analysis.

This amplification in pier displacement, and ultimately response forces, indicates that inertial

effects play an important role in driving the pier motion.

In comparing the low-energy cases (cases D and E for test P3T3, and cases G and H for

test B3T4) the opposite is observed. The case D dynamic analysis yields a maximum

displacement that is 27-percent smaller than that resulting from the case E static analysis.

Likewise, with test B3T4, the case G dynamic analysis yields a maximum displacement that is

14-percent smaller than that obtained from the case H static analysis. For both tests, the










predicted pile shears and moments exhibit a similar decrease in the dynamic cases, as is

expected.

Including the AASHTO results (i.e. cases C, F, and I), it can be seen that for the high-

energy impact of test PIT7, the AASHTO procedure predicts a load that is higher than the peak

dynamic load. As a result, the predicted pier displacements from the AASHTO load case are

360-percent larger than the maximum dynamic pier displacement, and consequently, the

resulting pile shears and moments are also much larger. In contrast, for cases F and I, the

ASSHTO procedure predicts loads that are smaller than the experimentally measured peak

dynamic loads for tests P3T3 and B3T4 respectively. However, when resulting pier

displacements and pile forces are compared, these quantities are only slightly smaller than the

maximum dynamic quantities for the respective tests.

For the cases in which piers are in isolation, the column shears and moments are negligible

for the static cases, as expected, considering there is no restraint at the top of the pier. However,

for the B3T4 cases, significant forces are developed in the columns in the static cases as load

transfers to the superstructure. Thus, in Figure 3-17 the maximum column shears and moments

and bearing shear forces are compared for cases G, H, and I. The column shear and moment, and

the bearing shear force predicted for case H are 26, 27, and 28-percent of those predicted from

the dynamic case; and those from case I are 20, 21, and 22-percent of the dynamic quantities.

Given that demands on the foundation, such as pile shears and moments, are quite similar for the

B3T4 cases, the disparity between the column force results of the static and dynamic cases is

attributable to amplification factors from the inertia forces generated in the superstructure.

Although momentary, the effects of these inertia forces can account for as much as 70 to 80-

percent of the column forces generated. Thus, failure to account for dynamic restraint, in










addition to the static resistance, at the top of a pier can lead to unconservative column design

forces.

3.5 Observations

Field testing of the old St. George Island Causeway Bridge was undertaken to investigate

impact loads, as well as resulting pier responses, generated during barge-to-bridge collisions.

Results from these experiments were then used to calibrate several finite element models, the

results of which were used to interpret the experimental results, as well as identify sources of

resistance that could not be measured during the experiments. Subsequently, the results from

these analyses were used to conduct a preliminary assessment of the current AASHTO load

determination procedure.

Results indicate that mass-proportional forces, known as inertial forces, constitute a

significant source of resistance to the motion of a pier during the early stages of a barge impact.

Inertial forces tend to mobilize earlier in an impact than static or damping forces (displacement

and velocity-proportional forces respectively). However, as the pier reaches its peak velocity

and begins to decelerate, these inertial forces change from a source of resistance to pier motion

to an effective load that actually drives pier motion.

The effect of inertia forces is even more pronounced in the analysis of bridge structures, as

opposed to isolated piers, as the inertial resistance and effective loading due to the presence of a

superstructure can significantly increase the design forces in pier columns. Given the important

influences that inertial forces have on structural response, it is recommended that dynamic

analysis procedures, which are capable of accounting for such phenomena, be used in designing

and assessing bridges that may be subj ected to vessel collision loading.










Table 3-1 Summary of forces acting on the pier during test PIT7
Force Approx. maximum (kips) Time to peak (sec)
Applied loading 850 0.15
Inertial forces on the pier 250 0.10
Sum of pile shears 275 0.27
Pressure forces on cap+seal 200 0.20
Friction forces on cap+seal 500 0.25


Table 3-2 Dynamic and static analysis cases
Impact Analysis Load Max. load Pier/pile Soil
Cas cndtin tpe decrption (kip) behavior b ehavi or


Dynamic
Static
Static
Dynamic
Static
Static
Dynamic
Static
Static


Time-varying PI T7
Peak PIT7 load
AASHTO
Time-varying P3T3
Peak P3T3 load
AASHTO
Time-varying B3T4
Peak B3T4 load
AASHTO


864
864
1788
516
516
398
328
328
276


PI T7
PIT7
PIT7
P3T3
P3T3
P3T3
B3T4
B3T4
B3T4


A
B
C
D
E
F
G
H
I


Linear
Linear
Linear
Linear
Linear
Linear
Linear
Linear
Linear


Nonlinear
Nonlinear
Nonlinear
Nonlinear
Nonlinear
Nonlinear
Nonlinear
Nonlinear
Nonlinear

















Concrete girder
simple spans


To Saint George Island, South

Pier-4 (water-1me footmng)

-Pier-3 (water-hne footing)


To Eastpont, NorthNagaonca el.... ..

Pier-1 (mud-hne footing) /

Pier-2 (mud-hne footing) -



Figure 3-1 Overview of the layout of the bridge













28'-1.5"


4'-7.5"


1"/12"


Pile cap


Tremie .
seal lr--r --r--r-





s" 7 @5'-2"18



Figure 3-2 Schematic of Pier-1













24'-0"


Batter:
0.25"/12"


Pile cap | I Strut 4'-0" 1\i

MSL nn 4-" MSL

Ba 2" B H


20" x 20" Concrete piles-


24" 24"
2 4" / 12'-6" 48"ee 4









Figure 3-3 Schematic of Pier-3
























Figure 3-4 Test barge with payload impacting Pier-1 in the series Pl tests



















Figure 3-5 Series B3 tests A) Empty test barge used in the series B3 tests, B) Test barge
impacting the bridge at Pier-3 during the series B3 tests



























Figure 3-6 Pier-3 in isolation for the series P3 tests

























Figure 3-7 Pier-1 FB-MultiPier model



























Figure 3-8 Pier-3 FB-MultiPier model
















Pier-2


Pier-5


Figure 3-9 Bridge FB-MultiPier model













Experimentally measured impact load


02 04 06 08 1 12 l
Time (sec)


.Cap+seal passive+active
force (sum of lead and
trail sides)


Figure 3-10 Schematic of forces acting on Pier-1












1000


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Time (sec)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (sec)


1.1 1.2 1.3 1.4


Figure 3-11 Resistance forces mobilized during tests PIT7 A) Forces acting on the pier structure,
B) Cap and seal pressure and friction forces















Experimentally measured impact load
1000
900
S800
S700
~600
Fsoo


Inertia force of
pier and caps


Tune (sec)


Figure 3-12 Schematic of forces acting on Pier-3















500[ -*-- Applied load
Effective load
400 1 Sum of pile shears
STotal pile damping
300~- Inertia of pier+ caps
200

100

r01

-100

-200

-300


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Time (sec)



Figure 3-13 Resistance forces mobilized during tests P3T3


















Experimentally measured impact load


900



400

2600




10 0 0 04 06 08 1 12 14 Ipcfre
Time (sec) Sum of mlle shear


superstructure






1110[118 foTCe Of
pier and caps


Sum of pile dampmg
forces (atpile heads)


Figure 3-14 Schematic of forces acting on Pier-3 during test B3T4














400 Et~ective load
SSum of pile shears
-e- Total pile damping
300- Inertia of pier+caps
-*- Sum of bearing shears
S200


C 100



-00


-100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Time (sec)


Figure 3-15 Resistance forces mobilized during tests B3T4



























_I ~ ~ II I I


Case A P1T7 dynamic Case B P1T7 static OCase C P1T7 staticAASHTO


e



~ ~
~L ~L
% %


1


h


M "


Max pier
displacement


Max pile
shear


Max pile
moment


Max applied
load


Max applied Max pier Max pile Max pile
load displacement shear moment


Max applied Max pier Max pile Max pile
load displacement shear moment


Figure 3-16 Comparison dynamic and static analysis results for foundation of pier A) PIT4, B)
P3T3, C) B3T4































Max applied Max pier Max pier Max total
load column shear column moment bearing shear


Figure 3-17 Comparison of dynamic and static analysis results for pier structure









CHAPTER 4
BARGE FORCE-DEFORMATION RELATIONSHIPS

4.1 Introduction

During a barge-bridge collision, the magnitude of impact force generated on the bridge

structure is strongly related to the stiffness of the barge bow among other factors. Barge bow

stiffness in both the linear elastic and nonlinear inelastic ranges may be characterized by a

nonlinear force-deformation relationship (also referred to as a barge crush-curve). In this chapter,

the basis for the current AASHTO crush-curve is reviewed, and new crush-curves, based on

work conducted in this study and in the prior St. George Island barge Impact study, are proposed.

4.2 Review of the Current AASHTO Load Determination Procedure

As described in Chapter 2, barge impact load calculations per the current AASHTO

specifications make use of an empirical load calculation procedure. The equivalent static load

determination equations presented in Chapter 2 are based upon an experimental study conducted

by Meier-Dornberg (1983). As previously noted, Meier-Doirnberg conducted both static and

dynamic impact tests on reduced-scale European Type IIa barge bow sections. The European

type IIa barge is similar in size and configuration to the jumbo hopper barges widely used

throughout the United States. Two dynamic tests, conducted using 2-ton pendulum hammers

and two different shapes of impact head, were conducted on 1:4.5-scale stationary (i.e. fixed)

barge bows. One dynamic test involved three progressive impacts using a cylindrical hammer

with a diameter of 1.7 m (67.0 in), whereas the other involved three progressive impacts using a

ninety-degree pointed hammer. A static test was also conducted on a 1:6 scale barge bow using

a 2.3 m (90.6 in) hammer. Results obtained from the dynamics tests are shown in Figures 4-la

and b and results from the static test are shown in Figure 4-1c.









Using the experimental data collected, Meier-Dirnberg developed mathematical

relationships between kinetic energy (EB), inelaStic barge deformation (aB), and dynamic and

static force (PB and PB respectively). These relationships are illustrated in Figure 4-2. As the

Eigure suggests, no maj or differences were found between the magnitude of dynamic and static

impact force. However, this fact is likely due to the stationary barge bow configuration used in

the testing. Omission of a flexible impact target and the corresponding barge-pier interaction

necessarily precludes the ability to measure and capture dynamic amplification effects.

Inelastic barge bow deformations were, however, accounted for in the Meier-Diirnberg

study. Results showed that once the barge bow yielding was initiated, at approximately 4 in. of

deformation (aB), the stiffness of the bow diminishes significantly (see Figure 4-2).

Additionally, Meier-Diirnberg recognized that inelastic bow deformations represent a significant

form of energy dissipation.

In the development of the AASHTO barge impact design provisions, the relationships

between initial barge kinetic energy (KE), barge deformation (aB) and equivalent static force (PB)

developed by Meier-Dirnberg, were adopted with minimal modifications:

C WV2
KE = H(4-1)
29.2
562KE \12 10.24 -
aB = 1+-1(42

(4112aBRB if aB < 0.34
B l(1349+110aB)RB if aB > 0.34 43


In these equations, aB is the depth of barge crush deformation (ft), KE is the barge kinetic energy

(kip-ft), and RB = BB/35 where BB is the width of the barge (ft). The only notable difference

between the expressions developed by Meier-Diirnberg and the AASHTO expressions given









above as Eqn. 2-7 2-9 is the use of a barge width correction factor (RB). While the AASHTO

utilize RB term to reflect the influence of barge width, no such factor has been included to

account for variations in either the size (width) or geometric shape of the bridge pier being

impacted. In Figure 4-3, the AASHTO barge force-deformation relationship (crush-curve) given

by Eqn. 2-9 is plotted for a hopper barge having a width BB=3 5 ft and for a tanker barge having a

width of BB=50 ft.

4.3 High-Fidelity Finite Element Barge Models

In this study, the development of updated barge bow force-deformation relationships was

carried out by developing high-resolution finite element barge models. These models were

created for the purpose of conducting quasi-static crushing analyses to obtain force-deformation

data. The nonlinear explicit finite element code, LS-DYNA (LSTC 2003), was chosen to

conduct the quasi-static barge crushing analyses. LS-DYNA is capable of analyzing large scale

nonlinear plastic deformations associated with extreme levels of barge bow crush (up to 200 in);

can account for global and local member buckling of barge bow components; and is capable of

modeling contact not only between the barge and the bridge, but also between internal

components within the barge itself.

Regarding the selection of an explicit finite element code over an implicit one, if time

intervals between contact detection events are consistently very small relative to the total

analysis time, then explicit analyses may be more efficient than implicit methods. Given that the

models developed in this study involve complex contact definitions, most notably the internal

contact between components inside the barge, it was determined that time intervals between

contact events would be small enough to warrant the use of an explicit method.

Two distinct barge models were created: a jumbo hopper barge (Figure 4-4) and an

oversize tanker barge (Figure 4-5). The jumbo hopper barge is the baseline vessel upon which










the AASHTO barge impact provisions were based (AASHTO 1991) and is the most common

type of barge found operating on the inland waterway system in the United States. Modeling and

analysis of a jumbo hopper barge allows for direct comparisons between Einite element analysis

results and the current AASHTO barge impact design provisions. To ensure that adequate ranges

of barge sizes and configurations were included in this study, the oversize tanker barge was also

modeled and analyzed. Tanker barges are also very commonly found operating on the inland

waterway system are considerably different from hopper barges in terms of geometric size and

mass.

4.3.1 Jumbo Hopper Barge Finite Element Model

For the purposes of generating a finite element model of a jumbo hopper barge, detailed

structural plans were obtained from a barge manufacturer. From the point of view of structural

configuration, the barge consists of two sections: 1.) the barge bow, and 2.) the hopper, or cargo

area (Figure 4-6). Since the focus here is on characterizing the force-deformation relationship

that is associated with the bow section of the barge, only this portion was modeled in the finite

element models. Development of the jumbo hopper barge model was part of a larger funded

research program; and thus, a number of people were involved in the development effort.

Internal stiffening elements on the starboard and port sides of the barge model (added by Mr.

Michael Davidson and Mr. Cory Salzano) permitted corner crush analyses to be conducted on

the barge model (Figure 4-7). Similarly, a hopper guard (Figure 4-7) was also added to the

model (by Mr. Michael Davidson) to improve its accuracy for large-deformation crush analyses.

With the exception of these components, all other aspects of the hopper barge modeling effort

were carried out by the author.

Further, it is noted that the range of crush deformations simulated in this study (16 ft.) is

greater than the range of deformations generated during the Meier-Dornberg test program (which










produced equivalent full-scale crush depths of less than approximately 12 ft). As will be shown

later, peak force levels are generated at relatively low deformation levels, therefore modeling

only the bow section of the barge is adequate.

Hopper barges are fabricated from steel plates and standard structural steel shapes

(channels, angles, etc.) in both the bow and hopper regions. The bow of the hopper barge

considered in this study is 27 ft 6 in. long by 35 ft -0 in. wide, and is composed of fourteen

internal rake trusses, transverse stiffening members, and several external hull plates of various

thicknesses (Figure 4-7). Structural steel members are welded together with gusset plates to

form the internal rake truss members (Figure 4-8).

All of the components in this model were modeled using four-node shell elements,

mimicking the actual geometric shape of the barge members (Figure 4-7 and 4-8). With regard

to modeling the rake trusses, the use of shell elements, as opposed to resultant beam elements,

was necessary to capture buckling of the barge bow components. By using a high resolution

mesh of shell elements, it was possible to capture not only global buckling, but also local

buckling effects. Thus, special attention was given to the number of elements used to model the

legs of the structural steel shapes, such that the legs could exhibit reverse curvature during

buckling. Additionally, the use of shell elements to discretely model the internal structural

members of the barge allows these components to exhibit a local material failure, which is

represented by element deletion in LS-DYNA. If resultant beam elements had been used to

model the internal members, material failure and local buckling modes would have been absent

from the simulations and the force-deformation data obtained would have been adversely

affected.









Individual steel components in the physical barge are inter-connected through a collection

of continuous and intermittent welds. Since the finite element model was intended to mimic the

physical barge with reasonable accuracy, approximation in the model of the weld conditions was

necessary (Figure 4-9). In the LS-DYNA model, this was accomplished using

CONSTRAINEDSPOTWELD constraints (LSTC 2003) which allow the user to define a massless

spotweld between two nodes. The spotweld is effectively treated as a rigid beam connecting the

two nodes. Although an option is available in LS-DYNA to define a spotweld failure criteria, the

welds within this model are not permitted to fail. Instead, failure is represented only by way of

element deletions when element strains reach the defined material failure criteria. Hence, by

using a sufficient density and distribution of spotwelds, a reasonable approximation of the

behavior of the continuous and intermittent welds present in the physical barge was achieved.

Since severe structural deformations (up to 16 ft) were to be analyzed using this model, it

was important to incorporate a material model capable of exhibiting nonlinearity and failure.

Therefore, the MAT PIECEWISE LINEAR PLASTICITY material model available in LS-DYNA

was used to model all barge components. This is an elastic-plastic behavior model that allows

the user to define an arbitrary true-stress vs. effective-plastic-strain relationship. This material

model also permits specification of an effective plastic failure strain at which element failure

(and subsequent element deletion from the model) occur. Additionally, this model allows the

user to define arbitrary strain-rate dependency using the Cowper-Symonds model. However, for

the purposes of this study, strain-rate effects were not included in the barge bow model since the

crush speeds analyzed (48 in./sec) did not result in extremely high strain-rates.

With regard to steel material properties, an A36 structural steel was selected to model the

barge components because most barges fabricated in the United States are constructed from this









material. Thus, material properties (Table 4-1) and a truss-stress vs. effective-plastic-strain

relationship for A36 steel (Figure 4-10) were specified for all barge components.

Finite element based determination of barge crush curves involves calculating the force

that is generated by an impactor (e.g. pier column) deforming the bow of the barge. Such

analyses are achieved by Eixing the rear section of the barge bow model and then pushing a rigid

impactor (having the shape of an pier column) into the bow model at a constant prescribed

velocity (Figure 4-1 1). Finite element models of several different sizes and shapes of pier

columns were developed to capture the effects that pier shape and size have on barge-bow force-

deformation relationships. Each pier was modeled using eight-node solid brick elements.

To simulate contact interaction between the barge bow and the impactor, a contact

definition was defined using the LS-DYNA CONTACTUTOMATI C_NODES_TOSURFACE option.

Static and dynamic coefficients of friction, 0.5 and 0.3 respectively, and a soft-constraint

formulation were specified in this contact definition. A soft-constraint option was also used

because this method is typically more effective when the contact entities have dissimilar stiffness

or mesh density (LS-DYNA 2003).

In addition to the barge-pier contact interface, a contact definition capable of detecting

self-contact between all components within the barge bow was also defined. This contact differs

from the barge-pier contact in that there are not two distinct surfaces between which contact can

be defined. Rather, it is not known in advance which surfaces of the barge will come into

contact with each other during crushing deformation. Therefore, in LS-DYNA, the CONTACT_-

AUTOMATICSINGLESURFACE definition was specified. By including all barge bow elements in

this contact definition, it was possible to detect internal self-contacts within the barge bow.










4.3.2 Tanker Barge Finite Element Model

In addition to the jumbo hopper barge model, a finite element model of a tanker barge was

also developed (primarily by Mr. Long Bui) as part of the larger funded research program.

Similar to hopper barges, tanker barges are constructed from steel plates and trusses, where the

trusses are each composed of welded structural steel sections. In contrast to the hopper barge,

the tanker barge bow has twenty-three internal rake trusses instead of fourteen, and is 49'-6"

wide instead of 35'-0".

Based on results obtained from hopper barge analyses, it was determined that the peak

contact force occurs within the first 12 to 24 in. of deformation. Consequently, it was only

necessary to discretely model a small region near the headlog of the tanker barge, rather than the

entire bow. For computational efficiency, the barge model was divided into three zones, each

represented by a different mesh resolution (Figure 4-12).

Zone-1 represents the first 8.75 ft of the barge bow, the area that was expected to undergo

the most significant permanent deformations. As such, this zone was modeled using discrete

shell elements for both the outer hull plates and internal rake trusses-much like the hopper

barge was modeled--in order to capture local inelastic buckling effects. The tanker barge model

used the same material model as well as the spot weld definitions that were used to model the

hopper barge.

Zone-2 of the tanker barge model represented the 19.75 ft portion of the raked bow that is

aft of Zone-1. Internal trusses extend throughout the entire raked barge bow; however, the truss

and hull components in Zone-2 were not expected to sustain buckling, inelastic deformation, or

experience internal contact during typical impact events. As a result, numerically efficient

resultant beam elements were used to model the stiffness of the internal trusses in Zone-2. The

outer hull was modeled with shell elements, but with a lower resolution than that used in Zone-1.









Zone-3 of the model represents the remaining 166.5 ft of the barge, and no significant

deformations were expected in this zone. This zone was modeled using a coarse mesh of stiff

brick elements.

4.4 High-Fidelity Finite Element Barge Crush Analyses

Using finite element models of the jumbo hopper and tanker barges, quasi-static barge bow

crush simulations were conducted to determine force-deformation relationships for the barge

bow models. Such analyses provide a means of characterizing barge force-deformation

relationships in a manner that is independent of AASHTO and Meier-Doirnberg.

4.4.1 Finite Element Barge Bow Crush Simulations

Contact force data and barge bow deformation data were extracted and combined for each

crush simulation to produce load-deformation relationships for the barge bow. Both the hopper

and tanker barges were crushed using both flat and round pier-impactors of varying widths.

Most crush simulations involved centerline crushing of the barge model; however, several crush

analyses were conducted involving corner-zone crushing of the barge model. Corner-zone crush

analyses were conducted to investigate the effect pier position has on the contact force generated.

Each crush case is designated by a specific identification tag composed of two letters and a

number. The first letter represents the barge type; the second letter designates the pier shape;

and the two digit number indicates the size (cross-sectional width or diameter) of the pier in units

of feet. For example, a simulation involving a hopper barge crushed by a 6 ft wide flat faced pier

(e.g., square or rectangular in cross-section) is designated HFO6. Hopper and tanker barges are

denoted by H and T respectively, and the flat and round pier impact face geometries are denoted

F and R respectively.

Force-deformation data for the hopper and tanker barges crushed by a flat-faced square

pier are presented in Figures 4-14, 4-15, and 4-16 respectively. The most notable observation










regarding the flat faced pier data is that the peak force values achieved are dependent upon the

widths of the piers. Both the hopper and tanker force-deformation results indicate that as the

width of the pier increases, so does the peak contact force. This trend is attributable to the

internal configuration of stiffening trusses inside the barge. The number of internal rake trusses

that are directly engaged during impact and deformation increases as the pier width increases

(Figure 4-17). Hence, the stiffness of the barge effectively increases as the number of trusses

that are directly engaged during crushing increases. In the current AASHTO provisions, the

effect of pier width on vessel impact forces is not taken into account.

Force-deformation data for the hopper barge crushed by a round pier are presented in

Figures 4-18 and 4-19. Unlike the flat faced pier results, the peak contact forces generated in

round pier simulations are not strongly influenced by the width of the pier. During impact by a

round pier, internal stiffening trusses inside the barge are engaged gradually and sequentially

(Figure 4-20) as the deformation level increases. This is in contrast to the behavior for impact by

a flat faced pier where all of the trusses within the impact zone width are immediately engaged at

the same point in time.

In the AASHTO and Meier-Doirnberg equations, the peak impact force is computed based

on the maximum expected barge bow deformation (which is itself computed based on initial

kinetic impact energy). However, the simulation results presented above indicate that the peak

force often does not occur at the maximum sustained level of barge bow deformation.

Additionally, the AASHTO and Meier-Doirnberg equations exhibit a work-hardening

phenomenon in which the impact force levels continue to rise with increasing deformation depth

(recall Figure 4-2). However, data from finite element simulations indicate that no such

hardening occurs, and in many cases a softening of the barge bow occurs after the peak impact









force has been reached. At significant deformation levels, widespread buckling of the internal

stiffening trusses, combined with fracturing of the outer hull plates, produces this structural

softening, particularly in impacts that involve flat faced piers. Based on these observations, it is

apparent that an elastic, perfectly-plastic force-deformation curve is adequate to conservatively

describe bow crushing behavior (Figure 4-21).

4.4.2 Development of Barge Bow Force-Deformation Relationships

Barge crush analyses of the type described above produce highly detailed force-

deformation relationships that are specific to vessel type, pier shape and size, and impact

location. In this section, the detailed data obtained from these finite element crush simulations

are used to develop simplified barge crush models that are suitable for use in bridge design.

Examination of Figures 4-14, 4-15, 4-16, 4-18, and 4-19 indicates that barge bow contact

force either remains constant or decreases after the peak contact force has been reached.

Therefore, in order to envelope both of these situations, it is assumed that the barge bow will

exhibit an elastic-perfectly plastic load-deformation behavior (recall Figure 4-21). Despite the

fact that current AASHTO provisions utilize a barge width correction factor (RB) to account for

the relationship between impact force magnitude and barge width, it is important to note that

crush data obtained in this study indicate that no strong relationship of this type exists. Because

the hopper and tanker barges have different widths (35 ft and 50 ft, respectively), based upon the

current AASHTO specifications, it would be expected that the impact forces for a given pier size

would be quite different if impact by a hopper versus a tanker barge. However, the impact force

magnitudes obtained for both barge types are in close agreement, indicating that a correction

factor for barge width is not appropriate.

A key element in developing a simplified barge bow behavior model is the formulation of

a relationship between peak impact force and pier width. Formation of this relationship is









accomplished by filtering the barge bow force history data--obtained from the crush simulations

described in the previous section--using low-pass filter. For each crush simulation, the time

history of contact force was transformed from the time domain to the frequency domain using a

fast Fourier transform (FFT). Once in the frequency domain, data above 100 Hz were discarded.

An inverse transformation was then performed to obtain a Einal filtered force time history. For

each crush simulation, the filtered force data were scanned and the maximum (peak) impact

force for each simulation quantified. In Figure 4-22, peak impact forces obtained from this

procedure are plotted as a function of pier column width for both flat and round columns cross-

sectional shapes.

Inspection of the crush data for flat-faced piers indicates that one of two distinct

mechanisms determines the magnitude of peak force that can be achieved. In Figure 4-23, it is

evident that for flat-faced pier with widths of less than approximately 9 ft, the contact force

reaches a temporary peak magnitude within the first 2 to 4 in. of barge bow deformation. With

continued bow deformation, the contact force gradually increases up to a global peak magnitude

at deformation levels of 16 to 20 in. Detailed inspection of finite element results for the

moderate-width flat-faced piers reveals that the initial local maxima are associated with initial

yielding of the internal rake trusses. As the barge bow deforms beyond this condition, the

internal trusses begin to buckle, and combined membrane action in the outer hull plates begins to

control. When the hull plates exhibit significant yielding, the global peak force has been

reached. Thus, for moderate-width flat-faced piers, the membrane yield capacity of the barge

hull plates governs the global contact force maxima.

For flat-faced pier with widths of greater than approximately 9 ft, a maximum of force

magnitude-representing yielding of the internal trusses-also occurs within the first 2 to 4 in of









barge bow deformation. However, in contrast to moderate-width flat-faced piers, the force

maximum at the 2 to 4 in. deformation level in large-width piers represents the global force

maximum that will be achieved. Instead of the force level continuing to increase with additional

deformation, the contact force decreases as the barge bow deforms further and the internal

trusses buckle. At large deformations (greater than 20 in), the trusses have buckled, and

membrane action of the outer hull plates determines the magnitude of load generated. Thus, for

large-width flat-faced piers, the yield capacity of the internal trusses in the barge governs the

global maximum of contact force (and therefore impact load).

Force-deformation data for a 9 ft wide flat-faced pier indicates that the force maximums

associated with truss yielding and outer hull plate yielding are approximately equal in magnitude

(Figure 4-23). This fact suggests that a pier width of 9 ft represents the approximate transition

point between force maximums that are controlled by truss-yielding and those that are controlled

by yielding of the outer hull plates of the barge.

In Figure 4-24, force maxima associated with both barge truss yielding and outer hull

plate yielding are plotted for flat-faced piers as a function of pier width. Distinct linear trends

associated with each of these behavioral modes are clearly identified. For each mode of barge

deformation behavior, a linear regression trend line was fit through the data (Figure 4-24). For

data points associated with truss yielding, a least-squares linear curve fit through the origin

yields the following relationship between maximum force and pier width:

Py = 138.73wP (4-4)

where PBY is the barge yield force in kips, and wP is the width of the flat-faced pier in ft. A

general least-squares line was fit through the data points associated with outer hull plate yielding

produced the relationship:










PBY= 963.84 + 52.27wP


(4-5)


Equating Eqns. 4-4 and 4-5 and solving for the pier width at which the two trend lines intersect,

the transition point between force maxima controlled by truss-yield and plate-yield was

determined to be 11.15 ft.

Eqns. 4-4 and 4-5 are based on analyses of hopper and tanker models that are

representative of typical vessel fabrication practices employed in the United States (in terms of

steel strength, structural configuration, and steel plate thicknesses). Outer hull plate thicknesses

of barges very commonly range between 0.25 in and 0.625 in, and 91-percent of barges have

plate thicknesses of 0.3 125 in (Nguyen 1993) and a steel strength of 36 ksi is often used. As

noted earlier, the models analyzed in this study employed a 3/8 in. (0.375 in.) hull plate thickness

and 36 ksi steel. However, some variability in barge fabrication practices is possible.

Additionally, the data points used to develop the Eqns. 4-4 and 4-5 trend lines include some

variability and are not perfectly linear in form. Therefore, in order to develop conservative

equations that can be recommended for general purpose use in bridge design, some modifications

to Eqns. 4-4 and 4-5 were made.

First, the lines represented by Eqns. 4-4 and 4-5 were vertically shifted (while maintaining

the best fit slopes) so that all of the data points are completely enveloped by the shifted lines.

After vertical shifting, the regression lines were scaled to account for the possibility of higher

steel strengths and/or thicker hull plates. Increases in either of these parameters would lead to an

increase in impact force levels. Steel having a yield strength of 50 ksi rather than 36 ksi

corresponds to a yield strength ratio of (50 / 36) = 1.389. Similarly, a hull plate thickness of

1/2 in. (0.5 in.) instead of 3/8 in. leads to a plate thickness ratio of (0.5 / 0.375) = 1.33. Hence,

after shifting, Eqns. 4-4 and 4-5 were each scaled by a factor of 1.33. Finally, the resulting









coefficients in the equations were rounded off, resulting in simplified design equations for the

peak impact force as a function pier width for flat-face pier columns:

Pgy=1300 + 80wP fP<10 ft (4-6)
Py = 300 + 180wP Pfw >10 ft (4-7)


where PBY is the peak impact force in kips. Eqns. 4-6 and 4-7 are graphically illustrated in Figure

4-25. By equating Eqns. 4-6 and 4-7 and solving for the pier width at which the equations

intersect, the transition point between the truss-yield and plate-yield dominated modes of barge

behavior was determined to be 10 ft.

Pier columns having round (circular) cross-sectional shapes also exhibited two distinct

mechanisms in terms of predicting maximum magnitude force levels (Figure 4-26). Moreover,

the transition point between these two modes was found to be located approximately at a pier

diameter of 9 ft, similar to the transition point for flat-faced pier columns. Hence, development

of a relationship between maximum impact force and pier column diameter was carried out in a

manner analogous that used for flat-faced piers.

Linear regression was used to fit trend lines through the moderate diameter and large

diameter round pier crush data (Figure 4-26). For moderate-diameter round pier columns, the

least-squares trend line was:

PBY = 975 +31.02wP (4-8)

A least-squares trend line fit through the large-diameter round pier data produces the following

relationship:

PBY = 1079.25 +18.54wP (4-9)


Equating Eqns. 4-8 and 4-9, and solving for the pier diameter at the intersection of the trend

lines, the pier diameter transition point was calculated to be 8.36 ft.









As with the flat-faced pier data, the round pier regression lines were first vertically shifted

so as to envelope all data points, and then scaled by a factor of 1.33. After rounding off the

coefficients, the resulting equations for barge yield force as a function of circular pier diameter

were :

PBY = 1300 +40wP Pfw <10 ft (4-10)
PBY = 1500 +20wP Pfw >10 ft (4-11)

These equations are graphically illustrated in Figure 4-27.

Construction of a complete design barge crush curve (recall Figure 4-21) requires not only

determination of the barge crush force (PBY) but also the barge yield deformation (aBY). The first

step in calculating the barge bow deformation at yield is to extract the initial barge bow stiffness

from the force-deformation data. The initial stiffness for each case was calculated by taking the

slope of the line connecting the first two data points of each curve in Figures 4-14, 4-15, 4-16, 4-

18, and 4-19. In Figure 4-28, the initial stiffnesses computed in this manner are plotted as a

function of pier width for all crush analysis cases.

Taking the peak force values from Figure 4-22, and dividing by the respective stiffnesses

from Figure 4-28 yields barge bow deformations that--in conjunction with the peak force

value--envelope the force-deformation data. In Figure 4-29, barge bow deformations at yield

are plotted as a function of pier width. In most design situations, pier columns will range from

4 ft to 12 ft in width. Consequently, based on the data show in Figure 4-29, barge bow yield

deformations of 0.5 in, and 2 in. were selected for flat-faced and round piers respectively.

4.4.3 Summary of Barge Bow Force-Deformation Relationships

Figure 4-30 presents a flowchart for calculating the force-deformation relationship for the

barge bow. Based upon the geometry and size of the pier column on the impact pier, the design

barge yield force is calculated. Additionally, based upon pier column geometry, the barge bow









deformation at yield is selected, and the barge bow force-deformation relationship is defined.

The resulting force-deformation relationship of Figure 4-30 is a necessary component of all of

the analysis methods presented in later chapters. This elastic, perfectly-plastic relationship

ultimately determines the magnitude and duration of the force imparted to a bridge structure

during an impact.










Table 4-1 Barge material properties
Parameter
Elastic modulus
Poisson's ratio
Unit weight
Failure strain
Yield stress
Ultimate stress


Value
29000 ksi
0.33
490 pcf
0.2
36 ksi
58 ksi (eng. stress)
69.8 ksi (true stress)











Bow Defonnation (m)
0 0.8 1.6 2.4 3.2
3,000
12

2,000 -


S1,000l es data "' 11


0 0
0 30 60 90 120 150
Bow Defonnation (in)A



Bow Defonnation (m)
0 0.8 1.6 2.4 3.2
3,000
12

2,000 ..--

;4" / Y I~ Test danta 4
1 ,000~J -V I Em elope 1 4


0 0
0 30 60 90 120 150
Bow Defonnation (in)B


Bow Defonnation (m)
0 0.8 1.6 2.4 3.2
3,000
Tet daa -12

2,000





0 0
0 30 60 90 120 150
Bow Defonnation (in)


Figure 4-1 Force-deformation results obtained by Meier-Doirnberg
(Adapted from Meier-Doirnberg 1983) A) Results from dynamic cylindrical impact
hammer test, B) Results from dynamic 900 pointed impact hammer test, and C)
Results from static impact hammer test












Bow defonnation (ft)
0 2.5 5 7.5
3,000




2,00


1,00


0 30 60 90 120 150
Bow defonnation (in)


Figure 4-2 Relationships developed from experimental barge impact tests conducted by Meier-
Doirnberg (1976) (Adapted from AASHTO 1991)











4,000


-2 3,000


S2,000


1,000 III Hopper barge (35 ft)
Y Tanker barge (50ft)


0 30 60 90 120 150
Bow deformation (in)


Figure 4-3 AASHTO barge force-deformation relationship for hopper and tanker barges











195'-0"






m aStarboard F

A


167'-6" 27'-6"





B


Figure 4-4 Hopper barge dimensions A) Plan view, B) Elevation
















Port




Starboard





255'-0" 35'-0"


290'-0"


Figure 4-5 Tanker barge dimensions A) Plan view, B) Elevation










195'-0"


Not modeled t Modeled


Figure 4-6 Hopper barge schematic































Figure 4-7 Hopper barge bow model with cut-section showing internal structure













with shell dentent


=Headl
-Ih.I-ntel


Figure 4-8 Internal rake truss model
























Figure 4-9 Use of spot weld constraints to connect structural components





0.05


0.1 0.15
Nominal strain


0.25


80


S60






S20


0.05


0.1 0.15
Effective true plastic strain


0.25


Figure 4-10 A36 stress-strain curve A) Nominal stress vs. nominal strain,
B) Effective true stress vs. effective true plastic strain

























Figure 4-11 Barge bow model with a six-foot square impactor



























109


aiR~M:lllll:ollirr
I;
11 j

'CI- I





























used for plates
in Zone-2


Shell elements


used in Zone-3 r- l1-~,1


ii__i i
I,


____-;I-L -I-~-~;


usd otti a
in Zone-2



Figure 4-12 Tanker barge bow model




















































110













Barge or pier) Barge
Models Model
~- .- . . ..- P .-.- ...... ........................ -- P ier
wPModel
Pier PB
Model

A B


Figure 4-13 Crush analysis models A) Centerline crushing of barge bow model,
B) Corner-zone crushing of barge bow model











5,000


4,000 1 HFO3 0f HFl8
M- HFO6 H- HF35
SHFO9
3,000


F~2,000

1,000


0 6 12 18 24
Bow defonnation (in)A



5,000

4,00011 1 HF01 HFl2
SHFO3 0f HFl8
S3,000 M HFO6 H HF35
;4~ ^HFO9

F~2,000

1,000


0 50 100 150 200
Bow defonnation (in)B


Figure 4-14 Hopper barge bow force-deformation data for flat piers subj ected to centerline
crushing A) Low deformation, B) High deformation











5,000


4,000

.9 3,000


F~2,000

1,000


0 6 12 18 24
Bow defonnation (in)A



5,000

4,000 H HFO6 Y HFl8

3,000


F~2,000

1,000


0 50 100 150 200
Bow defonnation (in)B


Figure 4-15 Hopper barge bow force-deformation data for flat piers subj ected to corner-zone
crushing A) Low deformation, B) High deformation











5,000


3,000

2,3000



1,000


0 6 12 18 24
Bow defonnation (in)


Figure 4-16 Tanker barge bow force-deformation data for flat piers
subj ected to centerline crushing
















I~i -A
1.,.lI:t-iiI



E ( IB1I~

Figure~~~~~~~~~~~~~~~~~1 4-17 Reainhpo irwdh oeggdtussA 6f iefa ir(F6
B) A12 t wde latpe H 2












H HRO3 0 HR12
0 HRO6 0 HR18
M- HRO9 0f HR35









D 6 12 18 24
Bow defonnation (in)





H HRO3 H HR12
0 HRO6 0 HR18
M- HRO9 0f HR35








D 50 100 150 20(
Bow defonnation (in)


5,000


4,000

.9 3,000


S2,000

1,000


5,000

4,000

83,000


S2,000

1,000


Figure 4-18 Hopper barge bow force-deformation data for round piers subj ected to centerline
crushing A) Low deformation, B) High deformation













4,000 IHe HnRO6

.9 3,000

F~2,000

1,000


0 6 12 18 24
Bow defonnation (in)A


5,000

4,000 IHe HnRO6


3,000

F~2,000

1,000


0 50 100 150 200
Bow defonnation (in)B


Figure 4-19 Hopper barge bow force-deformation data for flat piers subj ected to corner-zone
crushing A) Low deformation, B) High deformation


5,000

















1111





II-::




-


I jl
11~1~'~11
..'
1111
II


__

:
:
:'~~ .


'~ '


Figure 4-20 Gradual increase in number of trusses engaged with deformation in round pier

simulations A) A 6 ft diameter round pier at 10 in of deformation, B) A 6 ft diameter

round pier at 20 in of deformation


















































118














PBY






aB owdfrmto






Figur 4-21 Elsicpretlypastic ag o fredfrmto uv











5,000


0
O O Hopper, flat (HF) center crush
5 5 Ho er flat (HF) corner crush


4,000 IIrVIU 11jVIIIVUI
O O Tanker, flat (TF) center crush
a O O Hopper, round (HR) center crush
*Hopper, round (HR) corner crush
S3,000


S2,000
o Bo o




0 9 18 27 36
Pier width (ft)


Figure 4-22 Peak barge contact force versus pier width










5,000


4,0001 r H/ HFO3
0 HFO9

-A 3,000 M H3

0 ~Max. associated with plat
( 2,000


1,000



0 9 18 27
Bow defonnation (in)


Figure 4-23 Peak barge contact force versus pier width











5,000
O O Truss-yield controlled
4,0 O Plate-yield controlled
Tus-ildrersso
Plate-yield regression

S3,000


S2,000


1,000



0 9 18 27 36
Pier width (ft)


Figure 4-24 Comparison of truss-yield controlled peak force versus plate-yield controlled peak
force










6,000


5,000 1 wp = 10 ft


S4,000 1 1300 + 80w, .-


t:3,000 .*

8 ..* O O HF center
f 2,000.*
5 5 HF corner
..--ET" O TF center
1,000 El--- B ----------- Flat regression



0 6 12 18 24 30 36
Pier width (ft)


Figure 4-25 Design curve for peak impact force versus flat pier width











2,000


2 1,50 ---***


1,5000



8 O O HRdata
Low-diameter regression
& 500 Large-diameter regression




0 9 18 27 36
Pier diameter (ft)


Figure 4-26 Comparison of low diameter peak force versus large diameter peak force










2,500


2,000 |Iw, = IUnI


ai 1,500 ..--



O O O HR center
tfi* HR corner
P~500 Round regression
----------- Round design



0 6 12 18 24 30 36
Pier diameter (ft)


Figure 4-27 Design curve for peak impact force versus round pier diameter











10,000

0 O HF center
S HF corner
7,500 O O TF center
O O HR center
HR corner


OO


2,500 B

O O
OO O OO

0 9 18 27 36
Pier width (ft)


Figure 4-28 Initial barge bow stiffness as a function of pier width












O O
O O

O O O HF center
S HF corner
O O TF center
O O HR center
*HR corner

O Round pier





3 9 18 27 3(
Pier width (ft)


2




1 ,


0.


Figure 4-29 Barge bow deformation at yield versus pier width










..---------------------------------------- Barge Bow Force-Deformation ------------------------------------------
Impact Force I 'e ounsaeIImpact Force


aBY Bow deformation


Figure 4-30 Barge bow force-deformation flowchart














128









CHAPTER 5
COUPLED VESSEL IMPACT ANALYSIS AND SIMPLIFIED ONE-PIER TWO-SPAN
STRUCTURAL MODELING

5.1 Coupled Vessel Impact Analysis

An efficient coupled vessel impact analysis (CVIA) procedure involving coupling a single

degree-of-freedom (DOF) nonlinear barge model to a multi-DOF nonlinear bridge model was

developed by University of Florida researchers (Consolazio and Cowan 2005). As part of this

method, dynamic barge and bridge behavior are analyzed separately in distinct code modules that

are then linked together (Figure 5-1) through a common impact-force. The primary advantage of

this modular approach is that the barge behavior is encapsulated within a self-contained module

that can be integrated into existing nonlinear dynamic bridge analysis codes with relative ease.

For this research, the single-DOF barge module is integrated into the multi-DOF commercial pier

analysis program FB-MultiPier (BSI 2005).

5.1.1 Nonlinear Barge Bow Behavior

Analysis of barge-bridge collision events requires both the nonlinear barge bow behavior

and the inertial properties of the barge be modeled accurately. In this study, nonlinear barge bow

behavior is modeled using the force-deformation relationships developed in Chapter 4. The

entire mass of the barge is represented as a single degree-of-freedom (SDOF) mass which is

coupled to the bridge model at the point of contact through the contact force.

During barge collisions with bridge structures, the bow section of the barge will typically

undergo permanent plastic deformation (Figure 5-2). However, depending upon the barge speed

and type, and pier flexibility, dynamic fluctuations may occur that result in inelastic loading,

unloading, and subsequent reloading of the barge bow. This behavior is represented by tracking

the bow deformation of the barge throughout the impact analysis. Figure 5-3 illustrates the

various stages of barge bow loading, unloading, and reloading.









Whenever the barge is in contact with the pier, the barge bow deformation is computed as

the difference in the barge and pier displacements, (uB and uP TOSpectively):

aB UB UP (5-1)

Once contact between the barge and bridge models is detected, the barge bow loads

elastically until the barge bow deformation level exceeds the yield deformation of the barge

(aBY). As loading of the barge bow continues, the barge bow undergoes plastic deformation

(Figure 5-3a). Eventually, the velocity of the barge will drop below the velocity of the pier, and

the contact force (PB) between them will diminish. As this occurs, the bow deformation will also

begin to drop (Figure 5-3b) from the maximum barge bow deformation level (aemax) to the

residual plastic deformation level (aBP). Once the barge and pier are fully out of contact, the

contact force drops to zero, and all elastic deformation (aemax aBP) will have been recovered.

When the barge bow deformation level is below the plastic deformation level (aB < BP), the

contact force remains zero (Figure 5-3c). However, if reloading subsequently occurs, the contact

force will load back up the unloading-reloading portion of the force-deformation curve

(Figure 5-3d) to the original loading curve at maximum deformation. If loading continues

beyond the maximum deformation, plastic deformation will continue to accrue.

Algorithmically, the force-deformation relationship for the barge bow (Figure 5-3) is

modeled using a nonlinear compression-only spring. Data for this spring are obtained from the

relationships developed in Chapter 4. Regarding the unloading portion of the barge bow

relationship, unloading curves may consist of either linear curves equal in slope to the initial

slope of the loading curve (Figure 5-4a), or general nonlinear, deformation-dependent, unloading

curves (Figure 5-4b). In the latter case, each unloading curve must be specified with a

corresponding maximum barge bow deformation (aemax -










Computationally, the deformation (aemax) at the onset of unloading will not coincide with

the specified maximum deformation values given prior to analysis. Thus, to determine the

unloading relationship, it is necessary to find the two unloading curves, and associated maximum

barge bow deformation values, that bracket the desired unloading behavior. Using the bounding

curve data, the intermediate unloading curve is generated for the current maximum deformation

(aemax) through linear interpolation (Figure 5-5).

5.1.2 Time-Integration of Barge Equation of Motion

The analytical procedure presented here and in the next section is based on earlier work

conducted by Hendrix (2003), Consolazio et al. (2004) and Consolazio and Cowan (2005).

Given that the force-deformation behavior of the barge bow is nonlinear, the only practical

procedure to time-integrate the barge equation of motion is to use numerical procedures. In

development of this method, inertial effects from the barge are accounted for through use of the

SDOF barge mass. Additionally, energy dissipation as a result of inelastic deformation of the

barge bow is accounted for using the loading-unloading behavior described above. However,

any influence from the water in which the barge is suspended is neglected, as such effects are

beyond the scope of this study.

Recalling Figure 5-1, the equation of motion for the barge (single degree of freedom,

SDOF) is written as:

mBUB, = PB (5 -2)


where mB is the barge mass, tiB is the barge acceleration (or deceleration), and PB is the contact

force acting between the barge and pier. Evaluating Eqn. 5-2 at time-t--with the assumption that

the mass remains constant:

mB tuB= _PB (5-3)









where t iB, and tPB are the barge acceleration and force at time-t. The acceleration of the barge at

time-t is estimated using the central difference equation:


tuB_ thB-t thB (5-4)


where h is the time step size; and t+hUB, tuB, and t-hUB are the barge displacements at times t+h, t,

and t-h respectively. Substituting Eqn. 5-4 into Eqn. 5-3 yields the explicit integration central

difference method (CDM) dynamic update equation:

t~UB _T tP~h2mB)+2tu _t-UB (5-5)

which uses data at times t and t-h to predict the displacement (t+hUB) of the barge at time t+h.

Using the force-deformation relationships described earlier, the force tPB is computed.

5.1.3 Coupling Between Barge and Pier

As previously mentioned, coupling of the barge model to the pier model is accomplished

through the shared contact force between the two. Analysis of the two separate models is

handled in distinct code modules. Algorithmically, the overall time-integration process is

controlled by the pier module (Figure 5-6), while determination and convergence of the contact

force is controlled by the barge module (Figure 5-7).

At the beginning of each time step, the barge module is called to calculate an initial

estimate of the barge impact force. Taking the difference between the displacement of the barge

and the component of pier motion in the direction of barge motion, the crush depth is calculated:

aB = max(uB -UPB,aBP) (5-6)

where, uB is the displacement of the barge, uPB is the component of the impacted pier

displacement in the direction of barge motion, and aBP is the plastic deformation of the barge









bow. In Eqn. 5-6, the component of the pier displacement in the direction of barge motion is

calculated as:


uPB = UPx COS DB + UPy Sin BB) (5-7)


where uPx and uPy are the displacements of the impact point on the impacted pier in global x and

y-directions, and 8B is the angle of traj ectory of the barge relative to the global x-direction

(Figure 5-8a).

The crush depth in turn is used to calculate the barge impact force using the barge bow

force-deformation relationship. Using an iterative variation of the central difference method

(CDM), the displacement of the barge is calculated from the impact force and barge equation of

motion, and the barge impact force is updated to convergence and resolved into x and

y-components :

Pnx = PB COS DB) (5-8)
P~y = PB Sin BB) (5-9)

where PBx and PBy are the components of the impact force in the global x and y-directions.

These force components are returned to the pier module, and inserted into the proper location

within the external force vector.

Using the current estimate for the external load vector, the pier continues through time-step

integration to convergence. Upon convergence of the pier module, the x and y components of

the pier displacement are returned to the barge module. The barge module begins the

convergence procedure over again until the contact force is updated. The contact force for the

current cycle is compared to the force from the previous cycle, and if this difference is within the

convergence tolerance, the entire system has achieved convergence, and the program continues

to the next time step.









5.2 One-Pier Two-Span Simplified Bridge Modeling Technique

In concept, the CVIA procedure discussed above is easily extrapolated to use in barge

impacts of full bridge structures, as opposed to individual piers. However, based upon current

computer processing speeds, coupled vessel impact analyses of full bridge models may require

up to several hours of processing time. In a bridge design setting, analysis times of several hours

would generally be considered unacceptable. For this reason, a simplified one-pier two-span

(OPTS) structural modeling method that consists of the impact pier and the two adj acent

superstructure spans has been developed at the University of Florida (UF) (Davidson 2007).

In a barge-to-bridge collision, a significant fraction of the imparted lateral impact load is

transferred to the superstructure due to the displacement and acceleration proportional

resistances associated with the superstructure (discussed in Chapter 3). Any simplified bridge

modeling technique that is intended for use in dynamic analyses must therefore be able to

account for the complex interactions of inertial forces associated with the superstructure.

5.2.1 Effective Linearly Independent Stiffness Approximation

A key aspect of the OPTS modeling technique is the representation of the stiffness of the

bridge beyond the bounds of the OPTS model. In the OPTS model, this stiffness is

approximated using six linearly independent equivalent translational and rotational springs at the

far end of each of the two spans (Figure 5-9). Determination and application of these springs is

done using three distinct sections of the full bridge model: 1) the down-station piers and spans up

to the pier directly down-station from the impact pier, 2) the OPTS portion of the model, and 3)

the up-station piers and spans from the pier directly up-station from the impact pier (Figure 5-

10).

For both peripheral (i.e. the down and up-station) models, six unit forces are applied--one

in each of the six model degrees-of-freedom-at the point of separation from the OPTS model









(Figure 5-11). After analyzing each peripheral model statically, the three translational and three

rotational displacements at the application point are computed (Figure 5-12). Dividing each

applied load by the respective displacement, six linearly independent stiffnesses are obtained:

kx = Fx /u x (5-10)
k, = F, fu (5 11)
kz = Fz /uz (5-12)
krx = Mx /6x (5-13)
k, = M /6, (5-14)
krz = Mz /8z (5-15)

where k, F, and u are the translational stiffnesses, forces, and displacements respectively; kr, M,

and 6 are the rotational stiffnesses, forces, and displacements respectively; and the subscripts x,

y, and z represent the three orthogonal directions respectively. Using these stiffnesses, the six

springs may be added to each span of the OPTS model (Figure 5-9).

5.2.2 Effective Lumped Mass Approximation

An additional aspect of the OPTS modeling technique is the representation of inertial

properties of the bridge beyond the bounds of the OPTS model. The inertial properties of the

peripheral models are approximated by placing concentrated (i.e. lumped) masses at the far ends

of spans in each of the three global translational directions (Figure 5-13). These effective masses

are based on tributary lengths equal to the half superstructure span lengths adj acent to the OPTS

model (Figure 5-14):


mer = pAL (5-16)


where mesf is the effective lumped mass; and p, A, and L are the mass density, cross-sectional

area, and L is the length of the superstructure span.









5.3 Coupled Vessel Impact Analysis of One-Pier Two-Span Bridge Models

Both the coupled vessel impact analysis (CVIA) procedure and the one-pier two-span

(OPTS) model were implemented in the commercial bridge analysis software, FB-MultiPier (BSI

2005) in this research. Coupled impact analyses have been shown to produce force and

displacement time-histories that agree well with results obtained from both high-resolution finite

element simulations (Consolazio et al. 2004, and Consolazio and Cowan 2005), as well as

full-scale experimental data (Davidson 2007). Given the agreement between the CVIA method

and both high-resolution finite element and experimental results, CVIA techniques are able to

capture dynamic effects--effects precluded from a static analysis--in barge-to-bridge collisions.

With regards to the accuracy of OPTS models, excellent agreement has been observed

between coupled analysis results from full-resolution bridge models and respective OPTS

models during the collision phase in which the barge and pier are in contact (Davidson 2007).

During the free-vibration phase--once the barge and pier are no longer in contact--good

agreement is observed between the results from both models. Additionally, the time required to

analyze an OPTS model is significantly less than that required to analyze a corresponding

full-resolution multiple-span multiple-pier model. Therefore, OPTS models are significantly

more computationally efficient than full-resolution models while retaining the ability to capture

relevant dynamic effects.



















Barge trusnam~~are ouw~2L~ r
I I I I
Single DOF barge model Multi-DOF bridge model

Figure 5-1 Barge and pier modeled as separate but coupled modules (After Consolazio and
Cowan 2005)








































Figure 5-2 Permanent plastic deformation of a barge bow after an impact
























































138


~---E~fsi~~

;r



~3-~;~~ -44
i;''~'
_ET :~~
~r*

r*;*: IrY*Jr- i



.JI~P'LI jlL































pB pB


't.
''
''





i/I

C~P a
Bmax


c~ aB,,


Figure 5-3 Stages of barge crush A) Loading, B) Unloading,
D) Reloading and continued plastic deformation


C) Barge not in contact with pier,
(After Consolazio and Cowan 2005)
























Crush depth (aB )A


Crush depth (a,)B


Figure 5-4 Unloading curves A) Linear curves equal in slope to the initial slope of the loading
curve, B) General nonlinear, deformation-dependent, unloading curves













h
a~
v

o
F4


Pata
B Bmax


a Maximum crush deformation
Bm sustained prior to unloadmng

SExtrapolated portion of unloadmng curve
: / for deformation levelagmadz+7)


Intermediate unloadmng curve
Is generated by mnterpolatmg
at multiple load levels (PB)
between two boundmng curves


Crush depth (a,)


Figure 5-5 Generation of intermediate unloading curves by interpolation (After Consolazio and

Cowan 2005)













c---------------- ------ Pier/soil module **********************-


Imltlahze pler/soil module ...Barge module ..
mode= INIT*
"""""""""~~~~~~"""* Imltlahze state of barge
SInstruct barge module to perform imutlahzation a s (see barge module for details)
t=0


{R} = {} Imltlahze internal force vector
[K],[M],[C] form stiffness, mass, dampmg
[K]= fn?([K],[n],[[C]) form effective stiffness
For each time step I= 1,
Time for which a solution Is sought Is denoted as t+h
Form external load vector fih {F}

Extract displacements of pier lt mode = CALC, ux, up sent -- ag oue--
Impact point Ursus from {"u} Compute barge impact force
Pg based on displacements of
Barge module returns computed computed P, S, returned the pier Up and barge UB
barge impact forces PBx FBy (see barge module for details)

{F} = {F} tP~x F~y msert barge forces mnto external load vector|

{F}={F}- {R}+ ~fn([M],[C]) form effective force vector


For each Iteration} = 1,

t+h {u}= t {u}+ {au} update estimate of
[K]= f~t~h dsplacements at time t+h

[K]= ~fn([K],[M],[C])


{F}= {F}- {R}+f([M],[C])
{du} =[K] {F}



Check for convergence
of pier and soil

max({|8u|}) TOL

yes

max({| F }) TL ...........Barge module ..

yes Check for convergence of
mode = CONV coupled barge/pler/soil system
by checkmg convergence of
barge force predictions
Form new external (see barge module for details)
load vector with updated Fa Payretumed
updated P,.P b no |AP,| TOL
{F}= {F} PBx=FBy i yes


Record converged
pier, soil, barge data
and advance to next converged PBx FBy
time step are returned





Figure 5-6 Flow-chart for nonlinear dynamic pier/soil control module (After Consolazio and
Cowan 2005)













Entry pomt
-------- ------- Bargemodule ----------


mode=INIT
no





= 0 mtlahlze Internal barge module cycle counter


l~i~/Return to pier/soil module



= +1 merement mternal cycle counter


For each iteration k= 1, Iterative
t~hu -PBk) h mB+ 2 thuB central
difference
t~ha t~uB _t~hPB)method
P8'k) =fn(t+haB~p t agmax) update
MB= Bk)_(k-1) ye lotp

P, ~TOL damping
(relaxation)

AP(k B~) (k-1)
4(k) _(k-1) B(k)




mode=CALC
no
yes

pirmodule

mode=CONV

yes
If mode=CONV, then the pier/soil module has converged
Now, determine If the overall coupled barge/pler/soil
system has converged by exammg the difference between
barge forces for current cycle and previous cycle



aP," ~TOL
yes
no

Convergence not achieved cmuemrmna

M (e) = (1-p)AP, ) + p A, ,_) bargehfo ce usmng a

p (e) _p (-1) aP(e) compute updated barge

~ Return barge forces PBx ) and Pay () for current cycle


-Coupled barge/pler/soil system has converged, thus t+huB
Is now a converged value Update displacement data for
next time step
t-hu_ tu andrg tg tuhu

agb tagpmax, save convergedbarge crushparameters
~- Return to pier/soil module for next time step






Figure 5-7 Flow-chart for nonlinear dynamic barge module (After Consolazio and Cowan 2005)





143






















an= ecos(8)+ up,sin(8B)

.. A










ps?~p, cos(8B)-



Figure 5-8 Treatment of oblique collision conditions A) Displacement transformation,
B) Force transformation (After Consolazio and Cowan 2005)
































144



















Pier-3 1,





Figure 5-9 OPTS model with linearly independent springs





















Pier-1


Pier-2 ~


Up-station
model


Pier-3


Pier-4&


Pier-5


Figure 5-10 Full bridge model with impact pier

















Pier-1 ~ ; I" Pier-4


Pier-5
Pier-2 A B


Figure 5-11 Peripheral models with applied loads A) Down-station model, B) Up-station model


















Pie- /~ Pir-

Pier-12
Pier-2Pier-5~
A B


Figure 5-12 Displacements of peripheral models A) Down-station displacements, B) Up-station
displacements






















Figure 5-13 OPTS model with lumped mass


















Pier-1 ie-


Pier-5
Pier-2 A B


Figure 5-14 Tributary area of peripheral models for lumped mass calculation A) Down-station
model, B) Up-station model









CHAPTER 6
APPLIED VESSEL IMPACT LOAD HISTORY METHOD

6.1 Introduction

In previous chapters, a coupled vessel impact analysis (CVIA) procedure, which was

implemented in FB-MultiPier, was validated against experimental data, and demonstrated for

several different bridges. In a coupled analysis, the dynamic impact load is computed as part of

the analysis procedure. In this chapter, an alternative analysis approach is presented in which an

approximate time history of impact force is generated using characteristics of the vessel (barge

mass, initial velocity, and bow force-deformation relationship). The approximate time history of

impact load is then externally applied to the bridge structure, and a traditional (non-coupled)

dynamic analysis is performed to determine member design forces.

6.2 Development of Load Prediction Equations

The equations from which the load history is calculated are based upon the principles of

conservation of energy and conservation of linear momentum. Development of the applied

vessel impact load (AVIL) equations is based primarily on characteristics of the barge, although

basic bridge structure characteristics are also incorporated.

6.2.1 Prediction of Peak Impact Load from Conservation of Energy

The first step in calculating the impact load time history is to determine the peak dynamic

load to which the structure is subj ected using the principle of conservation of energy. Assuming

that the system is not sensitive to changes in temperature, conservation of energy for the system

can be expressed as follows:

AKEif + ADEif = 0 (6-1)


where AKEif is the change in kinetic energy of the barge, and ADEif is the change in total

deformation energy (i.e. the sum of the elastic and plastic deformation energies), associated with









the deformation of the barge bow, from the initial state (i) to the Einal state (f). Conservation of

energy is used to define a relationship between the peak impact load, and the barge parameters.

Assuming that the change in mass of the barge is negligible, the change in barge kinetic

energy can be expressed by the following relation:

AKEif =1/2m,(vb f i (6-2)

where mB is the constant (unchanging) mass of the barge and Vef and Vei are the magnitudes of

the barge velocities at the initial and Einal states respectively.

In general, the deformation energy for the barge can be described using the following

relation:


ADEif = as PB(aB)daB (6-3)


where PB RB) is the impact force as a function of barge crush depth (aB), and aei and aef are the

barge crush depths at the initial and final states respectively.

To calculate the peak impact load on the pier, the following assumptions are made: 1.) the

pier is assumed to be rigid and Eixed in space, 2.) the initial barge crush depth (aei) is assumed to

be zero, and 3.) the barge bow force-deformation relationship is assumed to be elastic perfectly-

plastic (Figure 6-1).

The first assumption implies that the initial kinetic energy of the barge is fully converted

into deformation energy of the barge bow during loading of the barge (Figure 6-2a). Thus, once

all of the barge initial kinetic energy has been converted into deformation of the barge bow (i.e.

the barge velocity becomes zero) the barge bow crush depth has reached its maximum value.

Additionally, when the barge bow recovers the elastic portion of its deformation energy through










unloading, this energy is then converted back into rebound motion of the barge (Figure 6-2b).

Final barge kinetic energy can then be determined from the recovered deformation energy.

If the barge bow remains linear and elastic, the conservation of energy up to the point of

maximum barge bow deformation can be represented by the following equation:

AKEim + ADEim = 1/2 mB Vdi + 1/2 PBm aBm = 0 (6-4)

where PBm is the maximum impact force observed during the impact, and aem is the maximum

barge bow deformation. The maximum impact force and barge bow deformation however,

remain undetermined up this point, and thus, an additional equation is required.

If the barge bow remains elastic, the maximum bow deformation can be defined as

follows:


aBm _BBm(6-5)


where aBY and PBY are the barge bow deformation and force at yield, respectively, and kB is the

initial elastic stiffness of the barge bow. Combining Equations 6-4 and 6-5, and then solving for

the peak load produces the following equation:


PBm = Vgi PB A = Vgi <, i~~B PBY (6-6)
aBY


Due to the elastic perfectly-plastic assumption for the barge bow force-deformation relationship,

the peak barge impact force is limited to the yield load of the barge bow.

To validate Equation 6-6, the central difference method was used to analyze a two degree-

of-freedom barge-pier-soil system (Figure 6-3) subjected to various impact conditions. For these

analyses, barge bow contact was modeled as a nonlinear spring using an elastic perfectly plastic

force-deformation relationship (Figure 6-1), and the pier-soil resistance was approximated using









a linear-elastic spring. For each analysis, the barge weight was varied from 250 tons to 8000

tons, and the initial velocity was varied from 0.5 knots to 10 knots. From the analysis results, the

maximum impact force generated between the barge and the pier was extracted for each case.

Additionally, Equation 6-6 was used to predict peak barge impact forces for each case.

As shown in Figure 6-4, although conservative, the estimate of peak forces as predicted by

Equation 6-6 are not in very good agreement with coupled analysis results. Referring to

Equation 6-6, it is noted that the ratio of the barge yield load to the barge bow yield deformation

represents the initial elastic stiffness of the system, assuming that the pier is rigid. However,

recalling Figure 6-3, the barge bow spring is in series with the pier-soil spring. Thus, if a linear

pier-soil spring is introduced and combined with the barge bow spring in series, an effective

barge-pier-soil spring stiffness can be defined as follows:

1-

skB kP PBY kP


where kP is the linear pier-soil spring stiffness.

Replacing the initial elastic barge stiffness (kB) in Equation 6-6 with the effective barge-

pier-soil series spring stiffness (ks) (Equation 6-7) produces the following equation:

PBm = Vgi J m = V~i CBP

where cBP is the barge-pier pseudo-damping coefficient, defined as follows:

cBP = km (6-9)


Using Eqn. 6-8 in place of Eqn. 6-6 to predict the peak impact load imparted to the pier,

and comparing the results to two-DOF coupled time-integrated dynamic analysis results, good

agreement between the two methods (Figure 6-5) is observed.









With the peak impact load calculated, the final barge velocity after impact can now be

calculated using the unloading characteristics of the barge bow. It is assumed that unloading of

the barge bow occurs along a path that has a slope equal to the initial elastic barge stiffness

(Figure 6-2b). Using this assumption, the conservation of energy equation from the point of

maximum barge bow deformation to the point at which the barge and pier are out of contact can

be written as follows:

AKEme,+A~DEme,= 1/2 -mB-v~,+ 1/2 -P,,m-(aBP a.,m)=0 (6-10)

where vef is the final barge velocity and aBP is the plastic barge bow crush depth (Figure 6-2b).

Note that in Equation 6-10, the final barge velocity and the plastic crush are unknown. However,

it is not necessary to actually calculate the plastic crush depth. It is only necessary to know that

the difference between the plastic and maximum barge bow deformations is the same as the

initial elastic portion of the barge-bow crush, and thus:


(at~m aBP= PBm (6-11)
kB

Then, combining Equations 6-10 and 6-11, and solving for the final barge velocity

produces the following:


V Bf = PBmB (6-12)


Replacing the initial elastic barge stiffness (kB) with the effective barge-pier-soil stiffness

(ks) and introducing the barge-pier-soil pseudo-damping coefficient (cBP= Jkm ), the final

barge velocity can be approximated as follows:

i 1 Pm
v~f = Pnm Bm~ (6-13)
SksmB BP,









For the situation in which the barge bow yields, the velocity at initial yield must be

calculated. Conservation of energy from initial state to yield can be expressed as:

AKEi, + ADEi, = 1/2 -mB b Y, V i)+1/2 -PBY -aBY = 0 (6-14)

where vBY is the barge velocity at yield. Solving for this velocity using Equation 6-14 the

velocity at yield can be expressed as:


VBY = /V2 BiB (6-15)
mB


Next, multiplying the numerator and denominator of the second term within the square-root, and

simplifying results in the following:


VB 2 _PBYaBY PBY = 2 P (616
mB PBY k~mB


Again, replacing the initial elastic barge stiffness (kB) with the effective barge-pier-soil stiffness

(ks) and introducing the barge-pier-soil pseudo-damping coefficient (cBP), the barge velocity at

yield can be approximated as follows:


VBY = /V = V PBY 2P (6-17)


6.2.2 Prediction of Load Duration from Conservation of Linear Momentum

The next step in calculating the impact load history is to determine the duration of time

that the load will act on the structure. This is accomplished by using the principle of

conservation of linear momentum:

(A~if = (I)(6-18)









where { Air is the vetor~n change of barge linear momentum from the initial state to the Einal

state, andl {} is the ]oloa imlpule Assuming that the mass of the barge does not change, the

change in barge linear momentum can be expressed as:

( if/ )=mBVsf \Vsi I) (6-19)

where {vef} and {V-\i}, are the ~ bare velocity vetorsT at the final and1 initial states TIrespectivly In


general, the load impulse for the barge can be defined as:


(I) = t, (PB(t)idt (6-20)

where {PB-t)} is the vetrnTmor ipc force as a fu~nction of time (t), t; is th nitial] time at~ which


impact occurs, and tr is the Einal time at which the impact ends and the impact force becomes

zero.

Although momentum and impulse are vector quantities, the analysis method used here is

one-dimensional. Therefore, the vector notation can be dropped by taking into consideration that

Einal barge velocity will have a negative sign when the barge moves away from the pier

following impact. Additionally, the impact force on the barge will have a negative sign since it

acts opposite to the direction of the initial barge motion. Taking these facts into account, the

conservation of linear momentum can be rewritten as:


mB V,,f + Vi)=S~ PBct~dt (6-21)


If the barge bow remains elastic for the duration of the analysis, it is assumed that the

elastic load history pulse takes the shape of a half-sine wave (Figure 6-6a). Assuming that the

analysis starts at 0 secs (ti = 0 secs), the impulse of the impact force between the barge and the

pier can be calculated as:











Ot I Sn X dt = -tE Pm (6-22)


where tE is the duration of loading for an elastic pulse.

Combining Equations 6-21 and 6-22, and solving for the elastic load duration:


tER _BVBf. + Vsi) (6-23)
2Pm


Comparing Eqn. 6-8 and 6-13, the final velocity of the barge is equal to the initial barge velocity

when the barge bow remains elastic, and thus Eqn. 6-23 becomes:

nm
tE BV~i (6-24)
Pm

For situations in which the barge bow yields, the load history is subdivided into three

distinct stages: 1.) elastic loading to yield, 2.) plastic loading, and 3.) elastic unloading (Figure 6-

6b). The time history of elastic loading to yield is assumed to take the shape of a quarter-sine

wave (Figure 6-7). For the purpose of calculating the time required to yield the barge bow, it is

assumed that the analysis starts at 0 secs (ti = 0 secs). The elastic loading portion of the load

impulse can be calculated as:


Stv PBY Sin t :dt = -t2 t_-PBY (6-25)
o 2t, n


where ty is the time required to yield the barge bow.

Using Equation 6-25 and the conservation of linear momentum the following relationship

can be defined:


mB(, V~- VBY 2~t -PBY (6-26)









This equation accounts for the fact that the impact force acts in the opposite direction of the

initial motion of the barge, and that the barge velocity at yield acts in the same direction as the

initial barge velocity. Solving Equation 6-26 for the time to yield gives:


ty= mBV~ -VY mBV s -V PBYCs (6-27)
2PBY 2PBY B


Following the initial elastic loading stage, an inelastic stage occurs. During this latter

stage, the plastic load is assumed to remain constant from the time at which the barge bow yields

to the time at which unloading of the barge bow begins. The load impulse for this stage is given

by:


PBdt = PBY P t (6-28)


where tP is the duration of plastic loading.

Taking into account the direction of the impact force, and assuming that the velocity of the

barge immediately before the barge bow unloads is zero, the conservation of linear momentum

can be defined as follows:

mB -VBY = PBY .P (6-29)

The assumption that the velocity of the barge immediately before the barge bow unloads is zero

is valid because, the deformation of the pier-soil system is much lower than the inelastic barge

bow deformation. Solving Equation 6-29 for the plastic load duration:


tP PB -VBY mB, V2s PBY (6-30)



Following the plastic loading stage, the time history of elastic unloading is assumed to take

the shape of the second quarter of a single sine wave cycle (Figure 6-8):











tPBY Sin Idt = -ttt o .PBY (6-31)


where to is the duration of unloading.

Assuming that the barge velocity at the beginning of the unloading stage is zero, and that

the impact load acts in the same direction as the final barge velocity, conservation of linear

momentum can be expressed as:


mB -VBf -U -PBY (6-32)


Rearranging Equation 6-32, the duration of unloading is given by:

nmB nmB PBY
to = (6-33)
2PBY 2PBY BP,


Additionally, summing Eqns. 6-27, 6-30, and 6-33, the total duration for inelastic loading

is defined as:


7:mg 2

2Pm BP \ /B


6.2.3 Summary of Procedure for Constructing an Impact Load History

Using key equations from the detailed derivations given in the sections above, a summary

of the complete procedure for constructing an impact load time-history function is presented

(Figure 6-9). The curves presented in Figure 6-9 are time-varying impact forces that may be

applied to a bridge structure in a dynamic sense. Although the AVEL method is not exact, it is

expected that it will approximate the CVIA well.









6.3 Validation of the Applied Vessel Impact Load History Method

To assess the accuracy of the AVEL method, three coupled dynamic analyses were

conducted on a one-pier two-span (OPTS) model of a new St. George Island Causeway Bridge

channel pier and the connecting two superstructure spans (Davidson 2007). The three cases

selected represent low, moderate, and high energy impacts respectively (Table 6-1). Comparing

each case to the current AASHTO provisions (Figure 6-10) indicates that the low-energy

analysis represents a situation in which the barge bow remains elastic throughout the duration of

impact; the moderate-energy case represents a situation in which the barge bow deformation is

slightly larger than the yield deformation (according to AASHTO); and the high-energy case

exhibits significant barge bow yielding.

The force-deformation relationship for the barge is dependent upon the size and shape of

the impact pier column. Pier columns for channel piers of the new St. George Island Causeway

are round with a 6-foot diameter. Thus, using Equation 4.8, the yield load and bow deformation

at yield for the barge are 1540 kips and 2 in respectively (Figure 6-11).

For each collision analysis, the impact load history computed using coupled analysis was

compared to the load history predicted by the AVEL method. Comparisons between the CVIA

force history results and the AVIL results are presented in Figure 6-12. For the low-energy case

(Figure 6-12a), the AVIL method predicts a slightly higher peak load and a slightly longer load

duration than is predicted by coupled analysis. Comparing the load impulses for each analysis-

42 kip-sec and 3 8 kip-sec for the AVIL and coupled analysis s methods respectively--reveal s that

the two differ by about ten-percent.

The moderate-energy force history comparison (Figure 6-12b) shows good agreement

between the coupled analysis and the AVIL method. Comparing the load impulses for this









case--733 kip-sec and 715 kip-sec for the AVIL and CVIA respectively-indicates that the

AVIL method over-predicts the load impulse by about two-percent.

Inspecting the results for the high-energy impact (Figure 6-12c), there is a negligible

discrepancy in the load histories predicted by the AVEL and CVIA methods. Load impulses for

the two methods--3449 kip-sec versus 3448 kip-sec for the AVIL and CVIA methods

respectively--reveal s that the two methods differ by less than one-percent.

Comparisons of the maximum bending moments from dynamic analyses performed using

the two methods are presented in Table 6-2 and Figure 6-13. Moments in Figure 6-13 represent

the maximum absolute bending moment throughout the duration of the analysis at a given

elevation across all pier columns or all piles, whereas values in Table 6-2 are the maximum

absolute bending moments for all piles or columns in the model, regardless of elevation.

Inspection of Table 6-2 and Figure 6-13 reveals that pile and column moments from the AVIL

analysis agree well with the CVIA results. Additionally, the AVIL method produces moments

that are slightly conservative in comparison to the coupled analysis moment results.










Table 6-1
Energy
Low
Moderate
High


Impact energies for AVIL validation
Barge weight (tons) Barge velocity (knot)
200 1
2030 2.5
5920 5


Impact-energy (kip-ft)
17.71
1123
13100


Table 6-2 Maximum moments in all pier columns and piles
Impact energy Location Moment (kip-ft)
Coupled A


668
603
4005
1805
4581
1826


~VIL


Percent
difference
5.2%
2.7%
1.8%
0.9%
0.0%
0.2%


Low

Moderate

High


Column
Pile
Column
Pile
Column
Pile


634
588
3934
1790
4580
1823



























a,, Barge Bow Deformation aB


Figure 6-1 Barge bow force-deformation relationship




























I I
ay am as A


Total deformation energy
is equal to the initial barge
kinetic energy


a,, a,, am as


Figure 6-2 Inelastic barge bow deformation energy A) Loading, B) Unloading













IBarge bow
force-deformation
relationship



k, (equivalent
linear stiffness
of pier and soil)


m, (mass
of pier)


Figure 6-3 Two degree-of-freedom barge-pier-soil model











1,600


1,200
.e oo

a4 O
8000


800


-Applied Load History Method




0 200 400 600 800
Barge Kinetic Energy (kip-ft)


Figure 6-4 Peak impact force vs. initial barge kinetic energy using a rigid pier assumption

















~rm ^"^^""^F~~^^^""~ ^


200 400 600 8
Barge Kinetic Energy (kip-ft)


1,600


1,200




800




400


Figure 6-5 Peak impact force vs. initial barge kinetic energy using an effective barge-pier-soil
stiffness


O O
O
O
O
O

O

O

O
0 O Dynamic Analysis Data
-Applied Load History Method






















/tE~ T" ime (sec)p ~t i Time (sec)
/A B

Figure 6-6 Impact load histories A) Elastic loading, B) Inelastic loading






















tynl t, to Time (sec)


Figure 6-7 Construction of loading portion of impact force






































170























tv t to Time (sec)


Figure 6-8 Construction of unloading portion of impact force






































171












/.---------------------------------- Applied Vessel Impact Load History Method t---------------------------------.,

Select barge type, and determine PBY, BY

Determine impact characteristics: mB, BI

Calculate pier-soil stiffness from static analysis: k,


ks B Calculate barge-pier-soil series stiffness


c,, =.km Calculate pseudo-damping coefficient


(VB1CBP~ PBY) (VBCBP >PBY
Elastic loading Inelastic loading








tE B1 EmB 2 PBY
Bm Y 2PBY Bl B BP


tp=mB PBYtomB Y
PBY CBP 2PBY CBP


t, = smB Bl BY 2 1v
2Pm CBP CBP






PBY S
PmSn71 2t,





-i, trt ,PBYSin 2to


Fiur 6- VLprcdr











3,000


2,000
a High energy


;40II Moderate energy
1,000



ni Low energy

0 4,000 8,000 12,000 16,000 20,000
Kinetic Energy (kip-ft)


Figure 6-10 AASHTO load curve indicating barge masses and velocities used in validating the
applied load history method











2,000


1,500


1,000o [2 in, 1540kips


500



0 5 10 15 20
Bow defonnation (in)


Figure 6-11 Barge bow force-deformation relationship for an impact
on a six-foot round column














450 H Coupled impact analysis
a ~ Applied vessel impact load
300





0 0.2 0.4 0.6 0.8 1
Time (sec)A



2,000

1,500

S1,000
;40 H Coupled impact analysis
500 0 Applied vessel impact load


0 0.2 0.4 0.6 0.8 1
Time (sec)



2,000






500 H Coupled impact analysis
Y Applied vessel impact load

0 0.5 1 1.5 2 2.5 3 3.5 4
Time (sec)


Figure 6-12 Impact load history comparisons A) Low-energy impact, B) Moderate-energy
impact, C) High-energy impact











Eley.
+53.9 ft

Eley.
+29.3 ft


Eley.
-62.4 ft


700
Moment (kip-ft)


1,400


N Coupled impact analysis
0t~ AVIL analysis


2,500
Moment (kip-ft)


5,000


0 2,500
Moment (kip-ft)


5,000


N Coupled impact analysis
Ht~ AVIL analysis


M Coupled impact analysis
0t~ AVIL analysis


Figure 6-13 Moment results profile for the new St. George Island Causeway Bridge channel pier
A) Channel pier schematic, B) Low-energy impact, C) Moderate-energy impact,
D) High-energy impact









CHAPTER 7
IMPACT RESPONSE SPECTRUM ANALYSIS

7.1 Introduction

In previous chapters, the dynamic response of the bridge structure (Figure 7-1) was

computing using time history impact analysis techniques. These techniques used numerical time-

step integration to obtain a solution to the equation of motion:

[M](ui(t>)+ [C] (u(t)}+ [K](u(t)}= (F(t)} (7-1)

In this equation, [M], [C], and [K] are the mass, damping, and stiffness matrices,

respectively, of the structure; {F Rt)} is the time varying external forrce vectnors andl {ut)},


{ u (t)}, and { u (t)} are the time varying displacement, velocity, and acceleration vectors,

respectively. An alternative dynamic analysis technique that does not require time-integration is

the response spectrum analysis technique. Response spectrum analysis is carried out using

structural vibration characteristics such as mode shapes and frequencies and involves estimating

the maximum dynamic structural response rather than computing the response at each point in

time. In this chapter, a response spectrum analysis technique that is suitable for analyzing

dynamic barge impact conditions is presented, validated, and demonstrated.

7.2 Response Spectrum Analysis

In order to conduct a response spectrum analysis, the numerical model must be

transformed from the structural system to the modal system. This process is generally achieved

through the use of an eigenanalysis, from which structural vibration characteristics are obtained.

Using the eigenanalysis results, the multiple degree-of-freedom (MDOF) structural system

matrices, forces, and displacements can be transformed into single degree-of-freedom (SDOF)

modal properties, forces (Pi), and displacements (qi) for each mode (Figure 7-2). Once the model









has been transformed into modal coordinates, an analysis on the modal equations of motion can

be conducted.

7.2.1 Modal Analysis

The generalized eigenproblem can be expressed by the following equation:

[K] Q]= h[MI[Q] (7-2)

where [K] and [M] are respectively, the stiffness and mass matrices of the system, h are the

eigenvalues of the system, and [O] is a matrix whose columns are the eigenvectors (i.e. modal

shapes of vibration) of the structure. Both the stiffness and mass matrices are square matrices

with dimensions equal to the number of degrees-of-freedom in the structural system model. In

general, the matrix of eigenvectors is also square with dimensions equal to the number of

degrees-of-freedom in the system. Each eigenvalue (hi) corresponds to a natural circular

frequency of the structure squared (ao where- t-i has:' units --- oftt rad/sec) whic intunanb


related to the period of vibration of the structure:


Ti = x(7-3)




structure assumes when it is excited (loaded) at its respective natural frequency (Figure 7-3).

The eigenvalue problem, Eqn. 7-2, can also be rewritten as follows:

[D]- 2 I])[Q]= [0] (7-4)

where [D] is the dynamic system matrix (defined as [D] = [M] [~K]), and [I] is the identity

matrix. To ensure a nontrivial solution to the above equation, the characteristic matrix--the

parenthetical quantity in Eqn. 7-4--must necessarily be singular. For a matrix to be singular, its









determinant must be zero. Imposing this condition leads to an nth degree polynomial, commonly

referred to as the characteristic polynomial:

p @2)= det [D]-02[I] = (7-5)

where p(mo2) iS the characteristic polynomial. Solutions for the eigenvalues of the system are

generally obtained by solving for the n-roots of the characteristic polynomial. Once the

eigenvalues are known, the corresponding eigenvectors can be solved for by rewriting Eqn. 7-4

for a specific mode i:

[D] f [I]) Qi) = (0) (7-6)

Further examination of Eqn. 7-6 shows that this equality holds true even if the eigenvector
{Oi}) is sIcalle by~ an arbtrary-t constnt~ (ci.\ Although the relative mag;+rnitd of each element

with respect to the other elements in an eigenvector is unique, the absolute magnitude of each

eigenvector is not.

The process of scaling the eigenvectors such that they each have a specific magnitude is

called normalization. A normalization method often used in structural analysis software is called

mass-normalization and involves scaling each eigenvector such that the following condition is

satisfied:

~i) [M] i = 1 or ~j[Mj = [] (7-7)

where IPi is a mass-normalized eigenvector. A mass-normalized eigenvector can be computed

from an arbitrarily normalized eigenvector as follows:


Ci i (7-8)








If the eigenvectors of a system have been computed, the model can be transformed into the

generalized modal system (recall Figure 7-2). Structural displacements of the system model may

be related to modal displacements through the eigenvectors as follows:

(u(t)}= [G](q(t)}l (7-9)
where {u(t)} andl {(t+)} are the time va~ryingr strctural andl moda~l displlacements, respectively.

Because the eigenvectors are not dependent upon time, taking the first and second derivatives of
{u(t)} with~ mnresec to time r yilds:

(u(t)}= [G]((t)} (7-10)
{uct)}= [e {qt)}t, (7-11)
where { u(t)} and { (t+)} are the stnrctural venlocityr and acceleration vectors, andl {, q(t) and




Eqn. 7-9 provides a means of transforming modal displacements into structural

displacements. An inverse process that transforms structural displacements into modal

displacements is achieved using the inverse of the eigenvector matrix:

(q\Lt)}= [G u\t)} (7-12)

However, inversion of the eigenvector matrix [G] in Eqn. 7-12 may be computationally

expensive. A more computationally efficient procedure for achieving the same transformation as

that described by Eqn. 7-12 is available and involves the use of both the eigenvector matrix and

the structural mass matrix. Assuming that the eigenvectors of the system have been mass-

normalized, and premultiplying each term in Eqrn. 7-9 by E61 [M] yields:


(P [Mj) (ut)} =(l ro [M O(q(t)}= I~l [M] I( q(t))= [I] k(qft (7-13)









Making use of Eqn. 7-7, Eqn. 7-13 is reduced to provide the transformation of modal

displacements into structural displacements:

(q(t))= P[M] (u(t)l (7-14)

Although Eqn. 7-14 is more computationally efficient than Eqn. 7-12, use of Eqn. 7-14 is

dependent upon the mass-normalization of the eigenvectors, and thus requires greater caution in

its use. Throughout the remainder of this chapter, it is assumed that the eigenvectors are mass-

normalized.

Typically, the equation of motion for the structural system is expressed as in Eqn. 7-1.

Introducing the displacement, velocity, and acceleration relationships (Eqns. 7-9, 7-10, and 7-

11), into Eqn. 7-1 produces:

[Mj[G](q(t)},+ [C][G](q(t>)+ [K<][G](q(t>)= (F(t)} (7-15)

Premultiplying each term in this equation by the transpose of the eigenvector matrix (i.e. [O]T)

yields:

[*lT [Mj[Q](q(t>) +'[c]T~~t) T~ [K][Q](q(t))= [G]T tF(t)} (7-16)

The above equation can be simplified by defining the following relationships:

[m]= [G]T [M] [G] (7-17)
[c]= [G]T [C] [G] (7-18)
[k]= [G]T [K] [G] (7-19)
(f(t)}= [G]T (F(t)} (7-20)

where [m], [c], and [k] are the modal mass, modal damping, and modal stiffness matrices, and

{f(t)} is the modal force vector.

The fact that eigenvectors are orthogonal with respect to the stiffness and mass matrices

results in the modal stiffness and modal mass matrices being diagonal (Chopra 2007). If an









orthogonal formulation is used to construct the structural damping matrix [C], then the modal

damping matrix [c] will also be diagonal. In this case, each diagonal term in [c] can be

expressed as a ratio of the critical damping for each mode:

ci = 2mitimi (7-21)

where Si is the ratio of critical damping for the ith mode. When [m], [c], and [k] are diagonal, the

coupled structural equation of motion (Eqn. 7-15) reduce to a series of uncoupled single DOF

modal equations of motion. (Note that the use of the term "coupled" here is different from that

discussed in Chapter 5. In Chapter 5, the coupling discussed was between the impacting vessel

and the bridge structure. The coupling being referred to in the present context, however, refers to

the linking of various DOF within the structural model to each other. Such linking occurs when

off-diagonal terms are present in the system matrices). For a given mode i, the SDOF modal

equation of motion is:

miii(t)+ 2mitimiqi(t)+ kiti (t)= fi (t) (7-22)

Normalizing each term by the modal mass yields:


4i(t)+ 2miti4i (t)+m qtit= \ijt) (7-23)
mi

Solving each SDOF modal equation of motion yields modal displacements, modal

velocities, and modal accelerations. With the time-varying modal quantities known, the

individual modal contributions to overall structural displacements, velocities, and accelerations

can be computed for each mode using Eqns. 7-9, 7-10, and 7-11. Structural response quantities

are thus determined through modal superposition in which the contributions from all modes are

added together.










7.2.2 General Response Spectrum Analysis

Time-step integration of the equations of motion, either structural (Eqn. 7-15) or modal

(Eqn. 7-22), generally requires hundreds or thousands of time steps to be analyzed in order to

evaluate the response of a structure. Subsequently, the results must be scanned to identify the

maximum absolute force values needed for design. In contrast, response spectrum analysis can

be used to estimate these maximums without the need for conducting time-history analysis. In a

response spectrum analysis, Eqns 7-22 and 7-23 are not directly time-integrated. Instead, the

maximum response contributed by each mode is determined using relationships that correlate

loading to peak response as a function of modal frequency (or period).

7.2.2.1 Modal Combination

To determine the maximum overall structural response, the individual modal contributions

must be combined. The simplest possible combination technique consists of direct superposition

of the maximum absolute response from each mode. This approach assumes that the peak

responses for each mode occur at the same point in time, which leads to overly-conservative

design forces. An alternative, and more reasonable, approach is to use a square-root-of-the-sum-

of-the-squares (SRSS) method. The SRSS method is based upon probabilistic theory and is

expressed as follows:


Ro~omb =1 (7-24)


where Roomb is the combined result of the response parameter under consideration (e.g.

displacement, force, etc.), and ri is the response parameter value for the i.th mode. Use of the

SRSS combination method tends to provide accurate approximations of structural response for

two-dimensional models in which the natural frequencies are well-spaced (Tedesco et al. 1999).









However, for three-dimensional models in which closely spaced natural frequencies may be

present, the complete-quadratic-combination (CQC) method yields more accurate results:


Room Piji j(7-25)
i=1 j=1


where pij is defined as follows:


p, (7-26)
ij-1u224iiu~ 2)+ 2


where Si and (3 are the modal damping ratios for modes i and j respectively, and a is the

frequency ratio (mi/" 3)

7.2.2.2 Mass Participation Factors

Although the total number of modes for a multiple degree-of-freedom structural model is

equal to the number of degrees-of-freedom, typically, only a relatively small number of these

modes are required in order to adequately describe the behavior of the system. Consequently,

eigenanalyses are typically conducted on a reduced system (or subspace), instead of the full

system. Subspace methods are advantageous in that significantly fewer computations are

required to obtain modal properties (shapes and frequencies) when only a small number of

modes are needed. However, a key step in using subspace methods is determining the number of

modes that are required to adequately capture the response of the system. Typically, this number

of modes is determined through the use of mass participation factors which represent the amount

of structural mass present in each mode of vibration.

To calculate the mass participation factor for a specific mode-i, the mass excitation factor

must be calculated as:


Li ~iT [M] (1F) (7-27)










where Li is the direction specific mass excitation factor for the it 1 e n lF) IS uni CILvector

where the subscript (F) indicates that the unit values are associated with degrees-of-freedom in

the direction of impact loading, and zero values are associated with degrees-of-freedom not

associated the direction of impact loading. The effective modal mass in each mode is calculated

as:


megf ~i (7-28)
mi,i


where mi~i is the diagonal term of modal mass matrix [m] associated with mode-i, and [m] is

defined in Eqn. 7-17:


[m]= 0 T [M] [G] (7-29)

The mass participation factor for each mode is defined as:


ri= =(7-30)
Total (mi~i)(Mtotal>


where Mtotal is the total mass of the structure being analyzed, and Li is the mass participation

factor for mode-i.

In seismic (earthquake) response spectrum analysis, design codes (ASCE-7 2005, FEMA

2003, etc.) require that sufficient modes be used such that the total mass participation of the

modes included adds up to at least ninety-percent (90%) of the total mass of the structure in two

orthogonal directions. For use in vessel impact response spectrum analysis (described in this

chapter) it is recommended that 99% mass participation in the direction of applied impact

loading be required.









7.3 Dynamic Magnification Factor (DMF)

For a single degree-of-freedom (SDOF) system, the maximum response of the system to a

time-varying dynamic loading condition can be determined through the combined use of static

response calculation and a dynamic magnification factor (DMF). The maximum dynamic

response is computed as the product of the static response and the DMF. When DMFs are

calculated for many different SDOF systems having different natural periods (and corresponding

natural frequencies), a DMF spectrum is produced. In this section, a DMF design spectrum

appropriate for use in barge impact analysis is developed.

Calculation of a single point on the DMF spectrum involves selecting an impact condition

(barge mass and speed), determining a time-history of impact loading, and analyzing an

equivalent SDOF bridge-pier-soil structural model subj ected to the loading condition using both

dynamic and static analysis procedures. For a given impact condition, the applied vessel impact

loading (AVIL) method, described earlier in this report, may be used to form a time-history of

impact loading. Using this loading, the SDOF structural system is dynamically analyzed and the

maximum dynamic displacement of the system (uD) is recorded. Subsequently, the peak

magnitude of the dynamic applied load is determined (Figure 7-4) and is applied to the SDOF

structural system as a static load. A static analysis is then performed to determine the maximum

static displacement of the system (us). The DMF is then given computed as the ratio of

maximum-dynamic and static displacements computed for the SDOF system (i.e.,

DMF = uD / us). Repeating this process for different SDOF structural models (having varying

natural periods and frequencies), but using the same time-history of impact loading, produces a

collection of points that constitute the DMF spectrum (Figure 7-5) for the specific loading

condition that has been used in the calculations.









If this process is repeated not only for different SDOF structural models but also for

different impact load histories, the result is a family of event-specific DMF spectra (Figure 7-6a).

The data in Figure 7-6a was generated by first varying the mass of the barge from 250 ton-mass

to 8000 ton-mass, and the initial velocity from 0.25 knots to 8 knots to generate various impact

load histories. A 250 ton-mass barge corresponds to a single jumbo hopper barge with 50 tons of

cargo, and the 8000 ton-mass is slightly larger than four fully-loaded jumbo hopper barges. The

upper bound on initial barge velocity, 8.0 knots, was selected based upon maximum barge tow

velocities reported at past-point data along Florida' s intracoastal waterway system. Overall,

1024 individual impact scenarios were investigated to generate the data in Figure 7-6a.

It is worth noting that the DMF data shown in Figure 7-6 never exceed a value of 2.0. A

DMF of 2.0 indicates that the magnitude of dynamic response is twice that of the corresponding

static response. Stated in other terms, for a DMF of 2.0, the dynamic response is 100% greater

than the corresponding static response.

In regard to the development of a design DMF spectrum, one of the simplest options is a

broad-banded design spectrum that envelopes all of the data generated (Figure 7-6b). While this

approach is simple and provides conservative estimates of dynamic response, in some cases, a

broad-banded design spectrum will yield results that are excessively conservative. For low-

energy impacts, a broad-banded design spectrum will grossly over-predict amplification effects

for long-period modes (modes with a natural structural period greater than one-second). In

Figure 7-7, the event-specific DMF spectrum for a low energy impact (200 ton barge drifting at

1.0 knots) is compared to a broad-banded design spectrum. For a mode with a structural period

of 5 sec, the broad-banded design envelope predicts a dynamic magnification factor of 2.0,

whereas the event-specific spectrum predicts a DMF of 0.09 (a DMF less than 1.0 indicates that









the dynamic response is less severe than the static response). Thus, in this instance, a broad-

banded design spectrum would over-predict the DMF by more than a factor 20.

An alternative to the broad-banded design spectrum is a DMF spectrum that evolves based

upon impact condition characteristics. This approach yields a design spectrum that is closer to

the event-specific spectrum. Inspection of Figure 7-8 shows that as the impact energy increases,

the width of the event-specific DMF spectrum increases by expansion in the short and long-

period ranges.

One of the key components of an evolving design DMF spectrum is to find a relationship

between the impact characteristics and the short and long-period transition points (Figure 7-9a),

which shift along the structural period axis, producing an expansion as the impact energy

increases. This is accomplished by defining the points that mark a transition from a constant

DMF of 2.0 to a sloping DMF. Based on a qualitative investigation of the event-specific DMF

data, the transition points are set at a dynamic magnification factor of 1.6 (Figure 7-9b). For

each event-specific DMF spectrum in Figure 7-6, the short and long-period transition points are

defined in this manner, and an expression for the structural period at each transition point is

defined. The expression for the short-period transition point is defined as the minimum of the

following two expressions:



Ts= i (7-31)

Ts = (7-32)
2.2

where Ts is the period for the short-period transition point in seconds, TI is the period of impact

loading (Figure 7-10)--calculated as twice the duration of loading predicted by the equations









from Chapter 6--and mB and Vei are the barge mass (kip-sec2/in) and initial velocity (in/sec)

respectively.

Figure 7-11 shows the short-period transition point as a function of barge kinetic energy.

Event-specific DMF data were calculated using the process described earlier, and short-period

expression data were calculated using Eqns 7-31 and 7-32. The data in Figure 7-11 are merely a

subset of all the data used to generate the expressions, reduced for visual clarity. Additionally,

each line indicates a change in barge mass with the initial barge velocity held constant. In

general, Eqns 7-31 and 7-32 predict short-periods that are lower than the data obtained from

event-specific DMF spectra generation. Referring back to Figure 7-9, this indicates that the

short-period transition point for the design DMF spectrum is slightly left of the corresponding

point in the event-specific DMF spectrum.

The expression for the long-period transition point is defined as the minimum of the

following two expressions:



TL= mV (7-33)

TL (7-34)
0.7

where TL is the period for the long-period transition point in seconds, and TI, mB, and Vei are in

seconds, kip-sec2/in, and in/sec respectively.

Long-period transition point data as a function of barge kinetic energy is shown in Figure

7-12. As with the short-period transition point data, event-specific DMF data were calculated

using the process illustrated in Figure 7-9b, and the long-period expression data were calculated

using Eqns 7-33 and 7-34. Generally, the long-period expression predicts a period that is higher

than the data generated from the event-specific DMF spectra; thus, indicating that the long-










period transition point for the design DMF spectrum is slightly right of the corresponding point

in the event-specific DMF spectrum (Figure 7-9).

With the transition points of the evolving design DMF spectrum quantified, expressions for

the design DMF spectrum as a function of structural period are defined. For impact response

spectrum analysis, the design DMF spectrum is assumed to be a piecewise linear function in log-

log space:


DMF = 2. -l" 2 1.2 if T < Ts (7-35)

DMF = 2.0 if Ts < T < TL (7-36)
.,;=~TT -0.95
DMF= 2 2 0.1 if T > TL (7-3 7)



where, recalling Eqns. 7-31 through 7-34, Ts is the short period transition point, the lesser of:


0.9 T, T
1300T
Ts = T, 2 n T (7-38)
mBV~i2.2


TL is the long-period transition point, defined as the lesser of:


TL = T, mB viz and TL TI (7-39)
100 0.7


Eqns 7-3 5 and 7-37 are applicable to structural periods outside of the transition points. For

structural periods between the transition points, the DMF is set to a value of 2.0 (Figure 7-13).

In Figure 7-14, event-specific DMF spectra are compared to design DMF spectra for four

different impact conditions that span a broad range of impact energies. The plots demonstrate

that the evolving design DMF spectrum equations adequately envelope the corresponding event-

specific DMF data over a broad range of impact energies. Furthermore, none of the design










spectra are excessively conservative as was the case for the broad-banded design spectrum

discussed earlier.

7.4 Impact Response Spectrum Analysis

Using the concept of a dynamic magnification factor (DMF), it is possible to approximate

the maximum dynamic response (displacement, internal member forces, etc.) of a SDOF system

using results from a static analysis of the same system. To achieve such an outcome, the DMF-

magnitude for the structural system and the applied load must be based upon vibration

characteristics of the system. As noted earlier, a modal analysis (e.g., eigen analysis) may be

used to obtain modal vibration characteristics and uncouple the MDOF dynamic equations of

motion--essentially transforming the MDOF system into many SDOF uncoupled modal

equations. The impact response spectrum analysis (IRSA) procedure proposed here uses modal

(eigen) analysis, in conjunction with a DMF spectrum, to simulate dynamic barge impacts on

MDOF bridge models. Key steps in the IRSA calculation process are

* Calculating the peak magnitude of impact load

* Applying the peak impact load to the structure in a static sense, and computing the
resulting static displacements

* Transforming the static structural displacements into static modal displacements

* Magnifying the static modal displacements--using a DMF spectrum-into dynamic modal
displacements

* Transforming the dynamic modal displacements into dynamic structural displacements for
each mode

* Recovering the internal member forces for each mode using the corresponding dynamic
structural displacements

* Combining the internal member forces and dynamic structural displacements from the
modal contributions using SRSS or CQC modal combination









A detailed flowchart for this process is illustrated in Figure 7-15. To calculate the static

displacements {us} o the1 structure, the peak va~lue o the~ dy~namic Irloa (PDm ;IS canlcunlae using


the applied vessel impact load (AVIL) equations developed of the previous chapter. Using

impact characteristics, a barge bow force-deformation relationship, and the stiffness of the pier,

the peak dynamic load can be calculated. Using AVEL equations, the period of loading--an

important component of DMF spectrum calculation--is calculated as twice the load duration

(recall Figure 7-10). For elastic loading:


T = Vgi (7-40)
Pm


and for inelastic loading:


T~ ~ = m iPY 2-1v B (7-41)
P~mBP \ /BP


Characteristics of the DMF spectrum are calculated from the impact vessel characteristics and

the loading period using Eqns. 7-31 through 7-37.

The peak dynamic load is applied to the structure at the impact point in a static sense, and a

static analysis is conducted to compute the resulting structural displacements {us} (Figure 7-16).

These static displacements can subsequently be amplified, in a modal sense, to obtain the

dynamic response of the system.

To transform the structural analysis into modal coordinates, an eigenanalysis is conducted

on the structure to determine the natural frequencies (and periods) and eigenvectors of the system

(recall Figure 7-3). Using the eigenvectors, the static displacements are transformed from

structural coordinates into modal coordinates using a variation of Eqn. 7-14 (Figure 7-17):

9s> 1T [M] (us) (7-42)









where {qs} andl {us}\ are vectnor of static displacepments in the modaol andl stnrctural cordnate~o

systems respectively. Each entry in the static modal displacement vector corresponds to a

specific mode shape and natural frequency (and period). Using the natural period for each mode,

and the design DMF spectrum (Eqns 7-35 and 7-37), a DMF value is computed for each mode

(Figure 7-18). The modal static displacement qsi for mode-i is then magnified by the DMF to

produce a dynamic modal displacement qDi (Figure 7-19). Using a modified version of Eqn. 7-9,

the contribution from mode-i to the overall dynamic structural displacements is then calculated

as:

U~i Ri i)(7-43)

Internal dynamic member forces {FDi) for each mode are determined by performing force

recovery on the structure using the dynamic displacements {uDi}. Maximum dynamic structural

displacements {uD)\ Of the system areP hobtanedl by combining the modaol contributino nst

displacement {uDi: Uing either SRSS or CQC combination techniques:


(uD >= SRS S((uD1, UD (u, ..., (U.n ) (7-44)
UD = UD1( D ,., >) (7-45)


The same process is then applied to the internal force vectors:

(FD ) = SRS S((FD1 (FD2 ),..., (Fn )) (7-46)
(FD/ = -\FD1),\ (FD2,, \Fn) (7-47)

to obtain the maximum dynamic internal member design forces where {FD) for the structural

sy stem.

7.5 Impact Response Spectrum Analysis for Nonlinear Systems

In the IRSA procedure presented above, it is assumed--for purposes of performing the

eigenanalysis--that the system matrices (stiffness and mass) correspond to a linear elastic

structure. Hence the procedure can only be used for the specific case in which the system is









approximated as being linear. Considering the fact that severe vessel-bridge collision events

generally exhibit nonlinear behavior, the linear IRSA procedure must be altered to account for

system nonlinearity-specifically the nonlinear stiffness matrix. Two iteration stages are

required to account for such nonlinearity: 1.) load determination iterations, and 2.) stiffness

linearization iterations for eigenanalysis.

7.5.1 Load Determination and DMF Spectrum Construction

For a nonlinear system, the effective secantt or tangent) pier-soil stiffness (kP) is dependent

upon the ultimate displacement level that is reached by the system, and is therefore unknown at

the beginning of the analysis. An iterative process (Figure 7-20) is therefore required to

calculate kP and the peak dynamic load (PBm). In Order to start the process, the pier is initially

assumed to be rigid (kP= cO), which causes the barge-pier-soil series stiffness to reduce to the

following:



(0ks= BY _1 aB 1 PY (7-48)
iilyPBY kP PBY m aB

where (0)ks is the initial estimate for the effective barge-pier series spring stiffness. The peak

dynamic load for the first iteration ((O)PBm) can then be calculated as:

(0cBP (0)'ks m (7-49)
(0) PBm = Vgi (0) CBP < PBY (7-50)

The initial load estimate ((0)PBm) is then applied to the structure in a static sense, and the

structural displacement at the load application point is computed. With both the load and the

displacement at the impact point known, the pier-soil secant stiffness for the next iteration (n)

can be calculated as:









(n-1) Pm
(n) k = Bm-1 (7-51)
UP

where (n)kP iS the pier-soil secant stiffness for the current iteration (n) and (n-i)PBm and (n-1 uP are,

respectively, estimates of the peak dynamic load and structural static displacement of the impact

point for the previous iteration (n-1). Furthermore, the effective barge-pier-soil series spring

stiffness ((n)ks) for the current iteration (n) can be updated as:



(n) k B (7-52)


With the effective barge-pier-soil series spring stiffness ((n)ks) known, the load ((n)PBm) for the

current iteration (n) can be updated accordingly:

(n) BP (nks -m (7-53)
(n) PBm = Vgi (n) CBP < PBY (7-54)

The incremental change in computed impact load from iteration (n-1) to iteration (n) is

then calculated as follows:

APBm _(n)PBm _(n-1) PBm (7-55)

If the incremental change in computed load (APBm) is Smaller than a chosen convergence

tolerance, then calculation of the peak load has converged to a solution. With the load (PBm)

determined, the design DIVF may be constructed using Eqns. 7-31 through 7-37.

7.5.2 Structural Linearization Procedure

For a linear system, eigenanalysis can be carried out directly using the system stiffness and

mass matrices. However, for a nonlinear system, the stiffness matrix itself is dependent upon the

displacements of the system. In such a case, an iterative process (Figure 7-21) must be used in

which the system secant stiffness matrix is updated using the dynamically magnified structural









displacements computed within each iteration. However, during the initial iteration, static

displacements are used to approximate the dynamically amplified displacements that will be used

in later iterations. Thus, to initialize the linearization process, the static external force vector

({Fs})whc contains the,;, peak impact,+ load (P.m),1 ,S well as, all; other stti lodsonth

structure (e.g. gravity loads)--is used to compute the external force vector for the first iteration:

'O0 (FE)= CFs) (7-56)

Additionally, the structural static displacements are used to estimate the dynamic

displacements for the initial iterations:

'0) (uD)= (us) (7-57)

These dynamic displacements are in turn used to compute the secant stiffness matrix for the first

iteration.

An eigenanalysis is then conducted using the mass matrix and the current secant stiffness

matrix. Eigenvectors and natural frequencies are extracted from the eigenanalysis results for use
in the IRSA. Static structural displacements {us}\ are transformedl innto stati modaol coordinates



(n]S (n) 1T [M] (us (-8


Using the estimate of structural period for mode-i obtained from the eigenanalysis at the n-th

nonlinear iteration:


(n)T 2 (7 59)


the DMF is calculated as:










(n)s DMi20 0. if ()Ti< T, (7-60)

(n DMhi = 2.0 if Ts < (n)Ti < TL (7-61)

TLj-0.95


(n) M i=. -( D0.1 if(nTiT (7-62)





and the dynamic displaced shape for each mode is obtained as:

(n) (uDi ~(nI) Di (n) Oi ) (7-64)

The overall structural displacement response of the system is updated by combining the

displaced mode shapes using the SRSS or CQC combination techniques:

(nI)(uD)= SRSS (n1)(uD1 >(n1) UD2 >***>(n) CUDn)) (7-65)
(nI)CuD) CQ(n) CUD1 X(nI) UD2 ***(n) UDn) (7-66)

The external force vector for each mode-i is updated by premultiplying the updated

displaced shape for each mode by the most recent estimate for the stiffness matrix:

(n) (F~i (n) [KI] (n) tu~i) (7-67)

Combining the external force vectors for each mode using either SRSS or CQC combination, the

updated estimate of external force is calculated:

'"n(FE )= SRSS (n1)(FE1~ (n1)CFE2 ),...,(n) Fn (7-68)
(n) CFE (n) (FE(n)nF1~I) CFE2 ** *>(nI) (F~n (7-69)









Using the modally (SRSS or CQC) combined external force vector, the incremental change in

force vector from the previous nonlinear iteration (n-1) the current iteration (n) is calculated:

{A\=)F (n E (-1 F (7-70)

If any term in the incremental external force vector (AFE: ;ES greater than the chosen


tolerance, then the process begins anew with recalculation of the secant stiffness matrix using the

updated displaced shape. Alternatively, if every term in (AFE) is less than or equal to the chosen

tolerance, then the linearization process has converged. In this case, internal dynamic member

forces (n)(FDi) for each mode are determined by performing force recovery on the structure using

the dynamic displacements (") {uDi). Modal internal member forces are then combined using

SRSS or CQC modal combination:

(FD )= SR SS((FD1 (FD2 ,..., (Fn ) (7-71 )
(FD = Q(tFD1 (FD2)..., (Fn )) (7-72)

to obtain the maximum overall dynamic structural internal member design forces (FD -

7.6 Validation and Demonstration of Impact Response Spectrum Analysis

Validation of the IRSA method is carried out by analyzing a series of jumbo hopper barge

impacts on bridge structures using both IRSA and coupled vessel impact analysis (described

earlier in Chapter 5) and subsequently comparing results. Since both methods are implemented in

same software analysis package, FB-MultiPier, any difference in analysis results are solely due

to differences in the analysis procedures themselves. Since the coupled vessel impact analysis

(CVIA) method was previously validated against both high-resolution finite element models and

experimental data (see Chapter 5), results obtained from this method are considered the

benchmark (or reference datum) against which the accuracy of the IRSA method is judged.









7.6.1 Event-Specific Impact Response Spectrum Analysis (IRSA)Validation

To validate the IRSA method, coupled dynamic time-history analyses were conducted on a

one-pier two-span (OPTS) model of the new St. George Island Causeway Bridge channel pier

and the two connected superstructure spans (Davidson 2007). Three impact cases (Table 7-1)

were analyzed: a low energy impact; a moderate energy impact; and a high energy impact. The

low-energy case represents a situation in which the barge bow remains elastic throughout the

duration of impact; the moderate-energy case represents a situation in which barge bow

deformation is slightly greater than the barge yield deformation; and the high-energy case

corresponds to a situation in which significant barge bow yielding is expected. (Note that these

are the same three cases previously used in the validation of the applied vessel impact load

(AVIL) method in Chapter 6).

Both the size and shape of the pier column that will be impacted affect the barge force-

deformation relationship that needs to be used. Pier columns for channel piers of the new St.

George Island Causeway Bridge are circular in cross-section and 6-ft in diameter. Using

equations presented earlier in Chapter 4, the yield load and yield deformation for a jumbo hopper

barge are determined to be 1540 kips and 2 in. respectively (Figure 7-22).

Each IRSA validation case used an event-specific DIVF spectrum that corresponded to the

impact energy of the collision being analyzed. Therefore, before each IRSA validation was

performed, a corresponding CVIA was conducted to establish a time-history of impact load

corresponding to the impact energy of interest. Each such impact load history was applied

dynamically to a large number of SDOF structural systems (having varying natural periods) and

the maximum dynamic displacement of each system (uD) TOCOVered. Additionally, for each case,

the peak magnitude of impact load was determined (recall Figure 7-4) and applied to the SDOF

structural system in a static sense so that the maximum static displacement of the system (us)









was recovered. For each structural period (i.e. each SDOF system) the DMF was then calculated

as the ratio of maximum dynamic to static displacement (i.e., DMF = uD / us). For the given

impact load history, the collection of all such DMF data points constitutes an event-specific

DMF spectrum (recall Figure 7-5).

For each case in Table 7-1, an event-specific spectrum was generated (Figure 7-23) and an

event-specific IRSA was conducted using the corresponding DMF spectrum. In each case, a

total of 12 eigen modes were used and combined together using the SRSS and CQC techniques.

The use of 12 eigen modes was necessary to ensure that the internal member force results from

the IRSA using a design DMF spectrum were conservative with respect to CVIA results.

Comparisons of the maximum absolute bending moments for all piles and columns in the

model from dynamic analyses performed using the IRSA and CVIA methods are presented in

Tables 7-2 and 7-3. The IRSA values presented in Table 7-2 were modally combined using the

SRSS technique, whereas corresponding values in Table 7-3 were combined using the CQC

technique. Additionally, Tables 7-2 and 7-3 present the total modal mass participation that was

achieved by using 12 modes for each IRSA.

Figure 7-24 shows a profile of bending moments, which represents the maximum absolute

bending moment throughout the duration of the analysis at a given elevation across all pier

columns or piles, for both the CVIA and the event-specific IRSA using the CQC modal

combination technique. Comparing values from Tables 7-2 and 7-3, the results from CQC modal

combination are slightly more conservative than the results obtained from SRSS combination.

Inspection of Tables 7-2 and 7-3, and Figure 7-24 reveals that the pile and column

moments from the event-specific IRSA cases (SRSS and CQC) generally agree well with the

CVIA results. In a few cases, the bending moments from the event-specific IRSA method are


200










slightly unconservative in comparison to the coupled analysis results. This is due to the

approximate nature of modal combination techniques such as SRSS and CQC. Such methods

attempt to quantify maximal dynamic responses by combining individual modal responses that

may maximize at different points in time during the vessel collision. Since response spectrum

techniques do not account for such timing issues, the results obtained are approximate in nature.

However, in practical design situations, a design DMF spectrum will be used instead of an event

specific DMF spectrum. Since a design DMF spectrum is inherently more conservative than an

event-specific spectrum, the use oflIRSA in design situations will generally lead to conservative

results, although this is not absolutely guaranteed. Additionally, using a CQC modal combination

rather than an SRSS combination can further increase the likelihood of obtaining conservative

design data.

7.6.2 Design-Oriented Impact Response Spectrum Analysis Demonstration

To demonstrate the IRSA method as it would be used in design, the three impact scenarios

considered above for validation purposes are re-analyzed. With the exception of the DMF spectra

used, all parameters and procedures in the demonstration analyses are the same as those used

during validation. In the demonstration cases, design DMF spectra (constructed using Eqns. 7-31

through 7-37) are used in place of the event-specific spectra that were used in the validation

process.

For each case in Table 7-1, an IRSA was conducted using the non-linear procedure

outlined in Figures 7-20 and 7-21. In each case, a total of 12 eigen modes were combined

together using the SRSS and CQC techniques. Comparisons of the maximum absolute bending

moments for all piles or columns in the model from dynamic analyses performed using the IRSA

and CVIA methods are presented in Tables 7-4 and 7-5. The IRSA values presented in Table 7-4

were modally combined using the SRSS technique, whereas corresponding values in Table 7-5









were obtained by modal combination using the CQC technique. Tables 7-4 and 7-5, also present

the total modal mass participation that was achieved by using 12 modes for each IRSA.

As stated above, inclusion of 12 eigen modes was necessary to ensure that the design IRSA

results were conservative with respect to CVIA results. Table 7-6 shows the modal mass

participation for each of the 12 eigen modes used and the cumulative mass participation.

Including 6 modes in the IRSA analyses yields 90% mass participation--as is required in

earthquake analysis. However, it was found that the use of 90% mass participation yielded IRSA

results that were unconservative with respect to CVIA results. Therefore, inclusion of mode 1 1

(the next maj or modal contribution) was investigated-bringing the cumulative mass

participation to between 94% and 98%. Inclusion of mode 1 1, however, still yielded pile force

results that were slightly unconservative with respect to CVIA pile force results. Inclusion of

mode 12--which corresponded to greater than 99% cumulative mass contribution--yiel ded

IRSA results that were conservative with respect to CVIA results. Therefore, it is recommended

that 99% modal mass participation be achieved to ensure conservative IRSA results.

Figure 7-26 shows profiles of bending moments, which represents the maximum absolute

bending moment throughout the duration of the analysis at a given elevation across all pier

columns or piles, for both the CVIA and the design IRSA using a CQC modal combination

technique. As with the validation cases discussed above, comparing values from Tables 7-4 and

7-5, results from CQC modal combination are slightly more conservative than results obtained

from SRSS combination.

The data presented in Tables 7-4 and 7-5, and Figure 7-26 reveal that pile and column

moments from the design-oriented IRSA agree well with the CVIA results. Additionally, as


202










expected, use of a design DMF spectrum added an extra level of conservatism, making the IRSA

results conservative with respect to CVIA results.


203









Table 7-1
Energy
Low
Moderate
High


Impact energies for IRSA validation
Barge weight (tons) B
200 1
2030 2
5920 5


~arge velocity (knot) Impact energy (kip-ft)
17.71
'.5 1123
13100


Table 7-2 Maximum moments for all columns and piles for event-specific IRSA validation with
SRSS combination
Impact Bridge Bending moment (kip-ft) IRSA total mass
energy component Coupled Event- Percent participation (12


impact species
analysis IRSA
886.2
596.5 5
3934
1863
4580
1895


difference modes)


Low Column
Pile
Moderate Column
Pile
High Column
Pile


899
;28.9
5267
1806
5425
1835


1%
-11%
34%
-3%
18%
-3%


99.8%

99.9%

99.9%


Table 7-3 Maximum moments for all columns and piles for event-specific IRSA validation with
CQC combination
Impact Bridge Bending moment (kip-ft) IRSA total mass
energy component Coupled Event- Percent participation


impact species
analysis IRSA
886.2 9
596.5 5
3934
1863
4580
1895 :


(12 modes)


difference


Low Column
Pile
Moderate Column
Pile
High Column
Pile


58.3
68.9
5362
1990
5520
2017


8%
-5%
36%
7%
21%
6%


99.8%

99.9%

99.9%


204

































Low Column 886.2 1190 34% 99.8%
Pile 596.5 692.7 16%
Moderate Column 3934 5921 51% 99.9%
Pile 1863 2150 15%
High Column 4580 5921 29% 99.9%
Pile 1895 2150 13%


Table 7-6 Mass participation by mode for design IRSA
Modes Low energy Moderate energy High energy
Mass Cumulative Mass Cumulative Mass Cumulative
Participation Participation Participation Participation Participation Participation
1 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
2 0.0% 0.0% 0.1% 0.1% 0.1% 0.1%
3 60.4% 60.5% 60.7% 60.8% 60.7% 60.8%
4 15.6% 76.1% 15.5% 76.3% 15.5% 76.3%
5 0.0% 76.1% 0.0% 76.3% 0.0% 76.3%
6 16.0% 92.1% 16.5% 92.8% 16.5% 92.8%
7 0.0% 92.1% 0.0% 92.8% 0.0% 92.8%
8 0.0% 92.1% 0.0% 92.8% 0.0% 92.8%
9 0.0% 92.1% 0.0% 92.8% 0.0% 92.8%
10 0.0% 92.1% 0.0% 92.8% 0.0% 92.8%
11 2.3% 94.4% 4.5% 97.3% 4.5% 97.3%
12 5.4% 99.8% 2.6% 99.9% 2.6% 99.9%


Table 7-4 Maximum moments for all columns and piles for design IRSA demonstration with
SRSS combination
Impact Bridge Moment (kip-ft) IRSA total
energy component Coupled Event- Percent mass


impact
analysis
886.2
596.5
3934
1863
4580
1895


specific
IRSA
1123
648.6
5813
1968
5813
1968


difference


participation
(12 modes)
99.8%

99.9%

99.9%


Column
Pile
Column
Pile
Column
Pile


27%
9%
48%
6%
27%
4%


Low

Moderate

High


Table 7-5 Maximum moments for all columns and piles for design IRSA demonstration with
CQC combination
Impact Bridge Moment (kip-ft) IRSA total
energy component Coupled Event- Percent mass


difference


impact
analysis


specific
IRSA


participation
(12 modes)


205










{u(t)}


{F(t)}








A B C

Figure 7-1 Time history analysis of a structure A) Structure, B) Finite element model of
structure, C) Displaced shape of structural model


206










{u(t)}


P (t)

Mode i


Rilf) q2(t
P (t) P P(t)
Mode 1 Mode 2


Figure 7-2 Time-history versus modal analysis A) Finite element model of structure, B) MDOF
time-history analysis of structural system, C) SDOF systems representing each mode
of vibration


207











{@ } : Mode 1 {@2) : MOde 2


{@3) : MOde 3














T3: per Od
3, irua
frequency


T,: period
m,: circular
frequency


Figure 7-3 Modal analysis A) Structural model,
D) Third mode shape


B) First mode shape, C) Second mode shape,


208


T2: periOd
m2: crua
frequency











f(t) m k

C~uD
SDOF dynamic system

peak

C~us
Corresponding SDOF
static system


Figure 7-4 Dynamic magnification of single degree-of-freedom system A) Impact force history,
B) Dynamic and static SDOF systems, C) Dynamic and static displacements


209















300


a,200


100


0
0 0.5 1
Time (sec)


1

-.




0.01


0.01


1.5 2


0.1 1
Structural Period (sec)


Figure 7-5 Dynamic magnification factor for a specific impact load history A) Impact load
history, B) Corresponding dynamic magnification factor


210












10 10


2 ----1 2
1I 1




0.1 0.1




0.01 0.01
0.01 0.1 1 10 0.01 0.1 1 10
Structural Period (sec) AStructural Period (sec)B


Figure 7-6 Dynamic magnification factor A) For a range of load histories, B) With a broad-
banded design spectrum





SLow-energy impact DMF
-Broad-banded design spectrum


1


0.01 1
0.0


0.1 1
Structural Period (sec)


Figure 7-7 Specific dynamic magnification factor for a low-energy impact vs. a broad-banded
design spectrum


212































0.01
0.01 0.1 1 10
Structural Period (sec)


Figure 7-8 Evolution of the dynamic magnification spectrum from short to long duration loading


213













Long-period transition points Long-period transition pomnt

1 ~1.61


S Short-period transition point Short-period transition point
0.1 0.1



0.01 0.01
0.01 0.1 1 10 0.01 0.1 1 10
Structural Period (sec) AStructural Period (sec)B


Figure 7-9 Definition of the short and long-period transition points A) Design DMF spectrum,
B) Event-specific DMF spectrmm


214














Duration
Sof Impact
Loading

E Periodof Impact Loading (T,) /






Time

Figure 7-10 Period of impact loading


215








































































216


H Event-specific data
I Design expression

0.01
1 10 100 1,000 10,000 100,000
Kinetic energy (kip-ft)


Figure 7-11 Short-period transition point data














H Event-specific data
I Design expression

10












0.1
1 10 100 1,000 10,000 100,000
Kinetic energy (kip-ft)


Figure 7-12 Long-period transition point data


217
















2.0 max '
DMF

/ 1.2 min
short-period
\ DMF
F~ Short-period
slope (0.6) Long-periodaL
slope (-0.95)

0.1

0.1 min
long-period
Short-period DMF Long-period
region region


0.01
0.01 0.1 1 10 100
Structural Period (sec)


Figure 7-13 Evolving design DMF spectrum


218






























0.1 1
Structural period (sec)


1<9




0.1




0.01
0.01




10









0.1




0.01


01


It




0.1




0.01
0.0


0.1 1
Structural period (sec)


0.1 1
Structural period (sec)


Figure 7-14 Event-specific and design DMF spectra for varying impact energies A) 200 ton
barge at 1.0 knots, B) 2030 ton barge at 2.5 knots, C) 5920 ton barge at 5.0 knots


219


















































T, =2x

Calculate DMF

DMF, = 2.0 T"21.2

DMF =2.0


,.-------------------------~--~ ~ Impact Response SpectLrum Analysis Procedure -----------------------------
Select barge type, and determine PBY" BY,

Determine impact characteristics: mB' BI,

Calculate pier-soil stiffness from static analysis: k,
-1\


"BI k,

Pam= v,,c, I P,

Calculate period of impact lo(

T, = B Bl
PBm




Calculate DMF characteristic
1300 T,
T,~~ aT m1

TL = TI m11vy, T
100 0.


Calculate barge-pier-soil series stitfness


cap = J m Calculate peak dynamic load

ding

Elastic loading BCPPB


17: 1 ') Inelastic loading (vBICBP >PBY


Short-period transition point

mB in kip-sec2 in, vBI 111 111Sec, and T, in sec
Long-period transition point


Apply static load (Pm) and conduct a static analysis to determine static displacements Cus!
Conduct eigenanalysis and recover mode shapes (CI,) and natural circular frequencies (oa,)

(qs '~T [M] {us} Transform structural static displacements into modal coordinates

For each mode (i)


Calculate structural period


(T,
(Ts & T, MTL


Short-period region

Intennediate periods


DMF =2.0 T 2 0.1 Long-period region (T, > TL

qDI = DMF qs, Magnify static modal displacement

"DI D1) 9 1~ Dynamic displaced mode shape
Recover internal member forces (FDI) for each mode from the displaced mode shapes

Combine modal quantities using SRSS or CQC combination:
{UD = SRSS({ID1, LID ., L\ or LID = 0 C LID1 LID2,, L\

{FD[=SRSS({F,~FD1 {F ,...,{F,,n) or F = {D1FD..Fn




Figure 7-15 Impact response spectrmm analysis procedure


220


CBP














{Fs} Pam Fs} = Pm









A B

Figure 7-16 Static analysis stage of IRSA A) Structural model with peak dynamic load applied
statically, B) Resulting statically displaced shape











cAS3 3


qs, { }


{Fs} = Pism










A B C D

Figure 7-17 Transformation of static displacements into modal coordinates A) Statically
displaced shape, B) Component of first mode in static displacement, C) Component
of second mode in static displacement, D) Component of third mode in static
displacement


222












DMF2
DMF -

DMF, 1





0.01


0.01 0.1 1 10
T3 T2 T,


Structural Period (sec)


Figure 7-18 Dynamic magnification factor as a function of structural period


223












or CQcC((11D D2,X


~"D~: = DMFq,, ~,,:
:U,,) = DMFBlf~,) fu,,)= DMFgf~,: ~ rm

O O O O 00
0~ O
FP F
;II ;I
~r ~r
Od 05 O


Figure 7-19 Combination of amplified dynamic modal displacements into amplified dynamic
structural displacements A) Dynamic displaced shape of mode 1, B) Dynamic
displaced shape of mode 2, C) Dynamic displaced shape of mode 3, D) Modally
combined dynamic displaced shape


224












,- ------------------------------------------- L oad D eterm nation ----------------------------------


Select barge type, and determine PBY" BY,
:Determine impact characteristics: mB' BI,

(o~ks = BY Estimate barge-pier-soil series stiffness
aBY
(0) PBm = VBp(0) CBP PBY (0)CBP = -Ok Estimate peak dynamic load

Apply static load (Pm) and conduct a static analysis to determine static displacement of the impact point (u )


For each iteration (n) ...

(nnk =Bm Undate detective
S(n-1)U_ pier-soil stitfness

'n'k aBY (n 1 Uodate barne-oier-soil
s-Psu (nk series stiffness

(n) PBm = VB p(n) CBP E PBY (n)CBP = Jnkm Update peak dynamic load
:Avoly static load (cn)P,.) and conduct a static analysis to determine static
displacement of the impact point (cn u,)
APm__(n)Pm_(n-1)Pm Calculate the incremental peak dynamic load

SAPm 5tol



Calculate period of impact loading

T, = ,m B Elastic loading BCPPB
PBm


T = amPsu( 2s -1vg P Inelastic loading (vBICBP >PBY

Calculate DMF characteristics
1300 T
Ts = T, m-~ 2. Short-period transition point
o 1 mB in kip-sec2/in, vBI in in/sec, and T, in sec
TL =T mBv T Long-period transition point
100 )0.7

DMF = 2. 1.2 Short-period region (T < Ts)

DMF = 2.0 Intermediate periods (Ts & T & TL)

: DM = 2 20.1Long-period region (T > TL





Figure 7-20 Nonlinear impact response spectrmm analysis procedure












225











,. -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- Linea rization -- -- -- -- - -- - -- -- -- --


Apply static load (Pm) and conduct a static analysis to detennine static displacements (us)

''FE )= (FsJ (11D = 11Sia 1111 Stillate foT eXter1181 foTCe YOCtOT
IU DI = Us(and structure displacentents
For each iteration (n) ...

Conduct eigenanalysis and recover mode shapes !*I: and natural circular tie~.....;Il.. : I .

For each mode (i) ...

(n)T, = 2x, Calculate structural period
Calculate DMF

(n ~ ~ TT DMj=2. Short-period region e") T, < Ts)

'"n DMF, = 2.0 Intermediate periods (Ts en)T, M TL

A TL


(n) 9D1 (n)DMF, en) qs Magnify static nxodal displacement

en) 11Dl (n) 9D, (n) CO ) Calculate dynamic displaced shapes

en) FE1 (n) [K] (n) C1DI External nxodal force

Combine nxodal quantities using SRSS or CQC combination:


en)(FE,)= SRSS en)(FE1 (n)(FE2 (n)(F,)n or "'(n~)(FE (n)(FE"1 (n(F2 n)F~

(aFE (n)" (FE (n-1) FE) C81CUlate incremental eXterna8 foTCe YOCtOT

SFor every term in {AFE) 101


:Recover internal nienber forces (F,,) for each mode frona the displaced mode shapes
Combine nxodal quantities using SRSS or CQC combination

(FD>= SRSS(CFD1 ZC,(F .... (Fn) or (FD = Q (CFD1 ZC,(F .... (F,,n




Figure 7-21 Nonlinear impact response spectrum analysis procedure


226











2,000


1,500


1 ,000 2 in, 1540 kipsip


500



0 5 10 15 20
Bow deformation (in)


Figure 7-22 Barge bow force-deformation relationship for an impact on a six-foot round column


227





0.01 0.1 1
Structural period (sec)




011 ---


0.1 1
Structural period (sec)


300



0




2,000




1,000
O


0.5 1 0.01 0.1 1
Time (sec) A Structural period (sec)


10



(1



0.1



0.01


0.5 1
Time (sec)


2,000




1,000


0.01 I
0.01


0 2 4
Time (sec)


Figure 7-23 Event-specific IRSA validation A) Low-energy impact load, B) Low-energy
event-specific DMF, C) Moderate-energy impact load, D) Moderate-energy
event-specific DMF, E) High-energy impact load, F) High-energy event-specific
DMF


228












Eley.
+53.9 ft

~Eley.
+29.3 ft

Eley.
+1.9 ft


M CVIA
-60 H IS

-80
0 600 1,200
Moment (kip-ft)


Eley.
-62.4 ft


N CVIA
Ht~ IRSA


0 3,000 6,000
Moment (kip-ft)


3,000
Moment (kip-ft)


6,000


Figure 7-24 Moment results profile for the new St. George Island Causeway Bridge channel pier
A) Channel pier schematic, B) Low-energy impact, C) Moderate-energy impact,
D) High-energy impact


229

















































Figure 7-25 Design-oriented IRSA demonstration A) Low-energy design DMF,
B) Moderate-energy design DMF, C) High-energy design DMF


230


1



0.1



0.01
0.01


000.1




0.01


1


0.1 1
Structural period (sec)


0.1 1
Structural period (sec)


0.01


001


0.1 1
Structural period (sec)












Eley.
+53.9 ft

~Eley.
+29.3 ft

Eley.
+1.9 ft


Eley.
-62.4 ft


600
Moment (kip-ft)


1,200


3,000
Moment (kip-ft)


6,000


3,000
Moment (kip-ft)


6,000


Figure 7-26 Moment results profile for the new St. George Island Causeway Bridge channel pier
A) Channel pier schematic, B) Low-energy impact, C) Moderate-energy impact,
D) High-energy impact









CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS

8.1 Concluding Remarks

Three distinct procedures for conducting dynamic analyses of barge impacts on bridge

piers have been presented in this dissertation. Two of the methods--coupled vessel impact

analysis (CVIA) and applied vessel impact load (AVIL)--use time-step integration to solve the

equation of motion. The other method--impact response spectrum analysis (IRSA)--is a

response spectrum analysis, which combines results from an eigenanalysis to approximate

maximum dynamic effects.

The CVIA technique, which involves coupling a single degree-of-freedom (SDOF) barge

model to a multiple degree-of-freedom (MDOF) bridge model, was developed by a research

team of which the author was a member. The CVIA method was validated against both

experimental data and high-resolution finite element analysis (FEA) results. Based on good

agreement between CVIA results, full-scale experimental data, and high-resolution FEA results,

CVIA was selected as the baseline analysis to which the two other dynamic analysis methods

(AVIL and IRSA) were compared in the present research.

The AVEL procedure uses barge impact characteristics and force-deformation behavior of a

barge in conjunction with the force-deformation behavior of the bridge to generate an applied

load-history. Using conservation of energy and momentum, the barge and pier characteristics

were correlated to load history characteristics (peak load and load duration), which in turn, were

used to develop expressions for time-varying impact loads. These loads are then applied to a

bridge model dynamically. Several analyses were conducted in which CVIA and AVIL analysis

results were compared. Using CVIA as the baseline comparison, results from the AVIL analysis

method were found to agree very well with those obtained from the CVIA method.


232









Both the CVIA and AVIL techniques are categorized as time-history analysis procedures,

which yield results for each time step in the analysis. Designers, however, are generally

interested in maximum response values obtained from results. Thus, once a time-history analysis

has been conducted, designers must scan the full time-history record for maximum response

parameters. As an alternative, maximum responses may be approximated, without requiring a

full time-history analysis, by using a response spectrum analysis technique.

The IRSA procedure is a response spectrum analysis procedure specifically tailored to

vessel impact loading. The IRSA procedure dynamically magnifies modal components of the

static displacement to approximate dynamic modal displacements. Magnification of the static

displacements is accomplished through the application of a dynamic magnification factor

(DMF), which is obtained from impact DMF spectra. After transforming the dynamic modal

response parameters into structural response parameters, internal structural member design

forces are recovered and modally combined for use in bridge design. Generation of a large

number of event-specific DMF spectra revealed that the DMF values do not exceed 2.0 for

impacted bridge systems having 5-precent modal damping. Furthermore, comparison of results

from the CVIA and IRSA techniques showed good agreement. Thus, it is concluded that the

IRSA method is capable of adequately modeling barge impact events.

Although AVEL equations are used as part of the IRSA method, the IRSA method is

generally considered more suitable for use as a design tool. An important aspect of the AVEL

equations is the calculation of the effective pier stiffness at the application of loading. The

general nonlinear IRSA method includes force determination iterations in which the peak

dynamic force and effective pier stiffness are updated to account for pier-soil model nonlinearity.









The AVEL method, however, does not incorporate nonlinear force determination iterations, but,

instead requires the designer to determine a suitable effective pier stiffness.

Furthermore, the IRSA has an extra degree of conservatism inherently built into the

procedure. As shown in Chapter 8, empirical DMF spectrum equations used in the IRSA method

produce a conservative spectrum in comparison to event-specific spectra, but without being

overly conservative. In the AVIL method, this extra level of conservatism is not included.

Additionally, analyses conducted in this research have primarily focused on barge impacts

that are perpendicular to the span direction of the bridge superstructure. However, if extended to

barge impacts occurring in the span direction, it is not known whether the AVIL method will be

able to adequately capture the complex effects of such an impact (e.g. torsional effects on the

impacted pier). In contrast, IRSA accounts for complex dynamic effects associated with

longitudinal impacts due to the inclusion of appropriate modes in the eigenanalysis stage.

A maj or aspect in the development of all of the dynamic methods (CVIA, AVIL, and

IRSA) was the development of force-deformation relationships for barge bows. To develop

these relationships, high-resolution models of hopper and tanker barges were developed and

subjected to crushing by flat-faced and round pier column impactors of various widths. Based on

force-deformation results obtained from the high-resolution models, it has been concluded that

barge bow crushing behavior may be adequately and conservatively modeled using an elastic,

perfectly-plastic representation. This conclusion differs substantially from the current AASHTO

relationship in which impact forces continue increasing with additional deformation beyond the

transition from elastic to inelastic behavior. Additionally, barge bow yield loads have been

found to be dependent on the shape and size of the pier column, and not a function of the barge

bow width, as the AASHTO provisions prescribe.


234









8.2 Recommendations


8.2.1 Recommendations for Bridge Design

* Elastic, perfectly-plastic barge bow model: It is recommended that barge bow force-
deformation relationships be represented using the elastic, perfectly-plastic model that was
presented in Chapter 4. The adequacy of using an elastic, perfectly-plastic relationship is
based upon results obtained from high-resolution static finite element crush analyses of the
two most common types of barges used in the United States.

* Dependence of barge bow yield force and deformation on pier width and geometry:
Based on results obtained from the high-resolution finite element analyses, it has been
found that the barge bow yield load is dependent upon the shape and width of the impacted
pier column. Additionally, it has been found that the barge bow deformation at yield is
dependent upon the geometry (flat-faced versus round) of the impacted pier column. It is
important to note that although AASHTO prescribes force-deformation relationships that
are dependent on barge width (i.e. inclusion of the RB factor in the AASHTO provisions),
no such correlation was observed in this research. Therefore, it is recommended that the
barge force-deformation relationship be formulated as a function of pier width and
geometry, as presented in Chapter 4, without dependence on barge bow width.

* Use of dynamic analysis techniques in lieu of static techniques: Both experimental data
and results from finite element analyses indicate that inertial-resistances and inertial-forces
significantly affect bridge response during a vessel collision. The presence of a
superstructure produces large inertial forces, which may in turn generate significant force
amplification effects in pier columns. Thus, it is recommended that dynamic analyses be
used to model barge impact events.

o If analysis software in which CVIA is implemented is available, it is
recommended that CVIA techniques be used to dynamically analyze a bridge
structure subj ected to barge impact.

o If software in which CVIA is implemented is not available, it is recommended
that the IRSA technique be used to conduct a barge impact analysis on a bridge
model .

o A maj or aspect of IRSA techniques is the calculation of peak dynamic load and
period of loading using the AVEL equations. However, for several reasons as
stated above, the IRSA technique is more accurate than the AVIL method.
Therefore, the AVIL method, as a stand-alone method, is not generally
recommended for use in design.

8.2.2 Recommendations for Future Research

* Improve probability of collapse: The AASHTO vessel collision specifications use a
probabilistic approach to bridge design, which requires designers to determine the return
period of collapse for bridge structures spanning navigable waterways. One aspect of this


235










probabilistic approach is calculation of the probability of collapse (PC) term. PC is the
probability that a bridge structure will collapse once a bridge component has been struck
by a vessel. This probability is a function of many factors: impact vessel size and shape,
vessel speed and displacement, ultimate pier strength, etc. However, the AASHTO PC
estimation equations were derived from research on damage observed between ship-to-ship
collisions at sea. Given that barge-to-bridge collisions are different from ship-to-ship
collisions, the accuracy of the AASHTO PC term should be investigated.

* Investigate barge impacts longitudinal to a bridge: In the current research study,
analysis of barge impacts on bridge structures has been restricted to impacts transverse to
the bridge span direction. However, barge-to-bridge collisions do not always occur
transverse to the bridge. Additionally, for design, the AASHTO provisions prescribe that
fifty-percent of the design impact load be applied in the direction longitudinal to the
bridge. AASHTO, however, prescribes static load cases for impact loading. Therefore, it
is recommended that further investigation of the longitudinal impact condition be studied
using dynamic techniques.

* Expand barge bow force-deformation database: In this study, the barge bow
force-deformation results from high-resolution crush simulations indicated that an elastic,
perfectly-plastic model is adequate to simulate barge bow crushing behavior. Further
investigation into the post-yield behavior of barge bow behavior should be conducted
using experimental testing so that the results obtained from high-resolution FEA can be
confirmed.


236










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Consolazio, G.R., and Cowan, D.R. (2005). "Numerically Efficient Dynamic Analysis of Barge
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Consolazio, G. R., Hendrix, J. L., McVay, M. C., Williams, M. E., and Bollman, H. T. (2004).
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Consolazio, G.R., Cook, R.A., McVay, M.C., Cowan, D.R., Biggs, A.E., Bui, L. (2006). "Barge
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Davidson, M.T. (2007). "Simplified Dynamic Barge Collision Analysis for Bridge Pier Design."
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237










Federal Emergency Management Agency (FEMA). (2003). NEHRP Reconanended Provisions
for Seismic Regulations for New Buildings and Other Structures, Washington, D.C.

Goble, G., Schulz, J., and Commander, B. (1990). "Lock and Dam #26 Field Test. Report for the
Army Corps of Engineers." Bridge Diagnostics Inc., Boulder, CO.

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239









BIOGRAPHICAL SKETCH

The author was born in Akron, Ohio, in 1979. In January, 1998, he began attending the

University of Florida, where he later obtained the degree of Bachelor of Science in civil

engineering in December 2002. After attending graduate school at the University of Florida, he

received the degree of Master of Engineering in December 2004. In December, 2007, the author

anticipates receiving the degree of Doctor of Philosophy from the University of Florida. Upon

graduation, the author plans to procure a position as an engineer in training at Finley Engineering

Group, located in Tallahassee, FL.


240





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1 DEVELOPMENT OF TIME-HISTORY AND RESPONSE SPECTRUM ANALYSIS PROCEDURES FOR DETERMINING BRIDGE RESPONSE TO BARGE IMPACT LOADING By DAVID RONALD COWAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 David Ronald Cowan

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3 To Zoey Elizabeth.

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4 ACKNOWLEDGMENTS Completion of this dissertation and the acco mpanying research would not have been feasible without the support and guidance of a number of individua ls. First, the author wishes thank Dr. Gary Consolazio for his continual suppor t in this endeavor. He has offered invaluable knowledge and insight throughout th e course of this research. The author also wishes to thank his superv isory committee: Dr. Ronald Cook, Dr. Kurtis Gurley, Dr. Trey Hamilton, Dr. Nam-Ho Kim, and Dr. Michael McVay, who have each contributed valuable insight into multiple aspects of this research. Furthermore, the author wishes to thank Mr. Henry Bollmann and Mr. Lex Collins for their continual leadership and support. Others deserving of thanks for their support and contributi ons include Alex Biggs, Long Bui, Michael Davidson, Daniel Getter, Jessica Hendrix, Ben Lehr, Cory Salzano, and Bibo Zhang. The author wishes to thank his friends and family for their support and encouragement.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............15 CHAPTER 1 INTRODUCTION................................................................................................................... ...16 1.1 Introduction............................................................................................................... ........16 1.2 Motivation................................................................................................................. ........16 1.3 Objectives................................................................................................................. ........19 1.4 Scope of Work.............................................................................................................. ....20 2 BACKGROUND..................................................................................................................... ...22 2.1 Vessel-Bridge Collision Incidents....................................................................................22 2.2 Review of Experimental Vessel Impact Tests..................................................................23 2.3 Design of Bridges According to th e AASHTO Barge Impact Provisions........................28 2.3.1 Selection of Design Vessel.....................................................................................29 2.3.2 Method II: Probability Based Analysis..................................................................30 2.3.3 Barge Impact For ce Determination........................................................................32 3 SUMMARY OF FINDINGS FROM ST. GEORGE ISLAND BARGE IMPACT TESTING........................................................................................................................ ........51 3.1 Introduction............................................................................................................... ........51 3.2 Overview of Experimental Test Program.........................................................................51 3.3 Overview of Analytical Research.....................................................................................53 3.3.1 FB-MultiPier Models..............................................................................................54 3.3.1.1 FB-MultiPier Pier-1 model...........................................................................54 3.3.1.2 FB-MultiPier Pier-3 model...........................................................................54 3.3.1.3 FB-MultiPier Bridge model.........................................................................54 3.3.2 Finite Element Simulation of Models.....................................................................55 3.3.2.1 Impact test P1T7...........................................................................................56 3.3.2.2 Impact test P3T3...........................................................................................57 3.3.2.3 Impact test B3T4..........................................................................................58 3.4 Comparison of Dynamic and Static Pier Response..........................................................59 3.5 Observations............................................................................................................... ......62 4 BARGE FORCE-DEFORMATION RELATIONSHIPS...........................................................81

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6 4.1 Introduction............................................................................................................... ........81 4.2 Review of the Current AASHTO Load Determination Procedure...................................81 4.3 High-Fidelity Finite Element Barge Models....................................................................83 4.3.1 Jumbo Hopper Barge Finite Element Model..........................................................84 4.3.2 Tanker Barge Finite Element Model......................................................................88 4.4 High-Fidelity Finite Elem ent Barge Crush Analyses.......................................................89 4.4.1 Finite Element Barge Bow Crush Simulations.......................................................89 4.4.2 Development of Barge Bow Force-Deformation Relationships............................91 4.4.3 Summary of Barge Bow For ce-Deformation Relationships..................................96 5 COUPLED VESSEL IMPACT ANALYSIS AND SIMPLIFIED ON E-PIER TWO-SPAN STRUCTURAL MODELING..............................................................................................129 5.1 Coupled Vessel Impact Analysis....................................................................................129 5.1.1 Nonlinear Barge Bow Behavior...........................................................................129 5.1.2 Time-Integration of Barge Equation of Motion...................................................131 5.1.3 Coupling Between Barge and Pier.......................................................................132 5.2 One-Pier Two-Span Simplified Bridge Modeling Technique........................................134 5.2.1 Effective Linearly Independent Stiffness Approximation....................................134 5.2.2 Effective Lumped Mass Approximation..............................................................135 5.3 Coupled Vessel Impact Analysis of One-Pier Two-Span Bridge Models......................136 6 APPLIED VESSEL IMPACT LOAD HISTORY METHOD..................................................151 6.1 Introduction............................................................................................................... ......151 6.2 Development of Load Prediction Equations...................................................................151 6.2.1 Prediction of Peak Impact Load from Conservation of Energy...........................151 6.2.2 Prediction of Load Duration from Conservation of Linear Momentum..............156 6.2.3 Summary of Procedure for Constr ucting an Impact Load History.......................160 6.3 Validation of the Applied Vessel Impact Load History Method....................................161 7 IMPACT RESPONSE SPECTRUM ANALYSIS...................................................................177 7.1 Introduction............................................................................................................... ......177 7.2 Response Spectrum Analysis..........................................................................................177 7.2.1 Modal Analysis.....................................................................................................178 7.2.2 General Response Spectrum Analysis..................................................................183 7.2.2.1 Modal Combination....................................................................................183 7.2.2.2 Mass Participation Factors.........................................................................184 7.3 Dynamic Magnification Factor (DMF)..........................................................................186 7.4 Impact Response Spectrum Analysis..............................................................................191 7.5 Impact Response Spectrum Analysis for Nonlinear Systems.........................................193 7.5.1 Load Determination and DMF Spectrum Construction.......................................194 7.5.2 Structural Linearization Procedure.......................................................................195 7.6 Validation and Demonstration of Impact Response Spectrum Analysis........................198 7.6.1 Event-Specific Impact Response Spectrum Analysis (IRSA)Validation.............199 7.6.2 Design-Oriented Impact Response Spectrum Analysis Demonstration...............201

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7 8 CONCLUSIONS AND RECOMMENDATIONS...................................................................232 8.1 Concluding Remarks......................................................................................................232 8.2 Recommendations...........................................................................................................235 8.2.1 Recommendations for Bridge Design..................................................................235 8.2.2 Recommendations for Future Research................................................................235 LIST OF REFERENCES.............................................................................................................237 BIOGRAPHICAL SKETCH.......................................................................................................240

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8 LIST OF TABLES Table page 3-1 Summary of forces acting on the pier during test P1T7..........................................................63 3-2 Dynamic and static analysis cases.......................................................................................... .63 4-1 Barge material properties.................................................................................................. .......98 6-1 Impact energies for AVIL validation.....................................................................................163 6-2 Maximum moments in all pier columns and piles.................................................................163 7-1 Impact energies for IRSA validation.....................................................................................204 7-2 Maximum moments for all columns and pile s for event-specific IRSA validation with SRSS combination............................................................................................................... .204 7-3 Maximum moments for all columns and pile s for event-specific IRSA validation with CQC combination................................................................................................................ .204 7-4 Maximum moments for all columns and pile s for design IRSA demonstration with SRSS combination.................................................................................................................... ......205 7-5 Maximum moments for all columns and pile s for design IRSA demonstration with CQC combination.................................................................................................................... ......205 7-6 Mass participation by mode for design IRSA.......................................................................205

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9 LIST OF FIGURES Figure page 2-1 Collapse of the Sunshine Skyway Bridge in Florida (1980) afte r being struck by the cargo ship Summit Venture .....................................................................................................35 2-2 Failure of the Big Bayou Canot railroad bridge in Alabam a (1993) after being struck by a barge flotilla............................................................................................................... ..........36 2-3 Collapse of the Queen Isabella Causeway Br idge in Texas (2001) after being struck by a barge flotilla................................................................................................................. ...........37 2-4 Collapse of an Interstate I-40 bridge in Oklahoma (2002) after being struck by a barge flotilla....................................................................................................................... ...............38 2-5 Reduced scale ship-toship collision tests con ducted by Woisin (1976).................................39 2-6 Instrumented full-scale ba rge-lock-gate collision tests...........................................................40 2-7 Instrumented 4-barge lo ck-wall collision tests........................................................................41 2-8 Instrumented 15-barge lo ck-wall collision tests......................................................................42 2-9 Barge tow configuration.................................................................................................... ......43 2-10 Design impact speed....................................................................................................... .......44 2-11 Bridge location correction factor......................................................................................... ..45 2-12 Geometric probability of collision........................................................................................ .46 2-13 Probability of collapse distribution...................................................................................... ..47 2-14 AASHTO relationship between kine tic energy and barge crush depth.................................48 2-15 AASHTO relationship between barg e crush depth and impact force...................................49 2-16 Relationship between kinetic energy and impact load..........................................................50 3-1 Overview of the layout of the bridge.......................................................................................64 3-2 Schematic of Pier-1........................................................................................................ .........65 3-3 Schematic of Pier-3........................................................................................................ .........66 3-4 Test barge with payload impacti ng Pier-1 in the series P1 tests.............................................67 3-5 Series B3 tests............................................................................................................ ..............68

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10 3-6 Pier-3 in isolation for the series P3 tests................................................................................ ..69 3-7 Pier-1 FB-MultiPier model.................................................................................................. ....70 3-8 Pier-3 FB-MultiPier model.................................................................................................. ....71 3-9 Bridge FB-MultiPier model.................................................................................................. ...72 3-10 Schematic of forces acting on Pier-1.....................................................................................73 3-11 Resistance forces mobilized during tests P1T7.....................................................................74 3-12 Schematic of forces acting on Pier-3.....................................................................................75 3-13 Resistance forces mobilized during tests P3T3.....................................................................76 3-14 Schematic of forces acting on Pier-3 during test B3T4.........................................................77 3-15 Resistance forces mobilized during tests B3T4.....................................................................78 3-16 Comparison dynamic and static anal ysis results for foundation of pier................................79 3-17 Comparison of dynamic and static an alysis results for pier structure...................................80 4-1 Force-deformation results obtained by Meier-Drnberg.........................................................99 4-2 Relationships developed from experime ntal barge impact tests conducted by MeierDrnberg (1976)................................................................................................................ ...100 4-3 AASHTO barge force-deformation re lationship for hopper and tanker barges....................101 4-4 Hopper barge dimensions.................................................................................................... ..102 4-5 Tanker barge dimensions.................................................................................................... ...103 4-6 Hopper barge schematic..................................................................................................... ...104 4-7 Hopper barge bow model with cut-se ction showing internal structure.................................105 4-8 Internal rake truss model.................................................................................................. .....106 4-9 Use of spot weld constraints to connect structural components............................................107 4-10 A36 stress-strain curve................................................................................................... .....108 4-11 Barge bow model with a six-foot square impactor..............................................................109 4-12 Tanker barge bow model.................................................................................................... .110 4-13 Crush analysis models..................................................................................................... ....111

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11 4-14 Hopper barge bow force-deformation data for flat piers subjected to centerline crushing....................................................................................................................... ..........112 4-15 Hopper barge bow force-deformation data for flat piers subjected to corner-zone crushing....................................................................................................................... ..........113 4-16 Tanker barge bow force-deformation data for flat piers subjected to centerline crushing....................................................................................................................... ..........114 4-17 Relationship of pier width to engaged trusses.....................................................................115 4-18 Hopper barge bow force-deformation data for round piers subjected to centerline crushing....................................................................................................................... ..........116 4-19 Hopper barge bow force-deformation data for flat piers subjected to corner-zone crushing....................................................................................................................... ..........117 4-20 Gradual increase in number of trusse s engaged with deformation in round pier simulations.................................................................................................................... ........118 4-21 Elastic-perfectly plastic barge bow force-deformation curve..............................................119 4-22 Peak barge contact fo rce versus pier width.........................................................................120 4-23 Peak barge contact fo rce versus pier width.........................................................................121 4-24 Comparison of truss-yield controlled peak force versus plate-yi eld controlled peak force.......................................................................................................................... ............122 4-25 Design curve for peak impact force versus flat pier width..................................................123 4-26 Comparison of low diameter peak for ce versus large diameter peak force.........................124 4-27 Design curve for peak impact force versus round pier diameter.........................................125 4-28 Initial barge bow stiffness as a function of pier width........................................................126 4-29 Barge bow deformation at yield versus pier width..............................................................127 4-30 Barge bow force-deformation flowchart.............................................................................128 5-1 Barge and pier modeled as separate but coupled modules....................................................137 5-2 Permanent plastic deformation of a barge bow after an impact............................................138 5-3 Stages of barge crush...................................................................................................... .......139 5-4 Unloading curves........................................................................................................... ........140

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12 5-5 Generation of intermediate unl oading curves by interpolation.............................................141 5-6 Flow-chart for nonlinear dynamic pier/soil control module..................................................142 5-7 Flow-chart for nonlinear dynamic barge module..................................................................143 5-8 Treatment of oblique collision conditions.............................................................................144 5-9 OPTS model with linear ly independent springs....................................................................145 5-10 Full bridge model with impact pier.....................................................................................146 5-11 Peripheral models with applied loads..................................................................................147 5-12 Displacements of peripheral models....................................................................................148 5-13 OPTS model with lumped mass..........................................................................................149 5-14 Tributary area of peripheral m odels for lumped mass calculation......................................150 6-1 Barge bow force-deformation relationship............................................................................164 6-2 Inelastic barge bow deformation energy...............................................................................165 6-3 Two degree-of-freedom barge-pier-soil model.....................................................................166 6-4 Peak impact force vs. initial barge ki netic energy using a ri gid pier assumption.................167 6-5 Peak impact force vs. initial barge kine tic energy using an effective barge-pier-soil stiffness...................................................................................................................... ...........168 6-6 Impact load histories...................................................................................................... ........169 6-7 Construction of loading portion of impact force...................................................................170 6-8 Construction of unloading portion of impact force...............................................................171 6-9 AVIL procedure............................................................................................................. ........172 6-10 AASHTO load curve indicating barge masse s and velocities used in validating the applied load history method..................................................................................................173 6-11 Barge bow force-deformation relationship for an impact on a six-foot round column......174 6-12 Impact load history comparisons.........................................................................................175 6-13 Moment results profile for the new St George Island Causeway Bridge channel pier........................................................................................................................... .............176

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13 7-1 Time history analysis of a structure.......................................................................................206 7-2 Time-history versus modal analysis......................................................................................207 7-3 Modal analysis............................................................................................................. ..........208 7-4 Dynamic magnification of si ngle degree-of-freedom system...............................................209 7-5 Dynamic magnification factor for a specific impact load history.........................................210 7-6 Dynamic magnification factor...............................................................................................211 7-7 Specific dynamic magnificati on factor for a low-energy impact vs. a broad-banded design spectrum................................................................................................................ ....212 7-8 Evolution of the dynamic ma gnification spectrum from shor t to long duration loading......213 7-9 Definition of the short and long-period tran sition points......................................................214 7-10 Period of impact loading.................................................................................................. ....215 7-11 Short-period tran sition point data........................................................................................216 7-12 Long-period transition point data........................................................................................217 7-13 Evolving design DMF spectrum..........................................................................................218 7-14 Event-specific and design DMF spect ra for varying impact energies.................................219 7-15 Impact response spectrum analysis procedure.....................................................................220 7-16 Static analysis stage of IRSA............................................................................................. ..221 7-17 Transformation of static displa cements into modal coordinates.........................................222 7-18 Dynamic magnification factor as a function of structural period........................................223 7-19 Combination of amplified dynamic moda l displacements into amplified dynamic structural displacements....................................................................................................... .224 7-20 Nonlinear impact response spectrum analysis procedure....................................................225 7-21 Nonlinear impact response spectrum analysis procedure....................................................226 7-22 Barge bow force-deformation relationship for an impact on a six-foot round column.......227 7-23 Event-specific IRSA validation...........................................................................................228

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14 7-24 Moment results profile for the new St George Island Causeway Bridge channel pier........................................................................................................................... .............229 7-25 Design-oriented IR SA demonstration.................................................................................230 7-26 Moment results profile for the new St George Island Causeway Bridge channel pier........................................................................................................................... .............231

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15 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEVELOPMENT OF TIME-HISTORY AND RESPONSE SPECTRUM ANALYSIS PROCEDURES FOR DETERMINING BRIDGE RESPONSE TO BARGE IMPACT LOADING By David Ronald Cowan December 2007 Chair: Gary Consolazio Major: Civil Engineering Bridge structures that span navigable waterways are inherent ly at risk for barge collision incidents, and as such, must be designed for im pact loading. Current design procedures for barge impact loading use an equivalent static load determination technique. However, barge impacts are fundamentally dynamic in nature, and a static analysis procedure may not be adequate in designing bridge structures to resist a barge co llision. Therefore, dynamic analysis methods for estimating the response of a bridge structure to barge collision are develo ped in this study. As part of this research, static barge bow crus h analyses were conducte dusing a general purpose finite element codeto define the barge bow fo rce-deformation relationship for various bridge pier column shapes and sizes. Using the ge nerated force-deformation relationships, in conjunction with barge mass and sp eed, relationships for the peak dynamic load and duration of loading are developed based upon principles of conservation of energy and linear momentum. The resulting relationships are then used to develop applied vessel impact load and impact response spectrum analysis procedures. Additionall y, both of these methods are validated against the previously validated coupled ve ssel impact analysis technique.

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16 CHAPTER 1 INTRODUCTION 1.1 Introduction Bridge structures that span navigable waterways are inhe rently susceptible to vessel collision. In the United States, vessels operate on a vast network of coas tal waterways and inland rivers, over which bridges span, to transpor t cargo between destin ations. Many of these waterways have relatively shallow channels, and therefore are unable to accommodate largedraft vessels. For this reason, la rge-draft cargo ships are restrict ed to operation in deep water routes and ports. Barges, however, have a sha llow draft, and are thus capable of operating on shallow coastal and inland waterways. Thus, heavily-loaded barge traffic routinely passes beneath highway bridge structures. Vessel collisions on bridge st ructures may occur when vessels veer off-course, becoming aberrant. Factors that affect vessel aberranc y include adverse weather conditions, mechanical failures, and human error. It has been noted in the literature that on average, at least one serious vessel collision occurs per year (Larsen 1993). Duri ng severe vessel collisi ons, significant lateral loads may be imparted to bridge structures. Engi neers must therefore account for lateral vessel impact loads when designing brid ge structures over na vigable waterways. If such bridges cannot adequately resist impact loading, vessel colli sions may result in failure and collapse of the bridge; leading to expensive repa irs, extensive traffic delays, and potentially, human casualties. 1.2 Motivation In 1980, the Sunshine Skyway Bridge over Ta mpa Bay was struck by a cargo ship, the Summit Venture, resulting in the collapse of a portion of the superstr ucture and thirty-five human casualties. Later, in 1988, prompted by the collapse of the Sunshine Skyway Bridge, a formal investigation was initiated to develop brid ge design specifications for vessel collision. As

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17 a result, the American Association of State Highway and Transportati on Officials (AASHTO) published the Guide Specification and Commentary for Vessel Collision Design of Highway Bridges in 1991 (AASHTO 1991). Additionally, sim ilar provisions were incorporated in the AASHTO Load and Resistance Factor Design (L RFD) Specifications (AASHTO 1994) a few years later. Publication of thes e design specifications represented a major step in improving the safety of bridge structures. At the time of the inception of the AASHTO vessel collision specifications, relatively few experimental studies had been conducted involvi ng barge collisions. Thus, very little data was available for the development of the AASHT O specifications. The data upon which the AASHTO specifications were based came from a single experimental st udy conducted by MeierDrnberg during the 1980s in Germany (Meier-D rnberg 1983). In these experiments, MeierDrnberg conducted pendulum drop-hammer imp acts on reduced-scale Eu ropean hopper barge bow models. Additionally, static load tests were also conducted on a similarly scaled barge bow model. One of the major findings of this study was that no significant diffe rences were observed between static and dynamic tests. Thus, relatio nships between barge ki netic energy, barge bow deformation, and static impact force were de veloped and recommended. W ith few modifications, the relationships developed by Meier-Drnbe rg were ultimately adopted by the AASHTO Specifications. Importantly, this implies that static loads are assumed to be sufficient in the prediction of structural cap acity during a vessel collision. Using reduced-scale models to describe a full -scale response may introduce an uncertainty in the accuracy of experimental results. This uncertainty is even more pronounced when fullscale experimental data are unavailable to va lidate results obtained fr om the reduced-scale

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18 models, as is the case with the Meier-Drnberg data. Therefore, the AASHTO vessel collision specifications necessarily incl ude the same degree of uncerta inty given that the AASHTO provisions are directly based upon the Meier-Drnberg results. Particularly, the nature of the dynamic te sts Meier-Drnberg conducted precludes the presence of significant dynamic effects that ar e generated between the barge and bridge in a collision event. The interaction between the barge, bridge, and soil significantly affects the loads, displacements, and stresses generated. One setbac k of the Meier-Drnberg experiments is the use of a pendulum drop-hammer. The interaction be tween the barge and the drop-hammer is not representative of barge-bridge -soil interaction. Additionally, for the Meier-Drnberg study, the barge bow models were fixed in a stationary co nfiguration, thereby simu lating impacts on a rigid pier structure. Hence, the fixed condition of th e barge bow models used in the Meier-Drnberg study prevents dynamic interaction between the barg e and drop-hammer. Use of static analysis and design procedures fails to account for importa nt dynamic effectssuch as inertial forces, damping forces, and rate effectspresent in an impact event. Omission of dynamic effects from analysis procedures may result in a non-uniform margin of safety against failure in a dynamic impact event. At the time that the AASHTO specifications were first published, typical analysis procedures use in bridge design practice were linear and static. Nonlinear and dynamic analyses required expensive, both with respect to cost a nd time, computational hardware and software. As a result, nonlinear dynamic analysis was primarily used in research, being too time-consuming for use in typical design practice. However, with technological advancements in both hardware and software over the past two decades, analysis t echniques that were hist orically in the domain of research have now become common place in design. Nonlinear analysis techniques are now

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19 commonly used in design of bridge foundati ons. Additionally, dynamic analyseseither modal or time-historyare commonly empl oyed in the design of structures to resist earthquake loads. Given the limitations of the Meier-Drnberg data, full-scale barge impact tests were initiated by the Florida Department of Transpor tation (FDOT) and researchers at the University of Florida (UF) in 2000 (Consolaz io et al. 2006). The old St. Ge orge Island Causeway Bridge was scheduled for demolition and eventual repl acement by a new structure. Located in the panhandle of Florida, approximately 5 miles east of Apalachicola, the bridge spanned, north to south, from Eastpoint to St. Geor ge Island. Due to environmental concerns, the alignment of the new replacement bridge deviated nearly 1500 ft from the old alignment in several locations, affording researchers the unique opportunity to conduct full-scale barge impact tests on the outof-commission structure without endangering the new bridge. During March and April of 2004, after the new St. George Island Causeway Bridge was opened to traffic, full-scale impact testing commenced. Two of the piers near the main navigation channel were selected fo r a total of fifteen impact tests. Instrumentation was installed on each pier, and during each test, barge impact fo rces, deformations, and decelerations; pier displacement and accelerations; and soil information were recorded. Once all of the experimental test data was reduced, analytical models of each pier were developed and validated using the experimental results. Based on the data recorded and insights gained from full scale testing of the old St. George Island Causeway Bridge, it was possible to initiate the development of updated design provisions for br idge structures subjected to barge impact loading. 1.3 Objectives The two primary objectives of this research were to develop: 1) improved methods for calculating maximum impact forces imparted to a bridge structure during a barge collision, and 2) improved procedures for determining barge impact load and corres ponding bridge response

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20 such loads. A major aspect of all of the proposed structural analysis methods is the ability to quantify important dynamic effects inherently present in a barge collision event. Most design provisions for impact loading use a static analysis to determine bridge response; however, these static methods fail to capture significant dynamic e ffects. Therefore, semi-empirical methods that can capture dynamic effects were de veloped as a part of this re search, each providing a more sophisticatedyet still design-orie nted (i.e., practical enough for use in bridge design)impact analysis. 1.4 Scope of Work Characterize barge bow force-deformation relationships: It was discovered during St. George Island full-scale testi ng that in situations in which the barge bow undergoes significant plastic deformation, the maximum imp act force that can be generated is limited by the load-carrying capacity of the barge. This implies that, as the barge bow crushes beyond yield and fracture, the impact force ge nerated does not increase with further bow deformation. The relationships of MeierDrnberg and the AASHTO specifications, however, prescribe that the impact force in creases monotonically with increasing barge bow deformation (crush depth). In this study, updated relationships between impact force and barge bow deformation for various barge ty pes have been developed independently of the AASHTO relationship. Develop dynamic analysis techniques for use in design: It was determined during St. George Island experimental test ing that dynamic effects have a significant effect on pier response during a barge impact. Early in an imp act event, inertial (mass-proportional) and damping (velocity-proportional) forces compri se more resistance to impact loading than the stiffness (displacement-proportional resistance ). However, at later stages in an impact, it has been shown that inertial forces actually change directions, a nd drive the motion of the bridge, effectively becoming a source of loading on the structure. This complex interaction of dynamic forces cannot adequate ly be captured through the use of static analysis, and thus dynamic anal ysis is required. In this study, three dynamic analysis techniques were developed and ar e available for use in design: o Coupled vessel impact analysis (CVIA): A time-history analysis that permits interaction between a single degree-of -freedom barge model and a numerical bridge. o Applied vessel impact analysis (AVIL): A time-history analysis technique that uses impact characteristics to generate an approximate time-varying impact load, which in turn is applied to a br idge structure in a dynamic sense.

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21 o Impact response spectrum analysis (IRSA): A response spectrum analysis method that uses impact characteristic s and dynamic structural characteristics (modal shapes and periods of vibration) to determine maximum dynamic forces in a bridge structure during an impact. Validate proposed dynamic analysis techniques: Using FB-MultiPier, the effectiveness of the proposed dynamic analysis methods are evaluated. Construction drawings for the new St. George Island Bridge were obtaine d, and used to develop numerical bridge models. This structure was selected becau se the new St. George Island Bridge is representative of current structures.

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22 CHAPTER 2 BACKGROUND 2.1 Vessel-Bridge Collision Incidents Designing bridge structures that span over vessel-navigable bodi es of water requires that careful consideration be given to the fact that cargo vessels may inadvertently collide with piers that support the bridge superstr ucture. Causes of such collis ions often involve poor weather conditions, limited visibility, strong cross-currents, poor navigational aids, failure of mechanical equipment, or operator error. Worldwide, vessel impacts occur frequently enough that, on average, at least one serious collision occurs e ach year (Larsen 1993). Within the United States, a succession of incidents involving ships and barg es impacting bridge structures clearly demonstrates that the potential for structural failure and loss of life exist. One of the most catastrophic incidents of vesse l-bridge collision in the United States was the 1980 collapse of the Sunshine Skyway Bri dge, which spanned over Tampa Bay in Florida. Navigating in poor weather conditions a nd limited visibility, the cargo ship Summit Venture collided with one of the anchor piers of the bridge causing the collapse of almost 1300 ft. of bridge deck (Figure 2-1) and the loss of thirty-five lives. Due in large part to this incident, comprehensive guidelines for designing bridge struct ures to resist ship and barge collision loads were later published, in 1991, as the AASHTO Guide Specification and Commentary for Vessel Collision Design of Highway Bridges (AASHTO 1991). While massive cargo ships clearly pose a significant threat to bridge structures, they are also limited in operation to relatively deep wa terways. Consequently, ships pose significant risks primarily to bridges near major shipping ports. In contrast, multi -barge flotillas are able to operate in much shallower waterways and thus po se a risk to a greater number of structures.

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23 Considering that each individual barge within a fl otilla might weigh as much as fifty fully-loaded tractor-trailers, the potential damage that can be caused by barges striking bridge piers is clear. In September 1993, a multi-barge tow navigating at night and in dens e fog collided with the Big Bayou Canot railroad bridge near M obile, Alabama resulting in a significant lateral displacement of the structure. Mo ments later, unaware that the br idge had just been struck by a barge, a passenger train attempted to cross th e structure at 70 mph, re sulting in catastrophic structural failure (Figure 2-2) and forty-seven fatalities (K nott 2000). In September 2001 a barge tow navigating near South Padre, Texas veered off course in strong currents and collided with piers supporting the Queen Isabella Causeway Bridge. As a result of this impact, three spans of the structure collapsed (Figure 2-3) and several people died (Wilson 2003). In May 2002, an errant barge tow struck a bridge on interstate I-40 near Webbers Falls, Oklahoma. On an average day, this structure carried appr oximately 20,000 vehicles across the Arkansas River. As a result of the impact, 580 ft. of superstructure collapsed (Figure 2-4), fourteen people died, and traffic had to be rerouted for approxi mately two months (NTSB 2004). 2.2 Review of Experiment al Vessel Impact Tests Despite the significant number of vessel-bridge collisions that have occurred in recent decades, only a small number of instrumented expe rimental tests have ever been performed to quantify vessel impact loading characteristics. Ge nerally, ship collision ev ents have been studied to a much greater extent than have barge collisio ns. Two key ship collision studies form the basis for most current theories rela ting to ship impact loading. Th e first was conducted by Minorsky (1959) to analyze collisions with reference to protection of nucle ar powered ships, and focused on predicting the extent of ve ssel damage sustained during a collision. A semi-analytical approach was employed using data from twenty -six actual collisions. From this data a relationship between the deformed steel volume a nd the absorbed impact energy was formulated.

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24 A second key ship collision study was that of Wo isin (1976) which also focused on deformation of nuclear powered ships during collisions. Data were collected from twenty-four collision tests (Figure 2-5) of reduced-scale (1:7.5 to 1:12) ship-bow models co lliding with ship-side-hull models. Relationships between impact energy, deformation, and force developed during this study were later used in the development of the AASHTO equations for calculating equivalent static ship impact forces (AASHTO 1991). In terms of quantifying the characteristics of barge impact loads, as opposed to ship impact loads, one of the most significant experimental studies conducted to da te is that of MeierDrnberg (1983). This research included both dyna mic and static loading of reduced-s cale (1:4.5 to 1:6) models of European Type IIa barges. In overall dimensions, European Type IIa barges are similar to the jumbo hopper barges that are commonl y found in the U.S. barge fleet. All tests in the Meier-Drnberg study were conducted on partial vessel models that co nsisted only of nose sections of barges. In conducting the dynamic tests in this study, the partial barge models were mounted in a stationary (fixed-boundary-condition) configur ation and then struck by a falling impact pendulum hammer. The amount of impact energy imparted to the barge model during each test was dictated by the we ight of the hammer and its drop height. Due to limitations of hammer drop height, repeated impacts were carried out on each partial barge model to accumulate both impact energy and impact damage (crushing deformation). From the experimental data collected, Meier-Drnberg de veloped relationships between kinetic impact energy, inelastic barge deformation, and force. More recently, experimental st udies have been conducted that overcome one of the key limitations of the Meier-Drnberg studyi.e., the use of reduced-scale models. In 1989, Bridge Diagnostics, Inc. completed a series of full-scale tests for the U.S. Army Corps of Engineers that

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25 involved a nine-barge flotilla impacting lock ga tes at Lock and Dam 26 on the Mississippi river near Alton, Illinois (Goble 1990). Ea ch of the impacts was performed at approximately 0.4 knots. Force, acceleration, and velocity time historie s for the impacting barge were recorded using commercially available sensors such as strain gages and accelerometers. In addition, custom manufactured and calibrated load cells, develope d by Bridge Diagnostics, were used to measure impact forces (Figure 2-6). Unfortunately, data obtained from this study are not directly applicable to bridge pier design because the sy stem struck by the barge was a lock gate, not a bridge pier. Lock gates and bri dge piers posses different struct ural characteristics which produce dissimilar impact loads. More importantly, the energy levels used during these tests were insufficient to cause significant inelastic barge deformation. Because inelastic barge deformations are common in head-on barge-pier collisions, and given that such deformations affect both barge stiffness and impact energy dissip ation, data obtained from this set of tests are not directly applicab le to bridge design. Several years later, full-scale barge impact tests on concrete lock walls were conducted by the U.S. Army Corps of Engineers. In 1997, a 4-ba rge flotilla was used to ram a lock wall at Old Lock and Dam 2, located north of Pittsburgh, Pennsylvania (Patev and Barker 2003). These experiments (Figure 2-7) were conducted to measure the stru ctural response of the lock wall at the point of impact and to quantify barge-to-b arge lashing forces dur ing impact. Strain gages were installed on the barge to record steel plate deformations at the point of impact. An accelerometer was used to capture the overall accele ration history of the flotilla, and clevis pin load cells quantified lashing for ces generated during impact. A total of thirty-six impact tests were successfully carried out on the lock wall.

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26 Following the 4-barge tests, larger impact e xperiments involving a 15barge flotilla were initiated in December of 1998 at the decommissione d Gallipolis Lock at Robert C. Byrd Lock and Dam in West Virginia (Arroyo et al. 2003). In contrast to the 4-ba rge tests, one of the primary goals of the 15-barge tests was to reco ver time-histories of impact force generated between the barge flotilla and the lock wall. To accomplish this goal, a load-measurement impact beam was affixed to the impact corner of the barge flotilla using tw o uniaxial high-capacity clevis-pin load cells (Figure 2-8). Additional instrumentation used during these tests included accelerometers, strain gages, water pressure tran sducers, and smaller capacity clevis-pin load cells that were installed in-line with the barge-to-b arge cable lashings. In total, forty-four impact tests were successfully carried out on the lock wall. In bridge pier design for barge collision lo ading, maximum impact forces are generally associated with head-on impact conditions, not obli que glancing blows of the type tested in the 4-barge and 15-barge tests performed by the Army Corps. Therefore, while the data collected during these tests could be useful in developing load prediction m odels for oblique side impacts on piers, the same data cannot be used to improve the AASHTO expressions for head-on impacts. Additionally, neither of the Army Corp s test series involved dynamic vessel-pier-soil interactions or significant crus hing deformation of the impacti ng barges. Data collected during the experimental tests were later used to deve lop several analytical methods for approximating impact load on structures have been developed based on the experimental test results (Patev 1999, Arroyo et al. 2003). In 2004, full-scale experimental barge impact tests on piers of the old St. George Island Causeway Bridge (Consolazio et al. 2006) were conducted by the Un iversity of Florida (UF) and the department of Transportati on (FDOT). The purpose of conducting these tests was to directly

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27 quantify barge impact loads and resulting pier, so il, and superstructure responses. The research revealed that the AASHTO static design forces currently employed in bridge design range broadly from being overly conservative in some instances to being unc onservative (due to dynamic amplification effects) in other cases. Th is variability stems from differences in the dynamic characteristics of varying bridge types an d in variations of design impact conditions. A detailed review of the St. George Island barge impact test program is provided in Chapter 3. Researchers at the University of Kentucky ( UK) conducted numerical finite element barge impact simulations to estimate forces imparted to bridge piers when impacted by a barge flotilla (Yuan 2005a, Yuan 2005b). High-resolution finite el ement models were developed of a single jumbo hopper barge and of multi-barge flotillas fo r the purpose of conducting collision analyses, using the finite element analysis code LSDYNA (LSTC 2003), on bri dge piers. From the analysis results, expressions were developed for predicting average barge impact force and duration of impact load. Additiona lly, the effects of pier shape a nd pier stiffness on barge impact force were investigated. A response spectrum an alysis technique involvi ng a four degree-offreedom pier model that is subjected to a re ctangular pulse impact load was also developed. Researchers at the University of Texas (U Tx) have recently investigated the AASHTO probabilistic analysis pr ocedures for bridge impact design under vessel impact loading (Manuel et al. 2006). Computer softwa re has been developed to determine the annual frequency of collapse, facilitating the AASHTO probabilistic de sign procedure. Addi tionally, high resolution finite element simulations were conducted to determine impact forces imparted to a pier during an impact event, as well as providing guidance to calculating the ultimate lateral strength of the pier.

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28 2.3 Design of Bridges According to the AASHTO Barge Impact Provisions A pooled fund research program sponsored by eleven states and the Federal Highway Administration (FHWA) was initiated in 1988 to develop methods of safeguarding bridges against collapse when impacted by ships or barg es. The findings of the research were adopted by AASHTO and published in the Guide Specification and Commentary for Vessel Collision Design of Highway Bridges and the Load and Resistance Factor De sign (LRFD) Specifications Provisions included in these public ations serve as a nationally adopted basis for bridge design with respect to vessel collision loads. The provisions allow two a pproaches to collision resistant bridge design. Either the struct ure can be designed to withstand the vessel impact loads alone, or a secondary protection system can be designed that will absorb the vessel impact loads and prevent the bridge structure itse lf from being struck. In additi on to providing design guidelines, the AASHTO provisions recommen d methodologies for placement of the bridge structure relative to the waterway as well as specifications for navigational aids. Both are intended to reduce the potential risk of a ve ssel collision with the bridge. Nonetheless, for a wide ranging set of reasonswindy high-current waterways; a dverse weather conditions; narrow or curved waterway geometrymost bridges that are accessib le to barge impact will likely be struck at some point during their lifetime (Knott and Prucz 2000). With this in mind, all bridges that span navigable waterways need to be designed with due consideration being given to vessel impact loading. The AASHTO Guide Specification and Commentary for Vessel Collision Design of Highway Bridges (1991) provides designers with three me thods by which piers may be designed: Methods I, II, and III. Method I uses a simple semi-deterministic procedure, calibrated to Method II criteria, for the selec tion of a design vessel. This method however, is less accurate than Method II, and as such, should not be us ed for complex structures. Method II is a

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29 probability based method for selecting an imp act design vessel. Method II is also more complicated than Method I; however, Method II is required if documentation on the acceptable annual frequency of collapse is necessary. Method III is a cost-eff ective vessel selection procedure that is permitted when complian ce with acceptable annual frequency of collapse provisions are neither economically nor technically feasible. This method is typically used in evaluation of vulnerability of existing bridges. 2.3.1 Selection of Design Vessel The first step in designing a br idge structure for barge impact loading is to determine the characteristics of the waterway over which the structure spans. Determination of waterway characteristics includes the geometric layout of th e channel (including the centerline) beneath the bridge, the design water depth for each com ponent, and the water cu rrents parallel and perpendicular to the motion of the barge. Next, impact vessel characteristicssuch as, ve ssel type, size, cargo weight, typical speed, and annual frequencymust be determined based upon actual vessels traversing the design waterway. Furthermore, vessel ch aracteristics must be calculated for waterway traffic in both directions along the channel. Using the characte ristics of individual barges, the overall length (length-overall, LOA) of the barge tow must be calculated as the total length of the barge tow plus the length of the tow vessel (Figure 2-9). The design impact velocity for each bridge el ement is calculated based upon the typical vessel transit speed, geometry of the channel, an d overall length of the barge tow. The design velocity for bridge elements w ithin the area extending from the cen terline of the channel to the edge of the channel is the t ypical vessel transit speed (VT) (Figure 2-10). From the edge of the channel (xC) to a distance equal to three ti mes the overall barge tow length (3LOA ) from the centerline (xL), the design speed linearly decreases to a minimum design speed (Vmin). Beyond a

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30 distance of 3LOA from the channel centerline, the design speed is held constant at a minimum design impact speed that corresponds to th e yearly mean curren t at that location. 2.3.2 Method II: Probability Based Analysis AASHTO Method II design requires the designer to calculat e the annual frequency of collapse resulting from vessel collisions as: PC PG PA N AF (2-1) where AF is the annual frequency of collapse, N represents the annual vessel frequency for each design element and vessel type and size, PA is the probability of vesse l aberrancy, PG is the geometric probability of impact, and PC is the pr obability of collapse due to impact. The annual frequency of collapse must be calculated for each bridge element and each type and size of vessel. Summing up all of the individual annua l frequencies of collapse yields the annual frequency of collapse for the total bridge. The an nual frequency of collapse determined from this process is required to be less than or equal to an acceptable annual frequency of collapse that depends upon the importance of the bridge: For critical bridges 0001 0 AF (i.e., a 1 in 10,000 probabi lity of failure each year) For regular bridges 001 0 AF (i.e., a 1 in 1000 probabi lity of failure each year) Probability of aberrancy (PA) represents the probability that a given vessel will deviate from its course, possibly imperiling the bridge. Based upon the proposed bridge site, the most accurate determination of probability of aberranc y is determined from historical data on vessel aberrancy. However, if appropr iate historical data are not available, ASSHTO permits an estimation for the probability of aberrancy: D XC C BR R R R BR PA (2-2)

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31 where BR is the base aberrancy rate (specified by AASHTO), RB is the bridge location correction factor, RC is the water current correction factor in the direction of barge travel, RXC is the water current correction factor transverse to barge travel, and RD is a vessel traffic density factor. The base aberrancy rate (BR) was determin ed from historical accident data along several U.S. waterways. For barge traffic, the base aberrancy rate specified by AASHTO is 1.2x10-4 (i.e., 0.00012, or 1 aberrancy in ever y 8333 barge transits). The bridge location correction factor (RB) was implemented to account for the added difficulty in navigating a ba rge tow around a bend (Figure 2-11): region bend 45 0 1 region nsition tra 90 0 1 region straight 0 1 RB (2-3) where is the angle of the bend in degrees. In order to account for the e ffects that water currents have on the navigation of a barge tow, correction factors for wa ter currents both parallel (RC) and transverse (RXC) to the motion of the barge tow are required: 10 V 1 RC C (2-4) XC XCV 1 R (2-5) where VC and VXC are the water current velocities parallel and transverse to the barge motion (in units of knots). Furthermore, depending upon the density of vessel traffi c in the waterway, a vessel traffic density factor is used to modify the base aberrancy rate: Low density traffic: RD = 1.0 Average density traffic: RD = 1.3 High density traffic: RD = 1.6

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32 Geometric probability of impact is defined as the probability that a vessel will strike a bridge element once it has become aberrant. Ba sed on historical barge collision data, AASHTO requires that a normal distribution be used to ch aracterize the locations of aberrant vessels in relation to the centerline of the channel (Figure 2-12). The normal distribution is then used to determine the chance that a given bridge elem ent will be struck by the aberrant barge. Furthermore, the mean value of the distribution is situated on the centerline of the channel and the standard deviation is assumed to be equal to the overall length of the barge tow (LOA). The effective width of impact for each bridge element is calculated as the width of the structural element plus half the width of the vessel on each side of the pier. Using this impact zone width, the geometric probability of imp act is calculated as the area under the normal distribution curve that is bounded by the effective imp act width and centered on the pier. Assuming a bridge element is struck by an er rant vessel, the probability that the element (e.g., a pier) will collapse must also be determined. AASHTO provides the following relationships for calculating the probability of collapse (PC) based upon the strength of the bridge element and the static vessel impact fo rce that is applied to the element (Figure 2-13): 0 1 P H if .0 0 1.0 P H 0.1 if 9 P H 1 0.1 P H if P H 0.1 9 .1 0 PC (2-6) In this equation, H is the ultimate strength of the pier element, and P is the design static vessel impact force. 2.3.3 Barge Impact Force Determination The AASHTO specifications use a kinetic en ergy based method to determine the design impact load imparted to a bridge element. Usin g the total barge tow weight and design velocity

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33 determined for each bridge element (e.g., pier) the kinetic energy of the barge is computed as follows: 2 29 WV C KE2 H (2-7) where KE is the barge kinetic energy (kip-ft), CH is a hydrodynamic mass co efficient, W is the weight of the vessel tow (in tones), and V is th e design speed of the vessel tow (ft/sec). The hydrodynamic mass coefficient (CH) is included to account for additional inertia forces caused by the mass of the water surrounding and moving w ith the vessel. Several variables may be accounted for in the determination of CH : water depth, underkeel clearances, shape of the vessel, speed, currents, dire ction of travel, and the clean liness of the hull underwater. A simplified expression has been adopted by AASHTO in the case of a vessel moving in a forward direction at high velocity (the worst-case scenario). Under su ch conditions, the recommended procedure depends only on the underkeel clearances: For large underkeel clearances H0.5 draft : C1.05 For small underkeel clearances H0.1 draft : C1.25 where the draft is the distance between the botto m of the vessel and the floor of the waterway. For underkeel clearances between the two limits cited above, CH is estimated by linear interpolation. Once the kinetic energy of the barge tow has be en determined, a two-part empirical loadprediction model is used to dete rmine the static-equivalent impact load. The first component of the model consists of an empirical relationship th at predicts barge crush deformation (inelastic deformation) as a function of kinetic energy:

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34 B 2 1 BR 2 10 1 5672 KE 1 a (2-8) where aB is the depth (ft.) of barge cr ush deformation (depth of penetration of the bridge pier into the bow of the barge), KE is the barge kinetic ener gy (kip-ft), and 35 B RB B ; where BB is the width of the barge (ft). Figure 2-14 graphically illustrates Eqn. 2-8. The second component of the load prediction m odel consists of an empirical barge crush model that predicts impact load s as a function of crush depth: 34 0 a if R a 110 1349 34 0 a if R a 4112 PB B B B B B B (2-9) where PB is the equivalent static barge impact load (kips) and aB is the barge crush depth (ft). The AASHTO barge force-deformati on relationship given in Eqn. 2-9 is illustrated in Figure 215. Furthermore, combining Eqns. 2-8 and 2-9, a relationship between barge impact force and initial barge kinetic energy may be defined (Figure 2-16). As will be discussed in additional detail in Chapter 4, Eqns. 2-8 and 2-9 above were both adopted from research conducted by Meier-Drnberg (1983).

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35 Figure 2-1 Collapse of the Sunshine Skyway Bridge in Florida (1980) after being struck by the cargo ship Summit Venture

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36 Figure 2-2 Failure of the Big Bayou Canot railroad bridge in Alabama (1993) after being struck by a barge flotilla

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37 Figure 2-3 Collapse of the Queen Isabella Causeway Bridge in Texas (2001) after being struck by a barge flotilla

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38 Figure 2-4 Collapse of an Intersta te I-40 bridge in Oklahoma (2002) after being struck by a barge flotilla (Source: Oklahoma DOT)

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39 A B Figure 2-5 Reduced scale ship-to-ship collisi on tests conducted by Woisin (1976) A) Ship bow model on inclined ramp prior to test, B) Permanent deformation of ship bow model after test

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40 A B Figure 2-6 Instrumented full-scale barge-lock-gat e collision tests (Source: Bridge Diagnostics, Inc.) A) Barge bow approaching lock gate, B) Load cells attached to barge bow

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41 A B Figure 2-7 Instrumented 4-barge lock-wall coll ision tests (Source: U.S. Army Corps of Engineers) A) Push boat and 4-barge flotilla B) Sensors at imp act corner of barge

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42 A B Figure 2-8 Instrumented 15-barge lock-wall co llision tests (Source: U.S. Army Corps of Engineers) A) Push boat and 15-barge flotilla, B) Force measurement beam attached to barge with clevis-pin load cells

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43 Figure 2-9 Barge tow configurati on (Source: AASHTO Figure 3.5.1-2)

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44 Distance from Centerline of ChannelImpact Speed VTVminxcxL 3 LOA Figure 2-10 Design impact speed

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45 A B Figure 2-11 Bridge location correct ion factor (Source: AASHTO Fi gure 4.8.3.2-1) A) Turn in channel, B) Bend in channel

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46 Figure 2-12 Geometric probability of coll ision (Source: AASHTO Figure 4.8.3.3-1)

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47 Figure 2-13 Probability of collapse distribution (Source: AASHTO Figure 4.8.3.4-1)

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48 0 2 4 6 8 10 12 02000400060008000 Kinetic Energy, KE (kip-ft) 10000 12000140001600018000Crush depth, (ft) aB Crush depth (a )B Figure 2-14 AASHTO relations hip between kinetic energy and barge crush depth

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49 024681012 0 500 1000 1500 2000 2500 3000 Crush depth, (ft) aBImpact Load P (kip)B Figure 2-15 AASHTO relations hip between barge crush depth and impact force

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50 02000400060008000 0 500 1000 1500 2000 2500 3000 Kinetic Energy, KE (kip-ft)Impact Load P (kip)10000 12000140001600018000B Figure 2-16 Relationship between ki netic energy and impact load

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51 CHAPTER 3 SUMMARY OF FINDINGS FROM ST. GEORGE ISLAND BARGE IMPACT TESTING 3.1 Introduction In 2004, the University of Florida (UF) a nd the Florida Department of Transportation (FDOT) conducted full-scale barge impact tests on the old St. Ge orge Island Causeway Bridge (Consolazio et al. 2006). Expe rimental results were obtaine d to quantify the loading and response of pier and bridge stru ctures subjected to a barge impact. To compliment the physical testing, numerical finite elem ent analysis (FEA) techniques were also employed to aid in interpretation of experimental test data. Comparisons of measur ed experimental data and FEA results substantiated the validity of the experimental data and provided additional insights into the nature of pier response to barge impact loading. 3.2 Overview of Experimental Test Program Impact tests were conducted on Piers 1 and 3 on the south side of the St. George Island Causeway Bridge, near the navigation channel (Figure 3-1). These piers were selected for the different structural configurat ions that they represented. Pier-1 (Figure 3-2), the more impact resistant pier, was a reinforced concrete pier composed of two pier columns, a pier cap, a shear wall for lateral resistance, and a massive concrete pile cap and tremie seal mud-line foo ting, all supported by fo rty HP 14x73 steel piles. This pier was adjacent to the navigation channel, and as such, was most prone to impact from an errant vessel. Therefore, Pier-1 was designed to be the most impact resistant pier supporting the bridge. From the perspective of the barge impact te st program, Pier-1 wa s expected to able to resist the highest loads as we ll as produce the largest amounts of barge bow deformation. In contrast to Pier-1, Pier-3 (Figure 3-3) was a more flexible, less impact resistant reinforced concrete structure co mposed of two pier columns, a pier cap, and a shear strut for

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52 lateral resistance. Unlike the massive pile and cap foundation sy stem supporting Pier-1, Pier-3 was supported by two small waterline pile caps, each on top of four 20 in. square prestressed concrete piles. Due to its flexibility, Pier-3 was not able to resist large impact forces. Pier-3, however, was selected to investig ate the effects low-energy impact s on secondary support piers. Furthermore, it was anticipated that the barge would sustain negligible permanent barge bow deformations during impacts on Pier-3. The overall experimental test program was conducted in three di stinct series, each with a different structural configuration. The first test series (Figure 3-4), denoted series P1, involved tests conducted on Pier-1 in isol ation, with the primary objectiv e of generating maximal impact loads and inelastic barge bow defo rmations. In order to achieve such loads and deformations, the mass of the barge was increased by placing two 55 ft. spans of concrete deck from a previously demolished section of the bridge on to p of the barge, for a total barge-plus-payload weight of 604 tons. Impact speeds in seri es P1 varied from 0.75 knots 3.45 knots. Instrumentation used during test series P1 incl uded load cells, acceleromet ers (on both the pier and the barge), displacement tr ansducers, optical break beams (used to trigger the data acquisition (DAQ) system and provi de precise barge velocity determination), and a pressure transducer (used to measure pressure changes in the water). See Consolazio et al. (2006) for a detailed discussion of the instrumentation systems used. Upon completion of the series P1 tests, the payload spans on the test barge were removed (Figure 3-5a), restoring the empty barg e weight to 275 tons. After removal of the spans, impact tests on Pier-3 began; the first series of which was denoted seri es B3. The B3 test series involved low-energy impacts on Pier-3 with superstr ucture spans still intact between piers 2, 3, 4, 5, and beyond (Figure 3-5b). Due to the relatively high flex ibility of Pier-3, impact velocities

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53 were limited so as to avoid dislodging the supe rstructure. With the reduced mass and lower velocities, the impact energies imparted to the pier during the B3 series of impacts were not substantial enough to generate noticeable permanent barge bow deformations. The instrumentation used during test series B3 in cluded load cells, accelerometers (including the addition of accelerometers on the superstructure), displacement transducer s, an optical break beam, and strain gages (attached to the piles). Following the B3 series tests, the superstructu re connecting piers 2, 3, and 4 was removed leaving Pier-3 isolated (Figure 3-6). Impact tests were then conducted on Pier-3 in isolation during tests series P3. The series P3 tests were very similar to the series B3 tests with respect to barge velocities and mass, instrumentation, and ba rge bow deformations. Data from the B3 and P3 series tests were used to compare the response of an isolated pier to a pier that is integrated within a bridge structure. A total of fifteen impact tests were conducted: eight in series P1, four in series B3, and three in series P3. Maximum allowable imp act speeds for each series were governed by equipment limitations, weather conditions, and sa fety concerns. For additional details, see Consolazio et al. (2006). 3.3 Overview of Analytical Research Finite element models were developed using the pier analysis program, FB-MultiPier (2005), which was selected for its ability to model both dynamic behavior and material nonlinearity. A brief description of the finite el ement models is presented below, however, for a more detailed discussion see Consolazio et al. (2006) and McVay et al. (2005).

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54 3.3.1 FB-MultiPier Models 3.3.1.1 FB-MultiPier Pier-1 model The FB-MultiPier Pier-1 model used frame elements to model the pier components, such as the columns, pier cap, shear wall, and piles; and shell elements to model the behavior of the cap and seal (Figure 3-7). Since shell elements were us ed to model the pile cap, the piles connected to the mid-plane of the cap. Howeve r, in the actual structure, the piles were embedded in both the 5 ft thick pile cap, and the 6 ft thick tremie seal. Within this embedment length, the piles were restrained against flexure. To correctly model this embedment, a network of cross bracing frame elements (Figure 3-7) was placed in the top 8.5 ft (half of the 5 ft cap plus the 6 ft seal) of the piles. The material behavior of all structural elements was chosen to be linear elastic because the phys ical tests were non-de structive in nature. 3.3.1.2 FB-MultiPier Pier-3 model All structural components of Pier-3pier co lumns, pier cap, shear strut, and piles excluding the pile caps, were mode led using frame elements (Figure 3-8). As with Pier-1, the pile caps were modeled using shell elements at th e mid-plane elevation of th e caps. In the Pier-3 model, it was determined that additional elements were necessary to co rrectly model the shear strut behavior. As such, cross bracing was provi ded to stiffen the shear strut. Furthermore, though not monolithically cast together, the strut and the pile caps were cast such that they could come in contact during impact. Therefore, a st rut was placed connecting the pier column to the pile caps to model this added stiffness. Similar to all previous models, the material model chosen was linear elastic. 3.3.1.3 FB-MultiPier Bridge model In addition to the Pier-3 model, a model of the bridge was also developed (Figure 3-9). This model included Piers 2, 3, 4, and 5 with supe rstructure elements connecting adjacent piers.

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55 Models of Piers 4 and 5 were developed from the Pier-3 model since these three piers have very similar structural layoutswith va riation only in the height of th e pier column. Pier-2 had a similar layout to Pier-1, in that it was composed of two pier columns, a pier cap, a shear wall, and a massive mudline footing with a pile cap res ting on twenty-one steel H-piles. All elements of each pier were modeled using frame elements with the exception of the pile caps, which were modeled using shell elements. The superstructure was modeled using frame elements with crosssectional properties calculated from the cross-se ctional properties of th e individual elements. 3.3.2 Finite Element Simulation of Models In order to understand the sour ces of resistance that are mobilized during a barge impact scenario, it was imperative to quantify all of the forces acting on the structure. However, sensors installed on the piers were unable to measure all of the forces gene rated during the impact tests. Thus, using experimental data, the numerical models discussed previously were calibrated to yield responses similar (e.g. pier displacem ent) to those measured experimentally. The general calibration procedure involved a pplying the impact loads measured from the experimental tests directly to the numerical mode ls of the piers. Various parameters (e.g. soil properties) were then calibrated such that th e numerical predictions of pier responsepier displacement, pile displacement, pile shears, pressure forces on the foundation, etc.agreed well with the measured experimental data. With c onfidence in the numerical models established, results from the analyses were then used to predict sources of resist ance that could not be measured during the experimental tests. This procedure was applied to three characte ristic impact tests from the experimental program. Each test in the expe rimental study was given a four-c haracter designation; the first two characters indicate the test series in which the impact test s were conducted, and the second two characters indicate the specifi c test within that series. Th e three tests analyzed are as

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56 follows: 1.) P1T7 (i.e. test series P1, test numbe r 7) 2.) P3T3, and 3.) B3T4. These three tests represent some of the more severe impacts conduc ted. The remainder of this section focuses on results from dynamic analyses of these three test s. Specifically, the sources of resistances are broken down and compared to the impact force (as well as each other) during each simulation. 3.3.2.1 Impact test P1T7 Figure 3-10 is a schematic of Pier-1 showing the forces acting upon the structure during test P1T7. Due to the embedment of the pier cap and tremie seal in the soil, a complex interactionboth static and dynamicbetween th e soil and the pier foundation occurred. Forces presented in the schematic that were experimentally measured either directly or indirectlyinclude the impact force, inertia forc e, shear force in the instrumented pile, and the cap+seal passive+active pressure force. The experi mentally measured impact load for test P1T7 was applied to the finite element mo del as an external force (Figure 3-10). Calibration of the model was then carried out to bring model and experimental re sults into agreement mainly through refinement of soil parameters. Figure 3-11a provides a comparison of all forces acting on the pier structure during the P1T7 impact, and Figure 3-11b compares the soil forces acting on the pile cap and tremie seal. Positive values indicate forces acting in the directions shown in Figure 3-10. The effective load presented in these plots is a combination of th e applied load and the negative portion of the inertial (acceleration-dependent) force. This appr oach was used because, as the inertial force changes from a positive valuerepresenting a resistance to the motion of the pierto a negative value, the force becomes a source of loadi ng, further driving the motion of the pier. Table 3-1 summarizes the maximum values of th e five major forces acting on the pier presented in Figures 3-11. Examining the data presented in th e table, it is eviden t that the inertial resistance is mobilized more rapidly than all other sources of resistance. Note that at the time at

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57 which the inertial resistance is fully mobilized (2 50 kips at 0.10 sec), the inertial resistance is approximately 40-percent of the applied load of 600 kips at the same point in time. Furthermore, the other three major sources of resistance indicated in Figure 3-11a are all less than 200 kips. This indicates that at an early stage of an imp act event, the mass propertie s of the pier structure comprise an important contribution to the total resistance developed. As the impact event continues, note that betw een 0.1 and 0.2 seconds, the inertial force has reversed direction, effectively providing an additional source of loading (Figure 3-11a). When combining this inertial load with the applied lo ad, the effective load maximizes at approximately 970 kips at 0.25 sec. Comparing this value to the major sources of resistance present, it is apparent that the sum of the pile shears (an indication of the soil forces acting on the piles) contributes only 30-percent to th e total resistance, whereas the soil forces acting on the cap and seal contribute to approximately 70-percent of th e total resistance, thus indicating that the cap and seal forces have a major eff ect on the total resistance (Figure 3-11). 3.3.2.2 Impact test P3T3 Figure 3-12 is a schematic of Pier-3 showing the forces acting upon the structure during test P3T3. One of the key differences between this test and test P1T7, is that the soil forces acting on the cap and seal during test P1T7 are not present in test P3T3, as the pile caps are above the mudline. Thus, looking at the forces acting on the pier structure, the inte raction of Pier-3 with the surrounding soil was far less comp lex than that of Pier-1 with its surrounding soil. As with test P1T7, the experimentally measured P3T3 load history was applied dynamically to the structure (Figure 3-12). Furthermore, results from the calibrated P3T3 model agreed well with the results from the experimental test (Consolazio et al. 2006). As with test P1T7, the inertial force ac ting on Pier-3 mobilizes earlierreaching a maximum value of approximately 370 kips at 0.08 secthan the sum of the pile shears

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58 maximizing at approximately 400 kips at 0.23 sec (Figure 3-13). Furthermore, this inertial resistance clearly dominates the total resistance of the pier at this early stage, representing approximately 75-percent of the total resistance at 0.08 sec, where as the pile shears (slightly larger than 100 kips at 0.08 sec) only represent 25-percent. Clearly, iner tial forces provide an important source of resistan ce early in the P3T3 test. 3.3.2.3 Impact test B3T4 Impact test B3T4 differed from tests P1T7 a nd P3T3 in that during test B3T4, portions of the superstructure connecting Pier-3 to adjacent piers were still intact. Comparing Figure 3-14 to Figure 3-12, it can be seen that the sources of resi stance present in test B3T4 are the same as those in P3T3 with the addition of load transferring through shear in the superstructure bearings to the superstructure and adjacent piers. Usi ng the bridge model discussed earlier, a dynamic FB-MultiPier analysis was conducted. As with tests P1T7 and P3T3, the experimentally measured B3T4 impact load was applied to Pier-3 in the numerical model (Figure 3-14). Forces acting on the pier were extracted from the FB-MultiPier analysis of test B3T4, and are presented in Figure 3-15. Like previous test s, pier inertia forces mo bilized earlier than any other forms of resistance, maximizing at 190 ki ps at approximately 0.08 seconds and accounting for approximately 70-percent of the 280-kip applie d load at this point in time, thus clearly dominating the resistance early in the impact. As the pier displaces further, shear forces (representing static and dynamic resistance from th e soil) start to mobilize, maximizing at about 260 kips at 0.20 seconds. Following the shear fo rces, forces in the bearing pads (representing resistance from adjacent piers) begin to mobili ze, maximizing at 130 kips at approximately 0.16 seconds. As with the P1T7 and P3T3 analyses, the iner tia resistance quickly decreases, and changes to a form of loading when it becomes negative, further driving pier motion. When the bearing

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59 shear force decreases and becomes negative, it cons titutes a form of loading on the structure also driving the motion of the pier. To illustrate this concept, the ma gnitudes of the inertial force and the bearing shear force (when negative) are added to the applied load to form an effective load history (Figure 3-15). Note that from 0.25 to 0.50 seconds this effective load is approximately equal to the shear in the piles (w ith the difference being attributable to structural damping), thus indicating that the effective load drives the motion of the pier, and the resistance to this motion is mobilized in the soil (acti ng through the pile shears). 3.4 Comparison of Dynamic and Static Pier Response Currently, design of bridges to re sist barge impact load involve s using an equivalent static load proceduresuch as the method outlined in AASHTOin which the calculated load is applied to a static pier model. However, as presented in the results outlined in previous sections, dynamic effects (such as damping and inertial forces) constitute a significant source of resistance, and in some cases additional loading. In the present section, comparisons between static and dynamic analyses for each of the th ree tests simulated (i.e. P1T7, P3T3, and B3T4) will be presented. Three separate analyses were conducted for each test: 1.) dynamic analysis using the experimentally measured load histories, 2.) static analysis using the peak load from the experimentally measured load histories applied statically, and 3.) static analysis using the AASHTO specified load based upon the initial impact energy measured for the tests. Each of these cases is summarized in Table 3-2. Comparisons of results for each of these three analyses will provide insight into the level of safety pres ent in static analysis procedures, such as the methods provided by AASHTO. Cases A, D, and G in Table 3-2 represent dynamic analyses in which the time-varying experimentally measured loads are applied to the piers in tests P1T7, P3T3, and B3T4 respectively. For cases B, E, and H, the peak lo ad from experimentally measured load histories

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60 (864 kips, 516 kips, and 328 kips for tests P1T7, P3T3, and B3T4 respecti vely) was applied to the structures statically. Finally, cases C, F, and I represent the cases in which the AASHTO equivalent static load is ca lculatedusing a hydrodynamic mass coefficient of 1.05from the measured impact energies (494 kip-ft, 108 kip-ft and 75 kip-ft for experiments P1T7, P3T3, and B3T4 respectively), and then applied to the stru cture in a static manner. All dynamic effects (such as structural and soil damping, inertial eff ects, and cyclic degrada tion behavior) are absent in the static analysis for each model. The experiments are broken down into two cat egories: 1.) high-ene rgy impacts (those in which there is enough impact energy to produce significant plastic defo rmation of the barge bow), and 2.) low-energy impacts (those in whic h negligible plastic deformation of the barge bow is observed). The P1T7 experimentalong with the majority of the P1 series testsfalls into the former category, where as the P3 and B3 series tests falls into the latter. Figure 3-16 presents key results for the foundation of each pier from each analysis. For the high-energy impact P1T7, a comparison between cases A and B shows that for the same peak load, the maximum dynamic pier displacement is 32-pe rcent larger than the static displacement. Furthermore, as expected, the pile shears and mo ments are also larger for the dynamic analysis. This amplification in pier displacement, and ultim ately response forces, indicates that inertial effects play an important role in driving the pier motion. In comparing the low-energy cases (cases D and E for test P3T3, and cases G and H for test B3T4) the opposite is observed. The case D dynamic analysis yields a maximum displacement that is 27-percent smaller than that resulting from the case E static analysis. Likewise, with test B3T4, the case G dynamic anal ysis yields a maximum displacement that is 14-percent smaller than that obtained from the case H static analysis. For both tests, the

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61 predicted pile shears and moments exhibit a similar decrease in the dynamic cases, as is expected. Including the AASHTO results (i.e cases C, F, and I), it can be seen that for the highenergy impact of test P1T7, the AASHTO procedure predicts a load that is higher than the peak dynamic load. As a result, the predicted pier displacements from the AASHTO load case are 360-percent larger than the maximum dynamic pier displacement, and consequently, the resulting pile shears and moments are also much larger. In contrast, for cases F and I, the ASSHTO procedure predicts load s that are smaller than the e xperimentally measured peak dynamic loads for tests P3T3 and B3T4 resp ectively. However, when resulting pier displacements and pile forces are compared, thes e quantities are only slightly smaller than the maximum dynamic quantities for the respective tests. For the cases in which piers are in isolation, the column shears and moments are negligible for the static cases, as expected, considering there is no restraint at the top of the pier. However, for the B3T4 cases, significant forces are developed in the columns in the static cases as load transfers to the superstructure. Thus, in Figure 3-17 the maximum column shears and moments and bearing shear forces are compared for cases G, H, and I. The column shear and moment, and the bearing shear force predicted for case H are 26, 27, and 28-percent of those predicted from the dynamic case; and those from case I are 20, 21, and 22-percent of the dynamic quantities. Given that demands on the foundation, such as pile shears and moments, ar e quite similar for the B3T4 cases, the disparity between the column fo rce results of the stat ic and dynamic cases is attributable to amplification f actors from the inertia forces generated in the superstructure. Although momentary, the effects of these inertia forces can account for as much as 70 to 80percent of the column forces generated. Thus failure to account for dynamic restraint, in

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62 addition to the static resistance, at the top of a pier can lead to unconservative column design forces. 3.5 Observations Field testing of the old St. George Island Ca useway Bridge was undertaken to investigate impact loads, as well as resulting pier respons es, generated during barge-to-bridge collisions. Results from these experiments were then used to calibrate several finite element models, the results of which were used to interpret the experi mental results, as well as identify sources of resistance that could not be measured during th e experiments. Subsequently, the results from these analyses were used to conduct a preliminary assessment of the current AASHTO load determination procedure. Results indicate that mass-proportional forces, known as inertial forces, constitute a significant source of resistance to the motion of a pier during the ear ly stages of a barge impact. Inertial forces tend to mobilize ea rlier in an impact than static or damping forces (displacement and velocity-proportional forces re spectively). However, as the pi er reaches its peak velocity and begins to decelerate, these inertial forces change from a s ource of resistance to pier motion to an effective load that act ually drives pier motion. The effect of inertia forces is even more pronoun ced in the analysis of bridge structures, as opposed to isolated piers, as the inertial resistance and effective loading due to the presence of a superstructure can significantly incr ease the design forces in pier columns. Given the important influences that inertial forces have on structural response, it is recommended that dynamic analysis procedures, which are capable of accoun ting for such phenomena, be used in designing and assessing bridges that may be s ubjected to vessel collision loading.

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63 Table 3-1 Summary of forces acti ng on the pier during test P1T7 Force Approx. maximum (kips) Time to peak (sec) Applied loading 850 0.15 Inertial forces on the pier 250 0.10 Sum of pile shears 275 0.27 Pressure forces on cap+seal 200 0.20 Friction forces on cap+seal 500 0.25 Table 3-2 Dynamic and static analysis cases Case Impact condition Analysis type Load description Max. load (kip) Pier/pile behavior Soil behavior A P1T7 Dynamic Time-varying P1T7 864 Linear Nonlinear B P1T7 Static Peak P1T7 load 864 Linear Nonlinear C P1T7 Static AASHTO 1788 Linear Nonlinear D P3T3 Dynamic Time-varying P3T3 516 Linear Nonlinear E P3T3 Static Peak P3T3 load 516 Linear Nonlinear F P3T3 Static AASHTO 398 Linear Nonlinear G B3T4 Dynamic Time-varying B3T4 328 Linear Nonlinear H B3T4 Static Peak B3T4 load 328 Linear Nonlinear I B3T4 Static AASHTO 276 Linear Nonlinear

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64 To: Saint George Island, South To: Eastpoint, North Continuous steel girder span (removed prior to impact tests) Pier-1 (mud-line footing) Pier-2 (mud-line footing) Pier-4 ( water-line footing) Pier-3 ( water-line footing) Concrete girder simple spans Island Area of bridge instrumented during barge impact tests Navigation channel Figure 3-1 Overview of th e layout of the bridge

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65 28'-1.5" 16'-4.5" 6'-0.25" 6'-0.25" 16'-4.5" 7'-0" 7'-0" 18" 4'-4.75" 4'-4.75" 18" 7'-0" 32'-0.375" 15'-0" 5'-0" 6'-0" Batter: 0.25"/12" Pier columns Bent Cap Pile cap Tremie seal HP 14x73 steel piles 18"18" 4'-0" 7'-2.5" 7'-2.5" 6'-7" 4'-4" 4'-7.5" Batter: 0.25"/12" 18" 18" 18" 18" 7 @ 5'-2" 4 @ 4'-6" Pile cap Tremie seal Batter: 1"/12" MSL Figure 3-2 Schematic of Pier-1

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66 MSL Batter: 1.5"/12" Batter: 1.5"/12" Batter: 0.25"/12" MSL 24'-0" 17'-0" 3'-6" 3'-6" 2'-3" 4'-0" 39'-5.8125" 4'-0" 4'-0" 2'-3" 3'-3" 3'-5" 2'-5.625" 2'-5.625" 2'-6" 5'-0.75" 24" 48" 24" 12'-6" 24" 48" 24" 24" 72" 20" x 20" Concrete piles Pier cap Pier columns Pile cap Strut Figure 3-3 Schematic of Pier-3

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67 Figure 3-4 Test barge with payload imp acting Pier-1 in the series P1 tests

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68 A B Figure 3-5 Series B3 tests A) Empty test barge used in the series B3 tests, B) Test barge impacting the bridge at Pier-3 during the series B3 tests

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69 Figure 3-6 Pier-3 in isolati on for the series P3 tests

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70 Figure 3-7 Pier-1 FB-MultiPier model

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71 Figure 3-8 Pier-3 FB-MultiPier model

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72 Pier-2 Pier-3 Pier-4 Pier-5 Figure 3-9 Bridge FB-MultiPier model

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73 Sum of pile shear forces (at pile heads) Cap+seal passive+active force (sum of lead and trail sides) Impact force Inertia force of pier, cap, and seal Cap+seal skin friction force (bottom and side surfaces of cap/seal) Instrumented pile Sum of pile damping forces (at pile heads) 0 100 200 300 400 500 600 700 800 900 1000 00.20.40.60.811.2 1.4 Impact force (kips)Time (sec) Experimentally measured impact load Figure 3-10 Schematic of forces acting on Pier-1

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74 Time (sec)Force (kip)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.31.4 -400 -200 0 200 400 600 800 1000 Applied load Effective load Sum of pile shears Total damping at pile heads Inertia of pier+cap+seal A Time (sec)Force (kip)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.31.4 -400 -200 0 200 400 600 800 1000 Applied load Effective load Cap+seal soil passive+active force Cap+seal soil friction force B Figure 3-11 Resistance forces mobili zed during tests P1T7 A) Forces acting on the pier structure, B) Cap and seal pressure and friction forces

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75 Impact force Inertia force of pier and caps Sum of pile shear forces (at pile heads) Sum of pile damping forces (at pile heads)Experimentally measured impact load 0 100 200 300 400 500 600 700 800 900 00.20.40.60.811.21.4Impact force (kips)Time (sec) 1000 Figure 3-12 Schematic of forces acting on Pier-3

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76 Time (sec)Force (kip) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.31.4 -400 -300 -200 -100 0 100 200 300 400 500 600 Applied load Effective load Sum of pile shears Total pile damping Inertia of pier+caps Figure 3-13 Resistance forces mobilized during tests P3T3

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77 Impact force Inertia force of pier and caps Sum of pile shear forces (at pile heads) Sum of pile damping forces (at pile heads) Bearing shear force from superstructure Experimentally measured impact load 0 100 200 300 400 500 600 700 800 900 00.20.40.60.811.21.4Impact force (kips)Time (sec) 1000 Figure 3-14 Schematic of forces act ing on Pier-3 during test B3T4

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78 Time (sec)Force (kip) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.31.4 -200 -100 0 100 200 300 400 500 Applied load Effective load Sum of pile shears Total pile damping Inertia of pier+caps Sum of bearing shears Figure 3-15 Resistance forces mobilized during tests B3T4

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79 Max. applied load Max. pier displacement Max. pile shear Max. pile moment864 kip 864 kip 1788 kip 0.62 in. 2.85 in. 0.47 in. 10 kip 8 kip 32 kip 65 kip-ft 49 kip-ft 251 kip-ft Case A: P1T7 dynamic Case B: P1T7 static Case C: P1T7 static AASHTOA Max. applied load Max. pier displacement Max. pile shear Max. pile moment516 kip 516 kip 398 kip 2.82 in. 3.86 in. 2.61 in. 52 kip 61 kip 47 kip 575 kip-ft 723 kip-ft 530 kip-ft Case D: P3T3 dynamic Case E: P3T3 static Case F: P3T3 static AASHTOB Max. applied load Max. pile shear Max. pile moment Case G: B3T4 dynamic Case H: B3T4 static Case I: B3T4 static AASHTO Max. pier displacement328 kip 328 kip 276 kip 1.58 in. 1.84 in. 1.47 in. 33 kip 37 kip 31 kip 438 kip-ft 495 kip-ft 410 kip-ft C Figure 3-16 Comparison dynamic and static analysis results for foundation of pier A) P1T4, B) P3T3, C) B3T4

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80 Case G: B3T4 dynamic Case H: B3T4 static Case I: B3T4 static AASHTO328 kip 328 kip 276 kipMax. applied load Max. pier column shear Max. total bearing shear Max. pier column moment74 kip 26 kip 22 kip 130 kip 52 kip 43 kip 1485 kip-ft 552 kip-ft 463 kip-ft Figure 3-17 Comparison of dynamic and static analysis results for pier structure

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81 CHAPTER 4 BARGE FORCE-DEFORMATION RELATIONSHIPS 4.1 Introduction During a barge-bridge collision, the magnitude of impact force generated on the bridge structure is strongly related to the stiffness of the barge bow among other factors. Barge bow stiffness in both the linear elastic and nonlin ear inelastic ranges may be characterized by a nonlinear force-deformation relationship (also referred to as a barge crush-curve). In this chapter, the basis for the current AASHTO crush-curve is reviewed, and new crush-curves, based on work conducted in this study and in the prior St. George Island barge Impact study, are proposed. 4.2 Review of the Current AASHTO Load Determination Procedure As described in Chapter 2, barge impact load calculations per the current AASHTO specifications make use of an empirical load cal culation procedure. The e quivalent static load determination equations presented in Chapter 2 are based upon an experimental study conducted by Meier-Drnberg (1983). As previously note d, Meier-Drnberg conducted both static and dynamic impact tests on reduced-scale European Type IIa barge bow sections. The European type IIa barge is similar in size and configuration to the ju mbo hopper barges widely used throughout the United States. Two dynamic test s, conducted using 2-ton pendulum hammers and two different shapes of impact head, were conducted on 1:4.5-scale st ationary (i.e. fixed) barge bows. One dynamic test i nvolved three progressive imp acts using a cylindrical hammer with a diameter of 1.7 m (67.0 in), whereas the other involved three progr essive impacts using a ninety-degree pointed hammer. A static test was also conducte d on a 1:6 scale barge bow using a 2.3 m (90.6 in) hammer. Results obtained fr om the dynamics tests are shown in Figures 4-1a and b and results from the sta tic test are shown in Figure 4-1c.

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82 Using the experimental data collected, Meier-Drnberg developed mathematical relationships between kinetic energy (EB), inelastic barge deformation (aB), and dynamic and static force (PB and BP respectively). These relationships are illustrated in Figure 4-2. As the figure suggests, no major differences were found between the magnitude of dynamic and static impact force. However, this fact is likely due to the stationary barge bow configuration used in the testing. Omission of a flex ible impact target and the co rresponding barge-pier interaction necessarily precludes the abil ity to measure and capture dyn amic amplification effects. Inelastic barge bow deformations were, how ever, accounted for in the Meier-Drnberg study. Results showed that once the barge bow yielding was initiat ed, at approximately 4 in. of deformation (aB), the stiffness of the bow dimi nishes significantly (see Figure 4-2). Additionally, Meier-Drnberg recognized that in elastic bow deformations represent a significant form of energy dissipation. In the development of the AASHTO barge im pact design provisions the relationships between initial barge kinetic en ergy (KE), barge deformation (aB) and equivalent static force (PB) developed by Meier-Drnberg, were adopted with minimal modifications: 2 29 WV C KE2 H (4-1) B 2 1 BR 2 10 1 5672 KE 1 a (4-2) 34 0 a if R a 110 1349 34 0 a if R a 4112 PB B B B B B B (4-3) In these equations, aB is the depth of barge crush deformatio n (ft), KE is the barge kinetic energy (kip-ft), and BBR = B/35 where BB is the width of the barge (ft). The only notable difference between the expressions devel oped by Meier-Drnberg and the AASHTO expr essions given

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83 above as Eqn. 2-7 2-9 is the use of a barge width correction factor (RB). While the AASHTO utilize RB term to reflect the influence of barge wi dth, no such factor has been included to account for variations in either the size (width) or geometric sh ape of the bridge pier being impacted. In Figure 4-3, the AASHTO barge force-deforma tion relationship (crush-curve) given by Eqn. 2-9 is plotted for a hopper barge having a width BB=35 ft and for a tanker barge having a width of BB=50 ft. 4.3 High-Fidelity Finite Element Barge Models In this study, the de velopment of updated ba rge bow force-deforma tion relationships was carried out by developing high-resolution fini te element barge models. These models were created for the purpose of conducting quasi-static cr ushing analyses to obtain force-deformation data. The nonlinear explicit finite element code, LS -DYNA (LSTC 2003), was chosen to conduct the quasi-static barge crus hing analyses. LS-DYNA is capab le of analyzing large scale nonlinear plastic deformations asso ciated with extreme levels of barge bow crush (up to 200 in); can account for global and local member buckling of barge bow components; and is capable of modeling contact not only between the barge and the bridge, but also between internal components within the barge itself. Regarding the selection of an explicit finite element code over an implicit one, if time intervals between contact detec tion events are consistently very small relative to the total analysis time, then explicit analyses may be more efficient than implicit methods. Given that the models developed in this study involve complex contact definitions, most notably the internal contact between components inside the barge, it was determined that time intervals between contact events would be small enough to warrant the use of an explicit method. Two distinct barge models were created: a jumbo hopper barge (Figure 4-4) and an oversize tanker barge (Figure 4-5). The jumbo hopper barge is the baseline vessel upon which

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84 the AASHTO barge impact provisions were based (AASHTO 1991) and is the most common type of barge found operating on th e inland waterway system in th e United States. Modeling and analysis of a jumbo hopper barge allows for direct comparisons be tween finite element analysis results and the current AASHTO barge impact de sign provisions. To ensure that adequate ranges of barge sizes and configurations were included in this study, the oversize tanker barge was also modeled and analyzed. Tanker ba rges are also very commonly found operating on the inland waterway system are considerably different fr om hopper barges in terms of geometric size and mass. 4.3.1 Jumbo Hopper Barge Finite Element Model For the purposes of generating a finite elem ent model of a jumbo hopper barge, detailed structural plans were obtained from a barge manufact urer. From the point of view of structural configuration, the barge consists of two sections: 1.) the barg e bow, and 2.) the hopper, or cargo area (Figure 4-6). Since the focus here is on charact erizing the force-deformation relationship that is associated with the bow section of the barge, only this portion was modeled in the finite element models. Development of the jumbo hop per barge model was part of a larger funded research program; and thus, a number of peopl e were involved in the development effort. Internal stiffening elements on the starboard and port sides of the barge model (added by Mr. Michael Davidson and Mr. Cory Salzano) permitte d corner crush analyses to be conducted on the barge model (Figure 4-7). Similarly, a hopper guard (Figure 4-7) was also added to the model (by Mr. Michael Davidson) to improve its accuracy for large-deformation crush analyses. With the exception of these components, all other aspects of the hopper barge modeling effort were carried out by the author. Further, it is noted that the ra nge of crush deformations simula ted in this study (16 ft.) is greater than the range of deformations generate d during the Meier-Drnberg test program (which

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85 produced equivalent full-scale crush depths of le ss than approximately 12 ft). As will be shown later, peak force levels are generated at rela tively low deformation le vels, therefore modeling only the bow section of the barge is adequate. Hopper barges are fabricated from steel plat es and standard stru ctural steel shapes (channels, angles, etc.) in both the bow a nd hopper regions. The bow of the hopper barge considered in this study is 27 ft 6 in. long by 35 ft -0 in. wide, and is composed of fourteen internal rake trusses, transverse stiffening memb ers, and several external hull plates of various thicknesses (Figure 4-7). Structural steel members are we lded together with gusset plates to form the internal rake truss members (Figure 4-8). All of the components in th is model were modeled using four-node shell elements, mimicking the actual geometric shape of the barge members (Figure 4-7 and 4-8). With regard to modeling the rake trusses, the use of shell elements, as opposed to resultant beam elements, was necessary to capture buckli ng of the barge bow components. By using a high resolution mesh of shell elements, it was possible to ca pture not only global buc kling, but also local buckling effects. Thus, special at tention was given to the number of elements used to model the legs of the structural steel shapes, such that the legs could exhibit reverse curvature during buckling. Additionally, the use of shell elements to discretely model the internal structural members of the barge allows th ese components to exhi bit a local material failure, which is represented by element deletion in LS-DYNA. If resultant beam elements had been used to model the internal members, material failure a nd local buckling modes wo uld have been absent from the simulations and the force-deformati on data obtained would have been adversely affected.

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86 Individual steel components in the physical barge are inter-connected through a collection of continuous and intermittent weld s. Since the finite element model was intended to mimic the physical barge with reasonable accuracy, approxim ation in the model of the weld conditions was necessary (Figure 4-9). In the LS-DYNA model, this was accomplished using CONSTRAINED_SPOTWELD constraints (LSTC 2003) which allo w the user to define a massless spotweld between two nodes. The spotweld is ef fectively treated as a ri gid beam connecting the two nodes. Although an option is av ailable in LS-DYNA to define a spotweld failure criteria, the welds within this model are not permitted to fail. Instead, failure is represented only by way of element deletions when element strains reach th e defined material failure criteria. Hence, by using a sufficient density and distribution of spotwelds, a reasonable approximation of the behavior of the continuous and intermittent we lds present in the physical barge was achieved. Since severe structural deforma tions (up to 16 ft) were to be analyzed using this model, it was important to incorporate a material model capable of exhibiting non linearity and failure. Therefore, the MAT_PIECEWISE_LINEAR_PLASTICITY material model available in LS-DYNA was used to model all barge components. This is an elastic-plastic behavior model that allows the user to define an arbitrary true-stress vs. e ffective-plastic-strain relationship. This material model also permits specification of an effective plastic failure strain at which element failure (and subsequent element deletion from the model) occur. Additionally, this model allows the user to define arbitrary strain -rate dependency using the Cowper-Symonds model. However, for the purposes of this study, stra in-rate effects were not included in the barge bow model since the crush speeds analyzed (48 in./sec) did not result in extremely high strain-rates. With regard to steel material properties, an A36 structural steel was selected to model the barge components because most barges fabricated in the United States are constructed from this

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87 material. Thus, material properties (Table 4-1) and a truss-stress vs. effective-plastic-strain relationship for A36 steel (Figure 4-10) were specified for all barge components. Finite element based determination of barge crush curves involves calculating the force that is generated by an impact or (e.g. pier column) deformin g the bow of the barge. Such analyses are achieved by fixing th e rear section of the barge bow model and then pushing a rigid impactor (having the shape of an pier column) into the bow model at a constant prescribed velocity (Figure 4-11). Finite element models of severa l different sizes and shapes of pier columns were developed to capture the effects th at pier shape and size have on barge-bow forcedeformation relationships. E ach pier was modeled using ei ght-node solid brick elements. To simulate contact interaction between the barge bow and the impactor, a contact definition was defined using the LS-DYNA CONTACT_AUTOMATIC_NODES_TO_SURFACE option. Static and dynamic coefficients of friction, 0.5 and 0.3 respectively, and a soft-constraint formulation were specified in this contact defi nition. A soft-constraint option was also used because this method is typically more effective wh en the contact entities have dissimilar stiffness or mesh density (LS-DYNA 2003). In addition to the barge-pier contact interf ace, a contact definition capable of detecting self-contact between all components within the barge bow was also defined. This contact differs from the barge-pier contact in that there are not two distinct surfaces between which contact can be defined. Rather, it is not known in adva nce which surfaces of the barge will come into contact with each ot her during crushing deformation. Therefore, in LS-DYNA, the CONTACT_AUTOMATIC_SINGLE_SURFACE definition was specified. By including all barge bow elements in this contact definition, it was possible to det ect internal self-contacts within the barge bow.

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88 4.3.2 Tanker Barge Finite Element Model In addition to the jumbo hopper barge model, a finite element model of a tanker barge was also developed (primarily by Mr. Long Bui) as pa rt of the larger funded research program. Similar to hopper barges, tanker ba rges are constructed from steel plates and trusses, where the trusses are each composed of weld ed structural steel sections. In contrast to the hopper barge, the tanker barge bow has twenty-three internal ra ke trusses instead of fourteen, and is 49-6 wide instead of 35-0. Based on results obtained from hopper barge an alyses, it was determined that the peak contact force occurs within the first 12 to 24 in. of deformation. Consequently, it was only necessary to discretely model a sm all region near the headlog of the tanker barge, rather than the entire bow. For computational efficiency, the barge model was divided into three zones, each represented by a different mesh resolution (Figure 4-12). Zone-1 represents the first 8.75 ft of the ba rge bow, the area that was expected to undergo the most significant permanent deformations. As such, this zone was modeled using discrete shell elements for both the outer hull plates and internal rake trussesmuch like the hopper barge was modeledin order to capture local inel astic buckling effects. The tanker barge model used the same material model as well as the spot weld definitions that were used to model the hopper barge. Zone-2 of the tanker barge model represented the 19.75 ft portion of the raked bow that is aft of Zone-1. Internal trusse s extend throughout the entire rake d barge bow; however, the truss and hull components in Zone-2 were not expected to sustain buckling, inel astic deformation, or experience internal contact during typical impact events. As a result, numerically efficient resultant beam elements were used to model the stiffness of the internal trusses in Zone-2. The outer hull was modeled with shell elements, but with a lower resolution than that used in Zone-1.

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89 Zone-3 of the model represents the remaini ng 166.5 ft of the barge, and no significant deformations were expected in this zone. This zone was modeled using a coarse mesh of stiff brick elements. 4.4 High-Fidelity Finite Element Barge Crush Analyses Using finite element models of the jumbo hoppe r and tanker barges, quasi-static barge bow crush simulations were conducted to determine force-deformation relationships for the barge bow models. Such analyses provide a means of characterizing barge force-deformation relationships in a manner that is inde pendent of AASHTO and Meier-Drnberg. 4.4.1 Finite Element Barge Bow Crush Simulations Contact force data and barge bow deformation data were extracted and combined for each crush simulation to produce load-deformation rela tionships for the barge bow. Both the hopper and tanker barges were crushed using both flat and round pier-i mpactors of varying widths. Most crush simulations involved centerline crus hing of the barge model; however, several crush analyses were conducted involving corner-zone crushing of the barge model. Corner-zone crush analyses were conducted to inves tigate the effect pier position has on the cont act force generated. Each crush case is designated by a specific id entification tag composed of two letters and a number. The first letter represen ts the barge type; the second letter designat es the pier shape; and the two digit number indicates the size (cross-sectional width or diameter) of the pier in units of feet. For example, a simulation involving a hoppe r barge crushed by a 6 ft wide flat faced pier (e.g., square or rectangular in cross-section) is designated HF06. Hopper and tanker barges are denoted by H and T respectively, and the flat an d round pier impact face geometries are denoted F and R respectively. Force-deformation data for the hopper and ta nker barges crushed by a flat-faced square pier are presented in Figures 4-14, 4-15, and 4-16 respectively. The most notable observation

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90 regarding the flat faced pier data is that the peak force values achieved are dependent upon the widths of the piers. Both the hopper and tanker force-deformation results indicate that as the width of the pier increases, so does the peak contact force. This trend is attributable to the internal configuration of stiffening trusses inside the barge. The number of internal rake trusses that are directly engaged during impact and defo rmation increases as the pier width increases (Figure 4-17). Hence, the stiffness of the barge e ffectively increases as the number of trusses that are directly engaged during crushing incr eases. In the current AASHTO provisions, the effect of pier width on vessel impact forces is not taken into account. Force-deformation data for the hopper barge crushed by a round pier are presented in Figures 4-18 and 4-19. Unlike the flat faced pier results, the peak contact forces generated in round pier simulations are not st rongly influenced by the width of the pier. During impact by a round pier, internal stiffening tr usses inside the barge are enga ged gradually and sequentially (Figure 4-20) as the deformation level increases. This is in contrast to the behavior for impact by a flat faced pier where all of th e trusses within the impact zone width are immediately engaged at the same point in time. In the AASHTO and Meier-Drnberg equations, the peak impact force is computed based on the maximum expected barge bow deformation (which is itself computed based on initial kinetic impact energy). However, the simulation results presented above i ndicate that the peak force often does not occur at the maximum su stained level of barge bow deformation. Additionally, the AASHTO and Meier-Drnberg equations exhibit a work-hardening phenomenon in which the impact force levels con tinue to rise with incr easing deformation depth (recall Figure 4-2). However, data from finite element simulations indicate that no such hardening occurs, and in many cases a softening of the barge bow occurs after the peak impact

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91 force has been reached. At significant deformati on levels, widespread buckling of the internal stiffening trusses, combined with fracturing of the outer hull plates, prod uces this structural softening, particularly in impacts that involve flat faced piers. Based on these observations, it is apparent that an elastic, perfec tly-plastic force-deformation curve is adequate to conservatively describe bow crushi ng behavior (Figure 4-21). 4.4.2 Development of Barge Bow Fo rce-Deformation Relationships Barge crush analyses of the type desc ribed above produce highly detailed forcedeformation relationships that are specific to vessel type, pier shape and size, and impact location. In this section, the detailed data obtai ned from these finite element crush simulations are used to develop simplified barge crush models that are suitable for use in bridge design. Examination of Figures 4-14, 4-15, 4-16, 4-18, and 4-19 indicates that barge bow contact force either remains constant or decreases afte r the peak contact force has been reached. Therefore, in order to envelope both of these situations, it is assumed that the barge bow will exhibit an elastic-perfect ly plastic load-deformation behavior (recall Figure 4-21). Despite the fact that current AASHTO provisions util ize a barge width co rrection factor (RB) to account for the relationship between impact fo rce magnitude and barge width, it is important to note that crush data obtained in this study indicate that no strong relationship of this type exists. Because the hopper and tanker barges have different widths (35 ft and 50 ft, respectively), based upon the current AASHTO specifications, it would be expected that the impact forces for a given pier size would be quite different if impact by a hopper vers us a tanker barge. However, the impact force magnitudes obtained for both barge types are in close agreement, indicating that a correction factor for barge widt h is not appropriate. A key element in developing a simplified barge bow behavior model is the formulation of a relationship between peak impact force and pier width. Formation of this relationship is

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92 accomplished by filtering the barge bow force hist ory dataobtained from the crush simulations described in the previous sectionusing low-pass filter. For each crush simulation, the time history of contact force was tr ansformed from the time domain to the frequency domain using a fast Fourier transform (FFT). Once in the fre quency domain, data above 100 Hz were discarded. An inverse transformation was then performed to obtain a final filtered force time history. For each crush simulation, the filtered force data were scanned and the maximum (peak) impact force for each simulation quantified. In Figure 4-22, peak impact forces obtained from this procedure are plotted as a func tion of pier column width for both flat and round columns crosssectional shapes. Inspection of the crush data for flat-faced piers indicates that one of two distinct mechanisms determines the magnitude of peak force that can be achieved. In Figure 4-23, it is evident that for flat-faced pier with widths of less than appr oximately 9 ft, the contact force reaches a temporary peak magnitude within the first 2 to 4 in. of barge bow deformation. With continued bow deformation, the contact force gr adually increases up to a global peak magnitude at deformation levels of 16 to 20 in. Detailed inspection of finite element results for the moderate-width flat-faced piers reveals that the in itial local maxima are associated with initial yielding of the internal rake trusses. As the barge bow de forms beyond this condition, the internal trusses begin to buckle, and combined me mbrane action in the oute r hull plates begins to control. When the hull plates exhibit signifi cant yielding, the global peak force has been reached. Thus, for moderate-width flat-faced pier s, the membrane yield capacity of the barge hull plates governs the global contact force maxima. For flat-faced pier with widths of greater than approximate ly 9 ft, a maximum of force magnituderepresenting yielding of th e internal trussesalso occurs within the first 2 to 4 in of

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93 barge bow deformation. However, in contrast to moderate-width flat-f aced piers, the force maximum at the 2 to 4 in. deformation level in large-width piers represents the global force maximum that will be achieved. Instead of the force level continuing to increase with additional deformation, the contact force decreases as th e barge bow deforms furt her and the internal trusses buckle. At large deformations (great er than 20 in), the trusses have buckled, and membrane action of the outer hull plates determines the magnitude of load generated. Thus, for large-width flat-faced piers, the yield capacity of the internal trusses in the barge governs the global maximum of contact force (and therefore impact load). Force-deformation data for a 9 ft wide flat-f aced pier indicates that the force maximums associated with truss yielding and outer hull plat e yielding are approximately equal in magnitude (Figure 4-23). This fact suggests that a pier width of 9 ft repres ents the approximate transition point between force maximums that are controlled by truss-yielding and thos e that are controlled by yielding of the outer hu ll plates of the barge. In Figure 4-24, force maxima associated with bo th barge truss yiel ding and outer hull plate yielding are plotted for flat-faced piers as a function of pier width. Distinct linear trends associated with each of these behavioral mode s are clearly identified. For each mode of barge deformation behavior, a linear regression tr end line was fit through the data (Figure 4-24). For data points associated with truss yielding, a least-squares linear curv e fit through the origin yields the following relationship betw een maximum force and pier width: BYPP = 138.73w (4-4) where PBY is the barge yield force in kips, and wP is the width of the flat-faced pier in ft. A general least-squares line was fit through the data points associated with outer hull plate yielding produced the relationship:

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94 BYPP= 963.84 + 52.27w (4-5) Equating Eqns. 4-4 and 4-5 and solving for the pier width at which the two trend lines intersect, the transition point between force maxima c ontrolled by truss-yield and plate-yield was determined to be 11.15 ft. Eqns. 4-4 and 4-5 are based on analyses of hoppe r and tanker models that are representative of typical vessel fabrication practices employed in the United States (in terms of steel strength, structural confi guration, and steel pl ate thicknesses). Outer hull plate thicknesses of barges very commonly range between 0.25 in and 0.625 in, and 91-percent of barges have plate thicknesses of 0.3125 in (N guyen 1993) and a steel strength of 36 ksi is often used. As noted earlier, the models analyzed in this st udy employed a 3/8 in. (0.375 in.) hull plate thickness and 36 ksi steel. However, some variability in barge fabrication practices is possible. Additionally, the data points used to develop the Eqns. 4-4 and 4-5 trend lines include some variability and are not perfectly linear in form Therefore, in order to develop conservative equations that can be recommende d for general purpose use in bri dge design, some modifications to Eqns. 4-4 and 4-5 were made. First, the lines represented by Eqns. 4-4 and 4-5 were vertically shifted (while maintaining the best fit slopes) so that all of the data poi nts are completely enveloped by the shifted lines. After vertical shifting, the regr ession lines were scaled to account for the possibility of higher steel strengths and/or thicker hull plates. Increases in either of th ese parameters would lead to an increase in impact force leve ls. Steel having a yield strength of 50 ksi rather than 36 ksi corresponds to a yield strength ra tio of (50 / 36) = 1.389. Simila rly, a hull plat e thickness of 1/2 in. (0.5 in.) instead of 3/8 in. leads to a pl ate thickness ratio of (0.5 / 0.375) = 1.33. Hence, after shifting, Eqns. 4-4 and 4-5 were each scaled by a factor of 1.33. Finally, the resulting

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95 coefficients in the equations were rounded off, resulting in simplified design equations for the peak impact force as a function pier width for flat-face pier columns: BYPPP=1300 + 80w if w<10 ft (4-6) BYPPP = 300 + 180w if w10 ft (4-7) where PBY is the peak impact force in kips. Eqns. 4-6 and 4-7 are graphically i llustrated in Figure 4-25. By equating Eqns. 4-6 and 4-7 and solving for the pier width at which the equations intersect, the transition point between the trussyield and plate-yield dominated modes of barge behavior was determined to be 10 ft. Pier columns having round (circu lar) cross-sectional shapes also exhibited two distinct mechanisms in terms of predicting maximum magnitude force levels (Figure 4-26). Moreover, the transition point between thes e two modes was found to be loca ted approximately at a pier diameter of 9 ft, similar to the transition point for flat-faced pier columns. Hence, development of a relationship between maximu m impact force and pier column diameter was carried out in a manner analogous that used for flat-faced piers. Linear regression was used to fit trend lines through the moderate diameter and large diameter round pier crush data (Figure 4-26). For moderate-diameter round pier columns, the least-squares trend line was: P BYw 02 31 975 P (4-8) A least-squares trend line fit through the larg e-diameter round pier da ta produces the following relationship: P BYw 54 18 25 1079 P (4-9) Equating Eqns. 4-8 and 4-9, and solving for the pier diamet er at the intersection of the trend lines, the pier diameter transition point was calculated to be 8.36 ft.

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96 As with the flat-faced pier data the round pier regression lines were first vertically shifted so as to envelope all data points, and then sc aled by a factor of 1.33. After rounding off the coefficients, the resulting equations for barge yiel d force as a function of circular pier diameter were: ft 10 w if w 40 1300 PP P BY (4-10) ft 10 w if w 20 1500 PP P BY (4-11) These equations are graphically illustrated in Figure 4-27. Construction of a complete desi gn barge crush curve (recall Figure 4-21) requires not only determination of the barge crush force (PBY) but also the barge yield deformation (aBY). The first step in calculating the barge bow de formation at yield is to extract the initial barge bow stiffness from the force-deformation data. The initial st iffness for each case was calculated by taking the slope of the line connecting the first two data points of each curve in Figures 4-14, 4-15, 4-16, 418, and 4-19. In Figure 4-28, the initial stiffnesses computed in this manner are plotted as a function of pier width for all crush analysis cases. Taking the peak force values from Figure 4-22, and dividing by the respective stiffnesses from Figure 4-28 yields barge bow deformations th atin conjunction with the peak force valueenvelope the force-deformation data. In Figure 4-29, barge bow deformations at yield are plotted as a function of pier width. In mo st design situations, pier columns will range from 4 ft to 12 ft in width. Consequent ly, based on the data show in Figure 4-29, barge bow yield deformations of 0.5 in. and 2 in. were select ed for flat-faced and round piers respectively. 4.4.3 Summary of Barge Bow Force-Deformation Relationships Figure 4-30 presents a flowchart for calculating th e force-deformation relationship for the barge bow. Based upon the geometry and size of th e pier column on the impact pier, the design barge yield force is calculated. Additionally, based upon pier column geometry, the barge bow

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97 deformation at yield is selected and the barge bow force-deforma tion relationship is defined. The resulting force-deformation relationship of Figure 4-30 is a necessary component of all of the analysis methods presented in later chapters This elastic, perfec tly-plastic relationship ultimately determines the magnitude and duration of the force imparted to a bridge structure during an impact.

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98 Table 4-1 Barge material properties Parameter Value Elastic modulus 29000 ksi Poisson's ratio 0.33 Unit weight 490 pcf Failure strain 0.2 Yield stress 36 ksi Ultimate stress 58 ksi (eng. stress) 69.8 ksi (true stress)

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99 Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data EnvelopeA Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data EnvelopeB Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data EnvelopeC Figure 4-1 Force-deformation resu lts obtained by Meier-Drnberg (Adapted from Meier-Drnberg 1983) A) Re sults from dynamic cylindrical impact hammer test, B) Results from dynamic 90 pointed impact hammer test, and C) Results from static impact hammer test

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100 Bow deformation (in) Bow deformation (ft)Contact force (kips) Impact energy (kip-ft) 0 30 60 90 120 150 0 2.5 5 7.5 10 00 1,0008,000 2,00016,000 3,00024,000 Average force Force envelope Impact energy Figure 4-2 Relationships devel oped from experimental barge im pact tests conducted by MeierDrnberg (1976) (Adapted from AASHTO 1991)

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101 Bow deformation (in)Impact force (kips) 0 30 60 90 120 150 0 1,000 2,000 3,000 4,000 Hopper barge (35 ft) Tanker barge (50 ft) Figure 4-3 AASHTO barge forc e-deformation relationship for hopper and tanker barges

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102 BowStarboard PortStern195'-0"35'-0"A 12'-6" 12' 2' 167'-6"27'-6"B Figure 4-4 Hopper barge dimensions A) Plan view, B) Elevation

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103 290'-0"50'-0" Stern BowPort StarboardA 255'-0"35'-0"12'-6" 12' 2' B Figure 4-5 Tanker barge dimensions A) Plan view, B) Elevation

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104 Hopper Region : 167'-6"Bow : 27'-6" 12'-0" 2'-0" Modeled Not modeled 16'-0" : Approx extent of crush 195'-0" Figure 4-6 Hopper barge schematic

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105 Internal Rake Trusses Top hull plate of barge bow Barge headlog Front plate of the hopper region Bottom hull plate of barge bow Internally stiffened side region Hopper guard plate Figure 4-7 Hopper barge bow model with cu t-section showing internal structure

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106 Gusset plates modeled with shell elements Truss members modeled with shell elements Headlog channel Figure 4-8 Internal rake truss model

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107 Regions of spotwelds used to tie barge components to eachother Headlog Figure 4-9 Use of spot weld constrai nts to connect structural components

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108 Nominal strainNominal stress (ksi) 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 A Effective true plastic strainEffective true stress (ksi) 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 B Figure 4-10 A36 stress-strain curve A) Nominal stress vs. nominal strain, B) Effective true stress vs. effective true plastic strain

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109 Barge bow model Pier column model Figure 4-11 Barge bow model with a six-foot square impactor

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110 Solid elements used in Zone-3 Beam elements used for trusses in Zone-2 Shell elements used in Zone-1 Shell elements used for plates in Zone-2 Figure 4-12 Tanker barge bow model

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111 Barge Model Pier Model C L PB wP (width of pier)A Barge Model Pier Model C L PB wPB Figure 4-13 Crush analysis models A) Cent erline crushing of barge bow model, B) Corner-zone crushing of barge bow model

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112 Bow deformation (in)Force (kips) 0 6 12 18 24 0 1,000 2,000 3,000 4,000 5,000 HF01 HF03 HF06 HF09 HF12 HF18 HF35A Bow deformation (in)Force (kips) 0 50 100 150 200 0 1,000 2,000 3,000 4,000 5,000 HF01 HF03 HF06 HF09 HF12 HF18 HF35B Figure 4-14 Hopper barge bow force-deformation da ta for flat piers subjected to centerline crushing A) Low deformati on, B) High deformation

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113 Bow deformation (in)Force (kips) 0 6 12 18 24 0 1,000 2,000 3,000 4,000 5,000 HF06 HF18A Bow deformation (in)Force (kips) 0 50 100 150 200 0 1,000 2,000 3,000 4,000 5,000 HF06 HF18B Figure 4-15 Hopper barge bow force-deformation da ta for flat piers subjected to corner-zone crushing A) Low deformati on, B) High deformation

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114 Bow deformation (in)Force (kips) 0 6 12 18 24 0 1,000 2,000 3,000 4,000 5,000 TF06 TF12 Figure 4-16 Tanker barge bow force-de formation data for flat piers subjected to cen terline crushing

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115 Engaged trusses A Engaged trusses B Figure 4-17 Relationship of pier width to engaged trusses A) A 6 ft wide flat pier (HF06), B) A 12 ft wide fl at pier (HF12)

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116 Bow deformation (in)Force (kips) 0 6 12 18 24 0 1,000 2,000 3,000 4,000 5,000 HR03 HR06 HR09 HR12 HR18 HR35A Bow deformation (in)Force (kips) 0 50 100 150 200 0 1,000 2,000 3,000 4,000 5,000 HR03 HR06 HR09 HR12 HR18 HR35B Figure 4-18 Hopper barge bow force-deformation da ta for round piers subjected to centerline crushing A) Low deformati on, B) High deformation

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117 Bow deformation (in)Force (kips) 0 6 12 18 24 0 1,000 2,000 3,000 4,000 5,000 HR06A Bow deformation (in)Force (kips) 0 50 100 150 200 0 1,000 2,000 3,000 4,000 5,000 HR06B Figure 4-19 Hopper barge bow force-deformation da ta for flat piers subjected to corner-zone crushing A) Low deformati on, B) High deformation

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118 Engaged trusses A Engaged trusses B Figure 4-20 Gradual increase in number of trusse s engaged with deformation in round pier simulations A) A 6 ft diameter round pier at 10 in of deform ation, B) A 6 ft diameter round pier at 20 in of deformation

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119 Bow deformationImpact force PBY aBY Figure 4-21 Elastic-perfectly plastic barge bow force-deformation curve

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120 Pier width (ft)Peak contact force (kips) 0 9 18 27 36 0 1,000 2,000 3,000 4,000 5,000 Hopper, flat (HF) center crush Hopper, flat (HF) corner crush Tanker, flat (TF) center crush Hopper, round (HR) center crush Hopper, round (HR) corner crush Figure 4-22 Peak barge contact force versus pier width

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121 Bow deformation (in)Force (kips) 0 9 18 27 36 0 1,000 2,000 3,000 4,000 5,000 Max. associated with plate yield Max. associated with truss yield HF03 HF09 HF35 Figure 4-23 Peak barge contact force versus pier width

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122 Pier width (ft)Peak contact force (kips) 0 9 18 27 36 0 1,000 2,000 3,000 4,000 5,000 Truss-yield controlled Plate-yield controlled Truss-yield regression Plate-yield regression Figure 4-24 Comparison of truss-yi eld controlled peak force versus plate-yield controlled peak force

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123 Pier width (ft)Peak contact load (kips) 0 6 12 18 24 30 36 0 1,000 2,000 3,000 4,000 5,000 6,000 HF center HF corner TF center Flat regression Flat design 1300 + 80wP 300 + 180wP PBY = 2100 kips wP = 10 ft Figure 4-25 Design curve for peak imp act force versus flat pier width

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124 Pier diameter (ft)Peak contact force (kips) 0 9 18 27 36 0 500 1,000 1,500 2,000 HR data Low-diameter regression Large-diameter regression Figure 4-26 Comparison of low di ameter peak force versus large diameter peak force

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125 Pier diameter (ft)Peak contact force (kips)061218243036 0 500 1,000 1,500 2,000 2,500 HR center HR corner Round regression Round design 1500 + 20wP 1300 + 40wP PBY = 1700 kips wP = 10 ft Figure 4-27 Design curve for peak impact force versus round pier diameter

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126 Pier width (ft)Bow stiffness (kip/in) 0 9 18 27 36 0 2,500 5,000 7,500 10,000 HF center HF corner TF center HR center HR corner Figure 4-28 Initial barge bow stiffn ess as a function of pier width

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127 Pier width (ft)Barge yield deformation (in) 0 9 18 27 36 0 0.5 1 1.5 2 2.5 HF center HF corner TF center HR center HR corner Flat pier Round pier Figure 4-29 Barge bow deformation at yield versus pier width

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128 Barge Bow Force-Deformation Pier column shape Flat-faced aBY = 0.5 in wP : Pier width (ft)PBY : Barge yield load (kips) 0 6 12 18 24 3036 0 2,000 4,000 6,000 8,000 10 F l a t 1 3 0 0 + 8 0 wP F l a t 3 0 0 + 1 8 0 wP R o u n d 1 3 0 0 + 4 0 wP R o u n d 1 5 0 0 +2 0 wP Round aBY = 2.0 in PBY = 1300+80wP PBY = 300+180wP PBY = 1300+40wP PBY = 1500+20wP wP Bow deformationImpact force PBY aBY wP Impact Force PB wP Impact Force PB Barge bow forcedeformation modelft 10 wP ft 10 wP ft 10 wP ft 10 wP ft 10 wP ft 10 wP Figure 4-30 Barge bow for ce-deformation flowchart

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129 CHAPTER 5 COUPLED VESSEL IMPACT ANALYSIS AND SIMPLIFIED ON E-PIER TWO-SPAN STRUCTURAL MODELING 5.1 Coupled Vessel Impact Analysis An efficient coupled vessel impact analysis (CVIA) procedure i nvolving coupling a single degree-of-freedom (DOF) nonlinear barge mode l to a multi-DOF nonlinear bridge model was developed by University of Florida researcher s (Consolazio and Cowan 2005) As part of this method, dynamic barge and bridge behavior are anal yzed separately in distinct code modules that are then linked together (Figure 5-1) through a common impact-force. The primary advantage of this modular approach is that the barge behavi or is encapsulated within a self-contained module that can be integrated into existing nonlinear dynam ic bridge analysis codes with relative ease. For this research, the single-DOF barge module is integrated into the multi -DOF commercial pier analysis program FB-MultiPier (BSI 2005). 5.1.1 Nonlinear Barge Bow Behavior Analysis of barge-bridge collision events re quires both the nonlinea r barge bow behavior and the inertial properties of the barge be modele d accurately. In this st udy, nonlinear barge bow behavior is modeled using the force-deformatio n relationships developed in Chapter 4. The entire mass of the barge is represented as a single degree-of-freedom (SDOF) mass which is coupled to the bridge model at the poin t of contact through the contact force. During barge collisions with bridge structures the bow section of the barge will typically undergo permanent plastic deformation (Figure 5-2). However, depending upon the barge speed and type, and pier flexibility, dyna mic fluctuations may occur that result in inelastic loading, unloading, and subsequent reloadi ng of the barge bow. This behavi or is represented by tracking the bow deformation of the barge th roughout the impact analysis. Figure 5-3 illustrates the various stages of barge bow load ing, unloading, and reloading.

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130 Whenever the barge is in contact with the pi er, the barge bow deformation is computed as the difference in the barge and pier displacements, (uB and uP respectively): P B Bu u a (5-1) Once contact between the barg e and bridge models is de tected, the barge bow loads elastically until the barge bow deformation leve l exceeds the yield deformation of the barge (aBY). As loading of the barg e bow continues, the barge bow undergoes plastic deformation (Figure 5-3a). Eventually, the velocity of the barg e will drop below the velocity of the pier, and the contact force (PB) between them will diminish. As this occurs, the bow deformation will also begin to drop (Figure 5-3b) from the maximum barge bow deformation level (aBmax) to the residual plastic deformation level (aBP). Once the barge and pier are fully out of contact, the contact force drops to zero, and all elastic deformation (aBmax aBP) will have been recovered. When the barge bow deformation level is below the plastic deformation level (aB < aBP), the contact force remains zero (Figure 5-3c). However, if reloading subsequently occurs, the contact force will load back up the unloading-reloading portion of the force-deformation curve (Figure 5-3d) to the original loading curve at ma ximum deformation. If loading continues beyond the maximum deformation, plastic deformation will continue to accrue. Algorithmically, the force-deformation re lationship for the barge bow (Figure 5-3) is modeled using a nonlinear compression-only spring. Data for this spring are obtained from the relationships developed in Chapter 4. Rega rding the unloading por tion of the barge bow relationship, unloading curves may consist of either linear curves equal in slope to the initial slope of the load ing curve (Figure 5-4a), or general nonlinear, de formation-dependent, unloading curves (Figure 5-4b). In the latter case, each unloadi ng curve must be specified with a corresponding maximum barge bow deformation (aBmax).

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131 Computationally, the deformation (aBmax) at the onset of unloading will not coincide with the specified maximum deformation values given prior to analysis. Thus, to determine the unloading relationship, it is necessary to find th e two unloading curves, and associated maximum barge bow deformation values, that bracket the desired unloading behavior. Using the bounding curve data, the intermediate unloading curve is generated for the current maximum deformation (aBmax) through linear inte rpolation (Figure 5-5). 5.1.2 Time-Integration of Barge Equation of Motion The analytical procedure presented here and in the next section is based on earlier work conducted by Hendrix (2003), Consolazio et al (2004) and Consolazio and Cowan (2005). Given that the force-deformation behavior of the barge bow is nonlinear, the only practical procedure to time-integrate the barge equation of motion is to use numer ical procedures. In development of this method, inertial effects fr om the barge are accounted for through use of the SDOF barge mass. Additionally, en ergy dissipation as a result of inelastic deformation of the barge bow is accounted for using the loading-unl oading behavior descri bed above. However, any influence from the water in which the barge is suspended is neglected, as such effects are beyond the scope of this study. Recalling Figure 5-1, the equation of motion for th e barge (single degree of freedom, SDOF) is written as: B B BP u m (5-2) where mB is the barge mass, Bu is the barge acceleration (or deceleration), and PB is the contact force acting between the barge and pier. Evaluating Eqn. 5-2 at time-twith the assumption that the mass remains constant: B t B t BP u m (5-3)

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132 where B tu and tPB are the barge acceleration and force at time-t. The accelerati on of the barge at time-t is estimated using the central difference equation: B h t B t B h t 2 B tu u 2 u h 1 u (5-4) where h is the time step size; and t+huB, tuB, and t-huB are the barge displacements at times t+h, t, and t-h respectively. Substituting Eqn. 5-4 into Eqn. 5-3 yields the explicit integration central difference method (CDM) dynamic update equation: B h t B t B 2 B t B h tu u 2 m h P u (5-5) which uses data at times t and t-h to predict the displacement (t+huB) of the barge at time t+h. Using the force-deformation relations hips described earlier, the force tPB is computed. 5.1.3 Coupling Between Barge and Pier As previously mentioned, coupling of the ba rge model to the pier model is accomplished through the shared contact force between the two. Analysis of the two separate models is handled in distinct code modul es. Algorithmically, the overall time-integration process is controlled by the pier module (Figure 5-6), while determination and convergence of the contact force is controlled by the barge module (Figure 5-7). At the beginning of each time step, the barg e module is called to calculate an initial estimate of the barge impact force. Taking the difference between the displacement of the barge and the component of pier motion in the direction of barge motion, the crus h depth is calculated: BP PB B Ba u u max a (5-6) where, uB is the displacement of the barge, uPB is the component of the impacted pier displacement in the directi on of barge motion, and aBP is the plastic deformation of the barge

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133 bow. In Eqn. 5-6, the component of the pier displacem ent in the direction of barge motion is calculated as: B Py B Px PBsin u cos u u (5-7) where uPx and uPy are the displacements of the impact point on the impacted pier in global x and y-directions, and B is the angle of trajectory of the barge relative to the global x-direction (Figure 5-8a). The crush depth in turn is us ed to calculate the barge impact force using the barge bow force-deformation relationship. Using an itera tive variation of the central difference method (CDM), the displacement of the ba rge is calculated from the imp act force and barge equation of motion, and the barge impact force is updated to convergence and resolved into x and y-components: B B Bxcos P P (5-8) B B Bysin P P (5-9) where PBx and PBy are the components of the impact for ce in the global x and y-directions. These force components are returned to the pier module, and inserted into the proper location within the external force vector. Using the current estimate for the external lo ad vector, the pier c ontinues through time-step integration to convergence. Upon convergence of the pier module, the x and y components of the pier displacement are returned to the barge module. The barge module begins the convergence procedure over again until the contact force is updated. The contact force for the current cycle is compared to the force from the prev ious cycle, and if this difference is within the convergence tolerance, the entire system has achieved convergence, and the program continues to the next time step.

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134 5.2 One-Pier Two-Span Simplified Bridge Modeling Technique In concept, the CVIA procedure discussed a bove is easily extrapolat ed to use in barge impacts of full bridge structures as opposed to individual pier s. However, based upon current computer processing speeds, coupled vessel impact analyses of full bri dge models may require up to several hours of processing time. In a bridge design setting, analysis times of several hours would generally be considered unacceptable. For this reason, a simplified one-pier two-span (OPTS) structural modeling method that consis ts of the impact pier and the two adjacent superstructure spans has been developed at the University of Florida (UF) (Davidson 2007). In a barge-to-bridge collision, a significant fr action of the imparted lateral impact load is transferred to the superstructure due to the displacement and acceleration proportional resistances associated with the superstructure (discussed in Chapter 3). Any simplified bridge modeling technique that is inte nded for use in dynamic analyses must therefore be able to account for the complex interacti ons of inertial forces associ ated with the superstructure. 5.2.1 Effective Linearly Independe nt Stiffness Approximation A key aspect of the OPTS modeling technique is the representation of the stiffness of the bridge beyond the bounds of th e OPTS model. In the OPTS model, this stiffness is approximated using six linearly i ndependent equivalent translationa l and rotational springs at the far end of each of the two spans (Figure 5-9). Determination and appl ication of these springs is done using three distinct sections of the full br idge model: 1) the down-station piers and spans up to the pier directly down-station from the impact pier, 2) the OP TS portion of the model, and 3) the up-station piers and spans from the pier directly up-station from the impact pier (Figure 510). For both peripheral (i.e. the down and up-statio n) models, six unit fo rces are appliedone in each of the six model degrees-of-freedomat the point of separation from the OPTS model

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135 (Figure 5-11). After analyzing each peripheral model statically, the three tr anslational and three rotational displacements at the application point are computed (Figure 5-12). Dividing each applied load by the respective displacement, six linearly independent s tiffnesses are obtained: x x xu F k (5-10) y y yu F k (5-11) z z zu F k (5-12) rxxxkM (5-13) ryyykM (5-14) rzzzkM (5-15) where k, F, and u are the translational stiffne sses, forces, and displa cements respectively; kr, M, and are the rotational stiffnesses, forces, and disp lacements respectively; and the subscripts x, y, and z represent the three ort hogonal directions respectively. Using these stiffnesses, the six springs may be added to each span of the OPTS model (Figure 5-9). 5.2.2 Effective Lumped Mass Approximation An additional aspect of the OPTS modeling t echnique is the representation of inertial properties of the bridge beyond the bounds of the OPTS model. The inertial properties of the peripheral models are approximate d by placing concentrated (i.e. lu mped) masses at the far ends of spans in each of the three glob al translational di rections (Figure 5-13). These effective masses are based on tributary lengths equal to the half superstructure span lengths adjacent to the OPTS model (Figure 5-14): eff1 mAL 2 (5-16) where meff is the effective lumped mass; and A, and L are the mass density, cross-sectional area, and L is the length of the superstructure span.

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136 5.3 Coupled Vessel Impact Analysis of One-Pier Two-Span Bridge Models Both the coupled vessel impact analysis (C VIA) procedure and th e one-pier two-span (OPTS) model were implemented in the commercia l bridge analysis software, FB-MultiPier (BSI 2005) in this research. Coupled impact anal yses have been shown to produce force and displacement time-histories that agree well with re sults obtained from bot h high-resolution finite element simulations (Consolazio et al. 2004, and Consolazio and Cowan 2005), as well as full-scale experimental data (Davidson 2007). Given the agreement be tween the CVIA method and both high-resolution finite element and experi mental results, CVIA techniques are able to capture dynamic effectseffects precluded from a static analysisin barge-to-bridge collisions. With regards to the accuracy of OPTS mode ls, excellent agreement has been observed between coupled analysis results from full-re solution bridge models and respective OPTS models during the collision phase in which the barge and pier are in contact (Davidson 2007). During the free-vibration phaseonce the barg e and pier are no longer in contactgood agreement is observed between the results from both models. Add itionally, the time required to analyze an OPTS model is signi ficantly less than that requi red to analyze a corresponding full-resolution multiple-span multiple-pier model. Therefore, OPTS models are significantly more computationally efficient than full-resolutio n models while retaining the ability to capture relevant dynamic effects.

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137 u BmBaBmPuP PB PB Pier structure Soil stiffness Crushable bow section of barge Barge Barge and pier/soil system are coupled together through a common contact force PB Single DOF barge modelMulti-DOF bridge model superstructure Figure 5-1 Barge and pier modeled as separate but coupled modules (After Consolazio and Cowan 2005)

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138 Figure 5-2 Permanent plastic deformati on of a barge bow after an impact

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139 PBaB aBYaBmaxA PBaB aBY aBp aBmaxB PBaBaBp aBmaxC PBaBaBp aBmaxD Figure 5-3 Stages of barge crush A) Loading, B) Unloading, C) Barge not in contact with pier, D) Reloading and continued plastic defo rmation (After Consolazio and Cowan 2005)

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140 Force ( )PB Crush depth ( )aB Unloading curves are linear and independent of max. deformation level that has been reached Initial slope of loading curve Slope of all unloading curves A Unloading curve for deformation level aBmax(i) Crush depth ( )aBForce ( )PB Unloading curve for deformation level aBmax(i+2) Unloading curve for deformation level aBmax(i+1) Unloading curves are nonlinear and dependent on max. deformation level that has been reached B Figure 5-4 Unloading curves A) Linear curves equa l in slope to the initial slope of the loading curve, B) General nonlinear, deformation-dependent, unloading curves

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141 aBmax aBmax(i)aBmax(i+1) Extrapolated portion of unloading curve for deformation levelaBmax(i+1) Crush depth ( )aBForce ( )PB Unloading curve for deformation level aBmax(i+1)Unloading curve for deformation level aBmax(i) Intermediate unloading curve is generated by interpolating at multiple load levels ( ) between two bounding curves PB P BaBmaxMaximum crush deformation sustained prior to unloadingat Figure 5-5 Generation of intermed iate unloading curves by inter polation (After Consolazio and Cowan 2005)

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142 form stiffness, mass, damping form effective stiffness Initialize pier/soil module For each time step i = 1, ... For each iteration j = 1, ... Check for convergence of pier and soil ... Form external load vector no no yes yes Check for convergence of coupled barge/pier/soil system by checking convergence of barge force predictions. (see barge module for details) no mode = CONV updated returned yes Form new external load vector with updated .... Record converged pier, soil, barge data and advance to next time step form stiffness, mass, damping initialize internal force vector ... insert barge forces into external load vector ... form effective force vector Extract displacements of pier at impact point from Barge module returns computed barge impact forces computedreturned mode = CALC; sent Compute barge impact force based on displacements of the pier and barge (see barge module for details) mode = INIT Instruct barge module to perform initialization Initialize state of barge (see barge module for details) Pier/soil module Barge module 4 3 2 1 Barge module Barge module Time for which a solution is sought is denoted as t+h converged are returned ... update estimate of displacements at time t+h000{},{},{}0 uuu 0 t1 {}[]{} uKF {}{}{}thtuuu []({})thKfnu []([],[],[]) KfnKMC {}({})th R fnu {}{}{}([],[]) FFRfnMC 1 {}[]{} uKF{}{}{} uuu max({||}) FTOL max({||}) uTOL {}{}, B xByFFPP {}{}, B xByFFPP BPTOL B xByP,P B xByP,P {}0 R {}{}{}([],[]) FFRfnMC B xByP,P P xPyu,u{}thu B xByP,P } { Fh t P u B P B u []([],[],[]) KfnKMC ,, KMC P xPyu,u B xByP,P Figure 5-6 Flow-chart for nonlinear dynamic pier /soil control module (A fter Consolazio and Cowan 2005)

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143 intialize internal barge module cycle counter For each iteration k = 1, ... yes Return current barge forces and to pier module. If mode=CONV, then the pier/soil module has converged. Now, determine if the overall coupled barge/pier/soil system has converged by examing the difference between barge forces for current cycle and previous cycle ... yes ... and ... Coupled barge/pier/soil system has converged, thus is now a converged value. Update displacement data for next time step :save converged barge crush parameters Return to pier/soil module for next time step. Return barge forces and for current cycle Convergence not achieved. ... compute incremental barge force using a weighted average ... compute updated barge forceincrement internal cycle counter mode=INIT no yes Return to pier/soil module mode=CALC yes mode=CONV no no yes no 2 4 1 3 Barge module Entry point Iterative central difference method update loop with damping (relaxation) 0000 max0 BBBpBuaaa0() th BBuuh()()0 BBPP0 ()2()2thktth B BBBBuPhmuu()ththth B BPBauu ()(,,)kthtt B BBpBmaxPfnaaa 11kkk BBBPPP BPTOL ()(1)kk BBBPPP() BxP()()(1) BBBPPP () BPTOLtht B Buutth B Buuttt B BpBmaxa,a,ath B u()()(1)1BBBPPP()(1)() BBBPPP() BxP ByP1 0BBPP 1 kkk BBBPPP k BBPP 00 P cos()sin()ththth P BPxBPyBuuu B yP Figure 5-7 Flow-chart for nonlinear dynamic ba rge module (After Consolazio and Cowan 2005)

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144 y x u B PxuPuPyu = u cos( ) + u sin( )Py Px BB PB mB A m x y B BPBPBP = P cos( )BB BxP = P sin( )BB ByB Figure 5-8 Treatment of oblique collision c onditions A) Displacement transformation, B) Force transformation (Aft er Consolazio and Cowan 2005)

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145 kzkykxkxkrxkzkrzkykryPier-3krxkrzkry Figure 5-9 OPTS model with li nearly independent springs

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146 Impact pier Up-station model Down-station model Pier-1 Pier-2 Pier-3 Pier-4 Pier-5 Figure 5-10 Full bridge m odel with impact pier

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147 FxFyMyMxPier-1 Pier-2 Fz MzA FxMyMxPier-4 Pier-5 Fy Fz MzB Figure 5-11 Peripheral models with applied load s A) Down-station model, B) Up-station model

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148 Pier-1 Pier-2 uxuyyx uz zA Pier-4 Pier-5 uxuyyx uz zB Figure 5-12 Displacements of peripheral models A) Down-station displacements, B) Up-station displacements

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149 Pier-3 meffimeffj Figure 5-13 OPTS model with lumped mass

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150 Pier-1 Pier-2 L L/2 A Pier-4 Pier-5 L L/2 B Figure 5-14 Tributary area of pe ripheral models for lumped mass calculation A) Down-station model, B) Up-station model

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151 CHAPTER 6 APPLIED VESSEL IMPACT LOAD HISTORY METHOD 6.1 Introduction In previous chapters, a coupled vessel imp act analysis (CVIA) procedure, which was implemented in FB-MultiPier, was validated agai nst experimental data, and demonstrated for several different bridges. In a coupled analysis, the dynamic impact load is computed as part of the analysis procedure. In this chapter, an alte rnative analysis approach is presented in which an approximate time history of impact force is genera ted using characteristics of the vessel (barge mass, initial velocity, and bow force-deformation relationship). The approximate time history of impact load is then externally applied to the bridge structur e, and a traditional (non-coupled) dynamic analysis is performed to determine member design forces. 6.2 Development of Load Prediction Equations The equations from which the lo ad history is calculated ar e based upon the principles of conservation of energy and conservation of lin ear momentum. Development of the applied vessel impact load (AVIL) equati ons is based primarily on charac teristics of the barge, although basic bridge structure characte ristics are also incorporated. 6.2.1 Prediction of Peak Impact Lo ad from Conservation of Energy The first step in calculating the impact load time history is to dete rmine the peak dynamic load to which the structure is subjected using th e principle of conservatio n of energy. Assuming that the system is not sensitive to changes in te mperature, conservation of energy for the system can be expressed as follows: 0 DE KEif if (6-1) where KEif is the change in kinetic energy of the barge, and DEif is the change in total deformation energy (i.e. the sum of the elastic and plastic deformation energies), associated with

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152 the deformation of the barge bow, from the initial st ate (i) to the final stat e (f). Conservation of energy is used to define a relationship between the peak impact load, and the barge parameters. Assuming that the change in mass of the barge is negligible, the change in barge kinetic energy can be expressed by the following relation: 2 Bi 2 Bf B ifv v m 2 1 KE (6-2) where mB is the constant (unchangi ng) mass of the barge and vBf and vBi are the magnitudes of the barge velocities at the initia l and final states respectively. In general, the deformation energy for the barge can be described using the following relation: Bf Bia a B B B ifda a P DE (6-3) where PB(aB) is the impact force as a f unction of barge crush depth (aB), and aBi and aBf are the barge crush depths at the initia l and final states respectively. To calculate the peak impact load on the pi er, the following assumptions are made: 1.) the pier is assumed to be rigid and fixed in space, 2.) the initial barge crush depth (aBi) is assumed to be zero, and 3.) the barge bow force-deformation relationship is assumed to be elastic perfectlyplastic (Figure 6-1). The first assumption implies that the initial ki netic energy of the barg e is fully converted into deformation energy of the barge bow during loading of the barge (Figure 6-2a). Thus, once all of the barge initial kinetic energy has been co nverted into deformation of the barge bow (i.e. the barge velocity becomes zero) the barge bow crush depth has reached its maximum value. Additionally, when the barge bow recovers the elastic portion of its deformation energy through

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153 unloading, this energy is then converted b ack into rebound motion of the barge (Figure 6-2b). Final barge kinetic energy can then be determ ined from the recovered deformation energy. If the barge bow remains linear and elastic, th e conservation of ener gy up to the point of maximum barge bow deformation can be represented by the following equation: 0 a P 2 1 v m 2 1 DE KEBm Bm 2 Bi B im im (6-4) where PBm is the maximum impact force observed during the impact, and aBm is the maximum barge bow deformation. The maximum impact force and barge bow deformation however, remain undetermined up this point, and t hus, an additional equation is required. If the barge bow remains elastic, the maxi mum bow deformation can be defined as follows: BY BY Bm B Bm Bma P P k P a (6-5) where aBY and PBY are the barge bow deformation and fo rce at yield, respectively, and kB is the initial elastic stiffness of the barge bow. Combining Equations 6-4 and 6-5, and then solving for the peak load produces the following equation: BY B B Bi B BY BY Bi BmP m k v m a P v P (6-6) Due to the elastic perfectly-plastic assumption for the barge bow force-deformation relationship, the peak barge impact force is limited to the yield load of the barge bow. To validate Equation 6-6, the central difference method was used to analyze a two degreeof-freedom barge-pier -soil system (Figure 6-3) subjected to various impact conditions. For these analyses, barge bow contact was modeled as a non linear spring using an el astic perfectly plastic force-deformation relationship (Figure 6-1), and the pier-soil resist ance was approximated using

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154 a linear-elastic spring. For each analysis, the barge weight was varied from 250 tons to 8000 tons, and the initial velocity was varied from 0.5 knots to 10 knots. From the analysis results, the maximum impact force generated between the barge and the pier was extracted for each case. Additionally, Equation 6-6 was used to predict peak ba rge impact forces for each case. As shown in Figure 6-4, although conservative, the estimate of peak forces as predicted by Equation 6-6 are not in very good agreement with c oupled analysis result s. Referring to Equation 6-6, it is noted that the ratio of the barge yield load to the barge bow yield deformation represents the initial elas tic stiffness of the system, assuming that the pier is rigid. However, recalling Figure 6-3, the barge bow spring is in series with the pier-soil spring. Thus, if a linear pier-soil spring is introduced and combined with the barge bow spring in series, an effective barge-pier-soil spring stiffne ss can be defined as follows: 1 P BY BY 1 P B Sk 1 P a k 1 k 1 k (6-7) where kP is the linear pier-soil spring stiffness. Replacing the initial elas tic barge stiffness (kB) in Equation 6-6 with the effective bargepier-soil series spring stiffness (kS) (Equation 6-7) produces the following equation: BY BP Bi B S Bi BmP c v m k v P (6-8) where cBP is the barge-pier pseudo-damping coefficient, defined as follows: B S BPm k c (6-9) Using Eqn. 6-8 in place of Eqn. 6-6 to predict the peak impact load imparted to the pier, and comparing the results to two-DOF coupled time-integrated dynamic analysis results, good agreement between the two methods (Figure 6-5) is observed.

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155 With the peak impact load calculated, the final barge velocity after impact can now be calculated using the unloading characteristics of the barge bow. It is assumed that unloading of the barge bow occurs along a path that has a sl ope equal to the initial elastic barge stiffness (Figure 6-2b). Using this assumption, the conser vation of energy equatio n from the point of maximum barge bow deformation to the point at wh ich the barge and pier are out of contact can be written as follows: 0 a a P 2 1 v m 2 1 DE KEBm BP Bm 2 Bf B mf mf (6-10) where vBf is the final barge velocity and aBP is the plastic barge bow crush depth (Figure 6-2b). Note that in Equation 6-10, the final barge velocity and th e plastic crush are unknown. However, it is not necessary to actually calc ulate the plastic crush depth. It is only necessa ry to know that the difference between the plastic and maximum barge bow deformations is the same as the initial elastic portion of the barge-bow crush, and thus: B Bm BP Bmk P a a (6-11) Then, combining Equations 6-10 and 6-11, and solving for the final barge velocity produces the following: B B Bm Bfm k 1 P v (6-12) Replacing the initial elas tic barge stiffness (kB) with the effective barge-pier-soil stiffness (kS) and introducing the barge-pier-s oil pseudo-damping coefficient (cBP= B Sm k), the final barge velocity can be approximated as follows: BP Bm B S Bm Bfc P m k 1 P v (6-13)

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156 For the situation in which the barge bow yields, the velocity at initial yield must be calculated. Conservation of energy from initial state to yield can be expressed as: 0 a P 2 1 v v m 2 1 DE KEBY BY 2 Bi 2 BY B iy iy (6-14) where vBY is the barge velocity at yield. Solving for this velocity using Equation 6-14 the velocity at yield can be expressed as: B BY BY 2 Bi BYm a P v v (6-15) Next, multiplying the numerator and denominator of the second term within the square-root, and simplifying results in the following: B B 2 BY 2 Bi BY BY B BY BY 2 Bi BYm k P v P P m a P v v (6-16) Again, replacing the initia l elastic barge stiffness (kB) with the effective barge-pier-soil stiffness (kS) and introducing the barge-pier-s oil pseudo-damping coefficient (cBP), the barge velocity at yield can be approximated as follows: 2 BP BY 2 Bi B S 2 BY 2 Bi BYc P v m k P v v (6-17) 6.2.2 Prediction of Load Duration from Conservation of Linear Momentum The next step in calculating the impact load history is to determine the duration of time that the load will act on the structure. This is accomplished by using the principle of conservation of linear momentum: I Lif (6-18)

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157 where { Lif} is the vector change of barge linear mo mentum from the initial state to the final state, and {I} is the load impulse. Assuming that the mass of the barge does not change, the change in barge linear momentum can be expressed as: Bi Bf B ifv v m L (6-19) where {vBf} and {vBi} are the barge velocity vectors at the fi nal and initial states respectively. In general, the load impulse for the barge can be defined as: f i t t Bdt t P I (6-20) where {PB(t)} is the vector impact for ce as a function of time (t), ti is the initial time at which impact occurs, and tf is the final time at which the imp act ends and the impact force becomes zero. Although momentum and impulse ar e vector quantities, the anal ysis method used here is one-dimensional. Therefore, the vector notation can be dropped by taking into consideration that final barge velocity will have a negative sign when the barg e moves away from the pier following impact. Additionally, the impact force on the barge will have a negative sign since it acts opposite to the direction of the initial barge motion. Taki ng these facts into account, the conservation of linear momentum can be rewritten as: f it t B Bi Bf Bdt t P v v m (6-21) If the barge bow remains elastic for the dura tion of the analysis, it is assumed that the elastic load history pulse takes the shape of a half-sine wave (Figure 6-6a). Assuming that the analysis starts at 0 secs (ti = 0 secs), the impulse of the impact force between the barge and the pier can be calculated as:

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158 Bm E t 0 E BmP t 2 dt t t sin PE (6-22) where tE is the duration of loading for an elastic pulse. Combining Equations 6-21 and 6-22, and solving for the elastic load duration: Bi Bf Bm B Ev v P 2 m t (6-23) Comparing Eqn. 6-8 and 6-13, the final velocity of the barge is equal to the initial barge velocity when the barge bow remains elastic, and thus Eqn. 6-23 becomes: Bi Bm B Ev P m t (6-24) For situations in which the barge bow yields the load history is subdivided into three distinct stages: 1.) elastic loading to yield, 2.) plastic loading, and 3.) elastic unloading (Figure 66b). The time history of elastic loading to yiel d is assumed to take the shape of a quarter-sine wave (Figure 6-7). For the purpose of calculating the ti me required to yield the barge bow, it is assumed that the analysis starts at 0 secs (ti = 0 secs). The elastic loading portion of the load impulse can be calculated as: BY Y t 0 Y BYP t 2 dt t 2 t sin PY (6-25) where tY is the time required to yield the barge bow. Using Equation 6-25 and the conservation of linear momentum the following relationship can be defined: BY Y BY Bi BP t 2 v v m (6-26)

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159 This equation accounts for the fact that the imp act force acts in the opposite direction of the initial motion of the barge, and that the barge velo city at yield acts in the same direction as the initial barge velocity. Solving Equation 6-26 for the time to yield gives: 2 BP BY 2 Bi Bi BY B BY Bi BY B Yc P v v P 2 m v v P 2 m t (6-27) Following the initial elastic load ing stage, an inelastic stag e occurs. During this latter stage, the plastic load is assume d to remain constant from the time at which the barge bow yields to the time at which unloading of the barge bow begi ns. The load impulse for this stage is given by: P BY t t t BYt P dt PP Y Y (6-28) where tP is the duration of plastic loading. Taking into account the direction of the impact force, and assu ming that the velocity of the barge immediately before the barge bow unloads is zero, the conservation of linear momentum can be defined as follows: P BY BY Bt P v m (6-29) The assumption that the velocity of the barge im mediately before the barge bow unloads is zero is valid because, the deformation of the pier-soi l system is much lower than the inelastic barge bow deformation. Solving Equation 6-29 for the plastic load duration: 2 BP BY 2 Bi BY B BY BY B Pc P v P m P v m t (6-30) Following the plastic loading stage, the time hi story of elastic unloading is assumed to take the shape of the second quarter of a single sine wave cycle (Figure 6-8):

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160 BY U t 0 U BYP t 2 dt t 2 t sin PU (6-31) where tU is the duration of unloading. Assuming that the barge velocity at the beginni ng of the unloading stage is zero, and that the impact load acts in the same direction as the final barge velocity, conservation of linear momentum can be expressed as: BY U Bf BP t 2 v m (6-32) Rearranging Equation 6-32, the duration of unloading is given by: BP BY BY B Bf BY B Uc P P 2 m v P 2 m t (6-33) Additionally, summing Eqns. 6-27, 6-30, and 6-33, the total duration for inelastic loading is defined as: 2 BP BY 2 Bi BP BY Bi Bm B Tc P v 1 2 c P v P 2 m t (6-34) 6.2.3 Summary of Procedure for Constr ucting an Impact Load History Using key equations from the detailed derivatio ns given in the sections above, a summary of the complete procedure for constructing an impact load time-history function is presented (Figure 6-9). The curves presented in Figure 6-9 are time-varying impact forces that may be applied to a bridge structure in a dynamic sens e. Although the AVIL method is not exact, it is expected that it will approximate the CVIA well.

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161 6.3 Validation of the Applied Vessel Impact Load History Method To assess the accuracy of the AVIL method, three coupled dynamic analyses were conducted on a one-pier two-span (OPTS) model of a new St. George Island Causeway Bridge channel pier and the connecting two superstruc ture spans (Davidson 2007). The three cases selected represent low, moderate, and high energy impacts respectively (Table 6-1). Comparing each case to the current AASHTO provisions (Figure 6-10) indicates that the low-energy analysis represents a si tuation in which the barge bow rema ins elastic throughout the duration of impact; the moderate-energy case represents a si tuation in which the barge bow deformation is slightly larger than the yield deformation (according to AASHTO); and the high-energy case exhibits significant barge bow yielding. The force-deformation relationship for the ba rge is dependent upon the size and shape of the impact pier column. Pier columns for cha nnel piers of the new St. George Island Causeway are round with a 6-foot diameter. Thus, using Eq uation 4.8, the yield lo ad and bow deformation at yield for the barge are 1540 kips and 2 in resp ectively (Figure 6-11). For each collision analysis, the impact load history computed using coupled analysis was compared to the load history predicted by th e AVIL method. Comparisons between the CVIA force history results and the AVIL results are presented in Figure 6-12. For the low-energy case (Figure 6-12a), the AVIL method predicts a slightly higher peak load and a s lightly longer load duration than is predicted by coupled analysis. Comparing the load impulses for each analysis 42 kip-sec and 38 kip-sec for the AVIL and coupled analysis methods resp ectivelyreveals that the two differ by about ten-percent. The moderate-energy force history comparison (Figure 6-12b) shows good agreement between the coupled analysis and the AVIL method. Compari ng the load impulses for this

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162 case kip-sec and 715 kip-sec for the AVIL and CVIA respectively indicates that the AVIL method over-predicts the load impulse by about two-percent. Inspecting the results for the high-energy impact (Figure 6-12c), there is a negligible discrepancy in the load histories predicted by the AVIL and CVIA methods. Load impulses for the two methods kip-sec versus 3448 ki p-sec for the AVIL and CVIA methods respectivelyreveals that the two me thods differ by less than one-percent. Comparisons of the maximum bending moment s from dynamic analyses performed using the two methods are presented in Table 6-2 and Figure 6-13. Moments in Figure 6-13 represent the maximum absolute bending moment throughout the duration of the analysis at a given elevation across all pier columns or all piles, whereas values in Table 6-2 are the maximum absolute bending moments for all piles or colu mns in the model, regardless of elevation. Inspection of Table 6-2 and Figure 6-13 reveals that pile and co lumn moments from the AVIL analysis agree well with the CVIA results. Additionally, the AVIL method produces moments that are slightly conservativ e in comparison to the coupled analysis moment results.

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163 Table 6-1 Impact energies for AVIL validation Energy Barge weight (tons) Barge velo city (knot) Impact-energy (kip-ft) Low 200 1 17.71 Moderate 2030 2.5 1123 High 5920 5 13100 Table 6-2 Maximum moments in all pier columns and piles Moment (kip-ft) Impact energy Location Coupled AVIL Percent difference Column 634 668 5.2% Low Pile 588 603 2.7% Column 3934 4005 1.8% Moderate Pile 1790 1805 0.9% Column 4580 4581 0.0% High Pile 1823 1826 0.2%

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164 PBaBPBYaBYYield Point Barge Bow DeformationImpact Force Figure 6-1 Barge bow force-deformation relationship

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165 PBaBPBYaBY aBmTotal deformation energy is equal to the initial barge kinetic energy A PBaBPBYaBY aBmRecoverable elastic deformation energy generates the final barge kinetic energy aBp B Figure 6-2 Inelastic barge bow deformation energy A) Loading, B) Unloading

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166 mB (mass of barge) mP (mass of pier) kP (equivalent linear stiffness of pier and soil) PBaB kB Barge bow force-deformation relationship Figure 6-3 Two degree-of-freedom barge-pier-soil model

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167 Barge Kinetic Energy (kip-ft)Peak Impact Force (kips) 0 200 400 600 800 0 400 800 1,200 1,600 Dynamic Analysis Data Applied Load History Method Figure 6-4 Peak impact force vs. initial barg e kinetic energy using a rigid pier assumption

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168 Barge Kinetic Energy (kip-ft)Peak Impact Force (kips) 0 200 400 600 800 0 400 800 1,200 1,600 Dynamic Analysis Data Applied Load History Method Figure 6-5 Peak impact force vs. initial barge ki netic energy using an e ffective barge-pier-soil stiffness

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169 Time (sec)Force (kips) tE A Time (sec)Force (kips) tP tUtY B Figure 6-6 Impact load hi stories A) Elastic loadi ng, B) Inelastic loading

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170 Time (sec)Force (kips) Time (sec)Force (kips) tP tUtY I Cycle tY Figure 6-7 Construction of load ing portion of impact force

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171 Time (sec)Force (kips) Time (sec)Force (kips) tP tUtY I Cycle 0 tU Figure 6-8 Construction of unloa ding portion of impact force

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172 Calculate barge-pier-soil series stiffness Calculate pseudo-damping coefficient Applied Vessel Impact Load History Method Select barge type, and determine PBY, aBYDetermine impact characteristics: mB, vBiCalculate pier-soil stiffness from static analysis: kP Elastic loading PBm = vBicBP PBm = PBY Inelastic loading Time (sec)Force (kips) tP tUtY PBY Y BYt 2 t sin P t 2 t t t t sin PU U P Y BY tT Time (sec)Force (kips) tE E Bmt t sin P 1 P BY BY Sk 1 P a k B S BPm k c Bi Bm B Ev P m t 2 BP BY 2 Bi Bi BY B Yc P v v P 2 m t 2 BP BY 2 Bi BY B Pc P v P m t BP BY BY B Uc P P 2 m t 2 BP BY 2 Bi BP BY Bi Bm B Tc P v 1 2 c P v P 2 m t BY BP BiP c v BY BP BiP c v Figure 6-9 AVIL procedure

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173 Kinetic Energy (kip-ft)Force (kips) 0 4,000 8,000 12,000 16,000 20,000 0 1,000 2,000 3,000 Low energy Moderate energy High energy AASHTO Figure 6-10 AASHTO load curve in dicating barge masses and veloci ties used in validating the applied load history method

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174 Bow deformation (in)Force (kips) 0 5 10 15 20 0 500 1,000 1,500 2,000 2 in, 1540 kips Figure 6-11 Barge bow force-deforma tion relationship for an impact on a six-foot round column

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175 Time (sec)Force (kips) 0 0.2 0.4 0.6 0.8 1 0 150 300 450 600 Coupled impact analysis Applied vessel impact loadA Time (sec)Force (kips) 0 0.2 0.4 0.6 0.8 1 0 500 1,000 1,500 2,000 Coupled impact analysis Applied vessel impact loadB Time (sec)Force (kips) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 500 1,000 1,500 2,000 Coupled impact analysis Applied vessel impact loadC Figure 6-12 Impact load history comparisons A) Low-energy impact B) Moderate-energy impact, C) High-energy impact

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176 Columns Pier Cap Piles Elev. +1.9 ft Elev. +29.3 ft Elev. +53.9 ft Elev. -62.4 ft Pile Cap AMoment (kip-ft)Elevation (ft) 0 700 1,400 -80 -60 -40 -20 0 20 40 60 Pile Cap Coupled impact analysis AVIL analysisB Moment (kip-ft)Elevation (ft) 0 2,500 5,000 -80 -60 -40 -20 0 20 40 60 Pile Cap Coupled impact analysis AVIL analysisCMoment (kip-ft)Elevation (ft) 0 2,500 5,000 -80 -60 -40 -20 0 20 40 60 Pile Cap Coupled impact analysis AVIL analysisD Figure 6-13 Moment results profile for the new St George Island Causeway Bridge channel pier A) Channel pier schematic, B) Low-ener gy impact, C) Moderate-energy impact, D) High-energy impact

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177 CHAPTER 7 IMPACT RESPONSE SPECTRUM ANALYSIS 7.1 Introduction In previous chapters, the dynamic res ponse of the bridge structure (Figure 7-1) was computing using time history impact analysis techniques. These te chniques used numerical timestep integration to obtain a solu tion to the equation of motion: t F t u K t u C t u M (7-1) In this equation, [M], [C], and [K] are the mass, damping, and stiffness matrices, respectively, of the structure; {F(t)} is the ti me varying external for ce vectors; and {u(t)}, {u(t)}, and {u (t)} are the time varying displacemen t, velocity, and acceleration vectors, respectively. An alternative dynamic analysis tec hnique that does not requ ire time-integration is the response spectrum analysis technique. Res ponse spectrum analysis is carried out using structural vibration characteristic s such as mode shapes and fr equencies and involves estimating the maximum dynamic structural response rather th an computing the respons e at each point in time. In this chapter, a respons e spectrum analysis technique th at is suitable for analyzing dynamic barge impact conditions is presen ted, validated, and demonstrated. 7.2 Response Spectrum Analysis In order to conduct a response spectrum an alysis, the numerical model must be transformed from the structural system to the mo dal system. This process is generally achieved through the use of an eigenanalysis, from which st ructural vibration charac teristics are obtained. Using the eigenanalysis results, the multiple degree-of-freedom (MDOF) structural system matrices, forces, and displacements can be tr ansformed into single degree-of-freedom (SDOF) modal properties, forces (Pi), and displacements (qi) for each mode (Figure 7-2). Once the model

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178 has been transformed into modal coordinates, an analysis on the modal equations of motion can be conducted. 7.2.1 Modal Analysis The generalized eigenproblem can be expressed by the following equation: M K (7-2) where [K] and [M] are respectively, the st iffness and mass matrices of the system, are the eigenvalues of the system, and [ ] is a matrix whose columns are the eigenvectors (i.e. modal shapes of vibration) of the stru cture. Both the stiffness and ma ss matrices are square matrices with dimensions equal to the num ber of degrees-of-freedom in the structural system model. In general, the matrix of eigenvectors is also s quare with dimensions equal to the number of degrees-of-freedom in the system. Each eigenvalue ( i) corresponds to a natural circular frequency of the structure squared (2 i where i has units of rad/sec) which in turn can be related to the period of vi bration of the structure: i i2 T (7-3) Likewise, each eigenvector { i} represents a shape (referred to as a mode shape) that the structure assumes when it is excited (loaded) at its respective natu ral frequency (Figure 7-3). The eigenvalue problem, Eqn. 7-2, can also be rewritten as follows: 0 I D2 (7-4) where [D] is the dynamic system matrix (defined as [D] = [M]-1[K]), and [I] is the identity matrix. To ensure a nontrivial solution to the above equation, the characteristic matrixthe parenthetical quantity in Eqn. 7-4must necessarily be singular. For a matrix to be singular, its

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179 determinant must be zero. Impos ing this condition leads to an nth degree polynomial, commonly referred to as the characteristic polynomial: 0 I D det p2 2 (7-5) where p( 2) is the characteristic polynomial. Soluti ons for the eigenvalues of the system are generally obtained by solving fo r the n-roots of the characte ristic polynomial. Once the eigenvalues are known, the corresponding eigenv ectors can be solved for by rewriting Eqn. 7-4 for a specific mode i: 0 I Di 2 i (7-6) Further examination of Eqn. 7-6 shows that this equality holds true even if the eigenvector { i} is scaled by an arbitrary constant (ci). Although the relative magnitude of each element with respect to the other elemen ts in an eigenvector is unique, the absolute magnitude of each eigenvector is not. The process of scaling the eigenvectors such th at they each have a sp ecific magnitude is called normalization. A normalizati on method often used in structural analysis software is called mass-normalization and involves scaling each eige nvector such that the following condition is satisfied: 1 M i T i or I M T (7-7) where i is a mass-normalized eigenvector. A massnormalized eigenvector can be computed from an arbitrarily normaliz ed eigenvector as follows: i T i i i M (7-8)

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180 If the eigenvectors of a system have been co mputed, the model can be transformed into the generalized modal system (recall Figure 7-2). Structural displacemen ts of the system model may be related to modal displacements through the eigenvectors as follows: t q t u (7-9) where {u(t)} and {q(t)} are the time varying stru ctural and modal displacements, respectively. Because the eigenvectors are not dependent upon tim e, taking the first and second derivatives of {u(t)} with respect to time yields: t q t u (7-10) t q t u (7-11) where {u (t)} and {u (t)} are the structural velocity and acceleration vectors, and {q(t)} and {q (t)} are the modal velocity and acceleration vectors. Eqn. 7-9 provides a means of transforming modal displacements into structural displacements. An inverse process that tran sforms structural displacements into modal displacements is achieved using the inverse of the eigenvector matrix: t u t q1 (7-12) However, inversion of the eigenvector matrix in Eqn. 7-12 may be computationally expensive. A more computationall y efficient procedure for achiev ing the same transformation as that described by Eqn. 7-12 is available and involves the us e of both the eigenvector matrix and the structural mass matrix. Assuming that the eigenvectors of the system have been massnormalized, and premultiplying each term in Eqn. 7-9 by M T yields: t q I t q M t q M t u M T T T (7-13)

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181 Making use of Eqn. 7-7, Eqn. 7-13 is reduced to provide the transformation of modal displacements into stru ctural displacements: t u M t qT (7-14) Although Eqn. 7-14 is more computationa lly efficient than Eqn. 7-12, use of Eqn. 7-14 is dependent upon the mass-normalization of the eigenv ectors, and thus requir es greater caution in its use. Throughout the remainder of this chapte r, it is assumed that the eigenvectors are massnormalized. Typically, the equation of moti on for the structural system is expressed as in Eqn. 7-1. Introducing the displacement velocity, and acceleration relationships (Eqns. 7-9, 7-10, and 711), into Eqn. 7-1 produces: t F t q K t q C t q M (7-15) Premultiplying each term in this equation by the transpose of the eigenvector matrix (i.e. []T) yields: t F t q K t q C t q M T T T T (7-16) The above equation can be simplified by defining the following relationships: M mT (7-17) C cT (7-18) K kT (7-19) t F t fT (7-20) where [m], [c], and [k] are the modal mass, m odal damping, and modal stiffness matrices, and {f(t)} is the modal force vector. The fact that eigenvectors are orthogonal with respect to the stiffness and mass matrices results in the modal stiffness and modal mass matrices being diagonal (Chopra 2007). If an

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182 orthogonal formulation is used to construct the structur al damping matrix [C], then the modal damping matrix [c] will also be diagonal. In this case, each diagonal term in [c] can be expressed as a ratio of the cr itical damping for each mode: i i i im 2 c (7-21) where i is the ratio of critical damping for the ith mode. When [m], [c], and [k] are diagonal, the coupled structural e quation of motion (Eqn. 7-15) reduce to a series of uncoupled single DOF modal equations of motion. (Note that the use of the term coupled here is different from that discussed in Chapter 5. In Chapter 5, the c oupling discussed was betw een the impacting vessel and the bridge structure. The coup ling being referred to in the pres ent context, however, refers to the linking of various DOF within the structural model to each other. Such linking occurs when off-diagonal terms are present in the system ma trices). For a given mode i, the SDOF modal equation of motion is: t f t q k t q m 2 t q mi i i i i i i i i (7-22) Normalizing each term by the modal mass yields: i i i 2 i i i i im t f t q t q 2 t q (7-23) Solving each SDOF modal equation of mo tion yields modal displacements, modal velocities, and modal accelerat ions. With the time-varyi ng modal quantities known, the individual modal contributions to overall struct ural displacements, veloci ties, and accelerations can be computed for each mode using Eqns. 7-9, 7-10, and 7-11. Structural response quantities are thus determined through modal superposition in which the contributio ns from all modes are added together.

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183 7.2.2 General Response Spectrum Analysis Time-step integration of the equations of motion, either structural (Eqn. 7-15) or modal (Eqn. 7-22), generally requires hundreds or thousands of time steps to be analyzed in order to evaluate the response of a struct ure. Subsequently, the results must be scanned to identify the maximum absolute force values needed for design. In contrast, response spectrum analysis can be used to estimate these maximums without the need for conducting time-history analysis. In a response spectrum analysis, Eqns 7-22 and 7-23 are not directly time-i ntegrated. Instead, the maximum response contributed by each mode is de termined using relationships that correlate loading to peak response as a functi on of modal frequency (or period). 7.2.2.1 Modal Combination To determine the maximum overall structural response, the individua l modal contributions must be combined. The simplest possible combina tion technique consists of direct superposition of the maximum absolute response from each m ode. This approach assumes that the peak responses for each mode occur at the same point in time, which leads to overly-conservative design forces. An alternative, and more reasonabl e, approach is to use a square-root-of-the-sumof-the-squares (SRSS) method. The SRSS method is based upon probabilistic theory and is expressed as follows: n 1 i 2 i combr R (7-24) where Rcomb is the combined result of the res ponse parameter under consideration (e.g. displacement, force, etc.), and ri is the response parameter value for the ith mode. Use of the SRSS combination method tends to provide accurate approximations of structural response for two-dimensional models in which the natural freq uencies are well-spaced (Tedesco et al. 1999).

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184 However, for three-dimensional models in wh ich closely spaced natural frequencies may be present, the complete-quadratic-combination (CQC) method yields more accurate results: n 1 i n 1 j j i ij combr r R (7-25) where ij is defined as follows: 2 2 j 2 i 2 j i 2 2 2 / 3 j i j i ij4 1 4 1 8 (7-26) where i and j are the modal damping ratios fo r modes i and j respectively, and is the frequency ratio ( j i ). 7.2.2.2 Mass Participation Factors Although the total number of mode s for a multiple degree-of-freedom structural model is equal to the number of degrees-o f-freedom, typically, only a relatively small number of these modes are required in order to ad equately describe the behavior of the system. Consequently, eigenanalyses are typically conduc ted on a reduced system (or s ubspace), instead of the full system. Subspace methods are advantageous in that significantly fewer computations are required to obtain modal properties (shapes a nd frequencies) when only a small number of modes are needed. However, a key step in usi ng subspace methods is determining the number of modes that are required to ade quately capture the response of th e system. Typically, this number of modes is determined through the use of mass participation factors wh ich represent the amount of structural mass present in each mode of vibration. To calculate the mass participation factor fo r a specific mode-i, the mass excitation factor must be calculated as: F T i i1 M L (7-27)

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185 where Li is the direction specific ma ss excitation factor for the ith mode, and {1F} is a unit vector where the subscript (F) indicates that the unit values are associat ed with degrees-of-freedom in the direction of impact loading, and zero valu es are associated with degrees-of-freedom not associated the direction of impact loading. Th e effective modal mass in each mode is calculated as: i2 i eff i,iL m= m (7-28) where mi,i is the diagonal term of modal mass matrix [m] associated with mode-i, and [m] is defined in Eqn. 7-17: M mT (7-29) The mass participation factor for each mode is defined as: i2 eff i i totali,itotalm L == M(m)(M) (7-30) where Mtotal is the total mass of the st ructure being analyzed, and Li is the mass participation factor for mode-i. In seismic (earthquake) response spectrum analysis, design codes (ASCE-7 2005, FEMA 2003, etc.) require that sufficient modes be used such that the total mass partic ipation of the modes included adds up to at l east ninety-percent (90%) of the total mass of the structure in two orthogonal directions. For use in vessel impact response spectrum analysis (described in this chapter) it is recommended that 99% mass partic ipation in the direction of applied impact loading be required.

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186 7.3 Dynamic Magnification Factor (DMF) For a single degree-of-freedom (SDOF) system the maximum response of the system to a time-varying dynamic loading condition can be dete rmined through the combined use of static response calculation and a dynamic magnifica tion factor (DMF). The maximum dynamic response is computed as the product of the st atic response and the DMF. When DMFs are calculated for many different SDOF systems ha ving different natural periods (and corresponding natural frequencies), a DMF spectrum is produ ced. In this section, a DMF design spectrum appropriate for use in barge im pact analysis is developed. Calculation of a single point on the DMF spect rum involves selecting an impact condition (barge mass and speed), determining a time-hist ory of impact loading, and analyzing an equivalent SDOF bridge-pier-soil structural model subjected to the loading condition using both dynamic and static analysis procedures. For a give n impact condition, the applied vessel impact loading (AVIL) method, described earlier in this report, may be used to form a time-history of impact loading. Using this loading, the SDOF st ructural system is dynamically analyzed and the maximum dynamic displacement of the system (uD) is recorded. Subs equently, the peak magnitude of the dynamic applied load is determined (Figure 7-4) and is applied to the SDOF structural system as a static load. A static anal ysis is then performed to determine the maximum static displacement of the system (uS). The DMF is then given computed as the ratio of maximum-dynamic and static displacements computed for the SDOF system (i.e., DMF = uD / uS). Repeating this process for different SDOF structural models (having varying natural periods and frequencies), but using the same time-history of impact loading, produces a collection of points that constitute the DMF spectrum (Figure 7-5) for the specific loading condition that has been us ed in the calculations.

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187 If this process is repeated not only for diff erent SDOF structural models but also for different impact load histories, the result is a family of event-specific DMF spectra (Figure 7-6a). The data in Figure 7-6a was generated by first varying th e mass of the barge from 250 ton-mass to 8000 ton-mass, and the initial velocity from 0.25 knots to 8 knots to generate various impact load histories. A 250 ton-mass barge corresponds to a single jumbo hopper barge with 50 tons of cargo, and the 8000 ton-mass is slightly larger th an four fully-loaded jumbo hopper barges. The upper bound on initial barge velocity, 8.0 knots, was selected based upon maximum barge tow velocities reported at past-point data along Floridas intracoastal waterway system. Overall, 1024 individual impact scenarios were inves tigated to generate the data in Figure 7-6a. It is worth noting that th e DMF data shown in Figure 7-6 never exceed a value of 2.0. A DMF of 2.0 indicates that the magnitude of dyna mic response is twice that of the corresponding static response. Stated in othe r terms, for a DMF of 2.0, the dynamic response is 100% greater than the corresponding static response. In regard to the development of a design DMF spectrum, one of the simplest options is a broad-banded design spectrum that envel opes all of the data generated (Figure 7-6b). While this approach is simple and provides conservative estimates of dynamic response, in some cases, a broad-banded design spectrum will yield results that are excessively conservative. For lowenergy impacts, a broad-banded design spectrum will grossly over-predict amplification effects for long-period modes (modes with a natural struct ural period greater than one-second). In Figure 7-7, the event-specific DMF spectrum for a lo w energy impact (200 ton barge drifting at 1.0 knots) is compared to a broad-banded design sp ectrum. For a mode with a structural period of 5 sec, the broad-banded design envelope pr edicts a dynamic magnification factor of 2.0, whereas the event-specific spectrum predicts a DM F of 0.09 (a DMF less than 1.0 indicates that

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188 the dynamic response is less severe than the static response). Thus, in this instance, a broadbanded design spectrum would over-predict the DMF by more than a factor 20. An alternative to the broad-banded design spec trum is a DMF spectrum that evolves based upon impact condition characteristics. This approach yields a design spectrum that is closer to the event-specific spectrum. Inspection of Figure 7-8 shows that as the impact energy increases, the width of the event-specific DMF spectrum increases by expansion in the short and longperiod ranges. One of the key components of an evolving design DMF spectrum is to find a relationship between the impact characteris tics and the short and long-pe riod transition points (Figure 7-9a), which shift along the structural period axis, producing an expansion as the impact energy increases. This is accomplished by defining the points that mark a transition from a constant DMF of 2.0 to a sloping DMF. Based on a quali tative investigation of the event-specific DMF data, the transition points are set at a dynamic magnification factor of 1.6 (Figure 7-9b). For each event-specific DMF spectrum in Figure 7-6, the short and long-period transition points are defined in this manner, and an expression for th e structural period at each transition point is defined. The expression for the short-period tran sition point is defined as the minimum of the following two expressions: 9 0 2 Bi B I Sv m 1300 T T (7-31) 2 2 T TI S (7-32) where TS is the period for the short-peri od transition point in seconds, TI is the period of impact loading (Figure 7-10)calculated as twice the duration of loading predicted by the equations

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189 from Chapter 6and mB and vBi are the barge mass (kip-sec2/in) and initial velocity (in/sec) respectively. Figure 7-11 shows the short-period tr ansition point as a function of barge kinetic energy. Event-specific DMF data were calculated using the process described ear lier, and short-period expression data were calculated using Eqns 7-31 and 7-32. The data in Figure 7-11 are merely a subset of all the data used to generate the expressions, reduced for visual clarity. Additionally, each line indicates a change in barge mass with the initial barge velocity held constant. In general, Eqns 7-31 and 7-32 predict short-periods that are lower than the data obtained from event-specific DMF spectra generation. Referring back to Figure 7-9, this indicates that the short-period transition point for the design DMF spectrum is slightly left of the corresponding point in the event-specific DMF spectrum. The expression for the long-period transition point is defined as the minimum of the following two expressions: 1 0 2 Bi B I L100 v m T T (7-33) 7 0 T TI L (7-34) where TL is the period for the long-period transition point in seconds, and TI, mB, and vBi are in seconds, kip-sec2/in, and in/sec respectively. Long-period transition point data as a functi on of barge kinetic energy is shown in Figure 7-12. As with the short-period transition point data, event-specific DMF data were calculated using the process illustrated in Figure 7-9b, and the long-period expres sion data were calculated using Eqns 7-33 and 7-34. Generally, the long-period expres sion predicts a period that is higher than the data generated from the event-specif ic DMF spectra; thus, indicating that the long-

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190 period transition point for the de sign DMF spectrum is slightly ri ght of the corresponding point in the event-specific DMF spectrum (Figure 7-9). With the transition points of the evolving de sign DMF spectrum quantified, expressions for the design DMF spectrum as a function of structur al period are defined. For impact response spectrum analysis, the design DMF spectrum is assu med to be a piecewise linear function in loglog space: S 6 0 ST T if 2 1 T T 0 2 DMF (7-35) L ST T T if 0 2 DMF (7-36) L 95 0 LT T if 1 0 T T 0 2 DMF (7-37) where, recalling Eqns. 7-31 through 7-34, TS is the short period transi tion point, the lesser of: 9 0 2 Bi B I Sv m 1300 T T and 2 2 T TI S (7-38) TL is the long-period transition point, defined as the lesser of: 1 0 2 Bi B I L100 v m T T and 7 0 T TI L (7-39) Eqns 7-35 and 7-37 are applicable to stru ctural periods outside of the transition points. For structural periods between the transition points, the DMF is se t to a value of 2.0 (Figure 7-13). In Figure 7-14, event-specific DMF spectra are comp ared to design DMF spectra for four different impact conditions that span a broad ra nge of impact energies. The plots demonstrate that the evolving design DMF sp ectrum equations adequately e nvelope the corresponding eventspecific DMF data over a broad range of impact energies. Furthermore, none of the design

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191 spectra are excessively conservative as was th e case for the broad-banded design spectrum discussed earlier. 7.4 Impact Response Spectrum Analysis Using the concept of a dynamic magnification factor (DMF), it is possible to approximate the maximum dynamic response (displacement, internal member forces, etc.) of a SDOF system using results from a static analysis of the same system. To achieve such an outcome, the DMFmagnitude for the structural system and the applied load must be based upon vibration characteristics of the system. As noted earlier, a modal analysis (e.g., eigen analysis) may be used to obtain modal vibrati on characteristics and uncouple th e MDOF dynamic equations of motionessentially transforming the MDOF system into many SDOF uncoupled modal equations. The impact response spectrum analysis (IRSA) procedure proposed here uses modal (eigen) analysis, in conjunction with a DMF sp ectrum, to simulate dynamic barge impacts on MDOF bridge models. Key steps in the IRSA calculation process are Calculating the peak magnitude of impact load Applying the peak impact load to the struct ure in a static sense, and computing the resulting static displacements Transforming the static structural displ acements into static modal displacements Magnifying the static modal displacement susing a DMF spectruminto dynamic modal displacements Transforming the dynamic modal displacements into dynamic structural displacements for each mode Recovering the internal member forces fo r each mode using the corresponding dynamic structural displacements Combining the internal member forces and dynamic structural displacements from the modal contributions using SR SS or CQC modal combination

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192 A detailed flowchart for this pr ocess is illustrated in Figure 7-15. To calculate the static displacements {uS} of the structure, the peak value of the dynamic load (PBm) is calculated using the applied vessel impact load (AVIL) equations developed of the previous chapter. Using impact characteristics, a barge bow force-deformat ion relationship, and the stiffness of the pier, the peak dynamic load can be calculated. Using AVIL equations, the period of loadingan important component of DMF spectrum calculation is calculated as twice the load duration (recall Figure 7-10). For elastic loading: Bi Bm B Iv P m 2 T (7-40) and for inelastic loading: 2 BP BY 2 Bi BP BY Bi Bm B Ic P v 1 2 c P v P m T (7-41) Characteristics of the DMF spectrum are calculate d from the impact vessel characteristics and the loading period using Eqns. 7-31 through 7-37. The peak dynamic load is applied to the structur e at the impact point in a static sense, and a static analysis is conducted to comput e the resulting structur al displacements {uS} (Figure 7-16). These static displacements can subsequently be amplified, in a modal sense, to obtain the dynamic response of the system. To transform the structural analysis into moda l coordinates, an eige nanalysis is conducted on the structure to determine the natural frequenc ies (and periods) and eigenvectors of the system (recall Figure 7-3). Using the eigenvectors, the sta tic displacements are transformed from structural coordinates into modal c oordinates using a variation of Eqn. 7-14 (Figure 7-17): S T Su M q (7-42)

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193 where {qS} and {uS} are vectors of static displacements in the modal and structural coordinate systems respectively. Each entr y in the static modal displace ment vector corresponds to a specific mode shape and natural frequency (and period). Using the natural period for each mode, and the design DMF spectrum (Eqns 7-35 and 7-37), a DMF value is computed for each mode (Figure 7-18). The modal static displacement qSi for mode-i is then magnified by the DMF to produce a dynamic modal displacement qDi (Figure 7-19). Using a modified version of Eqn. 7-9, the contribution from mode-i to the overall dynami c structural displacements is then calculated as: i Di Di q u (7-43) Internal dynamic member forces {FDi} for each mode are determined by performing force recovery on the structure using the dynamic displacements {uDi}. Maximum dynamic structural displacements {uD} of the system are obtained by co mbining the modal contributions to displacement {uDi} using either SRSS or CQ C combination techniques: Dn 2 D 1 D Du ,..., u u SRSS u (7-44) Dn 2 D 1 D Du ,..., u u CQC u (7-45) The same process is then applied to the internal force vectors: Dn 2 D 1 D DF ,..., F F SRSS F (7-46) Dn 2 D 1 D DF ,..., F F CQC F (7-47) to obtain the maximum dynamic intern al member design forces where {FD} for the structural system. 7.5 Impact Response Spectrum An alysis for Nonlinear Systems In the IRSA procedure presented above, it is assumedfor purposes of performing the eigenanalysisthat the system matrices (stiffness and mass) correspond to a linear elastic structure. Hence the procedure can only be used for the specific case in which the system is

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194 approximated as being linear. Considering the f act that severe vessel-br idge collision events generally exhibit nonlinear behavior, the linear IRSA procedure must be altered to account for system nonlinearityspecifically the nonlinear stiffness matrix. Two iteration stages are required to account for such nonlinearity: 1.) lo ad determination iterations, and 2.) stiffness linearization iterations for eigenanalysis. 7.5.1 Load Determination and DMF Spectrum Construction For a nonlinear system, the effective (s ecant or tangent) pier-soil stiffness (kP) is dependent upon the ultimate displacement level that is re ached by the system, and is therefore unknown at the beginning of the analysis. An iterative process (Figure 7-20) is therefore required to calculate kP and the peak dynamic load (PBm). In order to start the pr ocess, the pier is initially assumed to be rigid (kP= ), which causes the barge-pier-soil series stiffness to reduce to the following: BY BY 1 BY BY 1 P BY BY S ) 0 (a P 1 P a k 1 P a k (7-48) where (0)kS is the initial estimate for the effective ba rge-pier series spring stiffness. The peak dynamic load for the first iteration ((0)PBm) can then be calculated as: B S ) 0 ( BP ) 0 (m k c (7-49) BY BP ) 0 ( Bi Bm ) 0 (P c v P (7-50) The initial load estimate ((0)PBm) is then applied to the structure in a static sense, and the structural displacement at the load application point is computed. With both the load and the displacement at the impact point known, the pier-s oil secant stiffness for the next iteration (n) can be calculated as:

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195 P ) 1 n ( Bm ) 1 n ( P ) n (u P k (7-51) where (n)kP is the pier-soil secant stiffness for the current iteration (n) and (n-1)PBm and (n-1)uP are, respectively, estimates of the peak dynamic load and structural static di splacement of the impact point for the previous iteration (n -1). Furthermore, the effectiv e barge-pier-soil series spring stiffness ((n)kS) for the current iteration (n) can be updated as: 1 P ) n ( BY BY S ) n (k 1 P a k (7-52) With the effective barge-pier-soil series spring stiffness ((n)kS) known, the load ((n)PBm) for the current iteration (n) can be updated accordingly: B S ) n ( BP ) n (m k c (7-53) BY BP ) n ( Bi Bm ) n (P c v P (7-54) The incremental change in computed impact lo ad from iteration (n-1) to iteration (n) is then calculated as follows: Bm ) 1 n ( Bm ) n ( BmP P P (7-55) If the incremental change in computed load ( PBm) is smaller than a chosen convergence tolerance, then calculation of the peak load has converged to a solution. With the load (PBm) determined, the design DMF may be constructed using Eqns. 7-31 through 7-37. 7.5.2 Structural Linearization Procedure For a linear system, eigenanalysis can be carried out directly using th e system stiffness and mass matrices. However, for a nonlinear system, th e stiffness matrix itself is dependent upon the displacements of the system. In such a case, an iterative process (Figure 7-21) must be used in which the system secant stiffness matrix is upd ated using the dynamically magnified structural

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196 displacements computed within each iteration. However, during the initial iteration, static displacements are used to approximate the dynamically amplified displacements that will be used in later iterations. Thus, to ini tialize the linearization process, th e static external force vector ({FS})which contains the peak impact load (PBm), as well as all other static loads on the structure (e.g. gravity loads)is used to compute the external force vector for the first iteration: S E ) 0 (F F (7-56) Additionally, the structural static displ acements are used to estimate the dynamic displacements for the initial iterations: S D ) 0 (u u (7-57) These dynamic displacements are in turn used to compute the secant stiffness matrix for the first iteration. An eigenanalysis is then conducted using the mass matrix and the current secant stiffness matrix. Eigenvectors and natural frequencies are extracted from the eigenanalysis results for use in the IRSA. Static st ructural displacements {uS} are transformed into static modal coordinates {qS} using the current estimate of the ei genvectors for nonlinear iteration-n: S T ) n ( S ) n (u M q (7-58) Using the estimate of structural period for mode -i obtained from the eige nanalysis at the n-th nonlinear iteration: (n) i (n) i2 T= (7-59) the DMF is calculated as:

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197 0.6 (n) (n)(n) i iiS ST DMF=2.01.2 if T< T T (7-60) (n)(n) iSiLDMF2.0 if TTT (7-61) -0.95 (n) ()(n) i iiL LT DMF=2.00.1 if T>T Tn (7-62) Static modal displacements for mode-i are then magnified using the current estimate of the DMF to obtain dynamic modal displacements: Si ) n ( i ) n ( Di ) n (q DMF q (7-63) and the dynamic displaced shape for each mode is obtained as: i ) n ( Di ) n ( Di ) n ( q u (7-64) The overall structural displacement response of the system is updated by combining the displaced mode shapes using the SR SS or CQC combination techniques: Dn (n) 2 D (n) 1 D (n) D (n)u ,..., u u SRSS u (7-65) Dn (n) 2 D (n) 1 D (n) D (n)u ,..., u u CQC u (7-66) The external force vector for each mode -i is updated by premultiplying the updated displaced shape for each mode by the most recent estimate for the stiffness matrix: Di ) n ( ) n ( Ei (n)u K F (7-67) Combining the external force vectors for each mo de using either SRSS or CQC combination, the updated estimate of extern al force is calculated: En (n) 2 E (n) 1 E (n) E (n)F ,..., F F SRSS F (7-68) En (n) 2 E (n) 1 E (n) E (n)F ,..., F F CQC F (7-69)

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198 Using the modally (SRSS or CQC) combined exte rnal force vector, the incremental change in force vector from the previous nonlinear iteration (n-1) the current iteration (n) is calculated: (n)(n-1) EEE F=F F (7-70) If any term in the incremental external force vector { FE} is greater than the chosen tolerance, then the process begins anew with recalculation of the secant stiffness matrix using the updated displaced shape. Alterna tively, if every term in { FE} is less than or equal to the chosen tolerance, then the linearizati on process has converged. In this case, internal dynamic member forces (n){FDi} for each mode are determined by performing force recovery on the structure using the dynamic displacements (n){uDi}. Modal internal member forces are then combined using SRSS or CQC modal combination: Dn 2 D 1 D DF ,..., F F SRSS F (7-71) Dn 2 D 1 D DF ,..., F F CQC F (7-72) to obtain the maximum overall dynamic struct ural internal member design forces {FD}. 7.6 Validation and Demonstration of Impact Response Spectrum Analysis Validation of the IRSA method is carried ou t by analyzing a series of jumbo hopper barge impacts on bridge structures using both IRSA a nd coupled vessel impact analysis (described earlier in Chapter 5) and subsequently compari ng results. Since both methods are implemented in same software analysis package, FB-MultiPier, any difference in analysis results are solely due to differences in the analysis procedures themse lves. Since the coupled vessel impact analysis (CVIA) method was previously va lidated against both high-resolut ion finite element models and experimental data (see Chapter 5), results ob tained from this method are considered the benchmark (or reference datum) against which the accuracy of the IRSA method is judged.

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199 7.6.1 Event-Specific Impact Response Spectrum Analysis (IRSA)Validation To validate the IRSA method, coupled dynami c time-history analyses were conducted on a one-pier two-span (OPTS) model of the new St. George Island Causeway Bridge channel pier and the two connected superstr ucture spans (Davidson 2007). Three impact cases (Table 7-1) were analyzed: a low energy impact; a moderate energy impact; and a high energy impact. The low-energy case represents a situ ation in which the barge bow remains elastic throughout the duration of impact; the modera te-energy case represents a situation in which barge bow deformation is slightly greater than the barg e yield deformation; a nd the high-energy case corresponds to a situatio n in which significant barge bow yiel ding is expected. (Note that these are the same three cases previously used in th e validation of the applied vessel impact load (AVIL) method in Chapter 6). Both the size and shape of the pier column that will be impacted affect the barge forcedeformation relationship that needs to be used. Pier columns for channel piers of the new St. George Island Causeway Bridge are circular in cross-section and 6-ft in diameter. Using equations presented earlier in Chapter 4, the yi eld load and yield deformation for a jumbo hopper barge are determined to be 1540 ki ps and 2 in. respectively (Figure 7-22). Each IRSA validation case used an event-sp ecific DMF spectrum that corresponded to the impact energy of the collision being analyzed. Therefore, before each IRSA validation was performed, a corresponding CVIA was conducted to establish a time-history of impact load corresponding to the impact ener gy of interest. Each such imp act load history was applied dynamically to a large number of SDOF structur al systems (having varying natural periods) and the maximum dynamic displacement of each system (uD) recovered. Additionally, for each case, the peak magnitude of impact lo ad was determined (recall Figure 7-4) and applied to the SDOF structural system in a static sense so that th e maximum static displace ment of the system (uS)

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200 was recovered. For each structural period (i.e. each SDOF system) the DMF was then calculated as the ratio of maximum dynamic to static displacement (i.e., DMF = uD / uS). For the given impact load history, the collection of all such DMF data points constitutes an event-specific DMF spectrum (recall Figure 7-5). For each case in Table 7-1, an event-specific spec trum was generated (Figure 7-23) and an event-specific IRSA was conducted using the co rresponding DMF spectrum. In each case, a total of 12 eigen modes were used and combined together using the SRSS and CQC techniques. The use of 12 eigen modes was necessary to ensure that the internal member force results from the IRSA using a design DMF spectrum were c onservative with respect to CVIA results. Comparisons of the maximum absolute bending moments for all piles and columns in the model from dynamic analyses performed using th e IRSA and CVIA methods are presented in Tables 7-2 and 7-3. The IRSA values presented in Table 7-2 were modally combined using the SRSS technique, whereas corre sponding values in Table 7-3 were combined using the CQC technique. Additionally, Tables 7-2 and 7-3 present the total modal mass participation that was achieved by using 12 modes for each IRSA. Figure 7-24 shows a profile of bending moments, which represents the maximum absolute bending moment throughout the duration of the anal ysis at a given elev ation across all pier columns or piles, for both the CVIA and th e event-specific IRSA using the CQC modal combination technique. Comp aring values from Tables 7-2 and 7-3, the results from CQC modal combination are slightly more conservative than the results obtained from SRSS combination. Inspection of Tables 7-2 and 7-3, and Figure 7-24 reveals that the pile and column moments from the event-specific IRSA cases (S RSS and CQC) generally agree well with the CVIA results. In a few cases, the bending mo ments from the event-specific IRSA method are

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201 slightly unconservative in comparison to the co upled analysis results. This is due to the approximate nature of modal combination tec hniques such as SRSS and CQC. Such methods attempt to quantify maximal dynamic responses by combining individual modal responses that may maximize at different points in time during the vessel collision. Since response spectrum techniques do not account for such timing issues, the results obtained are approximate in nature. However, in practical design situations, a design DMF spectrum will be used instead of an event specific DMF spectrum. Since a design DMF spectrum is inherently more conservative than an event-specific spectrum, the use of IRSA in design situations will generally lead to conservative results, although this is not absolutely guarant eed. Additionally, using a CQC modal combination rather than an SRSS combination can further increase the like lihood of obtaining conservative design data. 7.6.2 Design-Oriented Impact Response Spectrum Analysis Demonstration To demonstrate the IRSA method as it would be used in design, the three impact scenarios considered above for validation purposes are re-a nalyzed. With the exception of the DMF spectra used, all parameters and procedures in the demo nstration analyses are th e same as those used during validation. In the dem onstration cases, design DMF spect ra (constructed using Eqns. 7-31 through 7-37) are used in place of the event-specific spectra that were used in the validation process. For each case in Table 7-1, an IRSA was conducted using the non-linear procedure outlined in Figures 7-20 and 7-21. In each case, a total of 12 eigen modes were combined together using the SRSS and CQC techniques. Comparisons of the maximum absolute bending moments for all piles or columns in the model from dynamic analyses performed using the IRSA and CVIA methods are presented in Tables 7-4 and 7-5. The IRSA values presented in Table 7-4 were modally combined using the SRSS techni que, whereas correspond ing values in Table 7-5

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202 were obtained by modal combination using the CQC technique. Tables 7-4 and 7-5, also present the total modal mass participation that wa s achieved by using 12 modes for each IRSA. As stated above, inclusion of 12 eigen modes wa s necessary to ensure that the design IRSA results were conservative with respect to CVIA results. Table 7-6 shows the modal mass participation for each of the 12 eigen modes used and the cumulative mass participation. Including 6 modes in the IRSA analyses yields 90% mass participationas is required in earthquake analysis. However, it was found that the use of 90% mass participation yielded IRSA results that were unconservative wi th respect to CVIA results. Therefore, inclusion of mode 11 (the next major modal contribution) was investigatedbringing the cumulative mass participation to between 94% and 98%. Inclusio n of mode 11, however, still yielded pile force results that were slightly unconservative with resp ect to CVIA pile force results. Inclusion of mode 12which corresponded to greater than 99% cumulative mass contributionyielded IRSA results that were conservative with respect to CVIA results. Theref ore, it is recommended that 99% modal mass participation be achieve d to ensure conservative IRSA results. Figure 7-26 shows profiles of bending moments, which represents the maximum absolute bending moment throughout the duration of the anal ysis at a given elev ation across all pier columns or piles, for both the CVIA and th e design IRSA using a CQC modal combination technique. As with the valid ation cases discussed above, co mparing values from Tables 7-4 and 7-5, results from CQC modal combination are sli ghtly more conservative than results obtained from SRSS combination. The data presented in Tables 7-4 and 7-5, and Figure 7-26 reveal that pile and column moments from the design-oriente d IRSA agree well with the CVIA results. Additionally, as

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203 expected, use of a design DMF spectrum added an extra level of conserva tism, making the IRSA results conservative with respect to CVIA results.

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204 Table 7-1 Impact energies for IRSA validation Energy Barge weight (tons) Barge velo city (knot) Impact energy (kip-ft) Low 200 1 17.71 Moderate 2030 2.5 1123 High 5920 5 13100 Table 7-2 Maximum moments for all columns and p iles for event-specific IRSA validation with SRSS combination Bending moment (kip-ft) Impact energy Bridge component Coupled impact analysis Eventspecific IRSA Percent difference IRSA total mass participation (12 modes) Column 886.28991% Low Pile 596.5528.9-11% 99.8% Column 3934526734% Moderate Pile 18631806-3% 99.9% Column 4580542518% High Pile 18951835-3% 99.9% Table 7-3 Maximum moments for all columns and p iles for event-specific IRSA validation with CQC combination Bending moment (kip-ft) Impact energy Bridge component Coupled impact analysis Eventspecific IRSA Percent difference IRSA total mass participation (12 modes) Column 886.2958.38% Low Pile 596.5568.9-5% 99.8% Column 3934536236% Moderate Pile 186319907% 99.9% Column 4580552021% High Pile 189520176% 99.9%

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205 Table 7-4 Maximum moments for all columns a nd piles for design IRSA demonstration with SRSS combination Moment (kip-ft) Impact energy Bridge component Coupled impact analysis Eventspecific IRSA Percent difference IRSA total mass participation (12 modes) Column 886.2112327% Low Pile 596.5648.69% 99.8% Column 3934581348% Moderate Pile 186319686% 99.9% Column 4580581327% High Pile 189519684% 99.9% Table 7-5 Maximum moments for all columns a nd piles for design IRSA demonstration with CQC combination Moment (kip-ft) Impact energy Bridge component Coupled impact analysis Eventspecific IRSA Percent difference IRSA total mass participation (12 modes) Column 886.2119034% Low Pile 596.5692.716% 99.8% Column 3934592151% Moderate Pile 1863215015% 99.9% Column 4580592129% High Pile 1895215013% 99.9% Table 7-6 Mass participati on by mode for design IRSA Low energy Moderate energy High energy Modes Mass Participation Cumulative Participation Mass Participation Cumulative Participation Mass Participation Cumulative Participation 1 0.0% 0.0%0.0%0.0%0.0% 0.0% 2 0.0% 0.0%0.1%0.1%0.1% 0.1% 3 60.4% 60.5%60.7%60.8%60.7% 60.8% 4 15.6% 76.1%15.5%76.3%15.5% 76.3% 5 0.0% 76.1%0.0%76.3%0.0% 76.3% 6 16.0% 92.1%16.5%92.8%16.5% 92.8% 7 0.0% 92.1%0.0%92.8%0.0% 92.8% 8 0.0% 92.1%0.0%92.8%0.0% 92.8% 9 0.0% 92.1%0.0%92.8%0.0% 92.8% 10 0.0% 92.1%0.0%92.8%0.0% 92.8% 11 2.3% 94.4%4.5%97.3%4.5% 97.3% 12 5.4% 99.8%2.6%99.9%2.6% 99.9%

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206 A {F(t)}B {u(t)}C Figure 7-1 Time history analysis of a structur e A) Structure, B) Finite element model of structure, C) Displaced shape of structural model

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207 {F(t)}A {u(t)} {F(t)}B P1(t) q1(t) P2(t) Pi(t) q2(t) qi(t) Mode 1Mode 2Mode i ...C Figure 7-2 Time-history versus modal analysis A) Finite element model of structure, B) MDOF time-history analysis of structural system, C) SDOF systems representing each mode of vibration

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208 A {1} : Mode 1 T1: period1: circular frequencyB {2} : Mode 2 T2: period2: circular frequency C {3} : Mode 3 T3: period3: circular frequency D Figure 7-3 Modal analysis A) Stru ctural model, B) First mode shape, C) Second mode shape, D) Third mode shape

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209 f(t) t fpeak A m k k SDOF dynamic system Corresponding SDOF static system uD uSf(t) fpeakB u(t) t uSuD C Figure 7-4 Dynamic magnification of single degree-of-freedom system A) Impact force history, B) Dynamic and static SDOF systems, C) Dynamic and static displacements

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210 Time (sec)Force (kips) 0 0.5 1 1.5 2 0 100 200 300 400 AStructural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 SDOF with period of 1 sec SDOF with a period of 0.1 secB Figure 7-5 Dynamic magnification f actor for a specific impact load history A) Impact load history, B) Corresponding dyna mic magnification factor

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211 Structural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 2 AStructural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 2 Design DMFB Figure 7-6 Dynamic magnification f actor A) For a range of load histories, B) With a broadbanded design spectrum

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212 Structural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Low-energy impact DMF Broad-banded design spectrum Figure 7-7 Specific dynamic magnifi cation factor for a low-energy impact vs. a broad-banded design spectrum

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213 Structural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Low energy impact High energy impact Short-period expansion Long-period expansion Figure 7-8 Evolution of the dynamic magnificati on spectrum from short to long duration loading

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214 Structural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Short-period transition point Long-period transition pointAStructural Period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 1.6 Short-period transition point Long-period transition pointB Figure 7-9 Definition of the shor t and long-period transition poi nts A) Design DMF spectrum, B) Event-specific DMF spectrum

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215 Impact LoadTime Duration of Impact Loading Period of Impact Loading (TI) Figure 7-10 Period of impact loading

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216 Kinetic energy (kip-ft)Short-period transition (sec) 1 10 100 1,000 10,000 100,000 0.01 0.1 1 Event-specific data Design expression Figure 7-11 Short-period transition point data

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217 Kinetic energy (kip-ft)Long-period transition (sec) 1 10 100 1,000 10,000 100,000 0.1 1 10 100 Event-specific data Design expression Figure 7-12 Long-period transition point data

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218 Structural Period (sec)DMF 0.010.1110100 0.01 0.1 10 1 Long-period region Short-period region Long-period transition point (TL,2.0) Short-period transition point (TS,2.0) Long-period slope (-0.95) Short-period slope (0.6) 2.0 max DMF 1.2 min short-period DMF 0.1 min long-period DMF Figure 7-13 Evolving design DMF spectrum

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219 Structural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Event-specific DMF Design DMFAStructural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Event-specific DMF Design DMFB Structural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 Event-specific DMF Design DMFC Figure 7-14 Event-specific and design DMF spect ra for varying impact energies A) 200 ton barge at 1.0 knots, B) 2030 ton barge at 2.5 knots, C) 5920 ton barge at 5.0 knots

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220 Apply static load (PBm) and conduct a static analysis to determine static displacements {uS} Conduct eigenanalysis and recover mode shapes { i} and natural circular frequencies ( i) Calculate barge-pier-soil series stiffness 1 P BY BY Sk 1 P a k Calculate peak dynamic load BY BP Bi BmP c v P B S BPm k c Calculate period of impact loading Elastic loading Bi Bm B Iv P m 2 T BY BP BiP c v Inelastic loading 2 BP BY 2 Bi BP BY Bi Bm B Ic P v 1 2 c P v P m T BY BP BiP c v Calculate DMF characteristics Short-period transition point 2 2 T v m 1300 T TI 9 0 2 Bi B I S Long-period transition point 7 0 T 100 v m T TI 1 0 2 Bi B I L Transform structural static displacements into modal coordinates S T Su M q For each mode (i) Calculate structural period i i2 T Calculate DMF DMF i = 2.0 Intermediate periods L i ST T T Short-period region S iT T 2 1 T T 0 2 DMF6 0 S i i Long-period region L iT T 1 0 T T 0 2 DMF95 0 L i i Si i Diq DMF q Magnify static modal displacement Dynamic displaced mode shapei Di Di q u Recover internal member forces (F Di ) for each mode from the displaced mode shapes Combine modal quantities using SRSS or CQC combination: Dn 2 D 1 D Du ,..., u u SRSS u Dn 2 D 1 D Du ,..., u u CQC u Dn 2 D 1 D DF ,..., F F SRSS F Dn 2 D 1 D DF ,..., F F CQC F or or Impact Response Spectrum Analysis Procedure m B in kip-sec2/in, v Bi in in/sec, and T I in sec Select barge type, and determine P BY a BY Determine impact characteristics: m B v Bi Calculate pier-soil stiffness from static analysis: k P Figure 7-15 Impact response sp ectrum analysis procedure

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221 {FS} = PBmA {uS} {FS} = PBmB Figure 7-16 Static analysis stage of IRSA A) Structural model with peak dynamic load applied statically, B) Resulting st atically displaced shape

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222 {uS} {FS} = PBmA qS1{ 1}B qS2{ 2} C qS3{ 3} D Figure 7-17 Transformation of st atic displacements into modal coordinates A) Statically displaced shape, B) Component of first m ode in static displacement, C) Component of second mode in static displacement, D) Component of third mode in static displacement

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223 Structural Period (sec) 0.010.1110DMF0.01 0.1 1 10 T3T2T1 DMF2DMF1DMF3 mode 3 mode 2 mode 1 Figure 7-18 Dynamic magnification factor as a function of structural period

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224 {uD1} = DMF1qS1{ 1}A {uD2} = DMF2qS2{ 2}B {uD3} = DMF3qS3{ 3}C {uD} = SRSS({uD1},{uD2},...)or CQC({uD1},{uD2},...) D Figure 7-19 Combination of amplified dynamic modal displacements into amplified dynamic structural displacements A) Dynamic di splaced shape of mode 1, B) Dynamic displaced shape of mode 2, C) Dynamic displaced shape of mode 3, D) Modally combined dynamic displaced shape

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225 BY BY S ) 0 (a P k Estimate barge-pier-soil series stiffness BY BP ) 0 ( Bi Bm ) 0 (P c v P B S ) 0 ( BP ) 0 (m k c Estimate peak dynamic load Apply static load ( (n) P Bm ) and conduct a static analysis to determine static displacement of the impact point ( (n) u P ) P ) 1 n ( Bm ) 1 n ( P ) n (u P k Update effective pier-soil stiffness 1 P ) n ( BY BY S ) n (k 1 P a k Update barge-pier-soil series stiffnessBY BP ) n ( Bi Bm ) n (P c v P B S ) n ( BP ) n (m k c Update peak dynamic load Bm ) 1 n ( Bm ) n ( BmP P P Calculate the incremental peak dynamic load tol PBm For each iteration (n) ... Calculate DMF characteristics no yes Load Determination Calculate period of impact loading Elastic loading Bi Bm B Iv P m 2 T Inelastic loading 2 BP BY 2 Bi BP BY Bi Bm B Ic P v 1 2 c P v P m T Select barge type, and determine P BY a BY Determine impact characteristics: m B v Bi m B in kip-sec2/in, v Bi in in/sec, and T I in sec Short-period transition point 2 2 T v m 1300 T TI 9 0 2 Bi B I S Long-period transition point 7 0 T 100 v m T TI 1 0 2 Bi B I L BY BP BiP c v BY BP BiP c v Apply static load (P Bm ) and conduct a static analysis to determine static displacement of the impact point (u P ) Intermediate periods 0 2 DMF L ST T T Short-period region 2 1 T T 0 2 DMF6 0 S ST T Long-period region 1 0 T T 0 2 DMF95 0 L LT T Figure 7-20 Nonlinear impact respon se spectrum analysis procedure

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226 ) u ( function KD ) 1 n ( ) n ( Update secant stiffness matrix Conduct eigenanalysis and recover mode shapes {(n)i} and natural circular frequencies ((n)i) Transform static structural displacements into static modal displacementsS T ) n ( S ) n (u M q For each mode (i) ... Calculate structural period i ) n ( i ) n (2 T Calculate DMF Magnify static modal displacement Si ) n ( i ) n ( Di ) n (q DMF q Recover internal member forces {F Di } for each mode from the displaced mode shapes Calculate dynamic displaced shapes i ) n ( Di ) n ( Di ) n ( q u Calculate incremental external force vector External modal force Di ) n ( ) n ( Ei (n)u K F Combine modal quantities using SRSS or CQC combination For each iteration (n) ... tol FE For every term in no yes Linearization E ) 1 n ( E ) n ( EF F F S D ) 0 (u u Initial estimate for external force vector and structure displacements Combine modal quantities using SRSS or CQC combination: or or En (n) 2 E (n) 1 E (n) E (n)F ,..., F F SRSS F En (n) 2 E (n) 1 E (n) E (n)F ,..., F F CQC F Dn (n) 2 D (n) 1 D (n) D (n)u ,..., u u SRSS u Dn (n) 2 D (n) 1 D (n) D (n)u ,..., u u CQC u Dn 2 D 1 D DF ,..., F F SRSS F Dn 2 D 1 D DF ,..., F F CQC F or Intermediate periods 0 2 DMFi ) n ( L i ) n ( ST T T Short-period region 2 1 T T 0 2 DMF6 0 S i ) n ( i ) n ( S i ) n (T T Long-period region 1 0 T T 0 2 DMF95 0 L i ) n ( i ) n ( L i ) n (T T Apply static load (P Bm ) and conduct a static analysis to determine static displacements (u S ) S E ) 0 (F F Figure 7-21 Nonlinear impact respon se spectrum analysis procedure

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227 Bow deformation (in)Force (kips) 0 5 10 15 20 0 500 1,000 1,500 2,000 2 in, 1540 kips Figure 7-22 Barge bow force-deform ation relationship for an impact on a six-foot round column

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228 Time (sec)Load (kips) 0 0.5 1 0 300 600 AStructural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 B Time (sec)Load (kips) 0 0.5 1 0 1,000 2,000 CStructural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 D Time (sec)Load (kips) 0 2 4 0 1,000 2,000 EStructural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 F Figure 7-23 Event-specific IRSA validation A) Low-energy impact load, B) Low-energy event-specific DMF, C) Moderate-energy impact load, D) Moderate-energy event-specific DMF, E) High-energy impact load, F) High-energy event-specific DMF

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229 Columns Pier Cap Piles Elev. +1.9 ft Elev. +29.3 ft Elev. +53.9 ft Elev. -62.4 ft Pile Cap AMoment (kip-ft)Elevation (ft) 0 600 1,200 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSAB Moment (kip-ft)Elevation (ft) 0 3,000 6,000 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSACMoment (kip-ft)Elevation (ft) 0 3,000 6,000 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSAD Figure 7-24 Moment results profile for the new St George Island Causeway Bridge channel pier A) Channel pier schematic, B) Low-ener gy impact, C) Moderate-energy impact, D) High-energy impact

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230 Structural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 AStructural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 B Structural period (sec)DMF 0.01 0.1 1 10 0.01 0.1 1 10 C Figure 7-25 Design-oriented IRSA demons tration A) Low-energy design DMF, B) Moderate-energy design DMF, C) High-energy design DMF

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231 Columns Pier Cap Piles Elev. +1.9 ft Elev. +29.3 ft Elev. +53.9 ft Elev. -62.4 ft Pile Cap AMoment (kip-ft)Elevation (ft) 0 600 1,200 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSAB Moment (kip-ft)Elevation (ft) 0 3,000 6,000 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSACMoment (kip-ft)Elevation (ft) 0 3,000 6,000 -80 -60 -40 -20 0 20 40 60 Pile Cap CVIA IRSAD Figure 7-26 Moment results profile for the new St George Island Causeway Bridge channel pier A) Channel pier schematic, B) Low-ener gy impact, C) Moderate-energy impact, D) High-energy impact

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232 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 Concluding Remarks Three distinct procedures for conducting dyna mic analyses of barge impacts on bridge piers have been presented in this dissertation. Two of th e methodscoupled vessel impact analysis (CVIA) and applied vessel impact load (AVIL)use time-step integration to solve the equation of motion. The other methodimpact response spectrum analysis (IRSA)is a response spectrum analysis, which combines re sults from an eigenanalysis to approximate maximum dynamic effects. The CVIA technique, which involves coupli ng a single degree-of-freedom (SDOF) barge model to a multiple degree-of-freedom (MDOF) bridge model, was developed by a research team of which the author was a member. The CVIA method was validated against both experimental data and high-resolution finite element analysis (FEA) results. Based on good agreement between CVIA results, full-scale experimental data, and high-resolution FEA results, CVIA was selected as the baseline analysis to which the two other dynamic analysis methods (AVIL and IRSA) were compared in the present research. The AVIL procedure uses barge impact character istics and force-deformation behavior of a barge in conjunction with the for ce-deformation behavior of the br idge to generate an applied load-history. Using conservation of energy and momentum, the barge and pier characteristics were correlated to load history characteristics (peak load and load duration), which in turn, were used to develop expressions for time-varying impact loads. Thes e loads are then applied to a bridge model dynamically. Several analyses we re conducted in which CVIA and AVIL analysis results were compared. Using CVIA as the base line comparison, results from the AVIL analysis method were found to agree very well with those obtained from the CVIA method.

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233 Both the CVIA and AVIL techniques are catego rized as time-history analysis procedures, which yield results for each time step in the analysis. Designers, however, are generally interested in maximum response va lues obtained from results. Thus, once a time-history analysis has been conducted, designers must scan the full time-history record for maximum response parameters. As an alternative, maximum res ponses may be approximated, without requiring a full time-history analysis, by using a re sponse spectrum analysis technique. The IRSA procedure is a response spectrum an alysis procedure spec ifically tailored to vessel impact loading. The IR SA procedure dynamically magnifies modal components of the static displacement to approximate dynamic modal displacements. Magnification of the static displacements is accomplished through the app lication of a dynamic magnification factor (DMF), which is obtained from impact DMF sp ectra. After transforming the dynamic modal response parameters into structural response pa rameters, internal structural member design forces are recovered and modally combined for use in bridge design. Generation of a large number of event-specific DMF spectra revealed that the DMF values do not exceed 2.0 for impacted bridge systems having 5-precent modal damping. Furthermore, comparison of results from the CVIA and IRSA techniques showed good agreement. Thus, it is concluded that the IRSA method is capable of adequa tely modeling barge impact events. Although AVIL equations are used as part of the IRSA method, the IRSA method is generally considered more suitable for use as a design tool. An importa nt aspect of the AVIL equations is the calculation of the effective pier stiffness at the application of loading. The general nonlinear IRSA method includes force de termination iterations in which the peak dynamic force and effective pier stiffness are updat ed to account for pier-soil model nonlinearity.

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234 The AVIL method, however, does not incorporat e nonlinear force determination iterations, but, instead requires the designer to determine a suitable effective pier stiffness. Furthermore, the IRSA has an extra degree of conservatism inherently built into the procedure. As shown in Chapter 8, empirical DMF spectrum equations used in the IRSA method produce a conservative spectrum in comparison to event-specific spectr a, but without being overly conservative. In the AVI L method, this extra level of conservatism is not included. Additionally, analyses conducted in this research have primarily focused on barge impacts that are perpendicular to the span direction of the bridge superstr ucture. However, if extended to barge impacts occurring in the span direction, it is not known whether the AVIL method will be able to adequately capture the complex effects of such an impact (e.g. torsional effects on the impacted pier). In contrast, IRSA account s for complex dynamic effects associated with longitudinal impacts due to the inclusion of a ppropriate modes in the eigenanalysis stage. A major aspect in the development of all of the dynamic methods (CVIA, AVIL, and IRSA) was the development of force-deformati on relationships for barge bows. To develop these relationships, hi gh-resolution models of hopper and ta nker barges were developed and subjected to crushing by flat-faced and round pier column impactors of va rious widths. Based on force-deformation results obtained from the high -resolution models, it has been concluded that barge bow crushing behavior may be adequately and conservatively modeled using an elastic, perfectly-plastic representation. This conclusi on differs substantially from the current AASHTO relationship in which impact fo rces continue increasing with additional deformation beyond the transition from elastic to inelastic behavior. Additionally, barge bow yield loads have been found to be dependent on the shape and size of th e pier column, and not a function of the barge bow width, as the AASHTO provisions prescribe.

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235 8.2 Recommendations 8.2.1 Recommendations for Bridge Design Elastic, perfectly-plastic barge bow model: It is recommended that barge bow forcedeformation relationships be represented using the elastic, perfectly-plastic model that was presented in Chapter 4. The adequacy of usi ng an elastic, perfectly-plastic relationship is based upon results obtained from high-resolution static finite el ement crush analyses of the two most common types of barges used in the United States. Dependence of barge bow yield force and de formation on pier width and geometry: Based on results obtained from the high-resolu tion finite element analyses, it has been found that the barge bow yield load is depende nt upon the shape and width of the impacted pier column. Additionally, it has been found that the barge bow deformation at yield is dependent upon the geometry (flat-faced versus round) of the impacted pier column. It is important to note that although AASHTO prescrib es force-deformation relationships that are dependent on barge width (i.e. inclusion of the RB factor in the AASHTO provisions), no such correlation was observed in this rese arch. Therefore, it is recommended that the barge force-deformation relationship be form ulated as a function of pier width and geometry, as presented in Chapter 4, without dependence on barge bow width. Use of dynamic analysis technique s in lieu of static techniques: Both experimental data and results from finite element analyses indicat e that inertial-resistances and inertial-forces significantly affect bridge response during a vessel collision. The presence of a superstructure produces large in ertial forces, which may in turn generate significant force amplification effects in pier columns. Thus it is recommended that dynamic analyses be used to model barge impact events. o If analysis software in which CVIA is implemented is available, it is recommended that CVIA techniques be used to dynamically analyze a bridge structure subjected to barge impact. o If software in which CVIA is implemented is not available, it is recommended that the IRSA technique be used to c onduct a barge impact analysis on a bridge model. o A major aspect of IRSA techniques is the calculation of peak dynamic load and period of loading using the AVIL equations However, for several reasons as stated above, the IRSA technique is more accurate than the AVIL method. Therefore, the AVIL method, as a st and-alone method, is not generally recommended for use in design. 8.2.2 Recommendations for Future Research Improve probability of collapse: The AASHTO vessel collision specifications use a probabilistic approach to bridge design, which requires designers to determine the return period of collapse for bridge structures spanni ng navigable waterways. One aspect of this

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236 probabilistic approach is calculation of the proba bility of collapse (PC) term. PC is the probability that a bridge struct ure will collapse once a bridge component has been struck by a vessel. This probability is a function of many factors: im pact vessel size and shape, vessel speed and displacement, ultimate pier strength, etc. However, the AASHTO PC estimation equations were derived from research on damage observed between ship-to-ship collisions at sea. Given that barge-to-bridge collisions ar e different from ship-to-ship collisions, the accuracy of the AASHT O PC term should be investigated. Investigate barge impacts longitudinal to a bridge: In the current research study, analysis of barge impacts on bridge structures has been restricted to impacts transverse to the bridge span direction. However, barg e-to-bridge collisions do not always occur transverse to the bridge. Additionally, for design, the AASHTO provi sions prescribe that fifty-percent of the design impact load be applied in the direction longitudinal to the bridge. AASHTO, however, prescribes static lo ad cases for impact loading. Therefore, it is recommended that further investigation of the longitudinal impact condition be studied using dynamic techniques. Expand barge bow force-deformation database: In this study, the barge bow force-deformation results from high-resolution crush simulations indicated that an elastic, perfectly-plastic model is adequate to simu late barge bow crushing behavior. Further investigation into th e post-yield behavior of barge bow behavior should be conducted using experimental testing so that the resu lts obtained from high-resolution FEA can be confirmed.

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237 LIST OF REFERENCES American Association of State Highway a nd Transportation Offici als (AASHTO). (1991). Guide Specification and Commentary for Vessel Collision Design of Highway Bridges, Washington, D.C. AASHTO. (1994). LRFD Bridge Design Specifications and Commentary, Washington, D.C. American Society of Civ il Engineers (ASCE). (2005). Minimum Design Loads for Buildings and Other Structures, Reston, VA. Arroyo, J.R., Ebeling, R.M., and Barker, B.C. (2003). Analysis of Impact Loads from FullScale Low-Velocity, Controlled Barge Impact Experiments, December, 1998. United States Army Corps of Engin eers Report ERDC/ITL TR-03-3, April. Arroyo, J.R., and Ebeling, R.M. (2004). A Numerical Method for Computing Barge Impact Forces Based on Ultimate Strength of the Lashings Between Barges United States Army Corps of Engineers Report ERDC/ITL TR-04-2, August. Bridge Software Institute (BSI). (2005). FB-MultiPier Users Manual. Bridge Software Institute, University of Florida, Gainesville, FL. Chopra, A.K. (2007). Dynamics of Structures: Theory and Applications to Earthquake Engineering, Pearson Prentice-Hall, Englewood Cliffs, NJ. Consolazio, G. R., Cook, R. A., and Lehr, G. B. (2002). Barge Impact Testing of the St. George Island Causeway Bridge Phas e I : Feasibility Study. University of Florida Engineering and Industrial Experiment Station St ructures Research Report No. 783, University of Florida, Gainesville, FL, January. Consolazio, G.R., and Cowan, D.R. (2005). Numer ically Efficient Dynamic Analysis of Barge Collisions with Bridge Piers. ASCE Journal of Structural Engineering, 131(8), 12561266. Consolazio, G. R., Hendrix, J. L., McVay, M. C., Williams, M. E., and Bollman, H. T. (2004). Prediction of Pier Response to Barge Imp acts Using Design-Oriented Dynamic Finite Element Analysis. Transportation Research Record, Transportation Research Board, Washington, D.C., 1868, 177-189. Consolazio, G.R., Cook, R.A., Mc Vay, M.C., Cowan, D.R., Biggs, A.E., Bui, L. (2006). Barge Impact Testing of the St. George Island Causeway Bridge. University of Florida Engineering and Industrial Experiment Sta tion Structures Research Report No. 26868, University of Florida, Gainesville, FL, March. Davidson, M.T. (2007). Simplified Dynamic Barg e Collision Analysis for Bridge Pier Design. Masters Thesis, Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL.

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238 Federal Emergency Management Agency (FEMA). (2003). NEHRP Recommended Provisions for Seismic Regulations for Ne w Buildings and Other Structures, Washington, D.C. Goble, G., Schulz, J., and Commander, B. ( 1990). Lock and Dam #26 Field Test. Report for the Army Corps of Engineers. Bridge Diagnostics Inc., Boulder, CO. Hendrix, J.L. (2003). Dynamic Analysis Techni ques for Quantifying Bridge Pier Response to Barge Impact Loads. Masters Thesis, Depart ment of Civil and Coastal Engineering, University of Florida, Gainesville, FL. Knott, M., and Prucz, Z. (2000). Vessel Collision Design of Br idges: Bridge Engineering Handbook, CRC Press LLC. Larsen, O.D. (1993). Ship Collision with Bridge s: The Interaction Betw een Vessel Traffic and Bridge Structures. IABSE Structural Engineering Document 4, IASBE-AIPC-IVBH, Zrich, Switzerland. Liu, C., and Wang, T.L. (2001). Statewide Vessel Collision Design for Bridges. Journal of Bridge Engineering, May/June, 213-219. Livermore Software Technology Corporation (LSTC). (2003). LS-DYNA Keyword Users Manual: Version 970. Livermore Software Technology Corporation, Livermore, CA. Manuel, L., Kallivokas, L.F., Williamson, E.B., Bomba, M., Berlin, K.B., Cryer, A., and Henderson, W.R. (2006). A Proba bilistic Analysis of the Fre quency of Bridge Collapses due to Vessel Impact. University of Texas Center fo r Transportation Research Report No. 0-4650-1, University of Texas, Austin, TX. McVay, M.C., Wasman, S.J., and Bullock, P.J. (2005). St. George Geot echnical Investigation of Vessel Pier Impact. University of Florida Engineer ing and Industrial Experiment Station, University of Florida, Gainesville, FL. Meier-Drnberg, K.E. (1983). Ship Collisions, Safety Zones, and Loading Assumptions for Structures in Inland Waterways. Verein De utscher Ingenieure (Association of German Engineers) Report No. 496. Minorsky, V.U. (1959). An Analysis of Ship Collis ions with Reference to Protection of Nuclear Power Plants. Journal of Ship Research, 3, 1-4. National Transportation Safety Board (NTSB). ( 2004). U.S. Towboat Robert Y. Love Allision With Interstate 40 Highway Bridge Near Webbers Falls, Oklahoma, May 26, 2002. National Transportation Safety Board, Washington D.C. Nguyen, H. (1993). Buckling. Marine Safety Center, U.S. De partment of Homeland Security, United States Coast Guard. Patev, R.C. (1999). Full-Scale Barge Impact Experiments. Transportation Research Board Circular 491, Transportation Research Board, Washington D.C..

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239 Patev, R.C., and Barker, B.C. (2003). Prototype Barge Impact Experiments, Allegheny Lock and Dam 2, Pittsburgh, Pennsylvania. United States Army Corp s of Engineers Report ERDC/ITL TR-03-2, January. Tedesco, J.W., McDougal, W.G., and Ross, C.A. (1999). Structural Dynamics: Theory and Applications, Addison Wesley Longman, Inc., Menlo Park, CA. Wilson, J. (2003). Allison Involving the M/V Br ownwater V and the Queen Isabella Causeway Bridge. U.S. Department of Homeland Security, United States Coast Guard. Woisin, G. (1976). The Collision Tests of the GKSS. Jahrbuch der Schiffbautechnischen Gesellschaft, 70, 465-487. Yuan, P. (2005). Modeling, Simulation and Analysis of Multi-Barge Flotillas Impacting Bridge Piers. Doctoral Dissertation, College of Engineering, University of Kentucky, Lexington, KY. Yuan, P., Harik, I.E., and Davidson, M.T. (D raft) Multi-Barge Flot illa Impact Forces on Bridges, Kentucky Transportation Center. Research Report No. KTC-05 /SPR261-031F, University of Kentucky, Lexington, KY.

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240 BIOGRAPHICAL SKETCH The author was born in Akron, Ohio, in 1979. In January, 1998, he began attending the University of Florida, where he later obtaine d the degree of Bachelor of Science in civil engineering in December 2002. After attending gradua te school at the University of Florida, he received the degree of Master of Engineering in December 2004. In December, 2007, the author anticipates receiving the degree of Doctor of Philosophy from the University of Florida. Upon graduation, the author plans to procure a position as an engineer in training at Finley Engineering Group, located in Tallahassee, FL.


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