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Simplified Dynamic Barge Collision Analysis for Bridge Pier Design

Permanent Link: http://ufdc.ufl.edu/UFE0021472/00001

Material Information

Title: Simplified Dynamic Barge Collision Analysis for Bridge Pier Design
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Davidson, Michael T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: barge, bridge, collision, design
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The American Association of State and Highway Transportation Officials barge impact provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to determine structural demand on bridge piers. However, conclusions drawn from experimental full scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand with consideration of additional forces generated from dynamic effects. Static quantification of bridge pier demand due to barge impact ignores mass related inertial forces generated by the superstructure which can amplify restraint of underlying pier columns. An algorithm for efficiently performing coupled nonlinear dynamic barge impact analysis on simplified bridge structure soil finite element models is presented in this thesis. The term ?coupled? indicates the impact of a finite element bridge model and a respective single degree of freedom barge model traveling at a specified initial velocity with a specified force deformation relationship. Coupled analysis is validated using experimental data. Also, results from simplified and full resolution analyses are compared for several cases to illustrate robustness of the algorithm for various barge impact energies and pier types. Simplified coupled dynamic analysis is shown to accurately capture dynamic forces and amplification effects.
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 Michael T Davidson.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Consolazio, Gary R.
Local: Co-adviser: Hoit, Marc I.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021472:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021472/00001

Material Information

Title: Simplified Dynamic Barge Collision Analysis for Bridge Pier Design
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Davidson, Michael T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: barge, bridge, collision, design
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The American Association of State and Highway Transportation Officials barge impact provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to determine structural demand on bridge piers. However, conclusions drawn from experimental full scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand with consideration of additional forces generated from dynamic effects. Static quantification of bridge pier demand due to barge impact ignores mass related inertial forces generated by the superstructure which can amplify restraint of underlying pier columns. An algorithm for efficiently performing coupled nonlinear dynamic barge impact analysis on simplified bridge structure soil finite element models is presented in this thesis. The term ?coupled? indicates the impact of a finite element bridge model and a respective single degree of freedom barge model traveling at a specified initial velocity with a specified force deformation relationship. Coupled analysis is validated using experimental data. Also, results from simplified and full resolution analyses are compared for several cases to illustrate robustness of the algorithm for various barge impact energies and pier types. Simplified coupled dynamic analysis is shown to accurately capture dynamic forces and amplification effects.
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 Michael T Davidson.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Consolazio, Gary R.
Local: Co-adviser: Hoit, Marc I.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021472:00001


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SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN


By

MICHAEL THOMAS DAVIDSON
























A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007



























2007 Michael Thomas Davidson



























To my wife, Kiristen









ACKNOWLEDGMENTS

This material is based on work supported under a National Science Foundation Graduate

Research Fellowship. However, this thesis would not have been completed without the support

of several individuals. First, the insight and guidance of Dr. Gary Consolazio has proven

invaluable. His willingness to invest time in helping graduate students become effective analysts

and independent researchers will undoubtedly garner countless and vast returns. The author also

wishes to thank Dr. Marc Hoit, Dr. Petros Christou, and Dr. Jae Chung for their assistance with

extending the capabilities of FB-MultiPier. A graduate student deserving of many thanks and

much future success is David Cowan, whose brilliance seems to be limitless. Finally, the author

wishes to thank his wife Kiristen, his family, and his friends for their enduring love and

fellowship.









TABLE OF CONTENTS

A CK N O W LED G M EN T S ............................................... .......................... ........................ 4

L IS T O F T A B L E S ........... ................... ...................................... ................ .. 7

LIST OF FIGURES .................................. .. ..... ..... ................. .8

A B S T R A C T ............ ................... ............................................................ 12

CHAPTER

1 INTRODUCTION ............................... ... ...... ... ................... 13

2 L ITE R A TU R E R E V IE W ......................................................................... ................... 17

2 .1 E xperim mental R research ........................................................................ .................. 17
2.2 A analytical R research ............................................ .. .. ............. .... ... ... 18

3 COUPLED BARGE COLLISION ANALYSIS..................... ...... ............... 20

3 .1 Intro du action ........................................................................................... ............... 2 0
3.1.1 Barge Loading and Unloading Behavior ............................................... 20
3.1.2 Coupled A analysis A lgorithm ................................................. .................21
3.1.3 Use of Experimental Data for Coupled Analysis Validation......................21
3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2 .....................22
3.3 Software Selection and M odel Development ................................... .................22
3.3.1 Coupled Analysis Module Parameters......................... ....................24
3.3.2 Accounting for Payload Sliding During Impact Testing ...........................25
3.4 Comparison of Analytical and Experimental Data.................................................26
3 .4 .1 C ase 1 ..........................................................................2 6
3 .4 .2 C ase 2 ............................................ ......... ................. 27

4 SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS....................................37

4 .1 O v erview ................. .... ................................ .............. ................. 37
4.2 Linearized Barge Force-Crush Relationship..................................... ............... 37
4.3 Reduction of the Bridge M odel ............................. ...................... ...................38
4.3.1 Uncoupled Condensed Stiffness M atrix ....................................... .......... 39
4.3.2 Lum ped M ass A pproxim action ........................................... .....................41
4.4 Multiple-Pier Coupled Analysis Simplification Algorithm....................................42









5 SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES ............................47

5 .1 In tro du ctio n .................. ..... ...... ... ...................................... ..... ............... 4 7
5.2 Geographical Information, Structural Configuration, and Impact Conditions ............48
5.2 .1 C ase 3 ........................................................................ ... .. .... 48
5.2 .2 C ase 4 ............................................ ......... ................. 49
5.2.3 Case 5....................................................... ..... ... ........ ............ 50
5.3 Comparison of Simplified and Full-Resolution Results............................................51
5.4 Conclusions from Simplified-Coupled Analysis Demonstrations.............................52
5.5 Dynamic Amplification of the Impacted Pier Column Internal Forces ..................53

6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ............62

6 .1 C o n clu sio n s ............................ ................. ................ ................ 6 2
6.2 Recommendations for Future Research........................ ................... ............... 63

APPENDIX

A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA.........................64

B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS ....................72

C SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT ........................................76

D ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS ............................95

R E F E R E N C E S ............................................................................... 100

B IO G R A PH IC A L SK E T C H ......... ................. ............................................. .......................... 102









LIST OF TABLES


Table page

1-1 Case descriptions: use, configuration, and impact data..................................................16









LIST OF FIGURES


Figure page

3-1 Coupling between barge and bridge (after Consolazio and Cowan 2005) ......................29

3-2 Stages of barge crush (after Consolazio and Cowan 2005).............................................30

3-3 Structural configurations analyzed (not to relative scale) ................ ..........................31

3-4 SDF barge force-crush relationship derived from experimental and analytical data .........32

3-5 Sliding criterion between payload and barge ............................... ................................. 33

3-6 Comparison of Case 1 coupled analysis output and P1T4 experimental data...................34

3-7 Comparison of Case 2 coupled analysis output and B3T4 experimental data...................35

3-8 Comparison of Case 2 coupled analysis output and B3T4 experimental data:
Im p u lse ........ ........ .. ......... ................. ................................................3 6

4-1 Derived and AASHTO SDF barge force-crush relationships
(unloading curves not show n) ............................ ........... ................................ .............43

4-2 Plan view of multiple pier numerical model and location of uncoupled springs
in tw o-span single-pier m odel.......................... .............. ................. ............... 44

4-3 Structural configuration analyzed in Case 3 ........................................... ............... 45

4-4 Plan view of multiple pier numerical model and location of lumped masses
in tw o-span single-pier m odel........................................................................ 46

5-1 Structural configuration analyzed in Case 4 ........................................... ............... 55

5-2 Structural configuration analyzed in Case 5 ........................................... ............... 56

5-3 Comparison of Case 3 simplified and full-resolution coupled analyses............................57

5-4 Comparison of Case 4 simplified and full-resolution coupled analyses............................58

5-5 Comparison of Case 5 simplified and full-resolution coupled analyses............................59

5-6 Time computation comparison of coupled analyses ......................................................60

5-7 Comparison of demonstration case simplified, full-resolution, and static analyses..........61

A-1 Analytical output comparison to experimental P1T4 barge impact data.........................65

A-2 Analytical output comparison to experimental P1T5 barge impact data.........................66









A-3 Analytical output comparison to experimental P1T6 barge impact data.........................67

A-4 Analytical output comparison to experimental P1T7 barge impact data.........................68

A-5 Analytical output comparison to experimental B3T2 barge impact data ........................69

A-6 Analytical output comparison to experimental B3T3 barge impact data ........................70

A-7 Analytical output comparison to experimental B3T4 barge impact data ........................71

C-1 Case 3 AASHTO curve coupled analysis output comparison at impact location..............77

C-2 Case 3 AASHTO curve coupled analysis output comparison at pier column top.............78

C-3 Case 3 AASHTO curve coupled analysis output comparison at pile head.....................79

C-4 Case 4 AASHTO curve coupled analysis output comparison at impact location..............80

C-5 Case 4 AASHTO curve coupled analysis output comparison at pier column top .............81

C-6 Case 4 AASHTO curve coupled analysis output comparison at pile head.....................82

C-7 Case 5 AASHTO curve coupled analysis output comparison at impact location..............83

C-8 Case 5 AASHTO curve coupled analysis output comparison at pier column top .............84

C-9 Case 5 AASHTO curve coupled analysis output comparison at pile head.....................85

C-10 Case 3 bilinear curve coupled analysis output comparison at impact location.................. 86

C-11 Case 3 bilinear curve coupled analysis output comparison at pier column top .................87

C-12 Case 3 bilinear curve coupled analysis output comparison at pile head..........................88

C-13 Case 4 bilinear curve coupled analysis output comparison at impact location.................. 89

C-14 Case 4 bilinear curve coupled analysis output comparison at pier column top ...............90

C-15 Case 4 bilinear curve coupled analysis output comparison at pile head..........................91

C-16 Case 5 bilinear curve coupled analysis output comparison at impact location..................92

C-17 Case 5 bilinear curve coupled analysis output comparison at pier column top ...............93

C-18 Case 5 bilinear curve coupled analysis output comparison at pile head............................94









LIST OF ABBREVIATIONS

L or L appended to symbol, indicates symbol exclusivity to left-flanking structure

R or R appended to symbol, indicates symbol exclusivity to right-flanking structure

[F] flexibility matrix

[Kcondensed condensed stiffness matrix

Kcouplng off-diagonal (coupling) stiffness term

KA translational stiffness term

K, plan-view rotational stiffness term

M"nt unit moment

mH mass of half-span of superstructure

mb mass of barge

mp mass of payload

u0 initial sliding velocity of payload

V""t unit shear force

Vcoupling shear due to coupled stiffness and plan-view rotation

VA shear due to translational stiffness and translation

Wp weight of payload

A translation

AM translation due to unit moment

A, translation due to unit shear

p static coefficient of friction between payload and barge









0 plan-view rotation

OM plan-view rotation due to unit moment

O6 plan-view rotation due to unit shear









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN

By

Michael Thomas Davidson

August 2007

Chair: Gary R. Consolazio
Cochair: Marc I. Hoit
Major: Civil Engineering

The American Association of State and Highway Transportation Officials barge impact

provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to

determine structural demand on bridge piers. However, conclusions drawn from experimental

full-scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand

with consideration of additional forces generated from dynamic effects. Static quantification of

bridge pier demand due to barge impact ignores mass related inertial forces generated by the

superstructure which can amplify restraint of underlying pier columns.

An algorithm for efficiently performing coupled nonlinear dynamic barge impact analysis

on simplified bridge structure-soil finite element models is presented in this thesis. The term

"coupled" indicates the impact of a finite element bridge model and a respective single

degree-of-freedom barge model traveling at a specified initial velocity with a specified

force-deformation relationship. Coupled analysis is validated using experimental data. Also,

results from simplified and full-resolution analyses are compared for several cases to illustrate

robustness of the algorithm for various barge impact energies and pier types. Simplified coupled

dynamic analysis is shown to accurately capture dynamic forces and amplification effects.









CHAPTER 1
INTRODUCTION

Potential loss of life and detrimental economic consequences due to bridge failure from

waterway vessel collision have been realized numerous times throughout modern history.

Catastrophic bridge failure events due to vessel collision, which occur approximately once a year

worldwide (Larsen 1993), led to the development of bridge design specifications for vessel

collision. The American Association of State and Highway Transportation Officials (AASHTO)

Guide Specification and Commentary for Vessel Collision Design ofHighway Bridges is used

along with characteristics of a given waterway and the accompanying waterway traffic to

determine static design loads, which are applied to respective piers for impact design purposes

(AASHTO 1991). Even though the AASHTO specifications are used for bridge pier design due

to ship and barge collision, limited barge impact data was available for use in their development.

In April 2004, Consolazio et al. (2006) conducted full-scale experimental barge impact

testing on bridge piers of the Old St. George Island Causeway Bridge located in Apalachicola,

Florida. Key findings from the experiments that are pertinent to the research presented in this

thesis include:

Inertial forces due to acceleration of bridge component masses can contribute
significantly to overall pier response during a collision event;
Significant portions of the impact load can transfer (or "shed") into the superstructure;
and,
Superstructure resistance is comprised of displacement-dependent and mass-dependent
(inertial) forces. Inertial forces can produce a momentary increase in pier restraint during
initial impact stages, and amplify structural demand on pier columns.
Restraint of a bridge pier due to acceleration of the mass of the overlying superstructure, and the

corresponding amplification of forces developed in the pier columns during initial stages of

barge collision events, are not addressed in current static design procedures. In contrast,

dynamic time-history analysis of bridges inherently accounts for such amplification effects.









However, due to the unique characteristics of each bridge, impact load time-histories vary from

bridge to bridge. Coupled dynamic analysis addresses this issue by employing a single

degree-of-freedom (SDF) barge mass, impact velocity, and vessel force-crush relationship to

simulate barge impact at a specified bridge pier location. This method enables efficient

time-history analysis that yields time-varying barge collision load and bridge response data

specific to each bridge structure. To validate the procedure, coupled analysis is performed and

compared with experimental data for single-pier and multiple-pier cases. A summary of all

analysis cases presented in this thesis is given in Table 1-1.

However, coupled full-resolution bridge finite element (FE) models are cumbersome to

analyze dynamically and time-history analysis of models of such size is not common in current

practice. To facilitate use of coupled analysis in design settings, simplifying modifications are

made to the barge and bridge structural models subject to impact. Specifically, to alleviate the

onus of developing an appropriate barge force-crush relationship for each of the possible barge

types, a simplified crush curve that is in accordance with current AASHTO design standards is

employed. Second, an algorithm is presented which incorporates coupled analysis but reduces a

multiple-pier model to essentially a pseudo-single pier model (with adjacent spans, springs, and

lumped masses) thereby significantly reducing required analysis time. Simplified-coupled

dynamic barge impact analysis is performed and compared to results from full-resolution models

for a range of bridge and collision configurations. In comparison to full bridge model coupled

time-history analysis, results from respective simplified models are sufficiently accurate for

design purposes. Comparisons are also made between static and dynamic analysis predictions of

bridge pier structural demand for each case. By employing coupled analysis with a simplified

crush curve and simplified bridge structural model, design-oriented software is produced that can









efficiently quantify collision induced bridge pier demand, including capture of dynamic

amplification effects.










Table. 1-1. Case descriptions: use, configuration, and impact data.
Barge impact parameters
Case Use a No. Piers Spans Weight Velocity Energy
1 V 1 0 5.37 MN (604 T) 1.33 m/s (2.59 knots) 0.484 MN-m (357 kip-ft)
2 V 4 3 3.06 MN (344 T) 0.787 m/s (1.53 knots) 0.097 MN-m (71.3 kip-ft)
3 U/D 5 4 1.78 MN (200 T) 1.03 m/s (2.00 knots) 0.096 MN-m (70.9 kip-ft)
4 D 5 4 18.0 MN (2020 T) 1.54m/s (3.00knots) 2.18MN-m (1610kip-ft)
5 D 5 4 68.7 MN (7720 T) 3.63 m/s (7.00 knots) 46.0 MN-m (34000 kip-ft)
aV = Validation; U = Uncoupled Condensed Stiffness Calculation; D = Demonstration









CHAPTER 2
LITERATURE REVIEW

2.1 Experimental Research

In 1983, Meier-Dornberg conducted reduced scale impact tests on barge bows using a

pendulum impact hammer. Static crush tests were also performed on reduced scale barge bows.

Results from this study were used to develop relationships between kinetic energy, barge bow

crush depth, and static impact force. These relationships comprise a major portion of the

collision-force calculation procedure adopted in the AASHTO specifications (1991). However,

this research did not address phenomena such as bridge superstructure effects and dynamic

amplification, nor did the tests involve pier or bridge response.

During this same time and afterward, full-scale experimental barge collision tests were

conducted in connection with the U.S. Army Corps of Engineers (USACE). In 1989, lock gate

impact tests were performed with a nine-barge flotilla traveling at low velocities at Lock and

Dam 26 near Alton, Illinois (Goble et al. 1990). In 1997, four-barge flotilla impact tests were

conducted on concrete lock walls at Old Lock and Dam 2, near Pittsburgh, Pennsylvania (Patev

et al. 2003). Additional lock wall tests were conducted with a fifteen-barge flotilla in 1998 at the

Robert C. Byrd Lock and Dam in West Virginia (Arroyo et al. 2003). All of these tests were

performed on lock walls and lock gates, which produce fundamentally different structural

responses to collision loading in comparison to that of bridge piers.

The impact testing (Consolazio et al. 2006) of the old St. George Island Bridge,

constructed in the 1960s, constitutes the only experimental research that explicitly measured

barge impact forces on bridge piers using full-scale tests. The experiments were divided into

three series of impact tests using a single barge and various pier/bridge structural configurations.

The first series (termed the P1 series) consisted of eight impacts on a single, stiff channel pier









(termed Pier 1-S) by a loaded barge with an impact weight of 5.37 MN (604 T) and impact

velocities approaching 1.8 m/s (3.5 knots). The second series of tests (termed the B3 series)

consisted of four impacts on a multi-span, multi-pier partial bridge structure by an empty barge

with an impact weight of 3.06 MN (344 T) and impact velocities approaching 0.78 m/s

(1.5 knots). The third series (termed the P3 series) consisted of three impacts on a single,

flexible pier (termed Pier 3-S) by an empty barge with an impact weight of 3.06 MN (344 T) and

impact velocities approaching 0.95 m/s (1.8 knots). These tests form an important dataset for

validating barge collision analysis methods.

2.2 Analytical Research

Development and analysis of very high-resolution contact-impact FE models (those with

tens to hundreds of thousands of elements) that simulate nonlinear dynamic barge impact on

bridge piers have been feasible as a research tool for approximately a decade. In preparation for

the full-scale St. George Island experimental barge impact testing, high-resolution FE pier

models were developed to determine appropriate experimental conditions with respect to barge

impact velocity and safety (Consolazio et al. 2002). Reanalysis of the models using

experimental data complimented the research findings from the experimental program

(Consolazio et al. 2006).

High-resolution FE models of single-barges and multi-barge flotillas were analyzed when

pier columns of various shape and dimension were subject to a variety of barge impact

simulations (Yuan 2005). These analytical results were used to develop a set of empirical

formulas for barge impact force quantification as an improvement to the current static design

procedures. Also, high-resolution FE single-barge models were developed and subjected to

quasi-static loading by various stiff impactors in an effort to better quantify barge force-crush

relationships (Consolazio and Cowan 2003).









As an alternative to very high resolution contact-impact FE analysis, coupled barge-pier

analysis was developed (Consolazio et al. 2004a, Consolazio et al. 2004b). Coupled analysis

simulates a SDF barge model (with specified mass, velocity, and force-crush relationship)

colliding with a multiple degree-of-freedom (MDF) bridge-pier-soil model. The coupled

analysis required the use of a barge force-crush relationship, which was developed for a common

barge type using high-resolution FE models. The force-crush curves encompass loading and

unloading behavior derived from quasi-static cyclic loading (Consolazio and Cowan 2005).









CHAPTER 3
COUPLED BARGE COLLISION ANALYSIS

3.1 Introduction

Within the context of coupled analysis, the term "coupled" refers to the use of a shared

contact force between the barge and impacted bridge structure (Fig. 3-1). The impacting barge is

assigned a mass, initial velocity, and bow force-crush relationship. Traveling at the prescribed

initial velocity, the barge impacts a specified location on the bridge structure and generates a

time-varying impact force in accordance with the force-crush relationship of the barge and the

relative displacements of the barge and bridge model at the impact location. The barge is

represented by a SDF model, and the pier structural configurations and soil parameters of the

impacted bridge structure constitute a MDF model. The MDF pier-soil model, subject to the

shared time-varying impact force, displaces, develops internal forces, and interacts with the SDF

barge model through the shared impact force during the analysis. Hence, coupled analysis

automatically generates the barge impact load time-history specific to each bridge structural

configuration and impacting barge type. This overcomes the challenge of pre-quantifying the

time-varying barge impact load as a necessary component of time-history analysis.

3.1.1 Barge Loading and Unloading Behavior

Barge behavior is represented by a force-crush relationship, consisting of a loading curve,

unloading curves, and a specified yield point (Fig. 3-2). The yield point represents the crush

depth beyond which plastic deformations occur. Any subsequent unloading beyond this point is

determined according to the specified unloading curves. Until the crush depth corresponding to

yield is reached, loading and unloading occurs elastically along the specified curve (Fig. 3-2A).

A series of unloading curves represent the unloading behavior at various attained maximum

crush depths (Fig. 3-2B). After unloading, if the barge is no longer in contact with the pier, no









impact force is generated (Fig. 3-2C). Alternatively, if reloading occurs (Fig. 3-2D), it is

assumed to occur along the previously traveled unloading curve. Plastic deformation subsequent

to complete reloading occurs along the originally specified loading curve (Fig. 3-2D).

Additional details of this model are given in Consolazio and Cowan (2005).

3.1.2 Coupled Analysis Algorithm

Algorithmically, the coupled analysis procedure involves a SDF barge code interacting

with a separate nonlinear dynamic pier-soil analysis code at a specified node of the MDF

pier-soil model. Specifically, coupled analysis utilizes an explicit time-step barge impact force

determination procedure and links the output, the resulting impact force, with a respective

numerical MDF pier-soil model analysis code (Hendrix 2003). The pier-soil analysis code then

responds to the impact force by generating iterative displacements and forces throughout the

MDF model.

3.1.3 Use of Experimental Data for Coupled Analysis Validation

Coupled analysis was previously developed and demonstrated as a proof-of-concept

using analytical data (Consolazio and Cowan 2005). Output from very high-resolution FE

models consisting of a MDF impacting barge and a MDF impacted pier were compared to output

obtained from coupled analysis of a SDF barge and MDF pier model. At present, experimental

data is now available for validation of the coupled analysis procedure. Using data from the

full-scale barge impact experiments (Consolazio et al. 2006), validation of the coupled analysis

procedure is carried out in four stages: select appropriate pier structures from the experimental

dataset; develop respective models in a nonlinear dynamic finite element analysis (NDFEA) code

capable of conducting coupled analysis; analyze the models using respective barge impact

conditions and coupled analysis; and, compare time-history results from the coupled analysis to

those obtained experimentally.









3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2

Data was collected more extensively from Pier 1-S than from any other pier in the 2004

full-scale experimental test set (Consolazio et al. 2006). Furthermore, a single pier is

representative of the type of structure often used in static design procedures for barge collision

analysis (Knott and Prucz 2000). Hence, a single pier (Pier 1-S) was selected for coupled

analysis validation using experimental data (Fig. 3-3A). Of the eight experimental tests

conducted on Pier 1-S, the fourth test (termed P1T4) consisted of a head-on impact at an

undamaged portion of the barge bow, as would be assumed in bridge design. Test P1T4, with

velocity and impact weight as specified in Table 1-1, was selected for Case 1.

In addition to validating the coupled analysis procedure for a single-pier, data from the

partial bridge (B3 series) tests were employed for validation purposes. Regarding impact

conditions used for validation, the fourth test (termed B3T4) generated the largest pier response

among the B3 test series. Hence, test B3T4, with velocity and impact weight as specified in

Table 1-1, was selected for Case 2 (Fig. 3-3B).

3.3 Software Selection and Model Development

Coupled analysis was previously implemented in the commercial pier analysis software,

FB-Pier (2003), and was shown to produce force and displacement time-histories in agreement

with those obtained from high-resolution contact-impact FE pier-soil model simulations.

Subsequent to implementation of the coupled analysis procedure in FB-Pier, an enhanced

package called FB-MultiPier (2007) was released. FB-MultiPier possesses the same analysis

capabilities as FB-Pier (including coupled analysis) but also has the ability to analyze bridge

structures containing superstructure elements. Therefore, FB-MultiPier was selected for all

model development and analysis conducted in this study.









FB-MultiPier employs fiber-based frame elements for piles, pier columns, and pier caps;

flat shell elements for pile caps; beam elements, based on gross section properties, for

superstructure spans; and, distributed nonlinear springs to represent soil stiffness. Transfer

beams transmit load from bearings, for which the stiffness and location are user-specified, to the

superstructure elements. FB-MultiPier permits Rayleigh damping, which was applied to all

structural elements in the models used for this study such that approximately 5% of critical

damping was achieved over the first five natural modes of vibration.

FB-MultiPier allows either linear elastic or material-nonlinear analysis of structural

elements. Linear elastic analysis was selected for all structural (non-soil) element components of

models used in this study. This approach was taken because the 2004 full-scale barge impact

experiments were non-destructive (Consolazio et al. 2006) and post-test inspection of the pier

structures subjected to collision loading indicated that the structural components had remained

largely in the elastic range.

Structural models of Case 1 and Case 2 (Fig. 3-3A and Fig. 3-3B, respectively) were

developed from original construction drawings and direct site investigation measurements. The

Case 2 structural model was limited to four piers, with springs representing the stiffness

contributions of piers beyond Pier 5-S (Fig. 3-3B), as contribution to structural response from

these piers was expectedly small (Consolazio et al. 2006). The soil model spring system for

Case 1 was developed based on boring logs and dynamic soil properties obtained from a

geotechnical investigation conducted in parallel with the 2004 full-scale barge impact testing

(McVay et al. 2005). For the development of the Case 2 soil-spring system, boring logs formed

the sole data source available.









For each model, a preliminary analysis was conducted in which the experimentally

measured time-history load was directly applied at the impact point for the specified test case.

The resulting displacement time-history of the structure was then compared to the experimentally

measured displacement time-history at the impact point. Output from the direct analysis and

comparison to experimental data aided in calibration of each model. Consequently, because

analytical application of the experimentally measured load time-history was shown to produce

pier response in agreement with that of the experimental data, the direct analysis comparison

provided a baseline means of judging the efficacy of the coupled analysis procedure.

3.3.1 Coupled Analysis Module Parameters

Within the coupled analysis procedure, the barge is modeled by a SDF point mass and

nonlinear compression spring. Barge impact conditions for the validation cases (P1T4 and

B3T4) were directly measured during the experimental tests. Thus, the experimental impact

weights and velocities were directly input into analytical Case 1 as 5.37 MN (604 T) traveling at

1.33 m/s (2.59 knots) and Case 2 as 3.06 MN (344 T) traveling at 0.79 m/s (1.53 knots),

respectively.

The loading portion of the barge force-crush relationship used for Case 1 and Case 2

(Fig. 3-4) was developed from impact-point force and displacement time-history data measured

during the P1T4 test; P1T4 was selected because of the undamaged bow impact location and

head-on nature of the collision event. The portion of the barge force-crush relationship up to the

peak force was obtained by performing coupled analysis using P1T4 impact conditions, and an

initially arbitrary force-crush relationship. After analysis completion, the coupled analysis

prediction of impact force was compared to that experimentally measured during the P1T4 test.

The analytical force-crush relationship was then adjusted to more closely match that measured

experimentally. After several iterations of this calibration process, a force-crush loading









relationship was obtained that produced force time-history data in agreement with the

experimental measurements of impact force.

The experimentally derived loading portion of the force-crush curve (Fig. 3-4) has a peak

impact force value of 5.74 MN (1065 kips) at a crush depth of 12.07 cm (4.75 in). Explicit

derivation of forces beyond this point, pertaining to the barge-bow impact force-crush

relationship, was not possible using the experimental dataset. However, barge bow force-crush

data are available in the literature that apply to the shape of the impacted pier in the P1T4 test;

specifically, a rectangular (flat) surface impactor. This data was obtained by subjecting a

high-resolution FE barge model to quasi-static crushing by square (flat) 1.8 m (6 ft) and 2.4 m

(8 ft) impactors (Consolazio and Cowan 2003). In the present study, barge force-crush

parameters pertaining to crush depths beyond that corresponding to the peak force were

proportioned from high-resolution FE force-crush data. Specifically, these parameters are: the

yield point, structural softening beyond the peak force, and the force plateau level beyond

softening (Fig. 3-4). The unloading curves (Fig. 3-4) chosen for Case 1 and Case 2 exhibit

steeper unloading paths at smaller crush-depths and shallower unloading paths at larger crush

depths. The unloading curves are consistent, with respect to qualitative shape, with those

employed in a prior study for a common barge type subject to quasi-static crush by square piers

(Consolazio and Cowan 2005).

3.3.2 Accounting for Payload Sliding During Impact Testing

During the Pier 1-S test series, payload in the form of 16.76 m (55 ft) reinforced concrete

bridge superstructure span segments was placed on the barge to simulate a loaded impact

condition. However, the payload was observed to slide during the collision events, implying the

development of frictional forces and dissipation of energy (Consolazio et al. 2006). In general

bridge design, the payload would not be assumed to slide. However, for the purpose of









validating the coupled analysis procedure as accurately as possible, enhancements were made to

the pre-existing coupled analysis procedure to numerically account for payload sliding (Fig. 3-5).

At each time-step and iteration, the ratio of barge acceleration (which, before sliding occurs, is

equal to the payload acceleration) to gravitational acceleration was computed and compared to

the static coefficient of friction (p/) between the barge and the payload. When the acceleration

ratio exceeded the static coefficient of friction, sliding was initiated (Fig. 3-5B). At sliding

initiation, the barge payload was assigned an initial velocity (u0) relative to the underlying

barge, equal to the corresponding current velocity of the barge-payload system. The payload was

assumed to continue sliding until the initial payload kinetic energy was completely dissipated

through friction. At all points in time during which sliding occurred, a constant frictional force,

equal to the product of the static coefficient of friction and the weight of the payload (Wp), was

applied to the barge. When the sliding kinetic energy of the payload barge was dissipated, the

payload mass (m,) and barge mass (mb) were assumed to rejoin as a single loaded

barge-payload system, as before sliding (Fig. 3-5A). For the P1T4 test, a sliding distance of

0.376 m (14.8 in) was predicted from the module modifications, which agreed very well with the

observed payload slide of approximately 0.38 m (15 in).

3.4 Comparison of Analytical and Experimental Data

3.4.1 Case 1

The Case 1 impact load time-history (Fig. 3-6A) is nearly identical to the respective

experimental curve up to the peak load, and expectedly so, because the portion of the barge

force-crush relationship (Fig. 3-4), up to the peak impact load, was derived from the impact force

and displacement data acquired during the Case 1 (P1T4) collision event. Additionally, the

analytical and experimental agreement for portions of the Case 1 force time-history curve









beyond the peak justifies the assumptions made during the development of the load softening,

load plateau, and unloading components of the force-crush curve (Fig. 3-4).

The analytically determined peak value of pier displacement exceeds the experimental

value by 16% (Fig. 3-6B). Supplementary coupled analyses of the Pier 1-S model were

conducted with impact velocities measured during similar and higher impact-energy P1 series

tests. Comparisons of displacement output from these analyses (Appendix A) to respective

experimental data show discrepancies of comparable or lesser magnitude to those of Case 1.

3.4.2 Case 2

Case 2, in direct contrast to Case 1, consists of a low-energy barge collision event on a

flexible pier with superstructure restraint. Case 1 and Case 2 share only the barge force-crush

relationship derived from the P1T4 experimental data. The Case 2 experimental and analytical

force time-histories (Fig. 3-7A) embody similar qualitative shapes; however, the analytical peak

force magnitude is larger than the experimental counterpart. Despite the disparity in magnitude,

numerical integration of the curves indicates that the shape and magnitude of the impulse, as a

function of time, agree well between the experimental and analytical results (Fig. 3-8). This

implies that the change in momentum of the barge was accurately predicted by the coupled

analysis and produced a pier response similar to that measured experimentally.

The concord of the analytical and experimental time-history of displacement (Fig. 3-7B)

demonstrates the proficiency of the coupled analysis procedure in adequately predicting barge

collision response for piers of varying stiffness. Accurate pier response predictions are

maintained while incorporating superstructure effects. Agreement of pier response is the most

important outcome of the coupled analysis procedure, as the accompanying internal forces

generated throughout the MDF pier-soil model ultimately govern the pier structural member

design. The coupled analysis procedure effectively shifts the analytical focus away from









determination of the barge impact force, and centers the emphasis on determining pier structural

demand.

Coupled analysis also inherently captures dynamic phenomena exhibited during

barge-bridge collisions. As evidenced by the time-history plots of Case 1 and Case 2 (Fig. 3-6

and Fig. 3-7), the peak impact force and displacement do not occur simultaneously for individual

experimental test cases involving appreciable impact-energies (Consolazio et al. 2006). Static

procedures do not account for peak load-displacement time disparity or the potential

amplification effects intrinsic to the early stages of collision events for bridge structures.

Coupled analysis automatically accounts for these effects.

















Barge
motion


Barge and bridge models
are coupled together through
a common contact force



r ....................................... F F


Barge Crushable bow
section of barge
"-----v---------- v----


SDF barge model


MDF bridge model


Figure 3-1. Coupling between barge and bridge (after Consolazio and Cowan 2005).


.super-
structure


Pier
Structure

Soil
stiffness














Impact
Force


Impact
Force


Impact
Force
4


Crush
Depth A














Crush
Depth C


,'\




Barge and bridge /
not in contact /
---" '- /


Crush
Depth B


Impact
Force


Plastic loading
'\ occurs along
\ loading curve

Reloading occurs
Along same path
as unloading

S Crush
Depth D


Figure 3-2. Stages of barge crush (after Consolazio and Cowan 2005).
A) Loading. B) Unloading. C) Barge not in contact with pier.
D) Reloading and continued plastic deformation.










Pier 1-S


Imp.


Pier 2-S


Pier 3-S


Pier 4-S


Pier 5-S


B

Figure 3-3. Structural configurations analyzed (not to relative scale).
A) Case 1: Single pier. B) Case 2: Four piers with superstructure.













Crush Distance (in)
4 6 8 10


0 50 100 150 200 250
Crush Distance (mm)


Figure 3-4. SDF barge force-crush relationship derived from experimental and analytical data.


5.5
5
4.5

S4
3.5
ao 3
3
S2.5
I 2

1.5
1
0.5


1200

1050

900 -

750
aC

600

450

300


150

0
300













No relative motion


b--- F





Barge Total barge-payload mass
contributes to impact force

Barge acceleration Static coefficient of friction
Gravitational acceleration between barge and payload A

m
p
Payload WI uo




Im 0W






Barge Barge mass and constant
payload frictional force
contribute to impact force

Barge acceleration > Static coefficient of friction
Gravitational acceleration between barge and payload B

Figure 3-5. Sliding criterion between payload and barge.
A) No sliding. B) Sliding.



















I -I 800

,)
600
0












II
400













,--- Coupled Analysis Output
10 / Experimental Data 0.40 ,
5 0200
400
0 0.25 0.5 0.75 1 1.25 1.5
Time (s) A







.5 0.50
S\Coupled Analysis Output
[0 I Experimental Data 0.40 ^

5 0.30 i

5 E020

.5 0.10



.5 -- -0.10


0 0.25 0.5 0.75 1 1.25
Time (s)


Figure 3-6. Comparison of Case 1 coupled analysis output and P1T4
A) Impact force. B) Pier displacement.


B

experimental data.


















300 .&

I I
1 1 0



0.5 -0
+ 1 -00



1 0
0 0.25 0.5 0.75 1 1.25 1.5
Time (s) A


50C I I 1


40- -- Coupled Analysis Output
--- Experimental Data
30 1.20 .

20 -- 0.80

O t \ 0.40

0 0.00



-20
0 0.25 0.5 0.75 1 1.25 1.5
Time (s) B


Figure 3-7. Comparison of Case 2 coupled analysis output and B3T4 experimental data.
A) Impact force. B) Pier displacement.













-4---. + ..1.. -+--1- 4 .1_;80

0.3
"60


0.2 -
__- 40
S" Coupled Analysis Output
S---- Experimental Data
0.1 20


III 0

0 0.25 0.5 0.75 1 1.25 1.5
Time (s)

Figure 3-8. Comparison of Case 2 coupled analysis output and B3T4 experimental data:
Impulse.









CHAPTER 4
SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS

4.1 Overview

At current computer processing speeds, barge impact time-history analysis of bridge

models can require between tens of minutes to several hours of processing time. Two

simplifications may be applied to the coupled analysis of bridge structural models to reduce

analysis time and facilitate its use in design settings. First, a simplified alternative to the

experimentally and analytically derived crush curve may be used in design when more detailed

barge force-crush behavior is not available. The bilinear curve found in the current static

AASHTO design specifications (Fig. 4-1) may be used for general barge-bridge collision design

applications. Second, multiple-pier models may be reduced to a pseudo-single pier model (with

two attached superstructure spans) and analyzed to produce results that match to a satisfactory

degree of accuracy, those obtained from corresponding full-resolution (multi-span, multi-pier)

models.

4.2 Linearized Barge Force-Crush Relationship

The nonlinear loading portion of the barge force-crush curve, developed from P1T4

experimental data (Fig. 4-1), is specific to the barge used in the 2004 impact experiments.

Phenomena such as structural-softening beyond the peak force level for each combination of

barge type and impactor shape are not well documented in the literature and further study is

warranted before these components of barge bow crushing behavior may be quantified for

general application. Hence, the use of a simple bilinear force-crush relationship, such as that

found in the AASHTO barge-collision specifications, is desirable at present as long as such a

curve produces reasonable results.









The AASHTO force-crush relationship is in reasonable agreement with the P1T4

experimentally determined based force-crush curve. The crush depth at which the AASHTO and

experimental curves shift from the initial linear portion to the subsequent linear portion occur at

103.63 mm (4.08 in) and 120.65 mm (4.75 in), respectively. For convenience, these locations

are termed the shift points. Note that the AASHTO force corresponding to the shift point,

6.17 MN (1386 kips), is significantly greater than that found in the experimentally based curve,

4.74 MN (1065 kips). Additionally, the AASHTO curve exhibits positive stiffness regardless of

crush depth, whereas the curve employed in the validation of the coupled analysis method is

assumed to exhibit perfectly plastic behavior at high crush depths (Fig. 4-1). Consequently, the

AASHTO curve yields higher impact forces than the experimental data for all barge crush

depths, and is therefore conservative.

4.3 Reduction of the Bridge Model

Barges impart predominantly horizontal forces to impacted bridge piers during collision

events. Displacement and acceleration based superstructure restraint (due to superstructure

stiffness and mass, respectively) can attract a significant portion of the horizontal forces and

cause the impact load to "shed" to the superstructure (Consolazio et al. 2006). The horizontal

force shed to the superstructure then propagates (initially) away from the impacted pier.

Consequently, lateral translational and plan-view rotational stiffnesses influence the structural

response as the force propagates through the superstructure from the impacted pier to adjacent

piers. Simultaneously, the distributed mass of the superstructure alternates between a source of

inertial resistance to a source of inertial load that respectively restrains or must be absorbed by

other portions of the bridge structure. Simplification of the multiple-pier structural model,

therefore, must adequately retain the influence of adjacent non-impacted (the lateral translational









and plan-view rotational stiffnesses of the adjacent piers; and, the dynamically participating mass

of the superstructure).

4.3.1 Uncoupled Condensed Stiffness Matrix

The stiffness DOF of a bridge model, beyond the superstructure spans that extend from

the impacted pier (Fig. 4-2), may be approximated by equivalent lateral translational springs and

plan-view rotational springs. These springs are linear elastic and represent the predominant DOF

of the linear elastic structural elements in the full-resolution model at piers adjacent to the

impacted pier. Soil nonlinearities at piers other than the impacted pier are ignored during

formation of the translational and rotational springs.

Replacement of numerous DOF from the flanking portions of a full bridge model

(Fig. 4-2) by two uncoupled springs at each end of a simplified two-span single-pier model may

be described in terms of a condensed stiffness matrix:


[Kcondensed KA Kco (4.1)
Kcouplng Ko


where [Koondeed] is the condensed stiffness matrix of the flanking bridge portion eliminated at

each side of the impacted pier; KA is the condensed lateral translational stiffness term; Kcoupng

is an off-diagonal stiffness term that couples the translational DOF to the rotational DOF; and

K, is the condensed stiffness plan-view rotational stiffness term. In the simplified model, the

diagonal terms KA and K, are represented by translational and rotational springs, respectively,

and the Kcouping terms are neglected. The exclusion of Kcouping in the simplified model is

justified by examining the forces generated by the condensed stiffness terms on one side of an

example five-pier model.









A channel pier was added to the previously discussed four-pier Case 2 model, using

bridge plans of the old St. George Island Bridge. This new five-pier model (Fig. 4-3) is referred

to as Case 3, as defined in Table 1-1. Through flexibility inversion (Fig. 4-2), the left-flanking

bridge structure in Case 3 (consisting of Pier 1-S to Pier 2-S), is reduced to the 2-DOF linear

elastic condensed stiffness matrix in Eq. (4.1), where KA = 97.0 MN/m (554 kip/in);

K = 2.58E+05 MN-m/rad (2.28E+09 kip-in/rad); and, Kouping = 398 MN/rad

(8.95+04 kip/rad). In row one of [Kcondened], the Kcouplng term may be interpreted as a horizontal

shear force generated when a unit rotation (1 rad) is induced at the right-most node of the

left-flanking structure. Static application of a load of 1.84 MN (414 kips) to the central pier of

the Case 3 five-pier model induces a plan-view rotation of 0 = 6.35E-06 rad at the location of

the condensed stiffness. The horizontal shear produced as a result of this rotation is:

ycouphng = K o (4.2)
0 couphng


where Vouplng is the shear produced from the coupling of rotational and translational DOF. In

this instance, Vgouphng = 2.53E-03 MN (0.568 kips). In comparison, the horizontal shear produced

as a result of diagonal lateral stiffness KA = 97.0 MN/m (554 kip/in) and lateral displacement

A = 4.62 mm (0.182 in) is:

V, = KA (4.3)


where VA is the shear produced directly from lateral translation. For this loading,

V = 0.448 MN (101 kips).


The amount of horizontal shear generated at the location of the condensed stiffness

matrix, due to the coupling stiffness term, is very small relative to the amount of horizontal shear









generated due to the diagonal stiffness term (V,"uphl is only 0.6% of V ). A similar examination

of the K, and Kcouphng terms yields ratios of comparable values (Appendix B). The large

difference in magnitude between the two shear forces demonstrates that the off-diagonal stiffness

terms of [Kcondeed ] generate negligible forces relative to those generated by the diagonal

stiffness terms. Uncoupling the condensed stiffness terms by applying two independent springs

is therefore warranted for design applications, as the uncoupled springs form a reasonable static

approximation of the stiffness of the excluded portions of the model.

As a further simplification to the full-bridge model, the diagonal stiffness terms KA and

K, may be approximated by direct inversion of the individual diagonal flexibility coefficients.

Specifically, this entails directly inverting the translational A, and rotational 0, displacements,

respectively, induced by the application of a unit shear force Vu"" and unit force-couple Mm""

on the applicable flanking structure (Fig. 4-2). This approximation produces only nominally

different magnitudes of stiffness with respect to that obtained by a flexibility matrix inversion

(Appendix B) and is simpler to carry out.

If significant nonlinear behavior is expected at non-impacted piers, then loads

representative of the forces that will be shed to the superstructure, and subsequently transmitted

into these piers, should be used to compute displacements (flexibility coefficients). Inversion of

flexibility coefficients formed in this manner yields a condensed secant stiffness that may then be

employed in the simplified model as described previously.

4.3.2 Lumped Mass Approximation

Mass is attributed to each node of the NDFEA models in this study, which consequently,

approximate a distributed mass system under dynamic loading. Therefore, a portion of mass of

the excluded structural components is assumed to contribute to the structural response of the









simplified models. This mass is assumed to fall within the tributary area (Fig. 4-4) extending

along the spans beyond the piers adjacent to the impacted pier of a given full-resolution model.

The mass is lumped and placed at respective ends of the simplified model. The lumped mass

simplification is combined with the stiffness approximation (Fig. 4-2) to complete the simplified

two-span single-pier model.

4.4 Multiple-Pier Coupled Analysis Simplification Algorithm

Simplified coupled analysis occurs in two stages. First, the two-span single-pier model is

assembled by replacing extraneous portions of the multiple-pier model with uncoupled linear

elastic springs and half-span lumped tributary masses. Coupled analysis is then performed, as

previously discussed, with the AASHTO bilinear crush-curve being employed for the barge.

The simplification algorithm automatically retains the ability to capture dynamic effects,

such as amplification, not addressed in static procedures. Furthermore, hundreds to thousands of

DOF are eliminated because the non-impacted piers and respective superstructure spans from the

full-resolution model are omitted from the model.











Crush Distance (in)
4 6 8 10


0 50 100 150 200
Crush Distance (mm)


Figure 4-1. Derived and


-1500


-1200


900 a
a

600


ntal data
300


S0
250 300


AASHTO SDF barge force-crush relationships (unloading curves not
shown).












P-1 P-2 P-3 P-4
,n n n n


P-5 P-6 P-7
n n n,


Impact location on full bridge model
Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans
Left-flanking structure Right-flanking structure
P-l P-2 P-3 P-5 P-6 P-7



Apply unit shear force at center of P-3 pile cap and center of P-5 pile cap


P-1 P-2 P-3
I -Q-Q


P-5 P-6 P-7

B--H--^


Record shear-induced translations and rotations at center of P-3 pile cap and center of P-5 pile cap

P-3 P-5

P-P-2 P-6 P-7

L VL
Apply unit moment at center of P-3 pile cap and center of P-5 pile cap


P-1 P-2 P-3


M Mt


P-5 P-6 P-7


M Mt


Record moment-induced rotations and translations at center of P-3 pile cap and center of P-5 pile cap
P-5


P-6 A A P-7
P-1 P-2 P-3 .i-i
IF 4 MR
AML

__ML

Form 2x2 left-flanking and right-flanking flexibility matrices using displacements and invert to form condensed stiffness

K L L LI F R R

VL ML coup/mg 0 VR coup/mg 0

Neglect off-diagonal stiffness and replace flanking-structures in full bridge model with diagonal stiffness as uncoupled springs


0

KR
K9


Figure 4-2. Plan view of multiple pier numerical model and location of uncoupled springs in
two-span single-pier model.







Pier 1-S Pier 2-S


Pier 3-S


Pier 4-S


i !


Figure 4-3. Structural configuration analyzed in Case 3.


Pier 5-S










N II II II II II II II k


Impact location on full bridge model
Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans


Left-flanking structure
P-2
n


Calculate mass of half-span beyond P-3
P-1 P-2 P-3
< n n n


Right-flanking structure
P-6
11


Calculate mass of half-span beyond P-5
P-5 P-6 P-7
n n n<


-m H


KMH+


Form lumped mass equal to m L


LHL


Form lumped mass equal to m HR


L HR


Apply lumped masses in place of flanking-structure masses in full bridge model


Impact location on two-span single-pier model


Figure 4-4. Plan view of multiple pier numerical model and location of lumped masses in two-
span single-pier mode.


P-1
'n









CHAPTER
SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES

5.1 Introduction

To illustrate the efficacy of the simplification algorithm, three demonstration cases

(FB-MultiPier bridge models) are presented. Each model was developed using methods

representative of those employed by bridge designers. Impact conditions prescribed for the

models are such that the range of scenarios encountered in practical bridge design for barge

impact loading is well represented. The cases employ the AASHTO bilinear barge crush-curve,

consist of impacted pier models of increasing impact resistance, and are subjected to impacts

with corresponding increases in impact energy. Time-history output of internal pier structural

member forces obtained from both full-resolution and simplified models are subsequently

compared for each case.

Each full-resolution model contains five piers: a centrally located impact pier and

additional structural components (soil, non-impacted piers, and superstructure spans) for a length

of two spans to either side of the central pier. A five-pier model contains a sufficient number of

piers and spans such that inclusion of additional piers would increase analytical computation

costs without appreciably improving the computed structural response. The appropriateness of

the decision to limit the full-resolution models to five piers is substantiated by the consistently

negligible acceleration response exhibited by the outer-most piers included in the five-pier

models. Alternatively stated, the added restraint provided by including additional piers is not

necessary, as the outer-most piers of the five-pier models are only nominally active throughout

the barge impact analysis.









A single time-step increment, 0.0025 sec, was employed for all demonstration analyses.

Each model also utilized Rayleigh damping, which is configured such that the first five vibration

modes undergo damping at approximately 5% of critical damping.

5.2 Geographical Information, Structural Configuration, and Impact Conditions

5.2.1 Case 3

The first demonstration case consists of analysis of the previously described Case 3

model (Fig. 4-3). This model was based on the old St. George Island Bridge from the

Apalachicola Bay area, linking St. George Island to mainland Florida, in the southeastern United

States. Apalachicola Bay is located approximately 80.5 km (50 mi) southwest of Tallahassee,

Florida in the "panhandle" portion of the state.

The structure of the old St. George Island Bridge, constructed in the 1960s, was detailed

in a prior report (Consolazio et al. 2006). Pertinent to demonstration Case 3, the superstructure

spanning from Pier 2-S to Pier 5-S (Fig. 4-3) consisted of 23 m (75.5 ft) concrete girder-and-slab

segments overlying concrete piers with waterline footings. Spanning the navigation channel and

one additional pier to either side, a 189 m (619.5 ft) continuous three-span steel girder and

concrete slab segment rested on Pier 1-S and Pier 2-S, each containing a mudline footing and

steel H-piles. The central pier in Case 3, Pier 3-S, contained two tapered rectangular pier

columns, with a 1.5 m (5 ft) wide impact face at approximately the same elevation as the top of a

small shear strut that spanned between the two 1.2 m (4 ft) thick waterline pile-cap segments.

The pier rested on eight battered 0.5 m (20 in) square prestressed concrete piles, each containing

a free length of approximately 3.7 m (12 ft).

The Case 3 FE model includes the southern channel pier and extends southward from the

centerline of barge traffic. The impacted pier, Pier 3-S, was constructed before the AASHTO

provisions were written (1991), and was flexible as it was not a channel pier. The pier was









located 115.8 m (380 ft) from the channel centerline, which was significantly closer to a distance

of three times the impacting vessel length, 138 m (450 ft), than the distance to the edge of the

navigation channel, 37.75 m (124 ft). Per the AASHTO specifications, the pier would be subject

to a reduced impact velocity, approaching that of the yearly mean current velocity

(Consolazio et al. 2002). The kinetic energy (Table 1-1) associated with an empty jumbo-hopper

barge drifting at the yearly mean current velocity for the Apalachicola Bay is representative of a

low-energy impact condition.

5.2.2 Case 4

Escambia Bay abuts Pensacola, Florida, in the southeastern United States. Case 4

(Fig. 5-1) consists of impact analysis of a model based on the Escambia Bay Bridge. Structural

components of this bridge model were derived from bridge plans developed in the 1960s. The

superstructure spanning from Pier 2-W to Pier 2-E consists of a 125 m (410 ft) continuous

three-span steel girder and concrete slab. A 28 m (92 ft) concrete girder-and-slab segment spans

the underlying concrete piers beyond Pier 2-E. All piers, except for the channel piers denoted as

Pier 1-E and Pier 1-W, contain two pier columns, a shear wall, pile cap, and waterline footing

foundation. The channel piers in Case 4 each contain two tapered rectangular pier columns, with

a 2.6 m (8.5 ft) wide head-on impact face at approximately the mid-height elevation of a 5.3 m

(17.5 ft) shear wall. The pier columns and shear wall overlie a 1.5 m (5 ft) thick mudline footing

and 1.8 m (6 ft) tremie seal. The channel pier foundations consist of eighteen battered and nine

plumb 0.6 m (24 in) square prestressed concrete piles.

The Case 4 FE model includes both of the channel piers and three auxiliary piers. The

impacted pier, Pier 1-E, was constructed before the AASHTO provisions were written (1991),

but contains large impact resistance relative to the impacted pier from Case 3, as Pier 1-E is a

channel pier. Impact on a channel pier with a relatively high impact resistance was chosen to









demonstrate the accuracy of the simplification algorithm for the medium-energy impact of a

fully-loaded jumbo-hopper barge and towboat, traveling at a higher speed than the mean

waterway velocity (Table 1-1).

5.2.3 Case 5

Case 5 (Fig. 5-2) consists of impact analysis of piers from the new St. George Island

Bridge, which replaced the old St. George Island Bridge in 2004. The structural model of the

new St. George Island Bridge was derived from construction drawings. Per these drawings,

Pier 46 through Pier 49 support five cantilever-constructed Florida Bulb-T girder-and-slab

segments at span lengths of 62.25 m (207.5 ft) for the channel and 78.5 m (257.5 ft) for the

flanking spans. Due to haunching, the depth of the post-tensioned girders vary from 2 m (6.5 ft)

at drop-in locations to 3.7 m (12 ft) at respective pier cap beam bearing locations. Simply

supported Florida Bulb-T beams with a depth equal to that of the launched beams at the drop-in

locations span either side of Pier 50. All piers included in this model contain two pier columns, a

shear strut centered near a respective pier column mid-height, a pile cap, and a waterline footing

system. The central pier in Case 5, Pier 48, contains two round 1.8 m (6 ft) pier columns, a

(6.5 ft) thick pile cap, and fourteen battered and one plumb 1.4 m (4.5 ft) diameter prestressed

cylinder piles with a 3 m (10 ft) concrete plug extending earthward from the pile cap.

The new St. George Island Bridge was designed in accordance with current AASHTO

barge collision design standards and provided a means of validating the simplification algorithm

for barge impact energies similar to those used in present day design. The Case 5 FE model

includes both of the channel piers and three auxiliary piers. The impacted pier, Pier 48 was

designed for a static impact load of 14.48 MN (3255 kips). With respect to the static AASHTO

design impact load, an energy equivalent impact condition (Appendix D) is employed in Case 5.

The prescribed vessel mass and velocity yields an impact kinetic energy equivalent to four









fully-loaded jumbo class hopper barges and a towboat traveling slightly above typical waterway

vessel speeds for the Apalachicola Bay waterway (Table 1-1).

5.3 Comparison of Simplified and Full-Resolution Results

In bridge design applications related to waterway vessel collision, the analytically

quantified internal forces in a given pier structure govern subsequent structural component

sizing. Hence, accurate determination of internal forces is a necessary outcome of a bridge

structural analysis method. To highlight the ability of simplified analysis to accurately quantify

design forces over the full range of impacted pier structures, time-histories of internal shear force

induced by the impact loading are shown for the top of the impacted pier column and an

underlying pile-head node for Case 3 through Case 5 shown in Fig. 5-3 through Fig. 5-5,

respectively (additional comparisons of the impact force, displacements, and internal moments

are documented in Appendix C).

The predictions of load duration (the time during which the barge and pier are in contact),

common to both simplified and full-resolution analyses, are 0.26 sec, 0.78 sec, and 2.9 sec,

respectively, for Case 3, Case 4, and Case 5. At points in time greater than the respective load

durations, each bridge is in an unloaded condition and undergoes damped free-vibration.

Accordingly, pier response to time-history barge collision analysis may be divided into two

phases: first a load-phase then a free-vibration phase. In all three demonstration cases, peak

internal pier forces occur during the load-phase (0.13 sec, 0.17 sec, and 2.1 sec for Case 3,

Case 4, and Case 5, respectively). Therefore, agreement between the simplified and

full-resolution models is most critical during the load-phase, as forces obtained during this phase

ultimately govern bridge pier member design. Simplified analysis retains the ability to

accurately capture forces during the load-phase of response (Fig. 5-3 through Fig. 5-5 for each

case, respectively). Peak shear forces generated by full-resolution and simplified analysis during









the load-phase for each case differ by less than 2%. Reduced, yet still reasonable, agreement

with respect to period of response and subsequent peak values of shear force occur during the

free-phase of response for each case, however, such agreement is less critical and typically

irrelevant for design purposes.

Case 3 through Case 5 were analyzed on a Dell Latitude D610 notebook computer using

a single 2.13 GHz Intel PentiumM CPU and FB-MultiPier. The computation times necessary for

analysis completion of the simplified models were only 8%, 7.5%, and 8.4% of those required

for the full-resolution models of Case 3 through Case 5, respectively (Fig. 5-6). All cases

required significantly less than an hour to complete 800, 800, and 1600 time-steps of analysis,

respectively. Engineering judgment is required to determine the appropriate amount of analysis

time specified. However, analysis generally need not be conducted beyond the end of

load-phase, as evidenced by forces during the load-phase for Case 3 through Case 5.

5.4 Conclusions from Simplified-Coupled Analysis Demonstrations

Excellent agreement is observed during the load-phase response of the full-resolution and

simplified test cases, especially with respect to peak internal forces generated at various locations

of the impacted piers. From a design perspective, reasonable agreement between full and

simplified analytical results is also observed during the free-phase portions of respective

time-history responses. Time-histories of internal shear force, moment, and displacement are

adequately captured by the simplification algorithm, despite the simplifying stiffness and mass

assumptions that are made.

The time necessary to analyze the simplified models is significantly less than one hour in

each case, which is in contrast to the several hours necessary to analyze respective full-resolution

models. It should be noted that all FB-MultiPier analyses were conducted in compilation debug









mode. Considerable additional reduction in analysis time is expected if the same analyses were

to be conducted in a release compilation or commercial version of FB-MultiPier.

5.5 Dynamic Amplification of the Impacted Pier Column Internal Forces

Application of the simplification algorithm to each of the demonstration cases inherently

incorporates mass and acceleration based inertial forces that emerge from integration of the

dynamic system equations of motion. The simplification algorithm accurately captures dynamic

amplification of forces generated in the pier columns that would be absent from static analysis

results. Dynamic amplification in each case may be quantified by considering the maximum pier

column shears developed in models subjected to static application of the peak impact load

predicted through the coupled analysis. The peak shear and moments developed in the pier due

to static loading are then compared to those from the simplified and full-resolution dynamic

analyses (Fig. 5-7)

With respect to peak pier column structural demand, the dynamic analyses are in

excellent agreement with each other for all cases. However, the peak magnitudes of the

statically generated shears and moments, respectively, correspond to 59% and 64% of the

magnitude of the dynamically obtained counterparts for Case 3; and, 38% and 37%, respectively,

for Case 4 (Fig. 5-7). In each of these cases, a static analysis employing a dynamically obtained

peak impact load leads to un-conservative predictions of peak pier column demand, as static

analysis only encompasses stiffness considerations. In contrast, dynamic analyses incorporate

both stiffness and inertial effects associated with the superstructure and therefore capture

dynamic amplification of pier column forces due to the mass of the superstructure. Furthermore,

the simplified procedure retains the ability to capture pier column force amplification as

evidenced by the agreement between the simplified and full-resolution output pertaining to peak

pier column demand.









The impact energy specified in Case 5 is of sufficient magnitude to cause the barge and

impacted pier to remain in contact for a time greater than several periods of the fundamental pier

vibration mode. Consequently, the inertial forces in the impacted pier begin to dissipate due to

damping effects. This is evidenced by attenuation of oscillation exhibited in the pile head shear

force time-history for Case 5 from 0.1 sec to 2.5 sec (Fig. 5-5B). Despite the continued dynamic

activity in the top of the Pier 48 pier columns throughout the analysis (Fig. 5-5A), the overall

pier behavior approaches that of a static response as the impact load approaches a maximum

value. Additionally, because the AASHTO barge bow force-crush relationship (Fig. 4-1)

maintains a positive stiffness regardless of crush depth, the Case 5 peak impact force occurs at a

time in which the dynamic component of behavior of Pier 48 has substantially diminished.

Therefore, the peak pier column demands are driven by a static response in this case. As a result,

there is not a great difference between dynamic and static response (Fig. 5-7).


























Pier 2-W Pier 1-W


Pier 1-E


Pier 2-E Pier 3-E


Figure 5-1. Structural configuration analyzed in Case 4.


IIill1Ia
I :


ii










Pier 46 Pier 47 Pier 48 Pier 49 Pier 50










Figure 5-2. Structural configuration analyzed in Case 5.

Figure 5-2. Structural configuration analyzed in Case 5.












600
-125
500 -
100
400 I Simplified Model
I Full-Resolution Model
75
300 -
50
i 200 e


0 0

-100 -25
-100 + -25

-200 -50

-300
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Time (s) A


200 1 1 1 1 1 1



S150 Simplified Model 30
S Full-Resolution Model
100 -

15
50
S 500




-50
-15

-100
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Time (s) B


Figure 5-3. Comparison of Case 3 simplified and full-resolution coupled analyses.
A) Pier column top node horizontal shear. B) Pile head node horizontal shear.
















I Simplified Model 125
-- Full-Resolution Model
S 100

S75 5

S50

) 1 + -\ -25
; V V: 0
L-25
-50

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Time (s) A




+ 45
---- implified Model
S "---- Full-Resolution Model
30 o

30
15s~


~h~RiC~~-cv.0


0 0.25 0.5 0.75 1
Time (s)


1.25 1.5 1.75 2


Figure 5-4. Comparison of Case 4 simplified and full-resolution coupled analyses.
A) Pier column top node horizontal shear. B) Pile head node horizontal shear.















1250 1 I 1 1 111
240
1000 -











Simplified Model
-500 Full-Resolution Model -1 1
750 180














0 0.5 1 1.5 2 2.5 3 3.5 4
Time (s) A


800 1-1-1-1l1l l ll-- 175
700 -









150
600 120
S250- 12560
750


















S Simplified Model
100 Full-Resolution Model 25
0 0.5 1 1.5 2 2.5 3 3.5 4
Time (s) A














Figure 5-5. Comparison of Case 5 simplified and full-resolution coupled analyses.
A) Pier column top node horizontal shear. B) Pile head node horizontal shear.
800

700

600
125
500


C) 75 o
300
200 0

Simplified Model
100 -- Full-Resolution Model 25


-100

0 0.5 1 1.5 2 2.5 3 3.5 4
Time (s) B


Figure 5-5. Comparison of Case 5 simplified and full-resolution coupled analyses.
A) Pier column top node horizontal shear. B) Pile head node horizontal shear.




















Simplified model
Full-resolution model


600

540

480

420

360

300

240

180

120

60

0


t l( I .


Case 3 Case4 Case 5

Figure 5-6. Time computation comparison of coupled analyses.










SFull-resolution dynamic analysis Static analysis


r- -


- e,


- e,


Case 3 Case 4 Case 5


SSimplified dynamic analysis Full-resolution dynamic analysis | Static analysis


00
2 &,


00
e i (


no
v(N


I
Case 5


Case 3 Case 4


Figure 5-7. Comparison of demonstration case simplified, full-resolution, and static analyses.
A) Peak pier column shear. B) Peak pier column moment.


,i


SSimplified dynamic analysis









CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

6.1 Conclusions

Numerical coupled analysis has been validated using experimental findings from the

2004 full-scale barge impact experiments. As one means of making coupled analysis feasible for

use in design settings, a simplified and standardized barge bow stiffness curve has been

recognized as desirable. Due to the scarcity of barge bow force-crush relationship data in the

literature, the AASHTO crush-curve has been selected. However, data specific to a particular

vessel type obtained by other means may easily be integrated into the coupled analysis

procedure. As an additional facilitation for the use of coupled analysis in design settings, an

algorithm has been presented that reduces a multi-span, multiple-pier model to a multi-span

single pier model with lateral and rotational springs, and lumped masses. With regard to the

stiffness approximation associated with the simplification algorithm, linear elastic lateral and

rotational springs have been shown to retain sufficient accuracy in respective simplified models

for design purposes despite the associated uncoupling of the respective DOF.

Three five-pier bridge analysis cases have been presented and subjected to the coupled

analysis procedure at simplified and full resolutions. Comparison of the results demonstrates the

ability of the simplification algorithm to predict time-history results in agreement with

full-resolution models for low, medium, and high-energy impact conditions through a range of

pier impact resistances. The simplified algorithm, used in conjunction with coupled analysis,

provides a feasible means of conducting barge-bridge collision analysis in design settings.

Required analyses times associated with simplified analysis are reduced to levels suitable for

design situations. Furthermore, the simplification algorithm retains analytical sophistication









sufficient to adequately quantify inertial bridge forces and the resulting distribution of internal

forces throughout a given pier.

Dynamic phenomena documented in previous barge-pier collision research, such as

dynamic amplification of pier column shear forces due to dynamic excitation of superstructure

elements, are quantified for three cases and compared to results obtained from a static analysis

procedure. Simplified coupled analysis is shown to adequately and efficiently capture such

effects and is found to be suitable for future incorporation into design provisions.

6.2 Recommendations for Future Research

Based on the advances made in this study, the following topics warrant additional future

investigation:

The development of experimental procedures leading to a standardized body of
crush-curves, including phenomena such as post-yield softening and unloading;
High-resolution modeling or experimental testing of multiple-barge flotilla impacts,
resulting in data sufficient to quantify any significant interactions between multiple barge
flotillas; this would be in relation to improving the state-of-the-art SDF impact model;
and,
Possible revision of the AASHTO Probability of Collapse term.









APPENDIX A
SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA

The 2004 full-scale experiments (Consolazio et al. 2006) consisted of three distinct
impact test setups, two of which are of interest in this study: the first impact tests were
conducted on the stiff channel pier, Pier 1-S; the second set of tests were conducted on a flexible
pier, Pier 3-S, with the superstructure intact for one span to the north and multiple spans to the
south. After development of the barge force-crush relationship, coupled analyses were
conducted on FB-MultiPier models of the Pier 1-S and Pier 3-S partial bridge structure at impact
energies corresponding to the impact test events.

The highest impact energies, and therefore the most appreciable impact loads and
structure response, occurred during tests four through seven on Pier 1-S (termed test P1T4
through P1T7). Due to the flexibility of Pier 3-S, and the non-destructive nature of the testing,
impact energies employed in the multi-span B3 bridge tests were considerably lower than that of
the P1 test series. Even so, the second through fourth tests (termed test B3T2 through B3T4)
generated considerable pier response and significant impact loads. Tests associated with
significant loading or pier response were selected for validation of the coupled analysis
procedure. Pertinent output from such analyses is included in this appendix. All P1 series
analyses included here were conducted using the payload modifications discussed in Chapter 3.











Impact Point Displacement Time History











AA-A


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)

Impact Force Time History
200

000

800

600

400

200

0 0.25 0.5 0.75 1 1.25 1.5


Time (sec)


Barge Force Crush Output


0 1 2 3 4 5 6


Crush (in)

- Analytical output
SA Experimental data
- Input loading curve


Figure A-1. Analytical output comparison to experimental P1T4 barge impact data.












Impact Point Displacement Time History









\ ,


i
A


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)

Impact Force Time History


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)


Barge Force Crush Output


0 1 2 3 4 5 6


Crush (in)

- Analytical output
SA Experimental data
- Input loading curve


Figure A-2. Analytical output comparison to experimental P1T5 barge impact data.


0.5

0.4

S 0.3




a
S 0.2

0.1
0


1200

1000

7 800

600

o 400

200


--A


A A .A


1\- ----- ^ ^ ^ ^ --











Impact Point Displacement Time History


-0.5'
0 0.25 0.5 0.75 1 1.25 1.5

Time (sec)

Impact Force Time History
1200

1000

800

600

400

200 A ~
-L -- -


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)

Barge Force Crush Output


0 1 2 3 4 5 6 7 8 9


Crush (in)

Analytical output
Experimental data
Input loading curve


Figure A-3. Analytical output comparison to experimental P1T6 barge impact data.











Impact Point Displacement Time History


-0.25'
0 0.25 0.5 0.75 1 1.25 1.5

Time (sec)

Impact Force Time History
1200

1000

800

600

400

200

0 0.25 0.5 0.75 1 1.25 1.5

Time (sec)

Barge Force Crush Output


0 1 2 3 4 5 6 7 8 9


Crush (in)

Analytical output
SExperimental data
Input loading curve


Figure A-4. Analytical output comparison to experimental P1T7 barge impact data.











Impact Point Displacement Time History


-0 .5 'L
0 0.25 0.5 0.75 1 1.25 1.5

Time (sec)

Impact Force Time History
250

200

150
i
100

50


0 0.25 0.5 0.75 1 1.25 1.5


Time (sec)


Barge Force Crush Output


0 0.5 1 1.5 2 2.5 3


Crush (in)

e- Analytical output
- i Experimental data
- -Input loading curve


Figure A-5. Analytical output in comparison to experimental B3T2 barge impact data.












Impact Point Displacement Time History


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)

Impact Force Time History


250

200

150

2 100
0
50


fl


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)


Barge Force Crush Output


1'
0
2 1(
0
F4


0 0.5 1 1.5 2 2.5 3


Crush (in)

S Analytical output
Si Experimental data
- Input loading curve


Figure A-6. Analytical output in comparison to experimental B3T3 barge impact data.


0.5

0.25
0.25
a 0

-0.25


A. '

A

4l `












Impact Point Displacement Time History


1.75
1.5
- 1.25
v 1
0.75
0.5
c 0.25
. 0
-0.25
-0.5
-0.75





450
400
350
300
S 250
8 200
0 150
100
50


0 0.25 0.5 0.75


1 1.25 1.5


Time (sec)


Barge Force Crush Output


0 0.5 1 1.5 2 2.5 3 3.5


Crush (in)

S Analytical output
Si Experimental data
Input loading curve


Figure A-7. Analytical output in comparison to experimental B3T4 barge impact data.


0.25 0.5 0.75 1 1.25 1.5

Time (sec)

Impact Force Time History


Ai





A
4


fl


0









APPENDIX B
CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS

Within the discussion of the condensed uncoupled stiffness matrix, presented in
Chapter 4, the condensed off-diagonal stiffness term that couples rotation and horizontal shear
force (Kcoup,, ) is shown to produce relatively negligible shear forces with respect to the applied
impact load. This affords the uncoupling of the condensed stiffness matrix of extraneous
non-impacted portions of a given bridge model. This appendix contains comparisons of the
same off-diagonal stiffness term (Kco,,,, ), alternatively viewed as a coupling between
horizontal translation and a vertical moment, and the moment produced by the diagonal
rotational stiffness term (K,) of the condensed stiffness matrix when the B3 numerical model
(Fig. 3-3) is subject to an arbitrary static load at the impact location. A comparison of the
diagonal and off-diagonal moments reveals that the off-diagonal stiffness of piers adjacent to the
impacted pier in a given full-resolution model may be neglected without sacrificing any
appreciable analytical accuracy of forces developed in the impacted pier.











Off-Diagonal stiffness quantification: Case 3 Numerical Model

1. Obtain condensed stiffness matrix of Pier 2S to Pier 1S portion of full-resolution model

1.1 Apply unit lateral load at location of stiffness condensation;
in this case, the center of the pier cap beam of Pier 2S

1.1.1 Store lateral translation and vertical rotation in appropriate entries
of condensed flexibility matrix

1.2 Apply unit vertical moment at location of stiffness condensation;
in this case, the center of the pier cap beam of Pier 2S

1.2.1 Store lateral translation and vertical rotation in appropriate entries
of condensed flexibility matrix

-8.5443-107
0.00181781
12
Condensed flexibility matrix: FlexP2:= in/kip and rad/kip-in
-8.5443-10-7 5.28799-10-9
12 12 J

The first row diagonal entry pertains to shear force per unit lateral translation; the second row
diagonal pertains to vertical moment per unit vertical rotation

2. Invert condensed flexibility matrix to obtain condensed stiffness matrix

-1
StiffP2:= FlexP2 1



553.616 8.945x 104
Condensed stiffness matrix: StiffP2= kip/in and kip-in/rad
8.945 x 104 2.284 x 10 9










Off-Diagonal stiffness quantification: Case 3 Numerical Model (Cont'd)


3. Apply peak static load at impact point of full-resolution model and record displacements

Static load applied at Node 109 of Pier 3-S:

PeakLoad:= 413.8 kips

Induced displacements at location of condensed stiffness:


Vertical rotation:

Oz := 1.994-10-4 rad

Horizontal translation:

Ax:= 0.1908 in

4. Calculate moment due to diagonal stiffness term and vertical rotation

Mzdiagonal:= StiffP2,2-z Mzdiagonal= 4.554x 105 kip-in

5. Calculate moment due to off-diagonal stiffness term and horizontal translation

Mzoffdiagonaf= StiffP1,2-Ax Mzoffdiagona 1.707 x 104 kip-in

6. Compare magnitudes of "diagonal" and "off-diagonal" moments


ratio:= Mdiagonal ratio = 26.681
Mzoffdiagonal

The "diagonal" moment is significantly larger than the "off-diagonal" moment.











Flexibility Approximation: Case 3 Numerical Model
7. Directly invert diagonal flexibility terms recorded in 1.2.1 of
Off-Diagonal Stiffness Quantification

7.1 Approximation of Translational Stiffness Term


I1
AppStift~rans,= Flx21


AppStifffrans= 550.112


kip/in


7.2 Approximation of Rotational Stiffness Term


AppStiffRot:=
FlexP22,2


AppStiffRot= 2.269


8. Calculate percent difference between approximated stiffness
terms and stiffness terms obtained by flexibility matrix inversion
(the latter terms being calculated in 2. of Off-Diagonal Stiffness Quantification)
8.1 Percent difference of translational stiffness term


AppStifffrans- StiffP21, 1
100 = -0.633
StiffP2, 1


percent


8.2 Percent difference of rotational stiffness term


AppStiffRot- StiffP2,2
-100 = -0.633
StiffP2 ,2


percent


The approximation yields nominally different values of stiffness.


kip-rad/in









APPENDIX C
SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT

To further bolster the assertion that the simplification algorithm predicts impacted pier
response with an accuracy that, within reason, matches that of full-resolution bridge coupled
analysis, additional time-history data from each of Case 3 through Case 5 are included in this
appendix. More specifically, time-histories of shear, moment, and displacement are provided at
the pier column top and pile head for each case. Additionally, barge force-crush data obtained
from simplified and full-bridge analyses are included. Accompanying this data are the impact
location displacement time-history and impact location force time-history for each of Case 3
through Case 5. Consequently, the data presented in Fig. C-l through Fig. C-9 were obtained
using the AASHTO barge bow force-crush relationship. Finally, data obtained from the same
pier models are presented when simplified and full-resolution analyses are conducted using a
bilinear barge bow force-crush relationship with an initial stiffness and "shift point" (see
Chapter 4) identical to that found in the AASHTO curve. Output pertaining to these analyses are
located in Fig. C-10 through Fig. C-18.











Impact Point Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Impact Force Time History
600

500

400

300

200

100

0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Barge Force Crush Output
500
7'--
/
200 /
/
900
7
600

300


0 1 2 3

Crush (in)

9-- Two-span single-pier
i- Five-pier
- Input loading curve


4 5


Figure C-1. Case 3 AASHTO curve coupled analysis output comparison at impact location.










Shear Force Time History


Time (sec)


Moment Time History


Time (sec)


Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)
Two-span single-pier
--A Five-pier


Figure C-2. Case 3 AASHTO curve coupled analysis output comparison at pier column top.










Shear Force Time History


Time (sec)

Moment Time History


Time (sec)

Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)
Two-span single-pier
-iA Five-pier


Figure C-3. Case 3 AASHTO curve coupled analysis output comparison at pile head.











Impact Point Displacement Time History


Time (sec)


Impact Force Time History
1600
1400
1200
1000
800
600
400
200


0.25 0.5 0.75


1 1.25 1.5 1.75 2


Time (sec)

Barge Force Crush Output


0 2 4 6 8 10 12 14 16

Crush (in)

** Two-span single-pier
Five-pier
- Input loading curve


Figure C-4. Case 4 AASHTO curve coupled analysis output comparison at impact location.










Shear Force Time History


0 0.25 0.5 0.75 1 1.25 1.5
0 0.25 0.5 0.75 1 1.25 1.5


Time (sec)

Moment Time History


Time (sec)


Displacement Time History


1.75 2


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)
Two-span single-pier
--A Five-pier


Figure C-5. Case 4 AASHTO curve coupled analysis output comparison at pier column top.


'A
AAAt


A AA
AJT\/V










Shear Force Time History


Time (sec)


Moment Time History


Time (sec)


Displacement Time History









A


0 0.25 0.5 0.75 1 1.25


1.5 1.75 2


Time (sec)
-- Two-span single-pier
-A-A Five-pier


Figure C-6. Case 4 AASHTO curve coupled analysis output comparison at pile head.










Impact Point Displacement Time History


Time (sec)


Impact Force Time History


0 0.5 1


1.5 2 2.5 3


3.5 4


Time (sec)

Barge Force Crush Output


0 25 50 75 100 125 150 175 200


Crush (in)
** Two-span single-pier
A-A Five-pier
Input loading curve


Figure C-7. Case 5 AASHTO curve coupled analysis output comparison at impact location.










Shear Force Time History


Time (sec)

Moment Time History


Time (sec)


Displacement Time History


0 0.5 1 1.5 2 2.5 3 3.5 4


Time (sec)
e-- Two-span single-pier
- A Five-pier


Figure C-8. Case 5 AASHTO curve coupled analysis output comparison at pier column top.











Shear Force Time History


-50

-100

-150

-200
0 0.5 1 1.5 2 2.5 3 3.5

Time (sec)

Moment Time History
,) Iz n--------------------


-250

-500

-750

-1000


0.5 1 1.5 2 2.5 3 3.5 4

Time (sec)

Displacement Time History


0 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec)

- Two-span single-pier
SA Five-pier


Figure C-9. Case 5 AASHTO curve coupled analysis output comparison at pile head.


Ar
A AA



], A


0











Impact Point Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Impact Force Time History
600

500

400

300

200

100

0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Barge Force Crush Output
1500

1200

900
/
600

300


0 1 2 3

Crush (in)

- Two-span single-pier
AA Five-pier
- Input loading curve


4 5


Figure C-10. Case 3 bilinear curve coupled analysis output comparison at impact location.










Shear Force Time History


Time (sec)


Moment Time History


Time (sec)


Displacement Time History


-1'
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)
Two-span single-pier
A-AA Five-pier


Figure C-11. Case 3 bilinear curve coupled analysis output comparison at pier column top.










Shear Force Time History


Time (sec)

Moment Time History


Time (sec)

Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)
- Two-span single-pier
-iA Five-pier


Figure C-12. Case 3 bilinear curve coupled analysis output comparison at pile head.











Impact Point Displacement Time History


0.75
o 0.5
'"1 0.25
0 A

-0.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Impact Force Time History
1600
1400
1200
.& 1000
800
S 600
400
200
0 ---- ------
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Barge Force Crush Output
1600
1400
,, 1200
._- 1000
800
600
S400
200

0 2 4 6 8 10 12 14 16

Crush (in)

o-* Two-span single-pier
A-A Five-pier
Input loading curve


Figure C-13. Case 4 bilinear curve coupled analysis output comparison at impact location.











Shear Force Time History


Time (sec)


2
C.
C

















C


Time (sec)


Displacement Time History


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

- Two-span single-pier
-iA Five-pier


Figure C-14. Case 4 bilinear curve coupled analysis output comparison at pier column top.


Moment Time History










Shear Force Time History


Time (sec)


Moment Time History


Time (sec)


Displacement Time History


Time (sec)
-- Two-span single-pier
-A-A Five-pier


Figure C-15. Case 4 bilinear curve coupled analysis output comparison at pile head.












Impact Point Displacement Time History


Time (sec)

Impact Force Time History


;0


10

10


0

0 \
0 1 2 3 4 5

Time (sec)

Barge Force Crush Output


1500

1250

1000

750

500

250

o0


0 30 60 90 120 150 180 210 240 270 300

Crush (in)

- Two-span single-pier
-- Five-pier
Input loading curve


Figure C-16. Case 5 bilinear curve coupled analysis output comparison at impact location.


0f

C,









0
C4











Shear Force Time History


Time (sec)


Moment Time History


2

C







C.)


Time (sec)


Displacement Time History


0 1 2 3 4 5 6


Time (sec)

--* Two-span single-pier
A Five-pier


Figure C-17. Case 5 bilinear curve coupled analysis output comparison at pier column top.











Shear Force Time History
25

0

-25

-50

0 -75

-100

-125
0 1 2 3 4 5 6

Time (sec)

Moment Time History
100


S-100
-200
-300
S-400 'A. ,
-500
-600
0 1 2 3 4 5 6

Time (sec)

Displacement Time History
0.6

0.5





C). 0.1

-0.1
0

0 1 2 3 4 5 6

Time (sec)

-- Two-span single-pier
A-A-A Five-pier


Figure C-18. Case 5 bilinear curve coupled analysis output comparison at pile head.









APPENDIX D
ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS

The new St. George Island Bridge was designed and constructed after the AASHTO vessel
collision specifications went in effect, hence, the piers of this bridge were designed to resist
barge impact loading. Furthermore, pier impact load data was contained within the bridge plans
used to develop the numerical model for Case 5 of this thesis. The initial kinetic energy
specified in the Case 5 coupled analyses was derived from the known design impact load and
pertinent equations found in AASHTO. Conversely, energy-equivalent static AASHTO impact
loads are calculated from the kinetic energies employed in Case 3 and Case 4. The barge width
modification factor is 1.0 in the following calculations as a jumbo-hopper is selected as the
impacting vessel in all demonstration cases.











Back calculation of AASHTO impact force using Case 3 Impact Energy



The Old St. George Island Bridge Pier 3-S was subject to an impact energy consisting of:

Barge weight


W := 1.78-MN

Barge velocity


W = 200.08T


W= 181.478tonne


m
V:= 1.03 -


V = 2.002knot


ft
V = 3.379-


Assume a hydrodynamic mass coefficient of Ch:= 1.05

Impact energy


1 Ch-W 2
KE:= -.- V
2 g


KE = 74.564kip.ft


KE = 0.101MN-m


From theAmerican Association of State and Highway Transportation Officials (AASHTO) Guide
Specification and Commentary for Vessel Collision Design of Highway Bridges, the equations for
barge crush depth and kinetic energy associated with impact are:


Barge crush depth:

ab := + 1 1 10.2
15672 1


ab = 0.067 ft


The energy equivalent AASHTO static impact force is:


Pb := Pb (1349+ 10-ab) if ab 2 0.34
Pb 4112ab if ab < 0.34
return Pb


Pb = 274.787











Back calculation of AASHTO impact force using Case 4 Impact Energy



The Escambia Bay Bridge Pier 1-E was subject to an impact energy consisting of:

Barge weight


W:= 18-MN

Barge velocity


W = 2.023 x 103T


W = 1.835 x 103tonme


m
V:= 1.54-


V = 2.994knot


ft
V = 5.052-


Assume a hydrodynamic mass coefficient of Ch:= 1.05

Impact energy


1 ChW .2
KE := -.-V
2 g


KE = 1.686 x 10 kipft KE = 2.285MN-m


From theAmerican Association of State and Highway Transportation Officials (AASHTO) Guide
Specification and Commentary for Vessel Collision Design of Highway Bridges, the equations for
barge crush depth and kinetic energy, and barge width associated with impact are:


Barge crush depth:

ab := 6 + 1 1 10.2
15672 1


ab = 1.417 ft


The energy equivalent AASHTO static impact force is:


Pb := Pb<- 1349 + 110-ab if ab > 0.34
Pb <-4112ab if ab < 0.34
return Pb


Pb = 1.505x 103










Back calculation of Case 5 impact energy using AASHTO impact force



From bridge plans of the New St. George Island Bridge Channel Pier, the design impact load is:


Pb := 3255


From theAmerican Association of State and Highway Transportation Officials (AASHTO) Guide
Specification and Commentary for Vessel Collision Design of Highway Bridges, the equations for
barge crush depth and kinetic energy associated with impact are:

Barge crush depth:


Pb- 1349 Pb- 1349
ab := ab- if 0.34
110 110
Pb Pb
ab <- if < 0.34
4112 4112
return ab


Kinetic energy associated with impact:


ab = 17.327


KE = 3.564 x 10


Sab
KE := -a + 1 1 5672
1(10.2 1.


kip-ft


Define flotilla design velocity as a function of Hydrodynamic Mass Coefficient and Flotilla
weight tonness)


V(CH,W) KE 29.2 E 02 5
CH.W


Assume Hydrodynamic Mass Coefficient is 1.05.

Define weight of barge as a function of the number of barges in the flotilla; assume towboat
weighs 120 tons (US, short)

n-(1700 + 200) + 120
W(n) :=
1.102311311










Back calculation of Case 5 impact energy using AASHTO impact force
(Cont'd)

Try using four fully loaded Jumbo Hopper barges and check that the accompanying velocity
is attainable within the waterway.

The weight of four fully loaded Jumbo Hopper barges and the tow boat is:


W(4) = 7.003 x 103 tonnes

The velocity of the flotilla, necessary to generate a static impact load of 3255 kips is:

V(1.05,W(4)) = 11.896 ft/sec

Conclusion: the four barge flotilla is a reasonable number of barges for use in a single column
flotilla in the southeastern United States, and 11.896 ft/sec is an attainable speed in the
St. George Island waterway as typical traveling speeds are: 10.13 ft/sec (Consolazio et al. 2002).









REFERENCES


AASHTO. (1991). Guide Specification and Commentaryfor Vessel Collision Design ofHighway
Bridges, American Association of State Highway and Transportation Officials,
Washington, D.C.

Arroyo, J. R., Ebeling, R. M., and Barker, B. C. (2003). "Analysis of Impact Loads from Full-
Scale Low-Velocity, Controlled Barge Impact Experiments, December 1998." US Army
Corps of Engineers Report ERDC/ITL TR-03-3, 2003.

Consolazio, G. R., Cook, R. A., and Lehr, G. B. (2002). "Barge Impact Testing of the St. George
Island Causeway Bridge Phase I : Feasibility Study." Structures Research Report No. 783,
Engineering and Industrial Experiment Station. University of Florida, Gainesville, Florida,
January.

Consolazio, G. R. and Cowan, D. R. (2003). "Nonlinear Analysis of Barge Crush Behavior and
its Relationship to Impact Resistant Bridge Design." Computers and Structures, Vol. 81,
Nos.8-11, pp. 547-557.

Consolazio, G. R., Lehr, G. B., and McVay, M. C. (2004a). "Dynamic Finite Element Analysis
of Vessel-Pier-Soil Interaction During Barge Impact Events." Transportation Research
Record: Journal of the Transportation Research Board. No. 1849, Washington, D.C., pp.
81-90.

Consolazio, G. R., Hendrix, J. L., McVay, M. C., Williams, M. E., and Bollman, H. T. (2004b).
"Prediction of Pier Response to Barge Impacts Using Design-Oriented Dynamic Finite
Element Analysis." Transportation Research Record: Journal of the Transportation
Research Board. No. 1868, Washington, D.C., pp. 177-189.

Consolazio, G. R. and Cowan, D. R. (2005). "Numerically Efficient Dynamic Analysis of Barge
Collisions with Bridge Piers." ASCE Journal of Structural Engineering, ASCE, Vol. 131,
No. 8, pp. 1256-1266.

Consolazio, G. R., Cook, R. A., and McVay, M. C. (2006). "Barge Impact Testing of the St.
George Island Causeway Bridge", Structures Research Report No. 2006/26868,
Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida,
March.

FB-MULTIPIER User's Manual. (2007). Florida Bridge Software Institute, University of
Florida, Gainesville, Florida.

FB-PIER User's Manual. (2003). Florida Bridge Software Institute, University of Florida,
Gainesville, Florida.

Goble, G., Schulz, J., and Commander, B. (1990). Lock andDam #26 Field Test Report for The
Army Corps ofEngineers, Bridge Diagnostics Inc., Boulder, CO.









Hendrix, J. L. (2003). "Dynamic Analysis Techniques for Quantifying Bridge Pier Response to
Barge Impact Loads." Masters Thesis, Department of Civil and Coastal Engineering, Univ.
of Florida, Gainesville, Fla.

Larsen, O. D. (1993). "Ship Collision with Bridges: The Interaction between Vessel Traffic and
Bridge Structures." IABSE Structural Engineering Document 4, IABSE

Knott, M., and Prucz, Z. (2000). Vessel Collision Design of Bridges: Bridge Engineering
Handbook, CRC Press LLC.

Meier-Dornberg, K. E. (1983). "Ship Collisions, Safety Zones, and Loading Assumptions for
Structures in Inland Waterways." Verein Deutscher Ingenieure (Association of German
Engineers) Report No. 496, 1983, pp. 1-9.

McVay, M. C., Wasman, S. J., Bullock, P. J. (2005). St. George Geotechnical Investigation of
Vessel Pier Impact, Engineering and Industrial Experiment Station, University of Florida,
Gainesville, Florida.

Patev, R. C., Barker, B. C., and Koestler, L. V., III. (2003). "Full-Scale Barge Impact
Experiments, Robert C. Byrd Lock and Dam, Gallipolis Ferry, West Virginia." United
States Army Corps of Engineers Report ERDC/ITL TR-03-7, December.

Yuan, P. (2005). "Modeling, Simulation and Analysis of Multi-Barge Flotillas Impacting Bridge
Piers." PhD dissertation, Dept. of Civil Engineering, Univ. of Kentucky, Lexington, Ky.









BIOGRAPHICAL SKETCH

Michael Davidson was born in Louisville, Kentucky. He enrolled at the University of

Kentucky in August 2000. After being awarded the National Science Foundation Graduate

Research Fellowship and obtaining his Bachelor of Science in civil engineering from the

University of Kentucky (summa cum laude) in May 2005, he began graduate school at the

University of Florida in the College of Engineering, Department of Civil and Coastal

Engineering. The author will receive his Master of Science degree in August 2007, with a

concentration in structural engineering. Upon graduation, the author will continue his education

at the University of Florida, ultimately earning a degree of Doctor of Philosophy with a

specialization in structural engineering.





PAGE 1

SIMPLIFIED DYNAMIC BARG E COLLISION ANALYSIS FO R BRIDGE PIER DESIGN By MICHAEL THOMAS DAVIDSON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007 1

PAGE 2

2007 Michael Thomas Davidson 2

PAGE 3

To my wife, Kiristen 3

PAGE 4

ACKNOWLEDGMENTS This material is based on work supporte d under a National Scien ce Foundation Graduate Research Fellowship. However, this thesis w ould not have been completed without the support of several individuals. First, the insight and guidance of Dr Gary Consolazio has proven invaluable. His willingness to invest time in helping graduate students become effective analysts and independent researchers will undoubtedly garner countless and vast return s. The author also wishes to thank Dr. Marc Hoit, Dr. Petros Christou, and Dr. Jae Chung for their assistance with extending the capabilities of FB-MultiPier. A graduate student deserving of many thanks and much future success is David Cowan, whose brilliance seems to be limitless. Finally, the author wishes to thank his wife Kiristen, his famil y, and his friends for th eir enduring love and fellowship. 4

PAGE 5

TABLE OF CONTENTS ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ........................................................................................................................... 7 LIST OF FIGURES ......................................................................................................................... 8 ABSTRACT ................................................................................................................................... 12 CHAPTER 1 INTRODUCTION ............................................................................................................. 13 2 LITERATURE REVIEW .................................................................................................. 17 2.1 Experimental Research ................................................................................................ 17 2.2 Analytical Research ..................................................................................................... 18 3 COUPLED BARGE COLLISION ANALYSIS ................................................................ 20 3.1 Introduction ................................................................................................................. 20 3.1.1 Barge Loading and Unloading Behavior 20 ...................................................... 3.1.2 Coupled Analysis Algorithm 21 ........................................................................ 3.1.3 Use of Experimental Data for Coupled Analysis Validation 21 ........................ 3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2 ....................... 22 3.3 Software Selection and Model Development .............................................................. 22 3.3.1 Coupled Analysis Module Parameters ..........................................................24 3.3.2 Accounting for Payload Slid ing During Impact Testing 25 .............................. 3.4 Comparison of Analytical and Experimental Data ...................................................... 26 3.4.1 Case 1 ............................................................................................................ 26 3.4.2 Case 2 27 ............................................................................................................ 4 SIMPLIFIED MULTIPLE-PI ER COUPLED ANALYSIS ............................................... 37 4.1 Overview ...................................................................................................................... 37 4.2 Linearized Barge Force-Crush Relationship ................................................................ 37 4.3 Reduction of the Bridge Model .................................................................................... 38 4.3.1 Uncoupled Condensed Stiffness Matrix 39 ....................................................... 4.3.2 Lumped Mass Approximation 41 ...................................................................... 4.4 Multiple-Pier Coupled Analysis Simplification Algorithm ......................................... 42 5

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5 SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES ............................ 47 5.1 Introduction .................................................................................................................. 47 5.2 Geographical Information, Structural Configuration, and Impact Conditions ............ 48 5.2.1 Case 3 48 ............................................................................................................ 5.2.2 Case 4 49 ............................................................................................................ 5.2.3 Case 5 50 ............................................................................................................ 5.3 Comparison of Simplified and Full-Resolution Results .............................................. 51 5.4 Conclusions from Simplified-Coupled Analysis Demonstrations ............................... 52 5.5 Dynamic Amplification of the Impacted Pier Column Internal Forces ....................... 53 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ............ 62 6.1 Conclusions .................................................................................................................. 62 6.2 Recommendations for Future Research ....................................................................... 63 APPENDIX A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA ........................... 64 B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS .................... 72 C SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT ................................................. 76 D ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS ................................ 95 REFERENCES ............................................................................................................................ 100 BIOGRAPHICAL SKETCH ....................................................................................................... 102 6

PAGE 7

LIST OF TABLES Table page 1-1 Case descriptions: use, configuration, and impact data.................................................... 16 7

PAGE 8

LIST OF FIGURES Figure page 3-1 Coupling between barge and bridge (after Consolazio and Cowan 2005) ........................29 3-2 Stages of barge crush (after Consolazio and Cowan 2005). ...............................................30 3-3 Structural configurations analyzed (not to relative scale).................................................. 31 3-4 SDF barge force-crush relationship de rived from experimental and analytical data......... 32 3-5 Sliding criterion between payload and barge .....................................................................33 3-6 Comparison of Case 1 coupled analysis output and P1T4 experimental data................... 34 3-7 Comparison of Case 2 coupled analysis output and B3T4 experimental data................... 35 3-8 Comparison of Case 2 coupled analysis output and B3T4 expe rimental data: Impulse........................................................................................................................ .......36 4-1 Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown)............................................................................................43 4-2 Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model............................................................................................ 44 4-3 Structural configuratio n analyzed in Case 3...................................................................... 45 4-4 Plan view of multiple pier numerical model and location of lumped masses in two-span single-pier model............................................................................................ 46 5-1 Structural configuratio n analyzed in Case 4...................................................................... 55 5-2 Structural configuratio n analyzed in Case 5...................................................................... 56 5-3 Comparison of Case 3 simplified and full-resolution coupled analyses............................57 5-4 Comparison of Case 4 simplified and full-resolution coupled analyses............................58 5-5 Comparison of Case 5 simplified and full-resolution coupled analyses............................59 5-6 Time computation comparison of coupled analyses ..........................................................60 5-7 Comparison of demonstration case simp lified, full-resolution, a nd static analyses.......... 61 A -1 Analytical output comparison to ex perimental P1T4 barge impact data ........................... 65 A -2 Analytical output comparison to ex perimental P1T5 barge impact data ........................... 66 8

PAGE 9

A -3 Analytical output comparison to ex perimental P1T6 barge impact data ........................... 67 A -4 Analytical output comparison to ex perimental P1T7 barge impact data ........................... 68 A -5 Analytical output comparison to ex perimental B3T2 barge impact data .......................... 69 A -6 Analytical output comparison to ex perimental B3T3 barge impact data .......................... 70 A -7 Analytical output comparison to ex perimental B3T4 barge impact data .......................... 71 C -1 Case 3 AASHTO curve coupled analysis output comparison at impact location .............. 77 C -2 Case 3 AASHTO curve coupled analysis output comparison at pier column top ............. 78 C-3 Case 3 AASHTO curve coupled analys is output comparison at pile head........................ 79 C -4 Case 4 AASHTO curve coupled analysis output comparison at impact location.............. 80 C-5 Case 4 AASHTO curve coupled analysis output comparison at pier column top............. 81 C -6 Case 4 AASHTO curve coupled analys is output comparison at pile head........................ 82 C -7 Case 5 AASHTO curve coupled analysis output comparison at impact location.............. 83 C-8 Case 5 AASHTO curve coupled analysis output comparison at pier column top............. 84 C-9 Case 5 AASHTO curve coupled analys is output comparison at pile head........................ 85 C-10 Case 3 bilinear curve coupled analys is output comparison at impact location.................. 86 C-11 Case 3 bilinear curve coupled analysis output comparison at pier column top................. 87 C-12 Case 3 bilinear curve coupled anal ysis output comparison at pile head............................ 88 C-13 Case 4 bilinear curve coupled analys is output comparison at impact location.................. 89 C-14 Case 4 bilinear curve coupled analysis output comparison at pier column top................. 90 C-15 Case 4 bilinear curve coupled anal ysis output comparison at pile head............................ 91 C-16 Case 5 bilinear curve coupled analys is output comparison at impact location.................. 92 C-17 Case 5 bilinear curve coupled analysis output comparison at pier column top................. 93 C -18 Case 5 bilinear curve coupled anal ysis output comparison at pile head ............................ 94 9

PAGE 10

LIST OF ABBREVIATIONS L or L appended to symbol, indicates symbol exclusivity to left-flanking structure R or R appended to symbol, indicates symbol exclusivity to right-flanking structure F flexibility matrix condensedK condensed stiffness matrix couplingK off-diagonal (coupling) stiffness term K translational stiffness term K plan-view rotational stiffness term unitM unit moment Hm mass of half-span of superstructure bm mass of barge pm mass of payload 0u initial sliding velocity of payload unitV unit shear force couplingV shear due to coupled sti ffness and plan-view rotation V shear due to translational stiffness and translation pW weight of payload translation M translation due to unit moment V translation due to unit shear static coefficient of fric tion between payload and barge 10

PAGE 11

plan-view rotation M plan-view rotation due to unit moment V plan-view rotation due to unit shear 11

PAGE 12

Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SIMPLIFIED DYNAMIC BARG E COLLISION ANALYSIS FO R BRIDGE PIER DESIGN By Michael Thomas Davidson August 2007 Chair: Gary R. Consolazio Cochair: Marc I. Hoit Major: Civil Engineering The American Association of State and Highway Transportation Officials barge impact provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to determine structural demand on bridge piers. However, conclusions drawn from experimental full-scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand with consideration of additional forces generated from dynamic effects. Static quantification of bridge pier demand due to barge impact ignore s mass related inertial forces generated by the superstructure which can amplify rest raint of underlying pier columns. An algorithm for efficiently performing coupl ed nonlinear dynamic ba rge impact analysis on simplified bridge structure-soil finite element models is presented in this thesis. The term coupled indicates the impact of a finite el ement bridge model a nd a respectiv e single degree-of-freedom barge model traveling at a specified initial velocity with a specified force-deformation relationship. Coupled analysis is validated using experimental data. Also, results from simplified and full-resolution analyses are compared for several cases to illustrate robustness of the algorithm for various barge impact energies and pier type s. Simplified coupled dynamic analysis is shown to accurately capture dynamic forces and amplification effects. 12

PAGE 13

CHAPTER 1 INTRODUCTION Potential loss of life and detrimental economic consequences due to bridge failure from waterway vessel collision have been realized numerous tim es throughout modern history. Catastrophic bridge failure events due to vessel collision, which occur approximately once a year worldwide ( Larsen 1993 ), led to the development of bridge design specifications for vessel collision. The American Association of State and Highway Transportation Officials (AASHTO) Guide Specification and Commentary for Vesse l Collision Design of Highway Bridges is used along with characteristics of a given waterway and the accompanying waterway traffic to determine static design loads, which are applie d to respective piers for impact design purposes ( AASHTO 1991 ). Even though the AASHTO specifications are used for bridge pier design due to ship and barge collision, limited barge impact data was available for use in their development. In April 2004, Consolazio et al. (2006) conducted full-scale expe rimental barge impact testing on bridge piers of the Ol d St. George Island Causeway Br idge located in Apalachicola, Florida. Key findings from the experiments that are pertinent to the research presented in this thesis include: Inertial forces due to acceleration of bridge component masses can contribute significantly to overall pier res ponse during a co llision event; Significant portions of the impact load can tr ansfer (or shed) into the superstructure; and, Superstructure resistance is comprised of displacement-dependent and mass-dependent (inertial) forces. Inertial forces can produ ce a momentary increase in pier restraint during initial impact stages, and amplify st ructural demand on pier columns. Restraint of a bridge pier due to acceleration of the mass of th e overlying superstructure, and the corresponding amplification of forces developed in the pier columns during initial stages of barge collision events, are not a ddressed in current static design procedures. In contrast, dynamic time-history analysis of bridges inheren tly accounts for such am plification effects. 13

PAGE 14

However, due to the unique characteristics of eac h bridge, impact load ti me-histories vary from bridge to bridge. Coupled dynamic analysis addresses this issue by employing a single degree-of-freedom (SDF) barge mass, impact velo city, and vessel force-crush relationship to simulate barge impact at a specified bridge pier location. This me thod enables efficient time-history analysis that yields time-varying barge collision load and bridge response data specific to each bridge structure. To validate the procedure, coupled an alysis is performed and compared with experimental data for single-pi er and multiple-pier cases. A summary of all analysis cases presented in th is thesis is given in Table 1-1 However, coupled full-resolution bridge finite element (FE) models are cumbersome to analyze dynamically and time-history analysis of models of such size is not common in current practice. To facilitate use of coupled analysis in design sett ings, simplifying modifications are made to the barge and bridge structural models s ubject to impact. Specif ically, to alleviate the onus of developing an appropriate barge force-crush relationship for each of the possible barge types, a simplified crush curve that is in accordance with current AASHTO design standards is employed. Second, an algorithm is presented whic h incorporates coupled analysis but reduces a multiple-pier model to essentially a pseudo-single pier model (with adjacent spans, springs, and lumped masses) thereby signifi cantly reducing require d analysis time. Simplified-coupled dynamic barge impact analysis is performed and co mpared to results from full-resolution models for a range of bridge and collis ion configurations. In comparison to full bridge model coupled time-history analysis, results from respective si mplified models are sufficiently accurate for design purposes. Comparisons are also made betw een static and dynamic analysis predictions of bridge pier structural demand for each case. By employing coupled analysis with a simplified crush curve and simplified bridge structural model, design-oriented software is produced that can 14

PAGE 15

efficiently quantify collision induced bri dge pier demand, including capture of dynamic amplification effects. 15

PAGE 16

Table. 1-1. Case descriptions: use, configuration, and impact data. Barge impact parameters Case Use a No. Piers Spans Weight Velocity Energy 1 V 1 0 5.37 MN (604 T) 1.33 m/s (2.59 knots) 0.484 MN-m (357 kip-ft) 2 V 4 3 3.06 MN (344 T) 0.787 m/s (1.53 knots) 0.097 MN-m (71.3 kip-ft) 3 U/D 5 4 1.78MN (200 T) 1.03 m/s (2.00 knots) 0.096 MN-m (70.9 kip-ft) 4 D 5 4 18.0 MN (2020 T) 1.54 m/s (3.00 knots) 2.18 MN-m (1610 kip-ft) 5 D 5 4 68.7 MN (7720 T) 3.63 m/s (7.00 knots) 46.0 MN-m (34000 kip-ft) a V = Validation; U = Uncoupled Condensed Stiffness Calculation; D = Demonstration 16

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CHAPTER 2 LITERATURE REVIEW 2.1 Experimental Research In 1983, Meier-Drnberg conducted reduced scale impact tests on barge bows using a pendulum impact hammer. Static crush tests were also performed on reduced scale barge bows. Results from this study were used to develop relationships between kinetic energy, barge bow crush depth, and static impact force. These relationships comprise a major portion of the collision-force calculation procedure adopted in the AASHTO specifications (1991) However, this research did not address phenomena such as bridge superstructure effects and dynamic amplification, nor did the tests invo lve pier or bridge response. During this same time and afterward, full-sc ale experimental barg e collision tests were conducted in connection with the U.S. Army Corp s of Engineers (USACE). In 1989, lock gate impact tests were performed with a nine-barge fl otilla traveling at low velocities at Lock and Dam 26 near Alton, Illinois ( Goble et al. 1990 ). In 1997, four-barge flotilla impact tests were conducted on concrete lock walls at Old Lock and Dam 2, near Pittsburgh, Pennsylvania ( Patev et al. 2003 ). Additional lock wall tests were conducted with a fifteen-barge fl otilla in 1998 at the Robert C. Byrd Lock and Dam in West Virginia ( Arroyo et al. 2003 ). All of these tests were performed on lock walls and lock gates, wh ich produce fundamentally different structural responses to collision load ing in comparison to that of bridge piers. The impact testing ( Consolazio et al. 2006 ) of the old St. George Island Bridge, constructed in the 1960s, constitu tes the only experimental resear ch that explicitly measured barge impact forces on bridge piers using full-sc ale tests. The experiments were divided into three series of impact tests usi ng a single barge and vari ous pier/bridge structural configurations. The first series (termed the P1 series) consisted of eight impacts on a single, stiff channel pier 17

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(termed Pier 1-S) by a loaded barge with an impact weight of 5.37 MN (604 T) and impact velocities approaching 1. 8 m/s (3.5 knots). The second series of tests (termed the B3 series) consisted of four impacts on a multi-span, multi-pier partial bridge structure by an empty barge with an impact weight of 3.06 MN (344 T) and impact velocities approaching 0.78 m/s (1.5 knots). The third series (t ermed the P3 series) consisted of three impacts on a single, flexible pier (termed Pier 3-S) by an empty barge with an impact weight of 3.06 MN (344 T) and impact velocities approaching 0.95 m/s (1.8 knots). These tests form an important dataset for validating barge collision analysis methods. 2.2 Analytical Research Development and analysis of very high-resolu tion contact-impact FE models (those with tens to hundreds of thousands of elements) th at simulate nonlinear dynamic barge impact on bridge piers have been feasible as a research t ool for approximately a decade. In preparation for the full-scale St. George Island experimental barge impact testing, high-resolution FE pier models were developed to determine appropriate experimental conditions with respect to barge impact velocity and safety ( Consolazio et al. 2002 ). Reanalysis of the models using experimental data complimented the research findings from the experimental program ( Consolazio et al. 2006 ). High-resolution FE models of single-barges a nd multi-barge flotillas were analyzed when pier columns of various shape and dimension were subject to a variety of barge impact simulations ( Yuan 2005 ). These analytical results were used to develop a set of empirical formulas for barge impact force quantification as an improvement to the current static design procedures. Also, high-resoluti on FE single-barge models were developed and subjected to quasi-static loading by various s tiff impactors in an effort to better quantify barge force-crush relationships ( Consolazio and Cowan 2003 ). 18

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As an alternative to very hi gh resolution contact-impact FE analysis, coupled barge-pier analysis was developed ( Consolazio et al. 2004a Consolazio et al. 2004b ). Coupled analysis simulates a SDF barge model (with specified ma ss, velocity, and force-crush relationship) colliding with a multiple degree-of-freedom (MDF) bridge-pier-soil model. The coupled analysis required the use of a barge force-crush relationship, which wa s developed for a common barge type using high-resolution FE models. The force-crush curves encompass loading and unloading behavior derived from quasi-static cyclic loading ( Consolazio and Cowan 2005 ). 19

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CHAPTER 3 COUPLED BARGE COLLISION ANALYSIS 3.1 Introduction Within the context of coupled analysis, the term coupled refe rs to the use of a shared contact force between the barge and impacted bridge structure (Fig. 3-1 ). The impacting barge is assigned a mass, initial velocity, and bow force-cr ush relationship. Traveling at the prescribed initial velocity, the barge impacts a specified location on the bridge st ructure and generates a time-varying impact force in accordance with th e force-crush relationship of the barge and the relative displacements of the barge and bridge model at the impact location. The barge is represented by a SDF model, and th e pier structural configurati ons and soil parameters of the impacted bridge structure constitute a MDF mode l. The MDF pier-soil model, subject to the shared time-varying impact force, displaces, devel ops internal forces, and interacts with the SDF barge model through the shared impact force dur ing the analysis. Hence, coupled analysis automatically generates the barge impact load ti me-history specific to each bridge structural configuration and impacting barge type. This overcomes the ch allenge of pre-quantifying the time-varying barge impact load as a nece ssary component of time-history analysis. 3.1.1 Barge Loading and Unloading Behavior Barge behavior is represented by a force-crus h relationship, consisting of a loading curve, unloading curves, and a speci fied yield point (Fig. 3-2 ). The yield point represents the crush depth beyond which plastic deformations occur. Any subsequent unloadi ng beyond this point is determined according to the specified unloading curves. Until the crush depth corresponding to yield is reached, loading and unloading occurs elastically along the specified curve (Fig. 3-2 A). A series of unloading curves represent the un loading behavior at va rious attained maximum crush depths (Fig. 3-2 B). After unloading, if the barge is no longer in contact with the pier, no 20

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impact force is generated (Fig. 3-2 C). Alternatively, if re loading occurs (Fig. 3-2 D), it is assumed to occur along the previously traveled un loading curve. Plastic deformation subsequent to complete reloading occurs along the or iginally specified loading curve (Fig. 3-2 D). Additional details of th is model are given in Consolazio and Cowan (2005) 3.1.2 Coupled Analysis Algorithm Algorithmically, the coupled analysis proce dure involves a SDF barg e code interacting with a separate nonlinear dynamic pier-soil analysis code at a specified node of the MDF pier-soil model. Specifically, coupled analysis utilizes an explicit time-s tep barge impact force determination procedure and links the output, th e resulting impact force, with a respective numerical MDF pier-soil model analysis code ( Hendrix 2003 ). The pier-soil analysis code then responds to the impact force by generating itera tive displacements and forces throughout the MDF model. 3.1.3 Use of Experimental Data for Coupled Analysis Validation Coupled analysis was previously develope d and demonstrated as a proof-of-concept using analytical data ( Consolazio and Cowan 2005 ). Output from very high-resolution FE models consisting of a MDF impacting barge and a MDF impacted pi er were compared to output obtained from coupled analysis of a SDF barge a nd MDF pier model. At present, experimental data is now available for validation of the coupl ed analysis procedure. Using data from the full-scale barge impact experiments ( Consolazio et al. 2006 ), validation of th e coupled analysis procedure is carried out in four stages: select appropriate pier structures from the experimental dataset; develop respective models in a nonlinear dynamic finite element analysis (NDFEA) code capable of conducting coupled an alysis; analyze the models using respective barge impact conditions and coupled analysis; and, compare timehistory results from the coupled analysis to those obtained experimentally. 21

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3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2 Data was collected more extensively from Pier 1-S than from any other pier in the 2004 full-scale experimental test set ( Consolazio et al. 2006 ). Furthermore, a single pier is representative of the type of st ructure often used in static desi gn procedures for barge collision analysis ( Knott and Prucz 2000 ). Hence, a single pier (Pier 1-S) was selected for coupled analysis validation using experimental data (Fig. 3-3 A). Of the eight experimental tests conducted on Pier 1-S, the fourth test (termed P1T4) consisted of a head-on impact at an undamaged portion of the barge bow, as would be assumed in bridge design. Test P1T4, with velocity and impact weight as specified in Table 1-1 was selected for Case 1. In addition to validating the c oupled analysis procedure for a single-pier, data from the partial bridge (B3 series) tests were employed for validation purposes. Regarding impact conditions used for validation, the fourth test (termed B3T4) gene rated the largest pier response among the B3 test series. Hence, test B3T4, with velocity and impact weight as specified in Table 1-1 was selected for Case 2 (Fig. 3-3 B). 3.3 Software Selection and Model Development Coupled analysis was previously implemented in the commercial pier analysis software, FB-Pier (2003) and was shown to produce force and disp lacement time-histories in agreement with those obtained from high-resolution cont act-impact FE pier-soil model simulations. Subsequent to implementation of the coupled an alysis procedure in FB-Pier, an enhanced package called FB-MultiPier (2007) was released. FB-MultiPier possesses the same analysis capabilities as FB-Pier (including coupled analysis) but also has the ability to analyze bridge structures containing superstructure elements. Therefore, FB-MultiPier was selected for all model development and analysis conducted in this study. 22

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FB-MultiPier employs fiber-based frame elements for piles, pier columns, and pier caps; flat shell elements for pile caps; beam el ements, based on gross section properties, for superstructure spans; and, distri buted nonlinear springs to represent soil stiffness. Transfer beams transmit load from bearings, for which the stiffness and location are user-specified, to the superstructure elements. FB-MultiPier permits Rayleigh damping, which was applied to all structural elements in the models used for th is study such that approximately 5% of critical damping was achieved over the first fi ve natural modes of vibration. FB-MultiPier allows either linear elastic or material-nonlinear analysis of structural elements. Linear elastic analysis was selected for all structural (non-soil) elemen t components of models used in this study. This approach wa s taken because the 2004 full-scale barge impact experiments were non-destructive ( Consolazio et al. 2006 ) and post-test inspection of the pier structures subjected to collision loading indicate d that the structural components had remained largely in the elastic range. Structural models of Case 1 and Case 2 (Fig. 3-3 A and Fig. 3-3 B, respectively) were developed from original construction drawings and direct site i nvestigation measurements. The Case 2 structural model was limited to four piers, with springs representing the stiffness contributions of piers beyond Pier 5-S (Fig. 3-3 B), as contribution to structural response from these piers was expectedly small ( Consolazio et al. 2006 ). The soil model spring system for Case 1 was developed based on boring logs and dynamic soil properties obtained from a geotechnical investigation conduc ted in parallel with the 2004 full-scale barge impact testing ( McVay et al. 2005 ). For the development of the Case 2 soil-spring system, boring logs formed the sole data source available. 23

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For each model, a preliminary analysis was conducted in which the experimentally measured time-history load was directly applied at the impact point for the specified test case. The resulting displacement time-history of the stru cture was then compared to the experimentally measured displacement time-history at the impact point. Output from the direct analysis and comparison to experimental data aided in calib ration of each model. Consequently, because analytical application of the experimentally measured load time-history was shown to produce pier response in agreement with that of the experimental data, the direct analysis comparison provided a baseline means of judging the efficacy of the coupled analysis procedure. 3.3.1 Coupled Analysis Module Parameters Within the coupled analysis procedure, the barge is modeled by a SDF point mass and nonlinear compression spring. Barge impact conditions for the validation cases (P1T4 and B3T4) were directly measured during the experi mental tests. Thus, the experimental impact weights and velocities were direc tly input into analytical Case 1 as 5.37 MN (604 T) traveling at 1.33 m/s (2.59 knots) and Case 2 as 3.06 MN (344 T) traveling at 0.79 m/s (1.53 knots), respectively. The loading portion of the barge force-crush relationship used for Case 1 and Case 2 (Fig. 3-4 ) was developed from impact-point force a nd displacement time-history data measured during the P1T4 test; P1T4 was selected beca use of the undamaged bow impact location and head-on nature of the collision event. The portion of the barge force-crush relationship up to the peak force was obtained by performing coupled an alysis using P1T4 impact conditions, and an initially arbitrary force-crush relationship. After analysis completi on, the coupled analysis prediction of impact force was compared to that experimentally measured during the P1T4 test. The analytical force-crush relationship was then adjusted to more closely match that measured experimentally. After several iterations of this calibration process, a force-crush loading 24

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relationship was obtained that produced force time-history data in agreement with the experimental measurements of impact force. The experimentally derived loading portion of the force-crush curve (Fig. 3-4 ) has a peak impact force value of 5.74 MN (1065 kips) at a crush depth of 12.07 cm (4.75 in). Explicit derivation of forces beyond this point, pertai ning to the barge-bow impact force-crush relationship, was not possible using the experiment al dataset. However, barge bow force-crush data are available in the literature that apply to the shape of the impacted pier in the P1T4 test; specifically, a rectangular (flat) surface imp actor. This data was obtained by subjecting a high-resolution FE barge model to quasi-static crushing by square (flat) 1.8 m (6 ft) and 2.4 m (8 ft) impactors ( Consolazio and Cowan 2003 ). In the present study, barge force-crush parameters pertaining to crus h depths beyond that corresponding to the peak force were proportioned from high-resolution FE force-crush data Specifically, these parameters are: the yield point, structural soften ing beyond the peak force, and the force plateau level beyond softening (Fig. 3-4 ). The unloading curves (Fig. 3-4 ) chosen for Case 1 and Case 2 exhibit steeper unloading paths at smalle r crush-depths and shallower unl oading paths at larger crush depths. The unloading curves are consistent, wi th respect to qualitative shape, with those employed in a prior study for a common barge type subject to quasi-static crush by square piers ( Consolazio and Cowan 2005 ). 3.3.2 Accounting for Payload Sliding During Impact Testing During the Pier 1-S test series, payload in the form of 16.76 m (55 ft) reinforced concrete bridge superstructure span segments was placed on the barge to simulate a loaded impact condition. However, the payload was observed to slide during the collision events, implying the development of frictional for ces and dissipation of energy ( Consolazio et al. 2006 ). In general bridge design, the payload would not be assumed to slide. However, for the purpose of 25

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validating the coupled analysis pr ocedure as accurately as possible, enhancements were made to the pre-existing coupled analysis procedure to numerically acc ount for payload sliding (Fig. 3-5 ). At each time-step and iteration, th e ratio of barge acceler ation (which, before sliding occurs, is equal to the payload acceleration) to gravitat ional acceleration was computed and compared to the static coefficient of friction ( ) between the barge and the payload. When the acceleration ratio exceeded the static coefficient of friction, sliding was initiated (Fig. 3-5 B). At sliding initiation, the barge payload was assigned an initial velocity ( ) relative to the underlying barge, equal to the corresponding current velocity of the barge-payload system. The payload was assumed to continue sliding until the initial payload kinetic energy was completely dissipated through friction. At all points in time during whic h sliding occurred, a constant frictional force, equal to the product of the static coefficient of friction and the weight of the payload ( ), was applied to the barge. When th e sliding kinetic energy of the payload barge was dissipated, the payload mass ( ) and barge mass ( ) were assumed to rejoin as a single loaded barge-payload system, as before sliding (Fig. 0u pW pm bm 3-5 A). For the P1T4 test, a sliding distance of 0.376 m (14.8 in) was predicted from the module modi fications, which agreed very well with the observed payload slide of approximately 0.38 m (15 in). 3.4 Comparison of Analytical and Experimental Data 3.4.1 Case 1 The Case 1 impact load time-history (Fig. 3-6 A) is nearly identical to the respective experimental curve up to the peak load, and expectedly so, because the portion of the barge force-crush relationship (Fig. 3-4 ), up to the peak impact load, was derived from the impact force and displacement data acquired during the Case 1 (P1T4) collision even t. Additionally, the analytical and experimental agreement for por tions of the Case 1 force time-history curve 26

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beyond the peak justifies the assumptions made during the development of the load softening, load plateau, and unloading components of the force-crush curve (Fig. 3-4 ). The analytically determined peak value of pier displacement exceeds the experimental value by 16% (Fig. 3-6 B). Supplementary coupled analys es of the Pier 1-S model were conducted with impact velocities measured durin g similar and higher impact-energy P1 series tests. Comparisons of displacement output from these analyses ( Appendix A ) to respective experimental data show discrepa ncies of comparable or lesser magnitude to those of Case 1. 3.4.2 Case 2 Case 2, in direct contrast to Case 1, consis ts of a low-energy barge collision event on a flexible pier with superstructure restraint. Case 1 and Case 2 share only the barge force-crush relationship derived from the P1T4 experimental data. The Case 2 experimental and analytical force time-histories (Fig. 3-7 A) embody similar qualitative shapes; however, the analytical peak force magnitude is larger than the experimental counterpart. Despite the disparity in magnitude, numerical integration of the curv es indicates that the shape and magnitude of the impulse, as a function of time, agree well between the e xperimental and analytical results (Fig. 3-8 ). This implies that the change in momentum of th e barge was accurately predicted by the coupled analysis and produced a pier response simila r to that measured experimentally. The concord of the analytical and experimental time-history of displacement (Fig. 3-7 B) demonstrates the proficiency of the coupled anal ysis procedure in adequately predicting barge collision response for piers of varying stiffness. Accurate pier response predictions are maintained while incorporating superstructure eff ects. Agreement of pier response is the most important outcome of the coupled analysis pr ocedure, as the accompanying internal forces generated throughout the MDF pie r-soil model ultimately govern the pier structural member design. The coupled analysis procedure effectiv ely shifts the analytical focus away from 27

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determination of the barge impact force, and centers the emphasis on determining pier structural demand. Coupled analysis also inherently cap tures dynamic phenomena exhibited during barge-bridge collisions. As evidenced by the ti me-history plots of Case 1 and Case 2 (Fig. 3-6 and Fig. 3-7 ), the peak impact force and displacement do not occur simultaneously for individual experimental test cases involving appreciable im pact-energies ( Consolazio et al. 2006 ). Static procedures do not account for peak load-displ acement time disparity or the potential amplification effects intrinsic to the early stag es of collision events for bridge structures. Coupled analysis automatically accounts for these effects. 28

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F F Pier structure Soil stiffness Crushable bow section of barge Barge Barge and bridge models are coupled together through a common contact force SDF barge model MDF bridge model Bridge motion Barge motion superstructure Figure 3-1. Coupling between barge and bridge (after Consolazio and Cowan 2005 ). 29

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Impact Force Crush Depth Yield point Elastic loading/unloading Loading curve A Impact Force Crush Depth Unloading curve Initiation of unloading B Impact Force Crush Depth Barge and bridge not in contact C Impact Force Crush Depth Plastic loading occurs along loading curve Reloading occurs along same path as unloading D Figure 3-2. Stages of barge crush (after Consolazio and Cowan 2005 ). A) Loading. B) Unloading. C) Barge not in contact with pier. D) Reloading and continued plastic deformation. 30

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ImpactPier 1-S A ImpactPier 2-SPier 3-SPier 4-S Pier 5-S Springs modeling additional spans beyond Pier 5-S B Figure 3-3. Structural c onfigurations analyzed ( not to relative scale). A) Case 1: Single pier. B) Case 2: Four pi ers with superstructure. 31

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Crushable barge bow SDF Barge MDF Pier Crush Distance (mm) Crush Distance (in)Impact Force (MN) Impact Force (kips) 0 50 100 150 200 250300 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 150 300 450 600 750 900 1050 1200 Loading Curve Unloading Curves Figure 3-4. SDF barge force-crus h relationship derived from expe rimental and analytical data. 32

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F Barge Payload Total barge-payload mass contributes to impact force No relative motion Gravitational acceleration < Static coefficient of friction between barge and payload Barge acceleration Wp mpmb A F Barge Barge mass and constant payload frictional force contribute to impact force u0 Payload WpGravitational acceleration > Static coefficient of friction between barge and payload Barge acceleration mpmb Wp B Figure 3-5. Sliding criteri on between payload and barge. A) No sliding. B) Sliding. 33

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Time ( s ) Impact Force (MN) Im p act Force ( ki p s ) 0 0.25 0.5 0.75 1 1.251.5 0 1 2 3 4 5 0 200 400 600 800 1000 Coupled Analysis Output Experimental Data A Time ( s ) Pier Displacement (mm) Pier Dis p lacement ( in ) 0 0.25 0.5 0.75 1 1.251.5 -5 -2.5 0 2.5 5 7.5 10 12.5 15 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 Coupled Analysis Output Experimental Data B Figure 3-6. Comparison of Case 1 coupled analysis output and P1 T4 experimental data. A) Impact force. B) Pier displacement. 34

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Time ( s ) Impact Force (MN) Im p act Force ( ki p s ) 0 0.25 0.5 0.75 1 1.251.5 0 0.5 1 1.5 2 0 100 200 300 400 Coupled Analysis Output Experimental Data A Time ( s ) Pier Displacement (mm) Pier Dis p lacement ( in ) 0 0.25 0.5 0.75 1 1.251.5 -20 -10 0 10 20 30 40 50 -0.40 0.00 0.40 0.80 1.20 1.60 Coupled Analysis Output Experimental Data B Figure 3-7. Comparison of Case 2 coupled anal ysis output and B3T4 experimental data. A) Impact force. B) Pier displacement. 35

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Time ( s ) Impulse (MN-sec) Im p ulse ( ki p -sec ) 0 0.25 0.5 0.75 1 1.251.5 0 0.1 0.2 0.3 0.4 0 20 40 60 80 Coupled Analysis Output Experimental Data Figure 3-8. Comparison of Case 2 coupled analysis output and B3T4 experimental data: Impulse. 36

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CHAPTER 4 SIMPLIFIED MULTIPLE-PI ER COUPLED ANALYSIS 4.1 Overview At current computer processing speeds, barge impact time-history analysis of bridge models can require between tens of minutes to several hours of processing time. Two simplifications may be applied to the coupled analysis of bridge structural models to reduce analysis time and facilitate it s use in design settings. First, a simplified alternative to the experimentally and analytically derived crush curve may be used in design when more detailed barge force-crush behavior is not available. The bilinear curve found in the current static AASHTO design specifications (Fig. 4-1 ) may be used for general barge-bridge collision design applications. Second, multiple-pier models may be reduced to a pseudo-single pier model (with two attached superstructure spans) and analyzed to produce results that match to a satisfactory degree of accuracy, those obtained from corre sponding full-resolution (multi-span, multi-pier) models. 4.2 Linearized Barge Force-Crush Relationship The nonlinear loading portion of the barge force-crush curve, developed from P1T4 experimental data (Fig. 4-1 ), is specific to the barge used in the 2004 impact experiments. Phenomena such as structural-softening beyond th e peak force level for each combination of barge type and impactor shape are not well docum ented in the literature and further study is warranted before these components of barge bow crushing behavior may be quantified for general application. Hence, the use of a simple bilinear force-cr ush relationship, such as that found in the AASHTO barge-collisio n specifications, is desirable at present as long as such a curve produces reasonable results. 37

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The AASHTO force-crush relationship is in reasonable agreement with the P1T4 experimentally determined based force-crush cu rve. The crush depth at which the AASHTO and experimental curves shift from the initial linear portion to the subseque nt linear portion occur at 103.63 mm (4.08 in) and 120.65 mm (4.75 in ), respectively. For c onvenience, these locations are termed the shift points. Note that th e AASHTO force corresponding to the shift point, 6.17 MN (1386 kips), is significantl y greater than that found in th e experimentally based curve, 4.74 MN (1065 kips). Additionally, the AASHTO curv e exhibits positive stiffness regardless of crush depth, whereas the curve employed in the validation of the coupled analysis method is assumed to exhibit perfectly plastic behavior at high cr ush depths (Fig. 4-1 ). Consequently, the AASHTO curve yields higher impact forces than the experiment al data for all barge crush depths, and is therefore conservative. 4.3 Reduction of the Bridge Model Barges impart predominantly horizontal forc es to impacted bridge piers during collision events. Displacement and acceleration based supers tructure restraint (due to superstructure stiffness and mass, respectively) can attract a significant porti on of the horizontal forces and cause the impact load to she d to the superstructure ( Consolazio et al. 2006 ). The horizontal force shed to the superstructure then propagate s (initially) away from the impacted pier. Consequently, lateral translationa l and plan-view rotational stiffnesses influence the structural response as the force propagates through the supers tructure from the impacted pier to adjacent piers. Simultaneously, the distri buted mass of the superstructure alternates between a source of inertial resistance to a source of in ertial load that respectively re strains or must be absorbed by other portions of the bridge structure. Simplif ication of the multiple-pier structural model, therefore, must adequately retain the influence of adjacent non-imp acted (the lateral translational 38

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and plan-view rotational stiffnesses of the adj acent piers; and, the dynamically participating mass of the superstructure). 4.3.1 Uncoupled Condensed Stiffness Matrix The stiffness DOF of a bridge model, beyond the superstructure sp ans that extend from the impacted pier (Fig. 4-2 ), may be approximated by equivalent lateral translational springs and plan-view rotational springs. Thes e springs are linear elastic a nd represent the predominant DOF of the linear elastic structural elements in th e full-resolution model at piers adjacent to the impacted pier. Soil nonlinearitie s at piers other than the imp acted pier are ignored during formation of the translational and rotational springs. Replacement of numerous DOF from the fl anking portions of a full bridge model (Fig. 4-2 ) by two uncoupled springs at each end of a simplified two-span single-pier model may be described in terms of a condensed stiffness matrix: K K KK Kcoupling coupling condensed (4.1) where is the condensed stiffness matrix of the flanking bridge portion eliminated at each side of the impacted pier; is the condensed lateral tr anslational stiffness term; is an off-diagonal stiffness term that couples the translational DOF to the rotational DOF; and is the condensed stiffness plan -view rotational stiffness term. In the simplified model, the diagonal terms and are represented by translational a nd rotational springs, respectively, and the terms are neglected. The exclusion of in the simplified model is justified by examining the forces generated by the condensed stiffness terms on one side of an example five-pier model. condensedK K couplingK K K K couplingK couplingK 39

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A channel pier was added to the previously discussed four-pier Case 2 model, using bridge plans of the old St. George Island Bridge. This new five-pier model (Fig. 4-3 ) is referred to as Case 3, as defined in Table 11 Through flexibility inversion (Fig. 4-2 ), the left-flanking bridge structure in Case 3 (cons isting of Pier 1-S to Pier 2-S) is reduced to the 2-DOF linear elastic condensed stiffness matrix in Eq. ( 4.1 ), where K 97.0 MN/m (554 kip/in); 2.58E+05 MN-m/rad (2.28E+ 09 kip-in/rad); and, K couplingK 398 MN/rad (8.95+04 kip/rad). In row one of the term may be interp reted as a horizontal shear force generated when a unit rotation (1 ra d) is induced at the right-most node of the left-flanking structure. Static application of a load of 1.84 MN (414 kips) to the central pier of the Case 3 five-pier model i nduces a plan-vie w rotation of condensedK couplingK 6.35E-06 rad at the location of the condensed stiffness. The horizontal shear produced as a result of this rotation is: coupling couplingKV (4.2) where is the shear produced from the coupling of rotational and translational DOF. In this instance, 2.53E-03 MN (0.568 kips). In compar ison, the horizonta l shear produced as a result of diagonal lateral stiffness couplingV couplingV K 97.0 MN/m (554 kip/in) and lateral displacement 4.62 mm (0.182 in) is: KV (4.3) where is the shear produced dir ectly from lateral transl ation. For this loading, 0.448 MN (101 kips). V V The amount of horizontal shear generated at the location of the condensed stiffness matrix, due to the coupling stiffness term, is very small relative to the amount of horizontal shear 40

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generated due to the diagonal stiffness term ( is only 0.6% of ). A similar examination of the and terms yields ratios of comparable values ( couplingV V K couplingK Appendix B ). The large difference in magnitude between th e two shear forces demonstrates that the off-diagonal stiffness terms of generate negligible forces relativ e to those generated by the diagonal stiffness terms. Uncoupling the condensed stif fness terms by applying two independent springs is therefore warranted for design applications, as the uncoupled sp rings form a reasonable static approximation of the stiffness of the excluded portions of the model. condensedK As a further simplification to the full-bri dge model, the diagonal stiffness terms and may be approximated by direct inversion of th e individual diagonal flexibility coefficients. Specifically, this entails directly inverting the translational K K V and rotational M displacements, respectively, induced by the appl ication of a unit shear force and unit force-couple on the applicable flanking structure (Fig. unitV unitM 4-2 ) and is simpler to carry out. ). This approximation produces only nominally different magnitudes of stiffness with respect to that obtained by a flexibility matrix inversion ( Appendix B If significant nonlinear behavi or is expected at non-impacted piers, then loads representative of the forces that will be shed to the superstructure, and subsequently transmitted into these piers, should be used to compute displ acements (flexibility coeffi cients). Inversion of flexibility coefficients formed in this manner yields a condensed secant stiffness that may then be employed in the simplified model as described previously. 4.3.2 Lumped Mass Approximation Mass is attributed to each node of the NDFEA models in this st udy, which consequently, approximate a distributed mass system under dynami c loading. Therefore, a portion of mass of the excluded structural component s is assumed to contribute to the structural response of the 41

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simplified models. This mass is assumed to fall within the tributary area (Fig. 4-4 ) extending along the spans beyond the piers adja cent to the impacted pier of a given full-resolution model. The mass is lumped and placed at respective ends of the simplified model. The lumped mass simplification is combined with the stiffness approximation (Fig. 4-2 ) to complete the simplified two-span single-pier model. 4.4 Multiple-Pier Coupled Analys is Simplification Algorithm Simplified coupled analysis occurs in two stages First, the two-span single-pier model is assembled by replacing extraneous portions of the multiple-pier model with uncoupled linear elastic springs and half-span lump ed tributary masses. Coupled an alysis is then performed, as previously discussed, with the AASHTO bilinea r crush-curve being employed for the barge. The simplification algorithm automatically reta ins the ability to capture dynamic effects, such as amplification, not addres sed in static procedures. Furt hermore, hundreds to thousands of DOF are eliminated because the non-impacted piers and respective superstructure spans from the full-resolution model are omitted from the model. 42

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Crush Distance ( mm ) Crush Distance (in)Impact Force (MN) Im p act Force ( ki p s ) 0 50 100 150 200 250300 0 2 4 6 8 10 0 1 2 3 4 5 6 7 0 300 600 900 1200 1500 Derived from experimental data AASHTO Figure 4-1. Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown). 43

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K = F =Impact location on full bridge model Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spansP-1 P-2 P-3 P-4 P-5 P-6 P-7 Apply unit shear force at center of P-3 pile cap and center of P-5 pile capVunit P-1 P-2 P-3 P-5 P-6 P-7 Record shear-induced translations and rotations at center of P-3 pile cap and center of P-5 pile cap Apply unit moment at center of P-3 pile cap and center of P-5 pile capMunit P-1 P-2 P-3 P-5 P-6 P-7 Munit Record moment-induced rotations and translations at center of P-3 pile cap and center of P-5 pile cap Form 2x2 left-flanking and right-flanking flexibility matrices using displacements and invert to form condensed stiffness Left-flanking structure Right-flanking structure P-1 P-2 P-3 P-5 P-6 P-7 VLP-1 P-2 P-3 P-7 P-6 P-5 VR Vunit VL VR P-4 Neglect off-diagonal stiffness and replace flanking-structures in full bridge model with diagonal stiffness as uncoupled spring s Impact location on two-span single-pier model VLML Kcoupling LK L LKLK L coupling 0K L L0K L K R LK R0 0LK LK L LK RK R L L condensed -1 -1K = F = Kcoupling RK R LKRK R coupling R R condensed -1 -1MR MR P-7 P-6 P-5 P-1 P-2 P-3 MLMLMLVLVRMRMRVR Figure 4-2. Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model. 44

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Pier 1-S Pier 2-S Pier 3-S Pier 4-S Pier 5-S Impact Figure 4-3. Structural configur ation analyzed in Case 3. 45

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Impact location on full bridge model Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spansP-1 P-2 P-3 P-4 P-5 P-6 P-7 Calculate mass of half-span beyond P-3 mHL P-1 P-2 P-3 P-5 P-6 P-7 P-4 Impact location on two-span single-pier model Calculate mass of half-span beyond P-5 Form lumped mass equal to mHLForm lumped mass equal to mHRApply lumped masses in place of flanking-structure masses in full bridge model Left-flanking structure Right-flanking structureP-1 P-2 P-3 P-5 P-6 P-7 mHR mHL mHR mHL mHR Figure 4-4. Plan view of multiple pier numerical model and location of lumped masses in twospan single-pier mode. 46

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CHAPTER SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES 5.1 Introduction To illustrate the efficacy of the simplifica tion algorithm, three demonstration cases (FB-MultiPier bridge models) are presented. Each model was developed using methods representative of those employed by bridge desi gners. Impact conditions prescribed for the models are such that the range of scenarios en countered in practical bridge design for barge impact loading is well represented. The cases employ the AASHTO bilinear barge crush-curve, consist of impacted pier models of increasing impact resistance, and are subjected to impacts with corresponding increases in impact energy. Time -history output of internal pier structural member forces obtained from both full-resolu tion and simplified models are subsequently compared for each case. Each full-resolution model contains five pier s: a centrally located impact pier and additional structural components (soil, non-impacted piers, and superstructure spans) for a length of two spans to either side of the central pier. A five-pier model contains a sufficient number of piers and spans such that inclusion of additiona l piers would increase analytical computation costs without appreciably improving the computed structural response. The appropriateness of the decision to limit the full-resolution models to five piers is substantiated by the consistently negligible acceleration response exhibited by th e outer-most piers included in the five-pier models. Alternatively stated, the added restra int provided by including additional piers is not necessary, as the outer-most piers of the five -pier models are only nominally active throughout the barge impact analysis. 47

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A single time-step increment, 0.0025 sec, was employed for all demonstration analyses. Each model also utilized Rayleigh damping, which is configured such that the first five vibration modes undergo damping at approximat ely 5% of critical damping. 5.2 Geographical Information, Structural Configuration, and Impact Conditions 5.2.1 Case 3 The first demonstration case c onsists of analysis of the previously described Case 3 model (Fig. 4-3 ). This model was based on the old St. George Island Bridge from the Apalachicola Bay area, linking St. George Island to mainland Florida, in the southeastern United States. Apalachicola Bay is located approximate ly 80.5 km (50 mi) southwest of Tallahassee, Florida in the panhandl e portion of the state. The structure of the old St. George Island Br idge, constructed in the 1960s, was detailed in a prior report ( Consolazio et al. 2006 ). Pertinent to demonstration Case 3, the superstructure spanning from Pier 2-S to Pier 5-S (Fig. 4-3 ) consisted of 23 m (75.5 ft ) concrete girder-and-slab segments overlying concrete piers with waterlin e footings. Spanning the navigation channel and one additional pier to either side, a 189 m (619.5 ft) continuous three-span steel girder and concrete slab segment rested on Pier 1-S and Pier 2-S, each containing a mudline footing and steel H-piles. The central pier in Case 3, Pi er 3-S, contained two tapered rectangular pier columns, with a 1.5 m (5 ft) wide impact face at approximately the same elevation as the top of a small shear strut that spanned be tween the two 1.2 m (4 ft) thick waterline pile-cap segments. The pier rested on eight battered 0.5 m (20 in) square prestressed c oncrete piles, each containing a free length of approximately 3.7 m (12 ft). The Case 3 FE model includes the southern ch annel pier and extends southward from the centerline of barge traffic. The impacted pier Pier 3-S, was construc ted before the AASHTO provisions were written (1991), a nd was flexible as it was not a channel pier. The pier was 48

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located 115.8 m (380 ft) from the channel centerlin e, which was significantly closer to a distance of three times the impacting vessel length, 138 m (450 ft), than the distance to the edge of the navigation channel, 37.75 m (124 ft). Per the AAS HTO specifications, the pi er would be subject to a reduced impact velocity, approaching th at of the yearly mean current velocity ( Consolazio et al. 2002 ). The kinetic energy (Table 1-1 ) associated with an empty jumbo-hopper barge drifting at the yearly mean current velocity for the Apalachicola Bay is representative of a low-energy impact condition. 5.2.2 Case 4 Escambia Bay abuts Pensacola, Florida, in the southeastern United States. Case 4 (Fig. 5-1 ) consists of impact analysis of a model based on the Escambia Bay Bridge. Structural components of this bridge model were derived from bridge plans developed in the 1960s. The superstructure spanning from Pier 2-W to Pier 2-E consists of a 125 m (410 ft) continuous three-span steel girder and conc rete slab. A 28 m (92 ft) concre te girder-and-slab segment spans the underlying concrete piers beyond Pier 2-E. All piers, except fo r the channel piers denoted as Pier 1-E and Pier 1-W, contain two pier columns, a shear wall, pile cap, and waterline footing foundation. The channel piers in Case 4 each contain two tapered rectangular pier columns, with a 2.6 m (8.5 ft) wide head-on impact face at appr oximately the mid-height elevation of a 5.3 m (17.5 ft) shear wall. The pier columns and shear wall overlie a 1.5 m (5 ft) thick mudline footing and 1.8 m (6 ft) tremie seal. The channel pier f oundations consist of eighte en battered and nine plumb 0.6 m (24 in) square prestressed concrete piles. The Case 4 FE model includes both of the ch annel piers and three a uxiliary piers. The impacted pier, Pier 1-E, was constructed be fore the AASHTO provision s were written (1991), but contains large impact resistance relative to the impacted pier from Case 3, as Pier 1-E is a channel pier. Impact on a channel pier with a relatively high impact resistance was chosen to 49

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demonstrate the accuracy of the simplification algorithm for the medium-energy impact of a fully-loaded jumbo-hopper barge and towboat, traveling at a higher speed than the mean waterway velocity (Table 1-1 ). 5.2.3 Case 5 Case 5 (Fig. 5-2 ) consists of impact analysis of piers from the new St. George Island Bridge, which replaced the old St. George Island Bridge in 2004. The structural model of the new St. George Island Bridge was derived from construction drawings. Per these drawings, Pier 46 through Pier 49 support fi ve cantilever-constructed Flor ida Bulb-T girder-and-slab segments at span lengths of 62.25 m (207.5 ft ) for the channel and 78.5 m (257.5 ft) for the flanking spans. Due to haunching, the depth of th e post-tensioned girders vary from 2 m (6.5 ft) at drop-in locations to 3.7 m (12 ft) at respec tive pier cap beam bear ing locations. Simply supported Florida Bulb-T beams with a depth equal to that of the haunched beams at the drop-in locations span either side of Pier 50. All piers included in this model contain two pier columns, a shear strut centered near a respective pier column mid-height, a pile cap, and a waterline footing system. The central pier in Case 5, Pier 48, c ontains two round 1.8 m (6 ft) pier columns, a (6.5 ft) thick pile cap, and fourteen battered a nd one plumb 1.4 m (4.5 ft ) diameter prestressed cylinder piles with a 3 m (10 ft) concrete plug extending earthward from the pile cap. The new St. George Island Bridge was de signed in accordance with current AASHTO barge collision design standards and provided a means of validat ing the simplification algorithm for barge impact energies similar to those used in present day design. The Case 5 FE model includes both of the channel piers and three auxiliary piers. Th e impacted pier, Pier 48 was designed for a static impact load of 14.48 MN ( 3255 kips). With respect to the static AASHTO design impact load, an energy equivalent impact condition ( Appendix D ) is employed in Case 5. The prescribed vessel mass and velocity yields an impact kinetic ener gy equivalent to four 50

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fully-loaded jumbo class hopper barges and a towboat traveling slightly ab ove typical waterway vessel speeds for the Apalachicola Bay waterway (Table 1-1 ). 5.3 Comparison of Simplified and Full-Resolution Results In bridge design applications related to waterway vessel collision, the analytically quantified internal forces in a given pier st ructure govern subsequent structural component sizing. Hence, accurate determination of internal forces is a necessary outcome of a bridge structural analysis method. To highlight the ability of simplified analysis to accurately quantify design forces over the full range of impacted pier st ructures, time-histories of internal shear force induced by the impact loading are shown for the top of the impacted pier column and an underlying pile-head node for Case 3 through Case 5 shown in Fig. 5-3 through Fig. 5-5 respectively (additional comparisons of the impact force, displ acements, and internal moments are documented in Appendix C ). The predictions of load duration (the time durin g which the barge and pier are in contact), common to both simplified and full-resolution an alyses, are 0.26 sec, 0.78 sec, and 2.9 sec, respectively, for Case 3, Case 4, and Case 5. At points in time greater than the respective load durations, each bridge is in an unloaded condition and underg oes damped free-vibration. Accordingly, pier response to time-history ba rge collision analysis ma y be divided into two phases: first a load-phase then a free-vibration phase. In all three demonstration cases, peak internal pier forces occur during the load-pha se (0.13 sec, 0.17 sec, and 2.1 sec for Case 3, Case 4, and Case 5, respectively). Theref ore, agreement between the simplified and full-resolution models is most cr itical during the load-phase, as forces obtained during this phase ultimately govern bridge pier member design. Simplified analysis retains the ability to accurately capture forces during the load-phase of response (Fig. 5-3 through Fig. 5-5 for each case, respectively). Peak shear forces generate d by full-resolution and simplified analysis during 51

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the load-phase for each case differ by less than 2%. Reduced, yet still reasonable, agreement with respect to period of response and subsequent peak values of shea r force occur during the free-phase of response for each case, however, su ch agreement is less critical and typically irrelevant for design purposes. Case 3 through Case 5 were analyzed on a Dell Latitude D610 notebook computer using a single 2.13 GHz Intel PentiumM CPU and FB-Mu ltiPier. The computation times necessary for analysis completion of the simplified models were only 8%, 7.5% and 8.4% of those required for the full-resolution models of Case 3 through Case 5, respectively (Fig. 5-6 ). All cases required significantly less than an hour to co mplete 800, 800, and 1600 time -steps of analysis, respectively. Engineering judgmen t is required to determine the appropriate amount of analysis time specified. However, analysis generally need not be conducted beyond the end of load-phase, as evidenced by forces during the load-phase for Case 3 through Case 5. 5.4 Conclusions from Simplified-Coupled Analysis Demonstrations Excellent agreement is observed during the load-phase response of the full-resolution and simplified test cases, especially with respect to peak internal forces generated at various locations of the impacted piers. From a design pers pective, reasonable agreement between full and simplified analytical results is also observ ed during the free-phase portions of respective time-history responses. Time-histories of intern al shear force, moment, and displacement are adequately captured by the simplification algor ithm, despite the simplifying stiffness and mass assumptions that are made. The time necessary to analyze the simplified mo dels is significantly less than one hour in each case, which is in contrast to the several hours necessary to analyze respective full-resolution models. It should be noted that all FB-MultiP ier analyses were conducted in compilation debug 52

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mode. Considerable additional reduction in analysis time is expected if the same analyses were to be conducted in a release compilation or commercial version of FB-MultiPier. 5.5 Dynamic Amplification of the Impa cted Pier Column Internal Forces Application of the simplification algorithm to each of the demonstra tion cases inherently incorporates mass and acceleration based inertial forces that em erge from integration of the dynamic system equations of motion. The simplif ication algorithm accurately captures dynamic amplification of forces generated in the pier columns that would be absent from static analysis results. Dynamic amplification in each case ma y be quantified by considering the maximum pier column shears developed in models subjected to static application of the peak impact load predicted through the coupled anal ysis. The peak shear and mome nts developed in the pier due to static loading are then compared to thos e from the simplified and full-resolution dynamic analyses (Fig. 5-7 ) With respect to peak pier column structural demand, the dynamic analyses are in excellent agreement with each other for all ca ses. However, the peak magnitudes of the statically generated shears and moments, re spectively, correspond to 59% and 64% of the magnitude of the dynamically obtained counterparts for Case 3; and, 38% and 37%, respectively, for Case 4 (Fig. 5-7 ). In each of these cases, a static analysis employing a dynamically obtained peak impact load leads to un-conservative predic tions of peak pier column demand, as static analysis only encompasses stiffness consideratio ns. In contrast, dynamic analyses incorporate both stiffness and inertial effects associated with the superstructure and therefore capture dynamic amplification of pier column forces due to the mass of the superstructure. Furthermore, the simplified procedure retains the ability to capture pier column force amplification as evidenced by the agreement between the simplified and full-resolution output pertaining to peak pier column demand. 53

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The impact energy specified in Case 5 is of sufficient magnitude to cause the barge and impacted pier to remain in contact for a time gr eater than several periods of the fundamental pier vibration mode. Consequently, the inertial forces in the impacted pier begin to dissipate due to damping effects. This is evidenced by attenuation of oscillation exhibited in the pile head shear force time-history for Case 5 from 0.1 sec to 2.5 sec (Fig. 5-5 B). Despite the continued dynamic activity in the top of th e Pier 48 pier columns th roughout the analysis (Fig. 5-5 A), the overall pier behavior approaches that of a static response as the impa ct load approaches a maximum value. Additionally, because the AASHTO ba rge bow force-crush relationship (Fig. 4-1 ) maintains a positive stiffness regardless of crush de pth, the Case 5 peak impact force occurs at a time in which the dynamic component of behavior of Pier 48 has substantially diminished. Therefore, the peak pier column demands are driven by a static response in th is case. As a result, there is not a great difference betw een dynamic and static response (Fig. 5-7 ). 54

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Pier 2-W Pier 1-W Pier 1-E Pier 2-E Pier 3-E Impact Figure 5-1. Structural configur ation analyzed in Case 4. 55

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Pier 46 Pier 47 Pier 48 Pier 49 Pier 50 Impact Figure 5-2. Structural configur ation analyzed in Case 5. 56

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Time ( s ) Force (kN) Force ( ki p s ) 0 0.25 0.5 0.75 1 1.25 1.5 1.752 -300 -200 -100 0 100 200 300 400 500 600 -50 -25 0 25 50 75 100 125 Simplified Model Full-Resolution Model A Time ( s ) Force (kN) Force ( ki p s ) 0 0.25 0.5 0.75 1 1.25 1.5 1.752 -100 -50 0 50 100 150 200 -15 0 15 30 Simplified Model Full-Resolution Model B Figure 5-3. Comparison of Case 3 simplifie d and full-resolution coupled analyses. A) Pier column top node horizontal shear B) Pile head node horizontal shear. 57

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Time ( s ) Force (kN) Force ( ki p s ) 0 0.25 0.5 0.75 1 1.25 1.5 1.752 -300 -200 -100 0 100 200 300 400 500 600 700 -50 -25 0 25 50 75 100 125 150 Simplified Model Full-Resolution Model A Time ( s ) Force (kN) Force ( ki p s ) 0 0.25 0.5 0.75 1 1.25 1.5 1.752 -50 0 50 100 150 200 250 0 15 30 45 Simplified Model Full-Resolution Model B Figure 5-4. Comparison of Case 4 simplifie d and full-resolution coupled analyses. A) Pier column top node horizontal shear B) Pile head node horizontal shear. 58

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Time ( s ) Force (kN) Force ( ki p s ) 0 0.5 1 1.5 2 2.5 3 3.54 -750 -500 -250 0 250 500 750 1000 1250 -120 -60 0 60 120 180 240 Simplified Model Full-Resolution Model A Time ( s ) Force (kN) Force ( ki p s ) 0 0.5 1 1.5 2 2.5 3 3.54 -100 0 100 200 300 400 500 600 700 800 0 25 50 75 100 125 150 175 Simplified Model Full-Resolution Model B Figure 5-5. Comparison of Case 5 simplifie d and full-resolution coupled analyses. A) Pier column top node horizontal shear B) Pile head node horizontal shear. 59

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Time (min) Time ( hrs ) 0 60 120 180 240 300 360 420 480 540 600 660 0 1 2 3 4 5 6 7 8 9 10 11 Case 3 Case 4 Case 5 Simplified model Full-resolution model Figure 5-6. Time computation co mparison of coupled analyses. 60

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Case 3 Case 4 Case 5 0.537 MN (121 kip) 0.539 MN (121 kip) 0.316 MN (71.0 kip) 0.691 MN (155 kip) 0.699 MN (157 kip) 0.259 MN (58.2 kip) 10.8 MN (2430 kip) 10.6 MN (2390 kip) 10.6 MN (2390 kip) Simplified dynamic analysis Full-resolution dynamic analysis Static analysis A Case 3 Case 4 Case 5 Simplified dynamic analysis Full-resolution dynamic analysis Static analysis 2.85 MN-m (2100 kip-ft) 2.86 MN-m (2110 kip-ft) 1.83 MN-m (1350 kip-ft) 4.28 MN-m (3150 kip-ft) 4.33 MN-m (3190 kip-ft) 1.62 MN-m (1190 kip-ft) 26.3 MN-m (19400 kip-ft) 25.1 MN-m (18500 kip-ft) 25.7 MN-m (19000 kip-ft) B Figure 5-7. Comparison of demonstration case simplified, full-resolution, and static analyses. A) Peak pier column shear. B) Peak pier column moment. 61

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CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 6.1 Conclusions Numerical coupled analysis has been validat ed using experimental findings from the 2004 full-scale barge impact experiments. As one means of making coupled analysis feasible for use in design settings, a simplified and sta ndardized barge bow stiffness curve has been recognized as desirable. Due to the scarcity of barge bow force-crush re lationship data in the literature, the AASHTO crush-curve has been selected. However, data specific to a particular vessel type obtained by other means may easily be integrated into the coupled analysis procedure. As an additional facilitation for the use of coupled analysis in design settings, an algorithm has been presented that reduces a mu lti-span, multiple-pier model to a multi-span single pier model with lateral a nd rotational springs, a nd lumped masses. W ith regard to the stiffness approximation associated with the simplification algorithm, linear elastic lateral and rotational springs have been shown to retain su fficient accuracy in resp ective simplified models for design purposes despite the associated uncoupling of the respective DOF. Three five-pier bridge analysis cases have been presented and subjected to the coupled analysis procedure at simplified and full resoluti ons. Comparison of the results demonstrates the ability of the simplification algorithm to pred ict time-history results in agreement with full-resolution models for low, medium, and hi gh-energy impact conditi ons through a range of pier impact resistances. The simplified algorithm, used in conjunction with coupled analysis, provides a feasible means of conducting barge-bri dge collision analysis in design settings. Required analyses times associated with simplifie d analysis are reduced to levels suitable for design situations. Furthermore, the simplification algorithm retains anal ytical sophistication 62

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sufficient to adequately quantify inertial bridge forces and the resulting distribution of internal forces throughout a given pier. Dynamic phenomena documented in previous barge-pier collision research, such as dynamic amplification of pier column shear forces due to dynamic excitation of superstructure elements, are quantified for three cases and compar ed to results obtained fr om a static analysis procedure. Simplified coupled analysis is show n to adequately and efficiently capture such effects and is found to be suitable for fu ture incorporation into design provisions. 6.2 Recommendations for Future Research Based on the advances made in this study, the following topics warrant additional future investigation: The development of experimental procedur es leading to a standardized body of crush-curves, including phenomena such as post-yield softening and unloading; High-resolution modeling or experimental te sting of multiple-barge flotilla impacts, resulting in data sufficient to quantify any significant interactions between multiple barge flotillas; this would be in relation to imp roving the state-of-the-art SDF impact model; and, Possible revision of the AASHTO Probability of Collapse term. 63

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APPENDIX A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA The 2004 full-scale experiments ( Consolazio et al. 2006 ) consisted of three distinct impact test setups, two of which are of interest in this study: the fi rst impact tests were conducted on the stiff channel pier, Pier 1-S; the second set of tests were conducted on a flexible pier, Pier 3-S, with the superstructure intact fo r one span to the north and multiple spans to the south. After development of the barge forc e-crush relationship, c oupled analyses were conducted on FB-MultiPier models of the Pier 1-S and Pier 3-S partial bridge structure at impact energies corresponding to the impact test events. The highest impact energies, and therefor e the most appreciable impact loads and structure response, occurred dur ing tests four through seven on Pier 1-S (termed test P1T4 through P1T7). Due to the flexibility of Pier 3-S, and the non-destructiv e nature of the testing, impact energies employed in the multi-span B3 bri dge tests were considerably lower than that of the P1 test series. Even so, the second through fourth tests (termed test B3T2 through B3T4) generated considerable pier re sponse and significant impact lo ads. Tests associated with significant loading or pier res ponse were selected for validation of the coupled analysis procedure. Pertinent output from such analyses is included in this appendix. All P1 series analyses included here were c onducted using the payload modifi cations discussed in Chapter 3 64

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00.250.50.7511.251.5 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 200 400 600 800 1000 1200 Time (sec)Force (kips)Impact Force Time History 0123456 0 200 400 600 800 1000 1200 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-1. Analytical output comparison to experimental P1T4 barge impact data. 65

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00.250.50.7511.251.5 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 200 400 600 800 1000 1200 Time (sec)Force (kips)Impact Force Time History 0123456 0 200 400 600 800 1000 1200 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-2. Analytical output comparison to experimental P1T5 barge impact data. 66

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00.250.50.7511.251.5 0.5 0.25 0 0.25 0.5 0.75 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 200 400 600 800 1000 1200 Time (sec)Force (kips)Impact Force Time History 0123456789 0 200 400 600 800 1000 1200 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-3. Analytical output comparison to experimental P1T6 barge impact data. 67

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00.250.50.7511.251.5 0.25 0 0.25 0.5 0.75 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 200 400 600 800 1000 1200 Time (sec)Force (kips)Impact Force Time History 0123456789 0 200 400 600 800 1000 1200 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-4. Analytical output comparison to experimental P1T7 barge impact data. 68

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00.250.50.7511.251.5 0.5 0.25 0 0.25 0.5 0.75 1 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 50 100 150 200 250 Time (sec)Force (kips)Impact Force Time History 00.511.522.53 0 50 100 150 200 250 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-5. Analytical output in comparison to experimental B3T2 barge impact data. 69

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00.250.50.7511.251.5 0.5 0.25 0 0.25 0.5 0.75 1 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 50 100 150 200 250 Time (sec)Force (kips)Impact Force Time History 00.511.522.53 0 50 100 150 200 250 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-6. Analytical output in comparison to experimental B3T3 barge impact data. 70

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00.250.50.7511.251.5 0.75 0.5 0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.5 0 50 100 150 200 250 300 350 400 450 Time (sec)Force (kips)Impact Force Time History 00.511.522.533.5 0 50 100 150 200 250 300 350 400 450 Analytical output Experimental data Input loading curve Analytical output Experimental data Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure A-7. Analytical output in comparison to experimental B3T4 barge impact data. 71

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APPENDIX B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS Within the discussion of the condensed uncoupled stiffness matrix, presented in Chapter 4 the condensed off-diagonal stiffness term that couples rotation and horizontal shear force ( ) is shown to produce relatively negligible shear forces with respect to the applied impact load. This affords the uncoupling of th e condensed stiffness matrix of extraneous non-impacted portions of a given bridge model. This appendix contains comparisons of the same off-diagonal stiffness term ( ), alternatively viewed as a coupling between horizontal translation and a vertical moment, and the moment produced by the diagonal rotational stiffness term ( ) of the condensed stiffness matr ix when the B3 numerical model (Fig. couplingK couplingK K 3-3 ) is subject to an arbitrary static load at the impact location. A comparison of the diagonal and off-diagonal moments re veals that the off-diagonal sti ffness of piers adjacent to the impacted pier in a given full-resolution mode l may be neglected w ithout sacrificing any appreciable analytical accuracy of fo rces developed in the impacted pier. 72

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Off-Diagonal stiffness quantification: Case 3 Numerical Model1. Obtain condensed stiffness matrix of Pier 2S to Pier 1S portion of full-resolution model 1.1 Apply unit lateral load at location of stiffness condensation; in this case, the center of the pier cap beam of Pier 2S 1.1.1 Store lateral translation and vertical rotation in appropriate entries of condensed flexibility matrix 1.2 Apply unit vertical moment at location of stiffness condensation; in this case, the center of the pier cap beam of Pier 2S 1.2.1 Store lateral translation and vertical rotation in appropriate entries of condensed flexibility matrix Condensed flexibility matrix: FlexP2 0.00181781 8.5443 107 12 8.5443 107 12 5.28799109 12 in/kip and rad/kip-in The first row diagonal entry pertains to shear force per unit lateral translation; the second row diagonal pertains to vertical moment per unit vertical rotation 2. Invert condensed flexibility matrix to obtain condensed stiffness matrix StiffP2FlexP21 Condensed stiffness matrix: StiffP2 553.616 8.945104 8.945104 2.284109 kip/in and kip-in/rad 73

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Off-Diagonal stiffness quantification: Case 3 Numerical Model (Cont'd)3. Apply peak static load at impact point of full-resolution model and record displacements Static load applied at Node 109 of Pier 3-S: PeakLoad413.8 kips Induced displacements at location of condensed stiffness: Vertical rotation: z1.994104 rad Horizontal translation: x0.1908 in 4. Calculate moment due to diagonal stiffness term and vertical rotation Mz diagonal StiffP222 z Mz diagonal 4.554105 kip-in 5. Calculate moment due to off-diagonal stiffness term and horizontal translation Mz offdiagonal StiffP212 x Mz offdiagonal 1.707104 kip-in 6. Compare magnitudes of "diagonal" and "off-diagonal" moments ratio Mz diagonal Mz offdiagonal ratio26.681 The "diagonal" moment is significantly larger than the "off-diagonal" moment. 74

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Flexibility Approximation: Case 3 Numerical Model7. Directly invert diagonal flexibility terms recorded in 1.2.1 of Off-Diagonal Stiffness Quantification 7.1 Approximation of Translational Stiffness Term AppStiffTrans 1 FlexP211 AppStiffTrans550.112 kip/in 7.2 Approximation of Rotational Stiffness Term AppStiffRot 1 FlexP222 AppStiffRot2.269109 kip-rad/in 8. Calculate percent difference between approximated stiffness terms and stiffness terms obtained by flexibility matrix inversion (the latter terms being calculated in 2. of Off-Diagonal Stiffness Quantification) 8.1 Percent difference of translational stiffness term AppStiffTransStiffP211 StiffP211 100 0.633 percent 8.2 Percent difference of rotational stiffness term AppStiffRotStiffP222 StiffP222 100 0.633 percent The approximation yields nominally different values of stiffness. 75

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APPENDIX C SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT To further bolster the assertion that the simp lification algorithm predicts impacted pier response with an accuracy that, within reason, ma tches that of full-resolution bridge coupled analysis, additional time-history data from each of Case 3 through Case 5 are included in this appendix. More specifically, tim e-histories of shear, moment, and displacement are provided at the pier column top and pile head for each case. Additionally, barge force-crush data obtained from simplified and full-bridge analyses are incl uded. Accompanying this data are the impact location displacement time-history and impact lo cation force time-history for each of Case 3 through Case 5. Consequently, the data presen ted in Fig. C-1 through Fig. C-9 were obtained using the AASHTO barge bow force-crush relations hip. Finally, data obtained from the same pier models are presented when simplified a nd full-resolution analyses are conducted using a bilinear barge bow force-crush relationship with an initial stiffness and shift point (see Chapter 4) identical to that f ound in the AASHTO curve. Output pertaining to these analyses are located in Fig. C-10 through Fig. C-18.

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2 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 2 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.51.752 0 100 200 300 400 500 600 Time (sec)Force (kips)Impact Force Time History 012345 0 300 600 900 1200 1500 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure C-1. Case 3 AASHTO curve coupled anal ysis output comparison at impact location. 77

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Shear Force Time History Moment Time History 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History 00.250.50.7511.251.51.752 2000 1500 1000 500 0 500 1000 1500 Time (sec)Moment (kip-ft) 00.250.50.7511.251.51.752 75 50 25 0 25 50 75 100 125 Time (sec)Force (kips) Figure C-2. Case 3 AASHTO curve coupled analys is output comparison at pier column top. 78

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00.250.50.7511.251.51.752 20 10 0 10 20 30 40 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 500 400 300 200 100 0 100 200 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 2 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-3. Case 3 AASHTO curve coupled an alysis output comparison at pile head. 79

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00.250.50.7511.251.51.752 0.25 0 0.25 0.5 0.75 1 1.25 1.5 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.51.752 0 200 400 600 800 1000 1200 1400 1600 Time (sec)Force (kips)Impact Force Time History 0246810121416 0 200 400 600 800 1000 1200 1400 1600 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure C-4. Case 4 AASHTO curve coupled anal ysis output comparison at impact location. 80

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00.250.50.7511.251.51.752 75 50 25 0 25 50 75 100 125 150 175 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 2500 2000 1500 1000 500 0 500 1000 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 0.5 0 0.5 1 1.5 2 2.5 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-5. Case 4 AASHTO curve coupled analys is output comparison at pier column top. 81

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00.250.50.7511.251.51.752 10 0 10 20 30 40 50 60 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 600 500 400 300 200 100 0 100 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 0.25 0 0.25 0.5 0.75 1 1.25 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-6. Case 4 AASHTO curve coupled an alysis output comparison at pile head. 82

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0255075100125150175200 0 500 1000 1500 2000 2500 3000 3500 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips) 00.511.522.533.54 0 500 1000 1500 2000 2500 3000 3500 Time (sec)Force (kips) 00.511.522.533.54 0.25 0 0.25 0.5 0.75 1 1.25 Time (sec)Displacement (in)Impact Point Displacement Time History Impact Force Time History Barge Force Crush Output Figure C-7. Case 5 AASHTO curve coupled anal ysis output comparison at impact location. 83

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00.511.522.533.54 150 100 50 0 50 100 150 200 250 Time (sec)Force (kips)Shear Force Time History 00.511.522.533.54 2500 2000 1500 1000 500 0 500 1000 1500 Time (sec)Moment (kip-ft)Moment Time History 00.511.522.533.54 0.5 0.25 0 0.25 0.5 0.75 1 1.25 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-8. Case 5 AASHTO curve coupled analys is output comparison at pier column top. 84

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00.511.522.533.54 200 150 100 50 0 50 Time (sec)Force (kips)Shear Force Time History 00.511.522.533.54 1000 750 500 250 0 250 Time (sec)Moment (kip-ft)Moment Time History 00.511.522.533.54 0.25 0 0.25 0.5 0.75 1 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-9. Case 5 AASHTO curve coupled an alysis output comparison at pile head. 85

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2 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 2 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.51.752 0 100 200 300 400 500 600 Time (sec)Force (kips)Impact Force Time History 012345 0 300 600 900 1200 1500 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure C-10. Case 3 bilinear curve coupled analysis output comparison at impact location. 86

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Shear Force Time History Moment Time History 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History 00.250.50.7511.251.51.752 2000 1500 1000 500 0 500 1000 1500 Time (sec)Moment (kip-ft) 00.250.50.7511.251.51.752 75 50 25 0 25 50 75 100 125 Time (sec)Force (kips) Figure C-11. Case 3 bilinear cu rve coupled analysis output co mparison at pier column top. 87

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00.250.50.7511.251.51.752 20 10 0 10 20 30 40 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 500 400 300 200 100 0 100 200 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 2 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-12. Case 3 bilinear curve coupled analysis output comparison at pile head. 88

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00.250.50.7511.251.51.752 0.25 0 0.25 0.5 0.75 1 1.25 1.5 Time (sec)Displacement (in)Impact Point Displacement Time History 00.250.50.7511.251.51.752 0 200 400 600 800 1000 1200 1400 1600 Time (sec)Force (kips)Impact Force Time History 0246810121416 0 200 400 600 800 1000 1200 1400 1600 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips)Barge Force Crush Output Figure C-13. Case 4 bilinear curve coupled analysis output comparison at impact location. 89

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00.250.50.7511.251.51.752 100 50 0 50 100 150 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 2500 2000 1500 1000 500 0 500 1000 1500 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 1 0.5 0 0.5 1 1.5 2 2.5 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-14. Case 4 bilinear cu rve coupled analysis output co mparison at pier column top. 90

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00.250.50.7511.251.51.752 10 0 10 20 30 40 50 60 Time (sec)Force (kips)Shear Force Time History 00.250.50.7511.251.51.752 600 500 400 300 200 100 0 100 Time (sec)Moment (kip-ft)Moment Time History 00.250.50.7511.251.51.752 0.25 0 0.25 0.5 0.75 1 1.25 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-15. Case 4 bilinear curve coupled analysis output comparison at pile head. 91

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0306090120150180210240270300 0 250 500 750 1000 1250 1500 Two-span single-pier Five-pier Input loading curve Two-span single-pier Five-pier Input loading curve Crush (in)Force (kips) 0123456 0 250 500 750 1000 1250 1500 Time (sec)Force (kips) 0123456 0.25 0 0.25 0.5 0.75 Time (sec)Displacement (in)Impact Point Displacement Time History Impact Force Time History Barge Force Crush Output Figure C-16. Case 5 bilinear curve coupled analysis output comparison at impact location. 92

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0123456 150 100 50 0 50 100 150 200 250 Time (sec)Force (kips)Shear Force Time History 0123456 2500 2000 1500 1000 500 0 500 1000 Time (sec)Moment (kip-ft)Moment Time History 0123456 0.25 0 0.25 0.5 0.75 1 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-17. Case 5 bilinear cu rve coupled analysis output co mparison at pier column top. 93

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0123456 125 100 75 50 25 0 25 Time (sec)Force (kips)Shear Force Time History 0123456 600 500 400 300 200 100 0 100 Time (sec)Moment (kip-ft)Moment Time History 0123456 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Two-span single-pier Five-pier Two-span single-pier Five-pier Time (sec)Displacement (in)Displacement Time History Figure C-18. Case 5 bilinear curve coupled analysis output comparison at pile head. 94

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APPENDIX D ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS The new St. George Island Bridge was desi gned and constructed after the AASHTO vessel collision specifications went in effect, hence, the piers of this bridge were designed to resist barge impact loading. Furthermore, pier impact load data was contained within the bridge plans used to develop the numerical model for Case 5 of this thesis. The initial kinetic energy specified in the Case 5 coupled analyses was derived from the known design impact load and pertinent equations found in AASHTO. Convers ely, energy-equivalent static AASHTO impact loads are calculated from the ki netic energies employed in Case 3 and Case 4. The barge width modification factor is 1.0 in the following calcu lations as a jumbo-hopper is selected as the impacting vessel in all demonstration cases. 95

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Back calculation of AASHTO impact force using Case 3 Impact EnergyThe Old St. George Island Bridge Pier 3-S was subject to an impact energy consisting of: Barge weight W1.78MN W200.08T W181.478tonne Barge velocity V1.03 m s V2.002knot V3.379 ft s Assume a hydrodynamic mass coefficient of Ch1.05 Impact energy KE 1 2 ChW g V2 KE74.564kipft KE0.101MNm From the American Association of State and Highway Transportation Officials (AASHTO) Guide Specification and Commentary for Vessel Collision Design of Highway Bridges the equations for barge crush depth and kinetic energy associated with impact are: Barge crush depth: ab KE 5672 1 0.51 10.2 ab0.067 ft The energy equivalent AASHTO static impact force is: PbPb1349110ab () ab0.34 if Pb4112ab ab0.34 if Pbreturn Pb274.787 kip 96

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Back calculation of AASHTO impact force using Case 4 Impact EnergyThe Escambia Bay Bridge Pier 1-E was subject to an impact energy consisting of: Barge weight W18MN W2.023103 T W1.835103 tonne Barge velocity V1.54 m s V2.994knot V5.052 ft s Assume a hydrodynamic mass coefficient of Ch1.05 Impact energy KE 1 2 ChW g V2 KE1.686103 kipft KE2.285MNm From the American Association of State and Highway Transportation Officials (AASHTO) Guide Specification and Commentary for Vessel Collision Design of Highway Bridges the equations for barge crush depth and kinetic energy, and barge width associated with impact are: Barge crush depth: ab KE 5672 1 0.51 10.2 ab1.417 ft The energy equivalent AASHTO static impact force is: PbPb1349110ab ab0.34 if Pb4112ab ab0.34 if Pbreturn Pb1.505103 kip 97

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Back calculation of Case 5 impact energy using AASHTO impact forceFrom bridge plans of the New St. George Island Bridge Channel Pier, the design impact load is: Pb3255 kips From the American Association of State and Highway Transportation Officials (AASHTO) Guide Specification and Commentary for Vessel Collision Design of Highway Bridges the equations for barge crush depth and kinetic energy associated with impact are: Barge crush depth: abab Pb1349 110 Pb1349 110 0.34 if ab Pb 4112 Pb 4112 0.34 if abreturn ab17.327 ft Kinetic energy associated with impact: KE ab 10.2 1 21 5672 KE3.564104 kip-ft Define flotilla design velocity as a function of Hydrodynamic Mass Coefficient and Flotilla weight (tonnes) VC H W KE29.2 C H W 0.5 Assume Hydrodynamic Mass Coefficient is 1.05. Define weight of barge as a function of the number of barges in the flotilla; assume towboat weighs 120 tons (US, short) Wn () n1700200 () 120 1.102311311 98

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Back calculation of Case 5 impact energy using AASHTO impact force (Cont'd)Try using four fully loaded Jumbo Hopper barges and check that the accompanying velocity is attainable within the waterway. The weight of four fully loaded Jumbo Hopper barges and the tow boat is: W4 ()7.003103 tonnes The velocity of the flotilla, necessary to generate a static impact load of 3255 kips is: V1.05W4 () ()11.896 ft/sec Conclusion: the four barge flotilla is a reasonable number of barges for use in a single column flotilla in the southeastern United States, and 11.896 ft/sec is an attainable speed in the St. George Island waterway as typical traveling speeds are: 10.13 ft/sec (Consolazio et al. 2002). 99

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REFERENCES AASHTO. (1991). Guide Specification and Commentary fo r Vessel Collision Design of Highway Bridges, American Association of State Hi ghway and Transportation Officials, Washington, D.C. 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. US Army Corps of Engineers Report ERDC/ITL TR-03-3 2003. Consolazio, G. R., Cook, R. A., and Lehr, G. B. (2002). Barge Impact Testing of the St. George Island Causeway Bridge Phase I : Feasibility Study. Structures Research Report No. 783 Engineering and Industrial Experime nt Station. University of Fl orida, Gainesville, Florida, January. Consolazio, G. R. and Cowan, D. R. (2003). Non linear Analysis of Barge Crush Behavior and its Relationship to Impact Resistant Bridge Design. Computers and Structures Vol. 81, Nos.8-11, pp. 547-557. Consolazio, G. R., Lehr, G. B., and McVay, M. C. (2004a). Dynamic Finite Element Analysis of Vessel-Pier-Soil Interaction During Barge Impact Events. Transportation Research Record: Journal of the Transportation Research Board. No. 1849, Washington, D.C., pp. 81-90. Consolazio, G. R., Hendrix, J. L., McVay, M. C., Williams, M. E., and Bollman, H. T. (2004b). Prediction of Pier Response to Barge Imp acts Using Design-Oriented Dynamic Finite Element Analysis. Transportation Research Record: Journal of the Transportation Research Board No. 1868, Washington, D.C., pp. 177-189. Consolazio, G. R. and Cowan, D. R. (2005). Numer ically Efficient Dynamic Analysis of Barge Collisions with Bridge Piers. ASCE Journal of Structural Engineering ASCE, Vol. 131, No. 8, pp. 1256-1266. Consolazio, G. R., Cook, R. A., and McVay, M. C. (2006). Barge Impact Testing of the St. George Island Causeway Bridge, Structures Research Report No. 2006/26868 Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida, March. FB-MULTIPIER Users Manual (2007). Florida Bridge Software Institute, University of Florida, Gainesville, Florida. FB-PIER Users Manual (2003). Florida Bridge Software In stitute, University of Florida, Gainesville, Florida. 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. 100

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Hendrix, J. L. (2003). Dynamic Analysis Techni ques for Quantifying Bridge Pier Response to Barge Impact Loads. Masters Thesis, Departme nt of Civil and Coastal Engineering, Univ. of Florida, Gainesville, Fla. Larsen, O. D. (1993). Ship Collision with Bridge s: The Interaction between Vessel Traffic and Bridge Structures. IABSE Structural Engineer ing Document 4, IABSE Knott, M., and Prucz, Z. (2000). Vessel Collision Design of Bridges: Bridge Engineering Handbook, CRC Press LLC. Meier-Drnberg, K. E. (1983). Ship Collisions, Safety Zones, and Loading Assumptions for Structures in Inland Waterways. Verein Deutscher Ingenieure (Association of German Engineers) Report No. 496, 1983, pp. 1-9. McVay, M. C., Wasman, S. J., Bullock, P. J. (2 005). St. George Geotech nical Investigation of Vessel Pier Impact, Engineering and Industrial Experiment Stat ion, University of Florida, Gainesville, Florida. Patev, R. C., Barker, B. C., and Koestler, L. V., III. (2003). Full-Scale Barge Impact Experiments, Robert C. Byrd Lock and Dam, Gallipolis Ferry, West Virginia. United States Army Corps of Engin eers Report ERDC/ITL TR-03-7, December. Yuan, P. (2005). Modeling, Simulation and Analysis of Multi-Barge Flotillas Impacting Bridge Piers. PhD dissertation, Dept of Civil Engineering, Uni v. of Kentucky, Lexington, Ky. 101

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BIOGRAPHICAL SKETCH Michael Davidson was born in Louisville, Kent ucky. He enrolled at the University of Kentucky in August 2000. After being awarde d the National Science Foundation Graduate Research Fellowship and obtaini ng his Bachelor of Science in civil engineering from the University of Kentucky (summa cum laude) in May 2005, he began graduate school at the University of Florida in the College of E ngineering, Department of Civil and Coastal Engineering. The author will receive his Ma ster of Science degree in August 2007, with a concentration in structural engineering. Upon graduati on, the author will continue his education at the University of Florida, ultimately earning a degree of Doctor of Philosophy with a specialization in stru ctural engineering. 102


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