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PAGE 1 A STATIC ANALYSIS METHOD FOR BARGE IMPACT DESIGN OF BRIDGES WITH CONSIDERATION OF DYNAMIC AMPLIFICATION By DANIEL JAMES GETTER 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 ENGINEERING UNIVERSITY OF FLORIDA 2010 1 PAGE 2 2010 Daniel James Getter 2 PAGE 3 To my fiance, Heather 3 PAGE 4 ACKNOWLEDGMENTS Completion of this thesis and the associat ed research relied strongly on the support, motivation, guidance, and patience provided by Dr Gary Consolazio. His steadfast insistence that the work could always be im proved greatly enhanced the quality of this thesis, and his keen ability to think critically about anything presented to him led to countless valuable conversations. The author would especially like to thank Mr Michael Davidson for his instrumental role in providing research advice during the course of this study. Michael is a seemingly endless source of insight and suggestions and never wave rs in providing assistance when needed. The author would also like to extend appreciation to Dr. David Cowan, w ho patiently answered questions in the midst of his own doctoral dissert ation work, and helped provide the practical and theoretical foundation on whic h this research is based. Lastly, the author thanks Dr. H.R. (Trey) Hamilton and Dr. Michael McVay for serving on his supervisory committee, and Mr. Henry Bo llmann, and Mr. Marcus Ansley for providing valuable suggestions and feedb ack throughout the research proce ss. Perhaps most importantly, the patience and support of the authors fian ce, Heather Malone, is greatly appreciated. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 LIST OF ABBREVIATIONS ........................................................................................................12 ABSTRACT ...................................................................................................................................15 CHAPTER 1 INTRODUCTION................................................................................................................. .17 Introduction .............................................................................................................................17 Objectives ...............................................................................................................................19 Scope of Work ........................................................................................................................20 2 BACKGROUND................................................................................................................... .21 Review of the Current AASHTO Load Determination Procedure .........................................21 Updated Barge Bow ForceDeformation Relationships .........................................................23 Coupled Vessel Impact Analysis (CVIA) ...............................................................................25 Dynamic Amplification of Pier Column Forces .....................................................................26 3 BRIDGE MODELING AND STUDY PRELIMINARIES....................................................35 Introduction .............................................................................................................................35 Superstructure .........................................................................................................................36 Substructure ............................................................................................................................38 Pier Modeling..................................................................................................................38 Soil Modeling ..................................................................................................................39 Bridge Inventory and Impact Conditions ...............................................................................41 Bridge Inventory ..............................................................................................................41 Barge Impact Conditions .................................................................................................42 4 STATIC BRACKETED IMPACT ANALYSIS (SBIA) METHOD......................................51 Introduction .............................................................................................................................51 Conceptual Overview .............................................................................................................51 Superstructure Modeling ........................................................................................................52 Static Impact Load Determination ..........................................................................................53 Potential Static Approximations of Superstructure Inertial Resistance ..................................56 Static Approximation of Inertial Re sistance by Direct Load Application ..............................58 5 PAGE 6 Factored Impact Load ......................................................................................................58 Determination of Ideal PierTop Loads ...........................................................................59 Correlation of PierTop Inertial Resistan ce Factors (IRFs) to Bridge Parameters ..........59 Substructure Considerations...................................................................................................63 Determination of Ideal ImpactPoint Loads ....................................................................63 Correlation of ImpactPoint DMF to Bridge Parameters ................................................63 Static Bracketed Impact Analysis (SBIA) Method .................................................................64 SBIA Overview ...............................................................................................................64 SBIA Demonstration .......................................................................................................65 SBIA demonstration with OP TS superstructure model ...........................................67 SBIA demonstration with sp ring superstructure model ...........................................69 Comparison of SBIA using OPTS vs. spring superstructure model ........................69 Parametric Study Results ........................................................................................................71 5 CONCLUSIONS AND RECOMMENDATIONS.................................................................90 Summary and Conclusions .....................................................................................................90 Recommendations...................................................................................................................91 APPENDIX A COMPARISON OF SBIA AND CVIA RESULTS...............................................................92 B DEMONSTRATION OF SBIA METHOD..........................................................................113 LIST OF REFERENCES.............................................................................................................120 BIOGRAPHICAL SKETCH.......................................................................................................122 6 PAGE 7 LIST OF TABLES Table page 31 Bridge pier configurations......................................................................................................44 32 Analysis matrix of barge impact energy conditions...............................................................45 43 SBIA demand prediction summary (OPTS superstructure)...................................................73 44 Comparison of demand predictionsC VIA vs. SBIA (OPTS superstructure).....................73 45 SBIA demand prediction summary (spring superstructure)...................................................73 46 Comparison of demand predictionsCVI A vs. SBIA (spring superstructure).....................73 47 Comparison of SBIA demand predic tionsOPTS vs. spring superstructure........................74 B1 SBIA demand prediction summary......................................................................................119 7 PAGE 8 LIST OF FIGURES Figure page 21 Forcedeformation results obtained by MeierDrnberg........................................................28 22 Relationships developed from experime ntal barge impact tests conducted by MeierDrnberg (1983).................................................................................................................29 23 AASHTO relationship between ba rge crush depth and impact force....................................29 24 Finite element simulation of barge bow crushing. 6 ft diameter round impact with jumbo hopper barge bow...................................................................................................30 25 Barge bow forcedeformation relationships...........................................................................31 26 Barge bow forcedeformation flowchart................................................................................32 27 Coupling between barge and bridge in CVIA........................................................................33 28 Analyses conducted to quantify dynamic amplification.........................................................33 29 Maximum pier column dynamic mome nts relative to static moments...................................34 31 Overview of superstructure model configuration in FBMultiPier........................................46 32 Cross section integration scheme for nonlinear frame elements............................................47 33 Nonlinear material models as implemented in FBMultiPier.................................................48 34 Typical soil resistance curves employed by FBMultiPier.....................................................49 35 Variation of row multipliers for differing pile group motion.................................................50 41 Static barge impact analysis.............................................................................................. .....74 42 Superstructure modeling techniques considered....................................................................75 43 Barge bow forcedeformation relationship.............................................................................76 44 Inelastic barge bow deformation energy................................................................................76 45 Statically approximating supe rstructure inertial resistance....................................................77 46 Impact load magnified by a factor of 1.45..............................................................................78 47 Determination of ideal piertop load and IRF for a given bridge pier (calibrated to column moment)................................................................................................................7 8 8 PAGE 9 48 Correlation between inertial resistance factor (IRF) and bridge parameters..........................79 49 Static loading to approximate superstructure dynamic amplification....................................80 410 Determination of ideal amplified impact lo ad and DMF for a bridge pier (calibrated to pile moment)................................................................................................................... ...80 411 Mean value and envelope of impactpoint DMF..................................................................81 412 Static bracketed impact analysis (SBIA) method.................................................................82 413 Structural configuration for Blount stown Bridge channel pier (BLTCHA).......................83 414 Loading conditions and maximum demand predictions for Load Case 1 with OPTS superstructure model..........................................................................................................8 3 415 Loading conditions and maximum demand predictions for Load Case 2 with OPTS superstructure model..........................................................................................................8 4 416 Loading conditions and maximum demand predictions for Load Case 1 with spring superstructure model..........................................................................................................8 4 417 Loading conditions and maximum demand predictions for Load Case 2 with spring superstructure model..........................................................................................................8 5 418 Comparison of CVIA and SBIA bridge responses for New Trammel Bridge at Blountstown (BLTCHA)..................................................................................................86 419 SBIA pier column moment demands relative to CVIA........................................................87 420 SBIA pier column shear demands relative to CVIA............................................................87 421 SBIA foundation moment de mands relative to CVIA..........................................................88 422 SBIA foundation shear demands relative to CVIA..............................................................88 423 SBIA total bearing shear demands relative to CVIA...........................................................89 A1 Comparison of CVIA, SBIAOPTS, and SB IAspring results. Bridge: Acosta Bridge (ACS) channel pier. Impact condition: 2030 tons at 2.5 knots..........................................93 A2 Comparison of CVIA, SBIAOPTS, and SB IAspring results. Bridge: Acosta Bridge (ACS) channel pier. Impact condition: 5920 tons at 5.0 knots..........................................94 A3 Comparison of CVIA, SBIAOPTS, and SB IAspring results. Bridge: Acosta Bridge (ACS) channel pier. Impact condition: 7820 tons at 5.0 knots..........................................95 A4 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: SR20 at Blountstown (BLT) channel pier. Impact condition: 2030 tons at 2.5 knots.....................96 9 PAGE 10 A5 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: SR20 at Blountstown (BLT) channel pier. Impact condition: 5920 tons at 5.0 knots.....................97 A6 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Eau Gallie Bridge (EGB) channel pier. Impact condition: 2030 tons at 2.5 knots..............................98 A7 Comparison of CVIA, SBIAOPTS, a nd SBIAspring results. Bridge: Melbourne Causeway (MBC) channel pier. Impact condition: 2030 tons at 2.5 knots.......................99 A8 Comparison of CVIA, SBIAOPTS, and SB IAspring results. Bridge: new St. George Island (NSG) channel pier. Impact condition: 2030 tons at 2.5 knots.............................100 A9 Comparison of CVIA, SBIAOPTS, and SB IAspring results. Bridge: new St. George Island (NSG) channel pier. Impact condition: 5920 tons at 5.0 knots.............................101 A10 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: new St. George Island (NSG) channel pier. Impact condition: 7820 tons at 7.5 knots.............................102 A11 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: new St. George Island (NSG) offchannel pier. Imp act condition: 200 tons at 1.0 knots.........................103 A12 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: old St. George Island (OSG) channel pier. Impact condition: 2030 tons at 2.5 knots.............................104 A13 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: old St. George Island (OSG) offchannel pier. Imp act condition: 200 tons at 1.0 knots.........................105 A14 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Pineda Causeway (PNC) channel pier. Impact condition: 2030 tons at 2.5 knots......................106 A15 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Ringling (RNG) channel pier. Impact condition: 2030 tons at 2.5 knots.......................................107 A16 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Ringling (RNG) channel pier. Impact condition: 5920 tons at 5.0 knots.......................................108 A17 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Ringling (RNG) offchannel pier. Impact condition: 200 tons at 1.0 knots...................................109 A18 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Ringling (RNG) offchannel pier. Impact condition: 2030 tons at 2.5 knots.................................110 A19 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Seabreeze (SBZ) channel pier. Impact condition: 2030 tons at 2.5 knots........................................111 A20 Comparison of CVIA, SBIAOPTS, and SBIAspring results. Bridge: Seabreeze (SBZ) channel pier. Impact condition: 5920 tons at 5.0 knots........................................112 B1 Structural configurati on for New Trammel Bridge.............................................................113 10 PAGE 11 B2 Barge yield load determination for 9ft round impact surface.............................................114 B3 Determination of pier stiffness (kP).....................................................................................115 B4 Determination of pier weight (WP) and pier height (hp)......................................................116 B5 Determination of superstructure stiffness (ksup)...................................................................117 B6 Loading conditions and maximum de mand predictions for Load Case 1...........................118 B7 Loading conditions and maximum de mand predictions for Load Case 2...........................119 11 PAGE 12 LIST OF ABBREVIATIONS a B barge bow crush depth a Bf barge bow crush depth after impact a Bi barge bow crush depth before impact a Bm maximum barge bow crush depth a BY barge bow crush depth at yield B B B barge width c BP bargepier pseudodamping coefficient C H hydrodynamic mass coefficient DMF dynamic magnification factor DMF ideal ideal dynamic magnification factor cf compressive strength of concrete '' cf effective compressive strength of concrete for FBMultiPier f r rupture strength of concrete in tension h P clear height of pier columns IRF inertial resistance factor b IRF inertial resistance factor calibrated to total bearing shear ideal bIRF ideal inertial resistance factor calibrated to total bearing shear idealIRF ideal inertial resistance factor (general) mIRF inertial resistance factor calibrated to pier moment ideal mIRF ideal inertial resistance fact or calibrated to pier moment vIRF inertial resistance factor calibrated to pier shear ideal vIRF ideal inertial resistance f actor calibrated to pier shear 12 PAGE 13 k B linear barge bow stiffness k P linear piersoil system stiffness k S series stiffness between barge bow and piersoil system k sup superstructure stiffness KE kinetic energy of barge flotilla L s superstructure span length m B mass of barge flotilla max dynM maximum dynamic pier moment max staticM maximum static pier moment P B barge impact force BP static barge impact force max BP maximum dynamic barge impact force ideal BampP ideal amplified barge impact force P BY barge impact force at yield P i generalized bridge parameter for i = 1 P I inertial force at top of pier r Pearson correlation coefficient R B barge width modification factor v Bf barge velocity after impact v Bi barge velocity before impact V barge impact speed W weight of barge flotilla W P weight of pier, including footing (p ile cap), shear wall(s), shear strut(s), pier column(s), and pier cap beam 13 PAGE 14 W sup weight of superstructure i parameter exponent for i = 1 DE if change in deformation energy fr om initial state to final state DE im change in deformation energy from initial state to maximum deformation KE if change in kinetic energy from initial state to final state KE im change in kinetic energy from initial state to maximum deformation product of bridge parameters (for IRF correlations) oblique barge impact angle 14 PAGE 15 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 Engineering A STATIC ANALYSIS METHOD FOR BARGE IMPACT DESIGN OF BRIDGES WITH CONSIDERATION OF DYNAMIC AMPLIFICATION By Daniel James Getter May 2010 Chair: Gary Consolazio Major: Civil Engineering Current practice with regard to designing bridge structures to resist impact loads associated with barge collisions relies upon the use of the AASHTO bridge design specifications. The AASHTO barge impact design provisions, which were developed from pendulum impacthammer testing of reducedscale barge models, employ a static analysis approach. However, recent studies have revealed that si gnificant dynamic amplifications of structural demands (pier design forces) are produced as the re sult of massrelated inertial forces associated with the bridge superstructure. These same st udies have also demonstrated that currently employed static analysis procedures fail to capture or account for such amplification effects. In the present study, an equivalentstatic analys is procedure is developed for use in barge impact design and assessment of bridge structur es. In contrast to the AASHTO static analysis procedure, the new method proposed here, called th e static bracketed impact analysis (SBIA) method, employs static loading conditions and st atic structural analyses, but produces design forces that conservatively approximate dynamic amplification effects associated with superstructure mass. The SBIA method produces de sign forces that are eq uivalent toor greater thanthose that would be predicted using more refined dynamic timedomain methods such as the previously developed coupled vessel impact analysis (CVIA) method. Due to its simplicity, 15 PAGE 16 SBIA is particularly appropriate for situations involving preliminary design of bridges or the design of relatively regular bridge structures for which timedomain dynamic analysis is not warranted. In this thesis, a detailed discussion of ma ssrelated dynamic amplifications in bridges subjected to barge impact loading is presente d. Based on insights gained through characterization of dynamic amplification modes, the static br acketed impact analysis (SBIA) method is developed and described in detail. A parametric study is then conducted using the SBIA method to demonstrate its ability to conservatively approximate dynamically amplified bridge design forces. 16 PAGE 17 CHAPTER 1 INTRODUCTION Introduction Design provisions such as the AASHTO LRFD Bridge Design Specifications and Commentary (2008) prescribe loading conditions that bridge struct ures must be adequately designed to resist. For bridges spanning over waterways that ar e navigable by barge traffic, design and vulnerability assessment calculations mu st consider the combined effects of lateral barge impact loading and vertical gravity load ing. Barge impact loading occurs when a moving barge flotilla (possessing initial kinetic energy) strikes a stationary bridge component (frequently a pier) and is rapidly redirected or brought to a stop. Given that kine tic energy affects the magnitudes of loads generated, barge collision events are fundamentally dynamic in nature. Dynamic sources of structural loading such as barge collis ion and earthquake loading are frequently assessed through the use equivalent st atic loading conditions and static structural analysis. In the case of barge collision loading, the AASHTO br idge design provisions permit designers to use a simplified static analysis proc edure to assess structural response in lieu of more complex fully dynamic methods. As detailed in past research reports (Cons olazio et al. 2006, Consol azio et al. 2008), the vessel collision components of the AASHTO bridge design provi sions include a static barge impact load prediction procedure that is based on tests conducted by MeierDrnberg (1983). In the MeierDrnberg study, both static and dynamicdrophammer tests were performed on reduced scale models of barge bow s to quantify impact loads. A key finding of the study was that no significant differences were observed between static and dynamic load tests. However, because the test procedures failed to include a moving barge striking a deformable bridge 17 PAGE 18 structure, dynamic amplification effects related to characteristics of the impacted bridge were omitted from the study. To overcome the limitations of the MeierDr nberg study (i.e., reduced scale and omission of pier characteristics) a fullscale barge im pact test program (Consolazio et al. 2006) was carried out on piers of the old St. George Isla nd Causeway Bridge. The St. George Island test program involved impacting a fullscale tanke r barge into three different bridge pier configurations, each having different structural characteristics. Over a series of eighteen (18) tests, conducted at varying impact speeds, direct measurements of impact loads and corresponding bridge responses were made. Subsequently, detailed dynamic structural analyses were conducted on models the bridge structure us ing the same impact conditions as those that were generated experimentally. The experimental test results and dynamic structural analysis results revealed that significant dynamic amplif ications of bridge pier design forces were produced by massrelated inertia l restraint from the bridge superstructure. The study also demonstrated that currently employed static an alysis procedures do not capture amplification effects caused by the weight (mass) of the bridge superstructure. In response to the discovery of superstructu reinduced dynamic amp lification effects, a followup study was conducted (Consolazio et al. 2008) to develop dynamic bargebridge collision analysis procedures capable of accounting for such dyna mic phenomena. A key result of this work was the development of a dynamic timedomain analysis procedure called coupled vessel impact analysis (CVIA) which numerically couples models of a barge, bridge pier, foundation, soil, and superstructure. Dynamic am plification due to inertial superstructure restraint is directly a ccounted for in this method through the inclusion of both superstructure (bridge deck) mass and stiffness. Used in conjunc tion with newly formulated barge crush curves 18 PAGE 19 (Consolazio et al. 2009a) and a simplified bridge modeling procedure (Consolazio and Davidson 2008, Davidson et al. 2010), coupled vessel impact an alysis permits bridge design forces to be quantified using a rational and accurate dynamic structural analysis procedure. However, while the CVIA method balances accur acy with numerical efficiency, it is still a timehistory analysis procedure. As such, anal ysis results such as impact loads, bridge displacements, and member forces are all function s of time that must be postprocessed in order to identify maximum design values of interest. If a transient dynamic assessment of structural adequacy is needed, CVIA is an ideal solu tion. However, in many cases, a simpler, yet conservative, analysis method invo lving a small number of discre te load cases (as opposed to hundreds or thousands of time st eps) is more desirable. Objectives The objectives of this study cen ter on the development of an equivalent static analysis procedure for barge impact design and assessment of bridge structures. In contrast to the AASHTO static barge impact analys is procedure, the equivalent static method developed here employs static loading conditions and static structural analyses, but produces structural demands (design forces) that conservatively approximate dynamic amplification effects associated with superstructure mass. The new method is intended to produce design forces that are equivalent toor greater thanthose that w ould be predicted using more re fined dynamic methods such as CVIA. Key objectives of this study are to ensure that the newl y developed equivalent static analysis method is both simple to use, conservative, and capable of accounting for dynamic amplification. Such a method will be particularly appropriate for situations involving preliminary design of bridges (during which fe w structural details are availabl e) or the design of relatively regular (nonlifeline) bridge structures for whic h the additional effort involved in conducting a timehistory analysis is not warra nted. An ideal bridge design pr ocess might involve the use of 19 PAGE 20 equivalent static analysis for preliminary design, followed by more refined timehistory analysis (e.g., CVIA) to maximize safety and minimize costs. Scope of Work Development of bridge models: Finite element bridge models are developed for a representative se t of bridges sampled from throughout the state of Florida. Each bridge model incorpor ates all necessary information that is needed to permit nonlinea r static and nonlinear dyna mic analyses to be performed. Nonlinearities are in corporated into the pier components piles, soil, and the barge. Development of an equivale nt static analysis method. Using insights gained through characterizat ion of dynamic amplification effects, an equivalent static analysis pr ocedure called static bracketed impact analysis (SBIA) is developed. The SBIA method utili zes two static analysis load cases to bracket (envelope) pier element design forces in such a ma nner that dynamic amplification effects are conservatively approximated. Demonstration parametric study: A comprehensive parametric study is conducte d using the SBIA method and two types of superstructure modeling. Parametric study results demonstrate the level of conservatism of the simplified static met hod relative to more refined dynamic CVIA analyses. 20 PAGE 21 CHAPTER 2 BACKGROUND Review of the Current AASHTO Load Determination Procedure For bridges that span over navigable waterw ays, the design specifications used in the United States include the American Association of State Highway and Transportation Officials (AASHTO) Guide Specification and Commenta ry for Vessel Collision Design of Highway Bridges (AASHTO 2009) and the AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications and Commentary ( AASHTO 2008). These documents, collectively referred to as the AASHTO provisions in this thesis, use an empirical load calculation procedure based upon an experimental study conducted by MeierDrnberg (1983). MeierDrnberg conducted both static a nd dynamic impact tests on reducedscale European Type IIa barge bow sections. The Eur opean type IIa barge is similar in size and configuration to the jumbo hopper barges wide ly used throughout the United States. Two dynamic tests, using 2ton pendulum hammers and two different shapes of impact head, were conducted on 1:4.5scale stationa ry (i.e., fixed) barge bows. One dynamic test involved three progressive impacts using a cylindrical hammer with a diameter of 67.0 in., whereas the other involved three progressive impacts using a 90 po inted hammer. A static test was also conducted on a 1:6 scale barge bow using a 90.6 in. hammer. Results obtained from the dynamics tests are shown in Figures 21 ab and results from the stat ic test are shown in Figure 21 c. Using the experimental data collected, MeierDrnberg developed mathematical relationships between kinetic energy (KE), inelastic barge deformation (a B ), and dynamic and static force (P B and BP respectively). These relationships are illustrated in Figure 22 As the figure suggests, no major differences were found between the magnitude of dynamic and static impact force. However, this observation was st rongly influenced by the stationary barge bow 21 PAGE 22 configuration used in the test ing. Omission of a flexible impa ct target and the corresponding bargepier interaction necessa rily precludes the ability to measure and capture dynamic amplification effects. Inelastic barge bow deformations measured by MeierDrnberg showed that once barge bow yielding was initiated, at appr oximately 4 in. of deformation (a B ), the stiffness of the bow diminishes significantly (see Figure 22 ). Additionally, MeierDrnberg recognized that inelastic bow deformations represent a significant form of energy dissipation during collision events. In the development of the AASHTO barge im pact design provisions, the relationships between initial barge kinetic en ergy (KE), barge deformation (a B ) and equivalent static force (P B ) developed by MeierDrnberg, were adopted with minimal modifications: 2.29 WVC KE2 H (21) B 21 BR 2.10 1 5672 KE 1a (22) 34.0a if Ra1101349 34.0a if Ra4112 PBBB B BB B (23) In Eqns. 21 22 KE is the barge kinetic energy (kipft), C H is the hydrodynamic mass coefficient, W is the vessel we ight (in tonnes where 1 tonne = 2205 lbs.), V is the impact speed (ft/sec), and R B = B B B B /35 where B B is the width of the barge (ft). The only notable difference between the expressions devel oped by MeierDrnberg and the AASHTO expr essions, given above as Eqn. 21 23 is the use of a barge width correction factor (R B). While the AASHTO specification utilizes the R B B term to reflect the influence of barge width, no such factor has been included to account for variations in either the size (width) or geom etric shape of the bridge pier being impacted. 22 PAGE 23 Note that Eqn. 23 is a barge forcedeformation relationship (i.e., a crush curve) that relates static barge impact force P B to barge deformation a B B The AASHTO barge crush model (Eqn. ) is illustrated graphically in Figure for a hopper barge having a width B 23 23 B B = 35 ft and therefore an R B = 1. B Updated Barge Bow ForceDeformation Relationships For dynamic structural analysis purposes, ba rge response during impact events may be characterized by appropriate forcedeformation relationships (crush cu rves) that describe nonlinear stiffness of the affected vessel por tions. As discussed above, development of the AASHTO crush curve (shown in Figure 23 ) relied upon scale model crushing tests. These experiments, however, were carried out usi ng reducedscale barge models of European, pontoonstyle construction, not typical of vessels navigating waterways in the United States. To address these limitations, studies were conduc ted (Consolazio et al. 2008; Consolazio et al. 2009a) to characterize barge bow crushing behavior. Highresolu tion finite element models of the bow sections of two common U.S. barg e typesa jumbo hopper barge and an oversize tanker bargewere developed. Over 120,000 elements were used to model each barge bow. During analysis, the barge bow models were subject ed to crushing by a wide variety of impactor shapes and sizes. Specifically, both round and fl atfaced impact surfaces were employed in the simulations, with impactor size s ranging from 1 ft to 35 ft. Forcedeformation relationships obtained from a multitude of simulations (a typical case is illustrated in Figure 24 ) were used to form an update d set of design forcedeformation relationships for barge bows. The study yielded the following findings: For typical design scenarios (headon impact conditions), barge bow forcedeformation can be idealized as an elastic, perfectlyp lastic relationship. Recall from Figure 23 that the AASHTO crush force continues to increase with increasing deformation. The simulations conducted by Consolazio et al. (2008; 2009a ) did not exhibit postyield hardening 23 PAGE 24 behavior. The AASHTO curve a nd a typical Consolazio et al. curve are compared in Figure 25 Due to the high degree of uniformity in barge internal structural configurations (Cameron et al. 2007), impact forces are not sensitiv e to the width of the barge. The AASHTO provisions employ a barge width correction factor, R B, to account for vessels with widths other than 35 feet. However, for a given pier shape and width, the finite element crushing simulations revealed no substantial differe nces between forces produced by crushing a 35foot wide jumbo hopper barge and crushing a 50foot wide tanker barge. B Maximum collision force is dependent, in part, on the shape of the impacted pier surface. The AASHTO crush model does not account fo r impact surface geometry; however, the finite element crushing simulations indicat e that rounded pier surfaces, as opposed to flatfaced surfaces, provide an effective mean s of mitigating the forces generated during barge impact events. Maximum barge impact force is related to the width of the impacted pier surface, particularly for flatf aced rectangular piers. Based on these findings, design barge bow forcedeformation relationships were developed [the formulation of these equations and an algorithm for determining the appropriate crush model for a given design scenario are detailed in Consolazio et al. 2008]. Additional simulations that were subsequently conducted by Consolazio et al. (2009a) resulted in minor changes being made to the proposed crush curves. The revised barg e crushmodel (forcedeformation behavior) is shown in Figure 26 and is used throughout the remainder of this study. All barge finite element simulations conducted as part of the 2008 and 2009a Consolazio et al. studies involved directly headon impact scenario s. However, the probability of perfectly headon collision between a barge and bridge is likely relatively small. A more likely scenario is that the barge collides with the pier at some small oblique angle. Thus, ongoing research is being conducted by the University of Florida (UF) and the Florida Department of Transportation (FDOT) to characte rize barge impact forces corresponding to oblique collision scenarios. Preliminary findings have suggested that oblique collision forces are significantly smaller even when impact occurs at relatively sm all angles. Specifically, at oblique angles of 2 24 PAGE 25 or greater, impact forces associ ated with very wide (near the fu ll barge width of 35 ft) flatfaced impact surfaces (e.g. pile caps placed at the wa terline) are at least 30% smaller than headon impact forces. Thus, for this study, barge co llision forces are reduced by 30% for impact scenarios involving waterline pile caps [see Consolazio et al. 2009b for details regarding the 30% reduction]. Coupled Vessel Impact Analysis (CVIA) Coupled vessel impact analysis (CVIA) i nvolves coupling (linking) a single degree of freedom (SDOF) nonlinear dynamic barge model to a multidegree of freedom (MDOF) nonlinear dynamic bridge analysis code. The term coupled refers to the use of a shared contact force between the barge and imp acted bridge structure (Figure 27 ). The impacting barge is defined by a mass, initial velocity, and bow forcedeformation (crush) relationship. Traveling at a prescribed initial velocity, the barge impacts a specified location on th e bridge structure and generates a timevarying impact force in accord ance with the crush curve of the barge and the relative displacements of the barge and bridge m odel at the impact location. The MDOF bridge model (pier, superstructure, and soil) is subjec ted to the timevarying dynamic impact force and consequently displaces, develops internal forc es, and interacts with the SDOF barge model through the shared impact force. The CVIA algorithm has been documented in detail in a number of previous publications (Consolazio and Cowan 2005; C onsolazio and Davidson 2008; Cons olazio et al. 2008) and has been implemented in the commercial pier anal ysis software package FBMultiPier (2009). Since barge, pier, and superstructure stiffness and mass related forces are all included in CVIA, the method is able to accurately predict pier a nd substructure design forces under dynamic barge impact conditions. CVIA has been validated ag ainst fullscale experimental impact data (Consolazio and Davidson 2008) and presently constitutes a stateoftheart computational tool 25 PAGE 26 for bargebridge collision anal ysis when numerical efficiency and direct consideration of dynamic effects are paramount. Because the CV IA procedure can accurately capture dynamic amplifications of pier design forces due to superstructure inertial effect s, it is used throughout this study when dynamic assessments of structural demand are required. Dynamic Amplification of Pier Column Forces A recent study by Davidson et al. (2010) quantif ied dynamic amplification of pier column internal forces for a wide range of pier and s uperstructure configurations. Dynamic amplification was assessed by comparing pier column desi gn forces predicted by dynamic CVIA (Figure 28 a) to forces predicted by static analyses in which peak dynamic impact loads P B max (Figure 28 b) were applied. This research uncovered two distinct dynamic amplification modes superstructure inertial restra int and superstructure momentum driven sway. Mixed cases were also identified in which both modes of amplification were present, i. e. inertial restraint controlled column moments, while sway controlled column shear forces Regardless of which dynamic amplification mode was most prominent, supers tructure mass was found to play an important role in driving bridge re sponse during impact events. Figure 29 illustrates the relative magnitude of dynamic amplification for a range of bridge pier configurations and impact conditions cas es total. Impact cases are denoted using a threeletter abbreviation corresponding to the bridge (e.g. ACS for the Acosta Bridge), a three letter abbreviation for the location of the impacted pier (CHA for a pier adjacent to the navigation channel and OFF for a pier located away from the na vigation channel), and a single letter denoting the impact energy (L, M, H, and S for low, moderate, high, and severe energy, respectively) [See Davidson et al. 2010 for detaile d descriptions of bridge configurations and impact energy cases]. 26 PAGE 27 Data presented in Figure 29 are ratios of maximum dynamic pier column moment divided by maximum static pier column mo ment, obtained as shown in Figure 28 Presented in this way, a ratio greater than 1.0 indicates that dynamic an alysis predicted a larger member demand than corresponding static analysis. In such cases, inertiarelated e ffects amplify member forces beyond the magnitude that typical static analys is methods can predict. As shown in Figure 29 ratios for most cases range from 1.5 to 2.0, indicating that superstructure inertia amplifies column moments at least 50100% for most impact scenarios, sometimes much more. Furthermore, for typical design situations, the magnitude of amplification for a given bridge configuration is not strongly sensitive to impact energy. For example, moment demands for the Acosta Bridge channel pier (ACSCHA) are amplified by approximately 500% when pier dynamics are considered, regardless of impact energy. This finding implies that dynamic amplification is primarily a func tion of structural characteris tics such as mass and stiffness distribution. 27 PAGE 28 Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data Envelope A Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data Envelope B Bow Deformation (in) Bow Deformation (m)Impact Force (kips) Impact Force (MN) 0 30 60 90 120 150 0 0.8 1.6 2.4 3.2 0 1,000 2,000 3,000 0 4 8 12 Test data Envelope C Figure 21. Forcedeformation results obt ained by MeierDrnberg (Adapted from MeierDrnberg 1983). A) re sults from dynamic cylindrical impact hammer test, B) results from dynamic 90 pointed impact hammer test and C) Results from static impact hammer test. 28 PAGE 29 Bow deformation (in) Bow deformation (ft)Contact force (kips) Impact energy (kipft) 0 30 60 90 120 150 0 2.5 5 7.5 10 00 1,000 8,000 2,000 16,000 3,000 24,000 Average force Force envelope Impact energy Figure 22. Relationships de veloped from experimental barge impact tests conducted by MeierDrnberg (1983) (Adapted from AASHTO 1991). Crush depth, aB (ft)Impact load, PB (kip) 0 2 4 6 8 1012 0 500 1000 1500 2000 2500 3000 Figure 23. AASHTO relati onship between barge crush depth and impact force. 29 PAGE 30 A B C D Figure 24. Finite element simulation of barge bow crushing. 6 ft diameter round impact with jumbo hopper barge bow. A) 0in. crush de pth, B) 60in. crush depth, C) 120in. crush depth and D) 180in. crush depth. 30 PAGE 31 Crush depth, aB (ft)Impact load, PB (kip) 0 2 4 6 8 1012 0 500 1000 1500 2000 2500 3000 A Crush depth, aB (ft)Impact load, PB (kip) 0 2 4 6 8 1012 0 500 1000 1500 2000 2500 3000 B Figure 25. Barge bow forcedeformation rela tionships. A) AASHTO crus h curve (independent of impact surface characteri stics) and B) Consolazio et al. (2008; 2009a) barge crush curve for 6foot diameter round impact surface. 31 PAGE 32 Barge Bow ForceDeformation Pier column shape wP : Pier width (ft)PBY : Barge yield load (kips) 0 6 12 18 24 3036 0 2,000 4,000 6,000 8,000 10 F l a t 1 5 0 0 + 6 0 wP F l a t 3 0 0 + 1 8 0 wP R o u n d 1 5 0 0 +3 0 wP PBY = 1500+60wP PBY = 300+180wP PBY = 1500+30wP wP Bow deformationImpact force PBY aBY wP Impact Force PB wP Impact Force PB Barge bow forcedeformation modelft 10 wP ft 10 wP ft 10 wP ft 10 wP Flatfaced Round 2 in. Figure 26. Barge bow for cedeformation flowchart. 32 PAGE 33 u BmBaBmPuP PB PB Pier structure Soil stiffness Crushable bow section of barge Barge Barge and pier/soil system are coupled together through a common contact force PB Single DOF barge model MultiDOF bridge model superstructure Figure 27. Coupling between barge and bridge in CVIA Pb(t) t Pb Pb(t) Pb max Dynamic CVIA A Pb max Static B Figure 28. Analyses conducted to quantify dynamic amplification. A) Dynamic CVIA and B) Static (using peak load, P B max from CVIA) 33 PAGE 34 0 1 2 3 4 5 6 7 8 NSGOFFL RNGOFFL ACSCHAM BLTCHAM EGBCHAM ESBCHAM GNDCHAM MBCCHAM NSGCHAM NSGOFFM RNGCHAM RNGOFFM ACSCHAH BLTCHAH NSGCHAH NSGOFFH RNGCHAH SRBCHAH ACSCHAS NSGCHAS ESBCHAM OSGCHAM PNCCHAM OSGOFFL SBZCHAM SRBCHAM GNDCHAH SBZCHAH Inertial restraint Superstructure momentum Mixed inertial restraint / superstructure momentum 23 24 22Maximum dynamic column moment divided by maximum static column moment Figure 29. Maximum pier column dynami c moments relative to static moments. (from Davidson et al. 2010) 34 PAGE 35 CHAPTER 3 BRIDGE MODELING AND STUDY PRELIMINARIES Introduction To maximize the accuracy and computational efficiency of the coupled vessel impact analyses conducted in this study, several modelin g and analysis procedures were employed, and, in selected cases, new features were develope d and implemented into the FBMultiPier finite element analysis code (FBMultiPier 2009). FBM ultiPier can be used to perform linear or nonlinear, static or dynamic analyses on single pier models or full bridge (multipier, multispan) models. As illustrated in Figure 31 FBMultiPier bridge models generally contain the following components: Superstructure: Modeled using resultant frame elements with linear elastic material behavior. Bridge pier: Modeled using cr osssection integrated frame elements in conjunction with nonlinear kinematic and constitutive material models. Pile cap: Modeled using thick shell elements with linear elastic material behavior. Foundation: Modeled using crosssection integrated frame elements in conjunction with nonlinear kinematic and constitutive material models. Soil: Modeled using nonlinear discrete spring elements, distributed along each embedded foundation element. Although FBMultiPier has the abil ity to directly analyze full multispan, multipier bridge structures, in this study, onepier twospan (OPTS) bridge models were used instead to increase computational efficiency (Consolazio and Da vidson 2008). Using FBMultiPier, an inventory was developed, consisting of twelve (12) finite element models of Florida bridges. All static and dynamic analyses conducted as part of this study utilized these twelve (12) models. 35 PAGE 36 Superstructure In an FBMultiPier bridge model, the supers tructure is represented using a series of linearelastic resultant frame elements. As a resu lt, the girders, deck, and other superstructure features are considered to act compositely during interpier force transmission. Rigid elements and multidegree of freedom springs are used to connect the primary superstructure elements to each pier. The superstructure modeling capabilitie s of FBMultiPier enable important static and dynamic interactions between bridge piers. The use of linear resultant frame elements for superstructure modeling in FBMultiPier necessitates the inclusion of additional elements to correctly distribute forces to the bearing locations of underlying piers. More specifically intermediate superstructure elements are employed to approximate the effect of the physi cal footprint of the s uperstructure and its interaction with the pier. As illustrated in Figure 31 the superstructuresubstructure interface model consists of four primary partsa rigid vertic al link, a rigid horizontal tran sfer beam, bearing springs, and horizontal bearing offset rigid links. The vertical link is used to produce the correct relative height between the superstructure center of gr avity and centerline of th e pier cap beam. This distance can be quite large (reaching heights grea ter than 10 ft), especially for bridges with longspan haunched girders or boxgirder superstr uctures. A horizontal tr ansfer beam, connected to the vertical link, serves to distribute superstructure load to physical bearing pad positions. For piers with two bearing rows, additional horizontal rigid elements act to offset the bearing locations from the pier cap centerline. Bearing pad elements are m odeled as discrete stiffne sses. Arbitrary, nonlinear loaddeformation relationships can be associated with each bearing DOF, providing the ability to model complex hyperelastic material behavior or finitewidth gapping. For the purposes of this 36 PAGE 37 research, this level of model sophistication was unwarranted. However, the use of nonrigid loaddeformation relationships for these elements did provide a more real istic representation of bearing behavior when compared to constraintbased pin or rolle r supports. Therefore, empirical loaddeformation relationships, as opposed to in finitely stiff bearing conditions, were employed for these model components. In the course of conducting this study, it was observed that constraint based (pin or roller) bearing conditions led to erratic distribution of bearing loads. For example, when constraints were employed, vertical bearing reactions under gravity loading could consist of widely varying compression and tension forces. For analysis cas es where constraintbased bearing conditions were employed, the sum of vertical bearing reac tions was consistently found to be statically equal to the superstructure d ead load, however, the conspicu ous load distribution led to undesired localized member fo rce concentrations and erro neous deformation patterns. To alleviate this problem, finite stiffness values were prescribed for each bearing spring DOF to simulate more realistic bearing conditi ons. To accomplish this, bridge models with constrained bearing conditions were first statically subjected to loading conditions similar to those associated with respective vessel impact events of interest. The stiffness of the rigid springs was incrementally reduced until the tota l bearing reaction was evenly distributed among the bearing elements. During this process, care was taken not to overly soften the bearing springs, as this would reduce the overall rigidity of the superstructure substructure interface. Reducing the relative stiffness of the superstructu re system could cause a larger portion of the vessel impact load to travel through the foundation, as opposed to the superstructure, resulting in unconservative pier column dema nd predictions. With this concern in mind, it was confirmed that the sum of the nonrigid bearing fo rces compared well with the sum of the 37 PAGE 38 constraintassociated bearing fo rces. Consequently, it was observe d that the softened bearings afforded a uniform distribution of bearing forc es to the pier while maintaining comparable superstructure rigidity. Substructure Bridge substructures, as modeled in this study, consist of one or more pier columns, a pier cap beam, a pile cap, multiple driven piles or drilled shafts, and a numerical representation of the soil. Horizontal and vertical soil stiffness is modeled using nonlinear spring elements. Linear elastic, thickformulation shell elements are used to model the pile cap. Pier Modeling Flexural substructure elementspier columns, pier cap beam, and either piles or drilled shaftsare modeled using nonlinear crosssection integrated frame elements. For relatively light loading, such as that associated with certain service loading conditi ons, bridge structural components are typically assumed to remain effec tively linear. As such, accurate predictions of pier behavior under this type of loading can be obtained using resultant frame elements with gross crosssectional properties a nd linearelastic material models. However, bargebridge collision is an extrem e event scenario, potentially resulting in permanent damage or even collapse of the br idge structure. As a result, robust nonlinear analytical tools are necessary to capture the e ffects of widespread concrete cracking, plastic hinge formation, and permanent member deformati on. Furthermore, bridge structural elements are commonly composed of multiple materials (i.e., reinforced or prestressed concrete). To address the composite nature of concrete pier elements, FBMultiPier employs crosssection integrated frame elements that permit the crosssectional shape and the locations of reinforcing bars or prestressing tendons to be modeled explicitly. When such elements are employed, the cross section is discretized into multiple regions of integration and rele vant material models 38 PAGE 39 (stressstrain relationships) are assigned to each di screte area (integration po int) as illustrated in Figure 32 As the section flexes in genera l biaxial bending and deforms axia lly, a planar (linear) strain field is generated across the section. Nonlinear stressstrain relationships for each material ( Figure 33 ) lead to a nonlinear distribu tion of stresses over the crosssection. Section axial force and moments are then computed by integrating (summing) the force and moment contributions from each integration point. FBMultiPier can employ either builtin stressstrain relationships or userdefined stressstrain relationships for concrete and steel. In Figure 33 builtin FBMultiPier stressstrain relationships for typical bridge pier materials are illustrated. Concrete is modeled using a modified Hogenstead parabola in compression, with strain softening described by a straight line. It is important to note that the maximum compressive stress ( ) is 0.85 times the specified cylinder compressive strength ( ). In tension, concrete is treated as linear until the cracking stress ( ) is reached. After cracking, a bilinear m odel is employed to account for tension stiffening. For 270ksi highstre ngth prestressing tendons, the st ressstrain model proposed by the Precast/Prestressed Concrete Institute (PCI 2004) is employed. Mild reinforcing steel is treated as elastic, perfectly plastic with a yield stress of 60 ksi. When combined with the crosssection integration scheme, these materi al models provide accurate predictions of momentcurvature and plastic hing ing behavior for reinforced and prestressed concrete members. cf'' cf' rf Soil Modeling With regard to representing soilstructure in teraction, FBMultiPier utilizes empirical soilstrength models, where these models correl ate pertinent soil parameters (e.g., internal friction angle, subgrade modulus) to deformation behavior under loading. Three primary modes 39 PAGE 40 of soilstructure interact ion are considered: latera l resistance (Py), axial skin friction along the pile or shaft ( z), and tip bearing resist ance (Qz). For each mode, nonlinear curves define relationships between resistance and displacement. Example soil resistance curves are illustrated in Figure 34 Due to the distributed nature of soil resistan ce, soilstructure interaction in each orthogonal direction is represented using nonlinear spring elements distribut ed along the embedded pile (or drilled shaft) length. One vertical spring at each nodal location models the tributary skin friction ( z). Additional spring elements are placed at each pile tip node to represent the tip resistance (Qz). Modeling lateral (horizontal) soil resistance requires a slightly more refined approach. FBMultiPier incorporates userdefined pile row multipliers (py multipliers) to account for pile group behavior. Each row of piles is assigned a factor by which the lateral resistance is scaled, depending on pile spacing and position within the group (Figure 35 ). However, these factors are dependent on the direction of pile group motion. For example, a pile on the leading row may be assigned a factor of 0.8. However, if the founda tion motion reverses direct ion, that same pile becomes part of a trailing row, for which a row multiplier of 0.3 is applied. Similarly, when group motion occurs in the transverse direction, unique multipliers are used. Consequently, at every node, four horizontal spring elements define the lateral soil resistance in two orthogonal directions. The nonlinear stiffness of each spri ng is scaled by the applicable row multiplier, providing a realistic representation of lateral pile group behavior. Behavior of soilspring elements under cyclic lateral loading can be considered either nonlinearelastic or nonlinearine lastic, depending on the soil type. Typically, for cohesionless soils, such as sands, the material exhibits nonlinearelastic behavior, returning to the original 40 PAGE 41 undeformed state when load is removed. In c ontrast, cohesive soils (e.g, clay) are often considered nonlinearinelastic a nd gaps may form between soil and pile surfaces during cyclic loading. FBMultiPier soil models can account for this type of behavior; however, a lack of available, dynamic soil data for many of the brid ges studied precluded the direct incorporation of such effects in the current study. Bridge Inventory and Impact Conditions Later in this thesis, a parametric study is conducted to facilitate the development of the proposed equivalent static analysis method a nd to assess the accuracy of the method when compared to dynamic analysis. The parametric study makes use of FBMultiPier numerical models (using the features previously described) of Florida bridges of varying age, pier configuration, superstructure type, foundation type and soil conditions subjected to barge impact scenarios of varying energy. Specific bridge structures and impact conditions that are considered in the parametric study are documented below. Bridge Inventory Twelve (12) models were developed for vari ous piers and spans of nine (9) different bridges ( Table 31 ) located in the state of Florida. For conciseness, and to simplify discussion, each bridge structure is assigne d a threeletter identification code. Specific piers within each bridge are further delineated by proximity (with respect to the vessel transit path): the letters CHA appended to a bridge identi fication code indicate that the pier is a channel pier, whereas the letters OFF indicate an offchannel pier (a pier not directly adjacent to the channel). Bridges listed in Table 31 selected from a larger Florid a Department of Transportation (FDOT) catalog of almost twohundred bridgesc onstitute a representa tive crosssection of bridge types currently in service in Florida. The nine (9) selected bridges also vary widely in age, with construction dates spanning from the late 1960s to 2004. 41 PAGE 42 Various past analytical studies (Consol azio et al. 2009a; Yuan et al. 2008) have demonstrated that the geometry (shape and width) of the impacted portion of a bridge pier affects the magnitude of impact loads generated during coll ision. Hence, in the parametric study, bridges were selected that vary in both pier column shape (flatfaced, round) and pier column width (ranging from 3.5 ft to 20 ft). Pier shape, foundation type (pileandcap or drilled shaft), and size data are summarized in Table 31 The extent to which load applied to an im pacted pier is transmitted to adjacent piers depends upon the type of superstructure th at connects the piers together. Three common superstructure types are included in the bri dge inventory employed in the parametric study ( Table 31 ): concrete slab on concrete girders; concrete slab on steel girders; and, segmental concrete box girder. Superstructu re span lengths included in th is study are representative of common, moderatespan bridges that span U.S. inland waterways as opposed to less common, longspan bridges. Barge Impact Conditions Jumbo hopper barges are the most common type of vessel found in the U.S. barge fleet, and additionally, constitute the baseline desi gn vessel for bargebridge collision in the AASHTO provisions. For these reasons, a jumbo hopper barg e was employed as the design vessel in the parametric study. Prescribed barge impact condi tions for bridges in the parametric study were chosen to span the range of collision events that are conceivable for navigable Florida waterways. Vessel weight (flotilla weight) and collision velocity were varied to produce a representative range of impact energy cases: low, moderate, high, and severe ( Table 32 ). Based on waterway traffic characteristics pier location (relative to th e navigation channel), and pier strength, one or more suitable impact energy cond itions were assigned to each pier in the study. 42 PAGE 43 43 The AASHTO provisions require that all bri dge components located in navigable water depths be designed for, at a minimum, imp act from a single empty hopper barge (200 tons) drifting at a velocity equal to the mean water current for the waterw ay (AASHTO 2009). This lowenergy drifting barge condition is meant to be representative of a barge that breaks loose during a storm and drifts into a pier. Such a c ondition is only relevant to the design of piers distant from the navigation channel (since near channel piers must be designed for greater impact energies, which are associated with errant tugpropelled barge flotillas). Therefore, the lowenergy impact condition was only applied to offchannel piers ( Table 32 ). Using watercurrent data for several waterways in Florid a, an approximate average current velocity of 1 knot was determined. Thus, the lowenergy case is defined as a 200ton barge drifting at a velocity of 1 knot. The majority of impact cases considered in this study fall into the categories of either moderate or high energy ( Table 32 ). A moderateenergy impact condition is defined as one fullyloaded hopper barge (2030 tons with tug) trav eling at 2.5 knots, and a highenergy impact condition is defined as a flotil la consisting of three fullyloa ded hopper barges (5920 tons with tug) traveling at 5.0 knots. These conditions cover the majority of possible impact energies that would be generated by collisions from typical Florida barge traffic (Liu and Wang 2001). For two of the piers considered in this study (c hannel piers of the Acos ta and New St. George bridges), barge traffic and waterw ay conditions warrant the defin ition of an additional severeenergy impact condition: a flotilla consisting of four fullyloaded hopper barges (7820 tons with tug) traveling at 7.5 knots. PAGE 44 44 Table 31. Bridge pier configurations. Pier column data Shaft/pile data Superstructure type Span lengths adjacent to pier (ft) Bridge ID Bridge name Cap a,b elevation Impact surface shape Impact surface width (ft) No. Width (ft) Height c (ft) Type No. shafts or piles Width (ft) Box girder Concrete slab on steel girders Concrete slab on concrete girders North or west South or east ACSCHA Acosta Waterline Flat 50.0 1 20 40 Drilled shaft 31 5.0 X 370 620 BLTCHA SR20 at Blountstown Waterline Round 9.0 2 5.5 37 Drilled shaft 2 9.0 X 280 220 EGBCHA Eau Gallie Mudline Flat 4.0 4 4.0 69 Steel pile 39 1.2 X 150 150 MBCCHA Melbourne Mudline Round 4.5 2 4. 5 72 Concrete pile 32 1.5 X 145 110 NSGCHA New St. George Island Waterline Flat 28.0 2 6.0 52 Concrete pile 15 4.5 X 250 260 NSGOFF New St. George Island Waterline Flat 28.0 2 5.5 52 Concrete pile 9 4.5 X 140 140 OSGCHA Old St. George Island Mudline Flat 5.7 2 5.5 47 Steel pile 40 1.2 X 250 180 OSGOFF Old St. George Island Waterline Flat 3.5 2 3.5 40 C oncrete pile 8 1.7 X 74 74 PNCCHA Pineda Mudline Round 4.5 2 4. 5 73 Concrete pile 30 1.7 X 120 68 RNGCHA John Ringling Waterline Round 13.0 1 13.0 40 Drilled shaft 2 9.0 X 300 300 RNGOFF John Ringling Waterline Round 13.0 1 13.0 25 Drilled shaft 2 9.0 X 300 300 SBZCHA Seabreeze Mudline Flat 8.0 1 8.0 58 Concrete pile 32 2.0 X 250 250 a Waterline footing indicates a foundation topsurface elevation near the mean high water level. b Mudline footing indi cates a foundation topsurface elevation near the soil surface. c Distance from top of foundation to bottom of pier cap. PAGE 45 Table 32. Analysis matrix of barge impact energy conditions. Impact energy condition Low (L) Moderate (M) High (H) Severe (S) Impact condition characteristics Velocity (knot) 1.0 2.5 5.0 7.5 Weight (tons) 200 2030 5920 7820 ID and location Cases analyzed ACSCHA X X X BLTCHA X X EGBCHA X MBCCHA X NSGCHA X X X NSGOFF X OSGCHA X OSGOFF X PNCCHA X RNGCHA X X RNGOFF X X SBZCHA X X 45 PAGE 46 P i e r c a p c e n t e r l i n eSuperstructure C.G. height Superstructure Bridge pier Superstructure C.G. P i e r c a p c e n t e r l i n eSuperstructure C.G. height Bearing springs Superstructure beam elements Rigid transfer beam Rigid vertical link Bearing offset rigid links Physical structure Numerical model Figure 31. Overview of superstructure model configuration in FBMultiPier. 46 PAGE 47 Each integration point assigned a material model Cross section discretized into integration points z y z y x dAi dFiN e u t r a l a x i s z y ziyii Concrete integration point Steel integration point z y Full cross section integrated to obtain section forces Concrete Steel Integration point i plane Integration point i Zi A i Yi A i i A iM= YdA YdA M= ZdA ZdA P= dA dA i i Figure 32. Cross section integratio n scheme for nonlinear frame elements. 47 PAGE 48 Tension Compression''' ccf0.85f '' cucf0.85f cE 2 '' cc 00ff2 r stf7.5 0.0001316 57000 0.0003 rstfrfr0.5f '' c 0 c2f E cu0.0038 c cf0.002 A sfssy0.00207 sE29,000ksi syf60ksi B py0.0086 0.025 p s p sf p spspsfE puf270ksi pyf245ksi psE28,500ksi ps ps0.04 f270 0.007 C Figure 33. Nonlinear material models as impl emented in FBMultiPier. A) concrete, B) mild steel and C) prestressing steel 48 PAGE 49 Lateral pile deflection (y)Lateral soil resistance (P) A Vertical pile deflection (z) Unit skin friction ( ) B Vertical tip deflection (z)Soil tip resistance (Q) C Figure 34. Typical soil resistance curves employed by FBMultiPier. A) lateral soil resistance (Py) curve, B) skin friction ( z) curve and C) tip resistance (Qz) curve. 49 PAGE 50 0.30.40.8 0.3 0.30.3 Group motion 0.30.30.3 0.3 0.40.8 Group motion 0.4 0.8 0.3 0.3 Group motion 0.3 0.3 0.4 0.8Group motion Figure 35. Variation of row multiplie rs for differing pile group motion. (Note: Specific multiplier values will vary based on pile spacing) 50 PAGE 51 CHAPTER 4 STATIC BRACKETED IMPACT ANALYSIS (SBIA) METHOD Introduction As illustrated by prior research (Consolaz io et al. 2006; Consolazio et al. 2008; Davidson et al. 2010), current sta tic analytical methods for bargebridge collision do not account for dynamic amplification of pier member forces exhibited during impact events. Consequently, design forces obtained using traditional static procedures can be markedly unconservative. The ideal means by which to simulate inertial forces stemming from an imp act eventand accurately predict member forcesis to use a fully dynamic tim ehistory structural an alysis procedure such as CVIA. However, such simulation procedures can be computationally expensive (relative to static analysis) and require a re latively detailed numerical description of the structure. These issues may be prohibitive in some design situations, particularly during preliminary bridge design when pertinent bridge details are subject to significant variation and frequent revision. To address the accuracy limitations of current static analysis and the computational requirements of dynamic timehistory analysis while still accounting for inertial effects manifested during bargebridge collision events, an equivalent static analysis method is developed, demonstrated, and verified in this chapter. The newly developed equivalent static analysis method is both simple to use and mini mally conservative when compared to dynamic analysis (i.e., it accounts for dynamic amplification). Conceptual Overview The primary limitation of static analysis me thods is the assumption that lateral pier resistance is provided only by so il and superstructure stiffness ( Figure 41 a). However, acceleration of superstructure mass results in significant inertial restraint at the top of the pier immediately after impact ( Figure 41 b). In many cases, superstructure inertial resistance will 51 PAGE 52 equal or greatly exceed stiffness based resistan ce. Consequently, for most bridges, momentary maximum pier forces are manifest ed during this early stage of the impact event (Davidson et al. 2010) In addition to inertial restraint based dynamic amplification, a second mode of amplification was also identified by Davidson et al. (2010): superstruc ture momentumdriven sway. In this case, maximum pier forces occur la ter in time, perhaps afte r the barge is no longer in contact with the pier. In this mode, oscillat ion of the superstructure mass acts as a source of loading, ultimately driving maximum pier forces However, even in cases where superstructure momentum is the dominant dynamic amplification mode, an initial spike in member forces occurs as a result of inertial restraint (resistanc e) at the pier top. For the swaycontrolled cases described in Davidson et al. (2010), this initial restraint based demand was 7080% of the maximum demands observed later, resulting from superstructure sway. As such, the superstructure inertial resi stance phenomenon is present and significant for all structural configurations considered by Da vidson et al. (2010) In additi on, no simplified means has been identified to predict which dynamic amplificatio n mode will dominate the response of a given bridge, aside from conducting a timehistory dynamic analysis. Th erefore, focus is given to developing static analysis proce dures that approximate superstruc ture inertial resistance, which can be scaled up to conservatively enve lope the swaydriven values as well. Superstructure Modeling Three primary superstructure stiffness modeli ng schemes were considered for use in this study: Full multiplepier, multiplespan bridge model Onepier, twospan (OPTS) bridge model in cluding impacted pier and adjacent spans Onepier model with a single lateral spring representing the superstructure stiffness 52 PAGE 53 However, it has previously been demonstrated (Consolazio and Davidson 2008) that onepier, twospan (OPTS) bridge descri ptions provides accurate approx imations of both static and dynamic behavior of full multiplepier multiples pan bridge models. As such, OPTS modeling was employed in development of the equivalent static analysis method, but full bridge models were not. To develop and verify the equi valent static analysis met hod, both OPTS and springbased superstructure models were considered ( Figure 42 ). While the OPTS modeling technique is more accurate, it is common practice in bridge desi gn to represent superstruc ture stiffness with a single equivalent spring. As such it is important to also consider this approach when assessing the suitability of proposed analys is methods for design practice. Static Impact Load Determination A critical step in any impact analysis is quantifying the maximum dynamic load to which the structure is subjected. In the equivalent static method developed here, maximum magnitude dynamic loads are determined, in part, by usi ng the principle of c onservation of energy (Consolazio et al. 2008) as follows: 0DE KEif if (41) where KE if is the change in kinetic energy of the barge, and DE if is the change in total deformation energy (i.e. the sum of the elastic and plastic deformation energies), associated with the deformation of the barge bow, from the initial state (i) to the final st ate (f). Conservation of energy is used to define a relationship betw een the maximum impact load, and the barge parameters. Assuming that barge mass does not change during the collision, the change in barge kinetic energy can be expressed as: 53 PAGE 54 2 Bi 2 BfB ifvvm21KE (42) where m B is the constant (unchangi ng) mass of the barge and v B Bf and v Bi are the magnitudes of the barge velocities at the initia l and final states respectively. In general, the deformation energy for the barge can be described using the following relation: Bf Bia a BBB ifdaaP DE (43) where P B(a B B B ) is the impact force as a f unction of barge crush depth (a B), and a B Bi and a Bf are the barge crush depths at the initial and final states respectively. To estimate the maximum impact load acting on the pier, the following assumptions were made: 1.) the initial barge crush depth (a Bi ) is assumed to be zero, and 2.) the barge bow forcedeformation relationship is assumed to be elastic perfectly plastic (Figure 43 ), and 3) the pier is initially assumed to be rigid and fixed in space (n ote that this assumption is ultimately removed, as described later). The first assumption implies that the initial kinetic energy of the barge is fully converted into deformation energy of the barge bow during loading of the barge (Figure 44 a). Thus, once all of the barge initial kinetic energy has been co nverted into deformation of the barge bow (i.e. the barge velocity becomes zero) the barge bow crush depth has reached its maximum value. Additionally, when the barge bow recovers the elastic portion of its deformation energy through unloading, this energy is then converted b ack into rebound motion of the barge (Figure 44 b). Final barge kinetic energy can th en be determined from the recovered deformation energy. If the barge bow remains linear and elastic, th e conservation of energy up to the point of maximum barge bow deformation can be represented by the following equation: 54 PAGE 55 2 imim BBiBBmKEDE12mv12Pa0 (44) where P B is the maximum impact force observed during the impact, and a B Bm is the maximum barge bow deformation. The maximum impact force and barge bow deformation however, remain undetermined up this point, and thus, an additional equation is required. If the barge bow remains elastic, the maxi mum bow deformation can be defined as follows: BB Bm BBYBYPP a kPa (45) where a BY and P BY are the barge bow deformation and fo rce at yield, respectively, and k B is the initial elastic stiffness of the barge bow. Combining Eqns. and and solving for the maximum load produces the following equation: B 44 45 BY BBiBBiBBBY BYP PvmvkmP a (46) Due to the elastic perfectlyplastic assumption for the barge bow forcedeformation relationship, the maximum barge impact force is limite d to the yield load of the barge bow. Validation of the analytical model in Eqn. 46 revealed that the ri gidpier assumption is overly conservative in cases where the barge deformation remains elas tic (Consolazio et al. 2008). This finding necessitates inclus ion of the finite stiffness of the impacted piersoil system into the formulation. The stiffness of the barge a nd piersoil system are then combined to form a series stiffness (k S ). 1 PBY BY 1 PB Sk 1 P a k 1 k 1 k (47) 55 PAGE 56 where k P is the linear stiffness piersoil system. Re placing the initial elastic barge stiffness (k B) in Eqn. with the effective bargepiersoil series spring stiffness (k B 46 S ) (Eqn. ) produces the following equation: 47 BBiSBBiBPBPvkmvcP Y (48) where c BP is the bargepier pseudodamping coefficient, defined as follows: BS BPmkc (49) By incorporating the series stiffness (k S ) in place of the barge stiffness (k B), the accuracy of the load prediction model is greatly enhan ced, providing a reliable means of estimating the peak vessel impact load. See Consolazio et al. (2008) for additiona l details and verification of the procedure. B Potential Static Approximations of Superstructure Inertial Resistance In previous research (Consolazio et al. 2006; Consolazio et al 2008; Davidson et al. 2010), superstructure inertial resist ance was identified as the domin ant source of dynamic pier force amplification during vessel impact events. Curre nt static analytical methods (e.g., AASHTO) do not account for such dynamic effects. To addre ss this important omission, three analytical schemes were assessed to statically approximate the additional source of superstructure resistance attributable to inertia: Restrain the pier top with an infi nitely stiff lateral boundary condition ( Figure 45 a) Amplify the lateral superstructure stiffness ( Figure 45 b) Directly apply an inertial load at the superstructure level ( Figure 45 c) Each of these approaches invol ved modifying one of three co mponents common to all static analyses: boundary conditions, stiffness, and loads. 56 PAGE 57 Approximating superstructure inertial resist ance by means of boundary conditions (i.e. fixing the piertop elevation) gene rally proved overly conservative; static predictions of pier demands greatly exceeded those quantified by dynam ic simulations. In addition, the level of conservatism varied greatly between various structural configurati ons. This result is, in part, predictable, as a fixed boundary condition allows no lateral deflection at the piertop elevation. Dynamic simulations consistently show positive, nonzero displacements (in the direction of impact) at this elevation during times at which maximum structural demands occur. As such, a fixed piertop boundary condi tion provides an unreasonable level of restraint. Given these findings, use of a finitestiffness boundary conditi on at the piertop elevation was considered. As noted previously, static s uperstructure stiffness alone does not provide sufficient resistance to adequate ly predict dynamic amplification effects. Thus, the lateral superstructure stiffness was amplified to accoun t for both static superstructure resistance and superstructure inertial resist ance. This approach proved reasonably effective in producing displaced pier shapes and member demand predictions that were c onsistent with dynamic analysis. However, a large degree of variabil ity in the stiffness magnification factors was observed across a range of struct ural configurations. Furthermor e, no correlation was observed between these magnification factors and corr esponding bridge parameters. Consequently, amplifying superstructure stiffness was deemed an impractical means of approximating of superstructure inertial resistance. Given the significant conservatism and variab ility associated with boundary condition and stiffnessbased approximations of inertial resistance, a third major static approach was consideredapproximating the superstr ucture inertial resisting for ce and applying it as a static load ( Figure 45 c). As was the case with magnified superstructure stiffnesses, static inertial 57 PAGE 58 forces varied substantially among differing structural configurations. However, correlations were identified between the static inertial loads and corresponding br idge parameters, thus providing an accurate static approximation of dynamic pier behavior. Static Approximation of Inertial Resi stance by Direct Load Application An empirical approach was employed to develop equivalent static loading conditions that provide accurate predictions of both maximu m dynamic member for ces and dynamic pier displacements. Ideal static loadi ng conditions were determined fo r twelve (12) Florida bridge piers, incorporating a wide range of common pier styles. In add ition, one or more representative impact energy conditions were considered for each bridge, resulting in twenty (20) pier/energy cases. Load prediction equations were then developed that correlate statically equivalent inertial loads to readily available bridge parameters. Factored Impact Load Static superstructure inertial forces were found to be strong ly sensitive to the choice of impact load. For any given impactpoint load (within a reasonable range), a corresponding piertop load can be identified that will provide an adequate prediction of dynamic member forces (shears and moments). However, if the im pact point load is too small, unrealistic pier displacements result. Specifically, if the maximum dynamic impact load is applied, as calculated from Eqn. 48 the corresponding piertop load s must be very large to obtain member forces similar to those predicted dynamically. In many cas es, the large piertop lo ad causes the pier to deflect into the negative range (opposite of th e impact direction). Nega tive displacements were not observed from dynamic analysis, thus this static response is undesirable. To counteract this tendency, a factored imp act load was employed. Use of an amplified impact load causes piers, as a whole, to deflect in the impact dire ction, avoiding possible unrealistic negative pier displacemen ts. Through an iterative process, an impact load factor of 58 PAGE 59 1.45 ( Figure 46 ) was found to be reasonably ideal (op timal) in terms of producing realistic displaced shapes, while still providing accura te estimations of dynamic member forces. Determination of Ideal PierTop Loads Having factored the peak impact load (P B) by 1.45, ideal static pi ertop inertial loads (P B I ) were determined by iteration. An ideal piertop load is defined as the load that generates a static column moment equal to the maximum dynamic co lumn moment. For each of the twenty (20) pier/energy cases studied, ideal piertop loads were determined utilizing both OPTS and springbased superstructure models (recall ). Ideal piertop inertial loads were normalized by corresponding impact loads (P Figure 42 B B ) to form ideal inertial resistance factors (IRF ideal ), as shown in Figure 47 Two additional sets of ideal inertial resistance factors (IRF) were analogously developed, corresponding to pier co lumn shear and total bearing shear. Correlation of PierTop Inertial Resistance Factors (IRFs) to Bridge Parameters In the interest of developing a universal lo ading condition for all br idge configurations, one possible approach is to use the maximum IRF value obser ved among all bridges types. However, significant variation in ideal IR F magnitudes was observed between differing structural configuratio nsIRF values ranged from a minimu m of 0.02 to a maximum of 1.17. Thus, applying an equivalent inertial load equal to B1.17P to a bridge for which the ideal IRF is 0.02 is unreasonably conservative. To mitigate ex cessive conservatism, ideal IRF values were correlated to key structural parameters, to form empirical IRF prediction equations. These equations allow for computation of bridgespeci fic IRFs, providing a more accurate result than simply utilizing a uniform load factor for all bridges. Additionally, IRF prediction equations were calibrated separately to each major pier demand typecolumn moment, column shear, and total bearing shearso that conservatism was minimized. 59 PAGE 60 Several bridge parameters were considered for possible correlation to inertial resistance. Quantities such as natural frequency or supers tructure acceleration generally control dynamic pier behavior. However, quantities of this na ture are not readily de termined without first conducting a dynamic analysis. Furthermore, duri ng the early portions of the bridge design process, sufficient information ma y simply not be available to estimate such dynamic properties. Instead, ideal IRF values were corr elated to parameters that can be easily quantified or estimated at any practical stage of design. Parameters were considered that involve geometry, mass distribution, and stiffn ess of the various bridge component sall of which influence dynamic pier behavior. The following quantities were considered: Pier height (h p ) Span length (L s ) Superstructure weight (W sup ) Superstructure stiffness (k sup ) Pier weight (W p ) Pier stiffness (k p ) Because these parameters were identified as lik ely predictors of inerti al bridge response, it was expected that a combination of these parameters would correlate well with computed ideal IRF values. To assess this expectation, an algorit hm was developed that systematically evaluated each of several thousand possible combinations of the six (6) bridge parameters listed above ( ), subject to the following functional form: 1p 6p 36 124 5 12346pppppp 6 (410) where, 1 1111 6 22441,1,,,, 60 PAGE 61 For each possible combination, the product of bridge parameters ( ) was computed for each bridge considered in the study. Li near regression was used to correlate values to observed ideal IRF factors for each pier /energy condition. To quantify th e level of dependency between and IRF, correlation coefficien ts (r) were computed for each of the several thousand trial correlations. This combinatorial analysis yielde d viable relationships between the six bridge parameters and IRF. Specifically, the following parameter combination was most meaningfully correlated with IRF: sup p pk 1 h W (411) By correlating each of the three sets of ideal IRF values to (as computed in Eqn. 411 ), the following linear regression equations result ( Figure 48 ): sup ideal m p pk 3.4 IRF0.123.40.12 h W (412) sup ideal v p pk 1.6 IRF0.241.60.24 h W (413) sup ideal b p pk 5.1 IRF0.205.10.20 h W (414) where ( ideal mIRF Figure 48 a), ( ideal vIRF Figure 48 b), and ( ideal bIRF Figure 48 c) are ideal IRF factors calibrated to pier mo ment, pier shear, and total bearing shear, respectively. Correlation coefficients (r) were computed for the correlations defined by Eqns. 412 413 and 414 (0.88, 0.59, and 0.85 respectively). Each of the correlation coefficients exceed the Pearsons critical r valu e of 0.561 (Pearson and Hartley 1958) for a sample size of 20 and 0.01 significanceimplying that the correla tions are statistically meaningful. 61 PAGE 62 Note that Eqns. 412 413 and 414 do not constitute a static analytical method that is conservative when compared to dynamic an alysis. These regression equations predict conservative values of IRF (and correspondi ng piertop forces) for only about 50% of conceivable cases. Consequently, an upper bound envelope is needed that has a greatly increased likelihood of conservatism. Thus, envelopes were developed for each correlation, corresponding to a 99% confidence level (using the Students tdistribution)implying th at the resulting static pier demands are 99% likely to exceed the corresponding dynamic demands. Envelopes of this form for each demand type are described by the following equations ( Figure 48 ): sup m p pk 4.5 IRF0.22 h W (415) sup v p pk 3.0 IRF0.36 h W (416) sup b p pk 7.0 IRF0.37 h W (417) Thus, dynamic column moment, column shear, and total bearing sh ear demands may be conservatively estimated by static ally analyzing the impacted pier under the load conditions illustrated in Figure 49 Note that three distinct static analys es must be conducted, each used to predict the maximum demand corresponding to the c hosen IRF. For example, when the structure is analyzed using the IRF corresponding to pier moment (IRF m ), the column shears and bearing shears predicted by this an alysis are not utilized. The static loading scheme illustrated in Figure 49 constitutes a conservativebut minimally sostatic analysis procedure for predic ting both direct load e ffects of barge impact and indirect dynamic amplifications caused by inertial superstructure resistance. Prediction 62 PAGE 63 equations for piertop inerti al resistance factors (Eqns. 415 416 and 417 ) include structural parameters that are important to dynamic amplif ication yet are readily quantified during design. Furthermore, conservatismrelative to dynamic analysisis minimized by separately analyzing for pier moment, pier shear, and total bearing shear. Substructure Considerations While the static loading conditions shown in Figure 49 provide adequate predictions of pier column and bearing shear demands, dynamic substructure forces (pile/shaft moments and shears) are consistently underestimated by this procedure. Because the static inertial load ( ) opposes the amplified impact load ( BIRFP B1.45P ), loads transmitted through the foundation to the soil are relieved. Thus, foundation demands must be considered separately, excluding a piertop iner tial load. An empirical static loading condition was developed for predicting pile/shaft design forces, implementi ng similar methodology to that described above. Determination of Ideal ImpactPoint Loads To develop a load model for predicting dyna mic foundation demands, ideal impactpoint loads were developed for each of the twenty (20) pier/impact energies cons idered. As illustrated in Figure 410 the ideal amplified impact load ( ) is defined as the load for which the static foundation moment ( ) equals the maximum moment predicted by dynamic analysis ( ). For each case, ideal amplified impact loads were normalized by corresponding approximate impact loads (P ideal BampP max staticM max dynM B) to form ideal impactpoint dynamic magnification factors (DMF B ideal ). Correlation of ImpactPoint DMF to Bridge Parameters Ideal impactpoint DMF values for the cas es considered ranged from 0.93 to 1.6 implying that the impact force (P B, calculated from Eqn. ) is too small to provide conservative B 48 63 PAGE 64 estimates of dynamic pile/shaft demands, in mo st cases. Thus, a dynamic magnification factor must be considered for impact loads. A combinatorial study was conducted using the methodology described previously to identify correlations between br idge structural parameters and impactpoint DMF. As before, severalthousand trial correlati ons were computed and assessed by means of correlation coefficients (r). In contrast to the previous inve stigation, no statistically meaningful correlations were observed between bridge parameters and impactpoint DMF. Consequently, the impactpoint DMF was treated as uniform across all structural configur ations and impact conditions. Among the twenty (20) cases studied, the mean DMF was 1.34. However, an upper bound envelope was desired to greatly increase the probability of conservatism. Thus, a uniform envelope of 1.85 was established ( Figure 411 )corresponding to a 99% confidence upper boundusing the Students tdist ribution. Thus, by amplifying the barge impact load (P B) by a factor of 1.85, conservative estimates of foundation design forces are produced. B Static Bracketed Impact Analysis (SBIA) Method The static bracketed impact an alysis (SBIA) method consists of bracketing (or enveloping) member design forces using the two static loading c onditions described above. In this manner, pier column, bearing shear, and foundation desi gn forces are conservatively quantified with regard to dynamic amplifications. The SBIA method is summarized and demonstrated in this section. SBIA Overview The proposed SBIA procedure ( Figure 412 ) consists of two primary load cases. Load Case 1 involves statically applying both an amplified impact load equal to and a statically equivalent superstructure inertial load equal to B1.45P BIRFP The magnitude of IRF d epends 64 PAGE 65 on both bridge structural parameters (h p k sup and W p ) and the desired pier demand type (pier moment, pier shear, or total bear ing shear). Typically, Load Case 1 controls the design of pier columns and bearing connections to resist vessel collision forces. Load Case 2 consists of a single amplified impact load equal to B1.85P and typically controls the design of foundation elements such as piles or drilled shafts. It should be noted that feasible structural configurations and impact conditions exist for which the typical controlling load case doe s not control for a given member type. Consequently, maximum pier de mands obtained between both load cases should be considered for design (i.e. the results must be bracketed by both load cases). This bracketing approach is consistent with widely accepted structural design practice concerning multiple loading conditions. Figure 412 provides a summary of the entire SBIA procedure as well as specific definitions for the quantities h p k sup and W p needed in calculating IRF values. SBIA Demonstration In this section, the SBIA method is demonstrated in de tail for one of the twenty (20) pier/energy cases considered in this studythe Blountstown Bri dge channelpier highenergy case (BLTCHAH). The bridge is analyzed using both OPTS and spring superstructure models (recall Figure 42 ). Refer to Table 31 and Table 32 in the previous chapter, for additional details of the structural configur ation for this bridge and for a de finition of the impact condition. For the Blountstown Bridgeformally known as the New Trammell Bridgehighenergy case (BLTCHAH), barge impact occurs on the channel pi er near the top of a 30.5 ft tall shear wall that connects two 9 ft diameter drilled shafts ( Figure 413 ). Two circular pier columns (5.5 ft diameter), which are axially collinear with each foundation shaft, span from the foundation elements to the top of the pier. 65 PAGE 66 Prior to constructing the SBIA load cases, th e barge impact force must be computed. For this bridge, impact occurs near the top of the sh ear wall, which has a 9ft diameter round impact surface ( ). Thus, the barge yield force is calculated as: pw9f t BY pP150030w150030(9)1770kips (418) According to Consolazio et al. ( 2009a), this yield force is exp ected to occur at a barge crush depth (a BY ) of 2 in. With the yield force quantifie d, the impact force correspond ing to the highenergy barge collision (5920ton flotilla, traveling at 5.0 knots) is computed. First, the series stiffness of the barge and pier/soil system (k S ) is calculated per Eqn. 47 Note that the stiffness of the pier/soil system for this bridge (k p ) is 963 kip/in. 1 1 BY S BYPa 121 k 461kip/in. Pk1770963 (419) Thus, the highenergy crush force is com puted given the barge tow velocity (v Bi ) of 5.0 knots (101 in/s) and mass (m B) of 5920 tons (30.7 kip/in/s ): B 2 BBiSB BY BBYPvkm10146130.712,015kips 12,015kipsP PP1,770kips (420) This calculation illustrates that the initial kineti c energy of the barge tow is more than sufficient to yield the barge bow, generating the maximu m crush force for this pier (1770 kips). With the barge impact load (P B) quantified, the two SBIA load cases are constructed. For Load Case 1, the amplified static impact load is computed: B B1.45P1.45(1770)2567kips (421) This amplified impact load is used for each part of Load Case 1, regardless of the demand type of interest. However, unique piertop loads are computed, corresponding to pier moment, pier 66 PAGE 67 shear, and total bearing shear (recall Figure 412 ). To quantify these lo ads, corresponding IRFs are calculated, based on bridge st ructural parameters. For this br idge, the height of the pier (h p ) is 37 ft, the lateral superstructure stiffness (k sup ) is 199 kip/in, and the tota l weight of the pier (W p ) is 1815 kips. Thus, sup m p pk 4.5 4.5(199) IRF0.22 0.22 0.48 h( 3 7 ) W (1815) (422) sup v p pk 3.0 3.0(199) IRF0.36 0.36 0.54 h( 3 7 ) W (1815) (423) sup b p pk 7.0 7.0(199) IRF0.37 0.37 0.78 h( 3 7 ) W (1815) (424) Lastly, the amplified impact load for Load Case 2 ( Figure 412 ) is calculated as: B1.85P1.85(1770)3275kips (425) With the SBIA load factors form ed, two approaches to statically analyzing bridge response to barge impact loading are now considered. SBIA demonstration with OPTS superstructure model The SBIA method is first demonstrated us ing a onepier, twospan (OPTS) bridge representation (recall Figure 413 ). Load Case 1 is analyzed by conducting three separate static analyses ( Figure 414 ). The amplified impact load (as computed in Eqn. 421 ) is applied at the impact location for all three analyses. For each of the three analyses, the corresponding IRF (as calculated in Eqns. 422 423 and 424 ) is multiplied by the impact force (P B), and applied at the superstructure center of gravity, in the opposite direction of impact ( ). B Figure 414 Having formed the complete loading condition fo r Load Case 1, the structure is statically analyzed. Predictions of pier member (colum n and foundation) moment, pier member (column 67 PAGE 68 and foundation) shear, and total bearing shear demands are qua ntified using the respective analyses (shown in Figure 414 ). Note that beamcolumn axial fo rces are obtained from the Load Case 1 moment analysis ( Figure 414 a), for use in loadmoment interaction calculations. SBIA Load Case 2 is also analyzed as shown in Figure 415 From this analysis, all pertinent member forces are quantifiedpier member (column a nd foundation) moments, pier member (column and foundation) shears, a nd total bearing shear. Design forces predicted by Load Ca ses 1 and 2 are summarized in Table 43 For each demand type, the maximum is selected for design. In this example, Load Case 1 controlled pier column and bearing design forces, while Load Ca se 2 controlled foundation design forces. This pattern is typical of the SBIA procedure; however, it is possible, given specific pier configurations and loading conditi ons, for either load case to dominate a given demand. Thus, the maximum demand predicted between both load cases must be considered for design. The SBIA results for the Blountstown Bridge are additionally compared to dynamic predictions of design forces. Specifically, a corresponding dynamic analysis was conducted using the coupled vessel impact analysis (CVIA) method. The br idge was analyzed considering identical impact conditionsa 5920ton barge tow, impacting th e pier at 5.0 knots. Design forces obtained from CVIA are compared to SBIA predictions in Table 44 The SBIA method conservatively predicts all relevant design forces, when compared to CVIA. For this example, SBIA predictions of pier column and bearing shear forces are within 2 to 8% of corresponding dynamic forces. The dispar ity is larger for foundation forces, at 27 to 35%. However, this level of conservatism is de emed acceptable, given the relative simplicity of the analysis procedure. 68 PAGE 69 SBIA demonstration with sp ring superstructure model In addition to demonstrating th e SBIA method using an OPTS bridge model, the procedure and results are assessed using a more simplistic spring superstructure model. Specifically, the superstructure is removed and replaced by an eq uivalent 199 kip/in lateral spring. This modeling approach is common in bridge de sign practice; thus, compatibility is assessed between OPTS and springbased superstructure modeling. As before, SBIA loads are calculated per Eqns. 421 422 423 and 424 While load magnitudes are identical to thos e applied to the OPTS model, placement of the piertop load differs. Using an OPTS model, this load is app lied at the superstructure center of gravity. However, without adding additional connecting elements and increasing model complexity, this loadapplication location is not available in the simplified numerical model. Instead, the piertop load is applied at the pier cap beam center of gravity ( Figure 416 ). Load Case 2 involves only replacing the OPTS superstructure with an equivalent spring. Th e amplified barge load is then applied at the impact location ( Figure 417 ). As before, maximum demands obtained between the two load cases are selected as design forces ( Table 45 ). Again, Load Case 1 controlled pier and bearing demands, while Load Case 2 controlled foundation demands. Static (SBIA) and dynami c (CVIA) demand predictions are compared in Table 46 Using an equivalent spring superstructu re model, SBIA column and b earing forces differ from CVIA by 2 to 9%, while foundation forces differ by 30 to 35%. Comparison of SBIA using OPTS vs. spring superstructure model Predictions of static (SBIA) design forces are compared in Table 47 for both OPTS and spring superstructure models. Observed disc repancies are smallless than 4%across all demand types. Additionally, forces predicted by the spring superstructure model were always 69 PAGE 70 conservative relative to the OPTS model, though this finding is specific to the demonstration case. In the next section, it will be shown that it is possible fo r spring superstructure models to produce lower force predictions th an OTPS models. In general, the small relative difference between OPTS and springbased design forces confirms the compatibility of either superstructure modeling technique fo r use with the SB IA procedure. In addition to quantifying ma ximum forces, deflected pier shapes and structural demand profiles (shear and moment) are compared in Figure 418 Displacement and force profiles are shown for CVIA (dynamic analys is), SBIA using an OPTS m odel, and SBIA using a spring superstructure model. Note that th e SBIA displacement profiles shown in Figure 418 were obtained from the Load Case 1 analysis calibrated to pier moment (recall Figure 414 and Figure 416 ). The displacement profile for th e Blountstown Bridge is repres entative of many of the cases studied in that the overall shape of the profile matches well with CVIA, but the pier displacements are smaller in magnitude, and some times negative. Negative pier displacements indicate that the net static lo ading pushed portions of the stru cture in the opposite direction of impact (typically near the piertop). Negativ e displacements are not indicative of dynamic inertial restraint behavior, making this prediction somewhat unrealistic. These unrealistic deflections are primarily a consequence of the conservative nature of SBIA. As previously discussed, this tendency is somewhat mitigated by scaling the impact load by 1.45. Using a scaled impact load, the ideal iner tial resistance static loading conditions did not produce negative pier displacements. However, when using the conservative 99% upper bound inertial resistan ce loads (recall Figure 48 ), negative pier displacem ents occur in ten of the twenty pier/energy cases consid ered (see Appendix A for displ acement profiles for all cases). 70 PAGE 71 With regard to pier forces, SBIA profiles match well with CVIA ( Figure 418 ). In this case, the overall shape of the profiles is consis tent with CVIA, though SB IA generally predicts larger magnitudesas is desired. Furthermore, in this case, all maximum SBIA pier demands occur in the same physical locations as predicte d dynamically. However, this is not universal for all bridges (see Appendix A) therefore it is recommended that the maximum SBIA force predictions be used to design an entire member (column or foundation) as opposed to using the detailed shear or moment profile. Parametric Study Results In addition to the demonstration case describe d in the previous sect ion, the SBIA method was assessed for a broad range of structural configurations and barge impact conditions by means of a parametric study. Specifically, the SB IA procedure was used to quantify pier design forcespier column moments and shears, founda tion moments and shears, and total bearing shear forcesfor twenty (20) pi er/impact energy conditions. As discussed previously, SBIA was additionally assessed using both OPTS and spring superstructure models. In the results summary plots ( Figure 419 through Figure 423 ), recall that each pier/energy case is denoted by a three part identi fier. The first portion is an abbreviation of the bridge name (e.g. ACS refers to the Acosta Br idge). The second portion re fers to the impacted pier locationCHA for a channel pier, and OFF for a pier located away from the navigation channel. The third portion refers to the imp act energy conditionL, M, H, and S for low, moderate, high, and severe impact energy, resp ectively. See Chapter 3 for additional details regarding the bri dges and energy conditions considered. For comparison, corresponding fully dynamic si mulations were conducted using the CVIA method, from which dynamic design force predictions were developed. To assess the suitability of SBIA for conservatively estimating dynamic design forces, SBIA demands are normalized by 71 PAGE 72 corresponding CVIA demands to form demand ratios. If a given demand ratio is equal to 1.0, this indicates that the SBIA demand exactly matches that obtained from CVIA. Ratios greater than 1.0 imply that SBIA is conservative compared to CVIA, and ratios less th an 1.0 imply that SBIA is unconservative. Based on the data presented Figure 419 through Figure 423 it is observed that SBIA is always conservative relative to CVIA for the ca ses studied. This is a natural consequence of utilizing upper bound envelopes (99% confidence) for the two SBIA load cases. Not only is SBIA conservative relative to CV IA, the overall level of conserva tism is reasonablewith mean values ranging from 1.3 to 1.6, depending on the demand type. Furthermore, nearly every relative demand is less than 2.0. Lastly, Figure 419 through Figure 423 illustrate that using a springbased superstructure model is compatible with SBIA. Specifically, observed discrepancies between OPTS and spring demands are typically smallranging from 0.01% to 36%, with an average of 7.0%. In general, the SBIA met hod provides adequate estimates of dynamic design forces across a wide spectrum of bridge configurations and impact conditions. 72 PAGE 73 Table 43. SBIA demand predictio n summary (OPTS superstructure). Load Case 1 Load Case 2 Maximum Calibrated to pier moment Calibrated to pier shear Calibrated to bearing shear Column moment (kipft) 7,826 5,559 7,826 Column shear (kips) 411 345 411 Foundation moment (kipft) 11,310 22,324 22,324 Foundation shear (kips) 904 1,638 1,638 Total bearing shear (kips) 891 640 891 Table 44. Comparison of demand predicti onsCVIA vs. SBIA (OPT S superstructure). Dynamic (CVIA) Equiv. static (SBIA) Percent difference Column moment (kipft) 7,281 7,826 7.5% Column shear (kips) 404 411 1.7% Foundation moment (kipft) 17,569 22,324 27.1% Foundation shear (kips) 1,216 1,638 34.7% Total bearing shear (kips) 883 891 0.9% Table 45. SBIA demand prediction summary (spring superstructure). Load Case 1 Load Case 2 Maximum Calibrated to pier moment Calibrated to pier shear Calibrated to bearing shear Column moment (kipft) 7,934 6,688 7,934 Column shear (kips) 413 343 413 Foundation moment (kipft) 11,596 22,753 22,753 Foundation shear (kips) 886 1,644 1,644 Total bearing shear (kips) 922 657 922 Table 46. Comparison of de mand predictionsCVIA vs. SBIA (spring superstructure). Dynamic (CVIA) Equiv. static (SBIA) Percent difference Column moment (kipft) 7,281 7,934 9.0% Column shear (kips) 404 413 2.2% Foundation moment (kipft) 17,569 22,753 29.5% Foundation shear (kips) 1,216 1,644 35.2% Total bearing shear (kips) 883 922 4.4% 73 PAGE 74 Table 47. Comparison of SBIA demand pr edictionsOPTS vs. spring superstructure. SBIA (OPTS) SBIA (spring) Percent difference Column moment (kipft) 7,826 7,934 1.4% Column shear (kips) 411 413 0.5% Foundation moment (kipft) 22,324 22,753 1.9% Foundation shear (kips) 1,638 1,644 0.4% Total bearing shear (kips) 891 922 3.5% Impact load Minimal column curvature A Impact load Superstructure inertial force Significant column curvature B Figure 41. Static barge impact analysis A) using existing methods (AASHTO 1991) and B) accounting for superstructu re inertial resistance. 74 PAGE 75 Uncoupled springs representing remaining bridge structure Superstructure frame elements Impact load A Impact load Single spring representing transverse superstructure stiffness B Figure 42. Superstructure mode ling techniques considered. A) OPTS superstructure model and B) spring superstructure model. 75 PAGE 76 PBaBPBYaBYYield Point Barge Bow DeformationImpact Force Figure 43. Barge bow for cedeformation relationship. PBaBPBYaBY aBmTotal deformation energy is equal to the initial barge kinetic energy A PBaBPBYaBY aBmRecoverable elastic deformation energy generates the final barge kinetic energy aBp B Figure 44. Inelastic barge bow deforma tion energy. A) loading and B) unloading. 76 PAGE 77 Impact load Pier is restrained by boundary condition to approximate superstructure inertial resistance A Superstructure stiffness is ampified to approximate superstructure inertial resistance Impact load B Impact load Approximate superstructure inertial force is applied directly as a load C Figure 45. Statically approximati ng superstructure inertial resist ance by: A) restraining the pier top with boundary conditions; B) amplifyi ng the superstructure stiffness and C) directly applying an inertial load. 77 PAGE 78 B1.45P Inertial load Figure 46. Impact load magnified by a factor of 1.45. Moment profile max staticM B1.45P IP Given:max dynM maximum column moment from dynamic analysis ( ) Iterate:IP whenmaxmax staticdynMM ideal IIPP thus ideal I ideal BP IRF P Figure 47. Determination of id eal piertop load and IRF for a gi ven bridge pier (calibrated to column moment). 78 PAGE 79 0.00 0.05 0.10 0.15 0.200.25 0.0 0.4 0.8 1.2 1.6 2.0 Ideal IRF values Linear regression 99% confidence bound IRF for pier moment (IRFm) p supp(1h)kW r 2 = 0.78 sup p p k 4.5 0.22 h W A 0.00 0.05 0.10 0.15 0.200.25 0.0 0.4 0.8 1.2 1.6 2.0 Ideal IRF values Linear regression 99% confidence bound IRF for pier shear (IRFv) r2 = 0.35 p supp(1h)kW sup p p k 3.0 0.36 h W B 0.00 0.05 0.10 0.15 0.200.25 0.0 0.4 0.8 1.2 1.6 Ideal IRF values Linear regression 99% confidence bound IRF for bearing shear (IRFb) r2 = 0.72 p supp(1h)kW sup p p k 7.0 0.37 h W C Figure 48. Correlation between inertial resistance factor (IRF) and bridge parameters. Calibrated to A) pier moment, B) pi er shear and C) total bearing shear. 79 PAGE 80 B1.45P IBPIRFP sup m p p k 4.5 IRF0.22 h W To obtain pier moments: sup v p p k 3.0 IRF0.36 h W To obtain pier shears: sup b p p k 7.0 IRF0.37 h W To obtain bearing shears: Figure 49. Static loadin g to approximate superstructure dynamic amplification. Moment profile Given:maximum pile moment from dynamic analysis( )Iterate: when thus max staticMmax dynMmaxmax staticdynMM ideal BampBampPP ideal Bamp ideal BP DMF P BampPBampP Figure 410. Determination of id eal amplified impact load and DM F for a bridge pier (calibrated to pile moment). 80 PAGE 81 p supp(1h)kW Impactpoint DMF 0.00 0.05 0.10 0.15 0.200.25 0.0 0.4 0.8 1.2 1.6 2.0 Ideal DMF values Mean DMF value 99% confidence bound1.85 Mean = 1.34 Figure 411. Mean value and e nvelope of impactpoint DMF. 81 PAGE 82 SBIA method aBYPBYaBPBObtain bow crush model per Figure 2.6 Pier impact surface shape/size Calculate impact load Barge mass (mB), Barge velocity (vBi) per AASHTO (1991) Obtain: Obtain: Obtain: Obtain: Load Case 1 Load Case 2 Design forces ksuphp Wp 1 BY s pBYa 1 k kP BBisBPvkm sup m p p k 4.5 IRF0.22 h W sup v p p k 3.0 IRF0.36 h W sup b p p k 7.0 IRF0.37 h W colfnd. max,1max,1MMcolfnd. max,1max,1VV b rg tot,1Vcolfnd. max,2max,2 b rg colfnd. max,2max,2 tot,2MM VVV col colcol um a x 1m a x 2 col colcol um a x 1m a x 2MmaxM,M VmaxV,V pile fnd.fnd. um a x 1m a x 2 pile fnd.fnd. um a x 1 m a x 2MmaxM,M VmaxV,V b rgbrg brg tot,u tot,1tot,2VmaxV,V B1.45P mBIRFP B1.45P vBIRFP B1.45P b BIRFP B1.85P h p : k sup : W p : Pier column height (ft) Superstructure stiffness (kip/in) Pier weight (kip) Figure 412. Static bracketed im pact analysis (SBIA) method. 82 PAGE 83 Impact location Uncoupled springs Uncoupled springs Figure 413. Structur al configuration for Blountstown Bridge channel pier (BLTCHA). Superstructure C.G. B850kips0.48P B1.45P 2567kips col max,1M7,826kipft fnd. max,1M11,310kipft A Superstructure C.G. B956kips0.54P B1.45P 2567kips col max,1V411kip fnd. max,1V904kip B Superstructure C.G. B1380kips0.78P B1.45P 2567kips brg tot,1V891kip C Figure 414. Loading conditions and maximum demand predictions for Load Case 1 with OPTS superstructure model. Calibrated to A) pi er moment, B) pier shear and C) total bearing shear. 83 PAGE 84 B1.85P 3275kips col max,2M6,669kipft fnd. max,2M22,324kipft col max,2V345kip fnd. max,2V1,638kip brg tot,1V640kip Figure 415. Loading conditions and maximum demand predictions for Load Case 2 with OPTS superstructure model. Pier cap C.G. B1.45P 2567kips col max,1M7,934kipft fnd. max,1M11,596kipft B850kips0.48P A Pier cap C.G. B1.45P 2567kips col max,1V413kip fnd. max,1V886kip B956kips0.54P B Pier cap C.G. B1.45P 2567kips col max,1V922kip B1380kips0.78P C Figure 416. Loading conditions and maximum demand predictions for Load Case 1 with spring superstructure model. Calibrated to A) pi er moment, B) pier shear and C) total bearing shear. 84 PAGE 85 B1.85P 3275kips col max,2M6,688kipft fnd. max,2M22,753kipft col max,2V343kipfnd. max,2V1,644kipbrg tot,1V657kip Figure 417. Loading conditions and maximum demand predictions for Load Case 2 with spring superstructure model. 85 PAGE 86 Displacement (in) Moment (kipin) Shear (kips) 0 200 400600 0 5 10 15 20 25 30 35 40 45 0 4000 8000 0 5 10 15 20 25 30 35 40 45 Height above impact point (ft) 1 0 12 0 5 10 15 20 25 30 35 40 45 CVIA SBIAOPTS SBIASPRING A Displacement (in) 0 1000 2000 150 120 90 60 30 0 Moment (kipft) 0 8000 1600024000 150 120 90 60 30 0 Shear (kips)Height below impact point (ft) 1 0 12 150 120 90 60 30 0 CVIA SBIAOPTS SBIASPRING B Figure 418. Comparison of CVIA and SBIA br idge responses for New Trammel Bridge at Blountstown (BLTCHA). A) pier column profiles a nd B) foundation profiles. 86 PAGE 87 Column moment relative to CVIA 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 ACSCHAM ACSCHAH ACSCHAS BLTCHAM BLTCHAH EGBCHAM MBCCHAM NSGCHAM NSGCHAH NSGCHAS NSGOFFL OSGCHAM OSGOFFM PNCCHAM RNGCHAM RNGCHAH RNGOFFL RNGOFFM SBZCHAM SBZCHAH OPTS superstructure Spring superstructure (Mean: 1.31) (Mean: 1.29) Figure 419. SBIA pier column moment demands relative to CVIA. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Column shear relative to CVIA ACSCHAM ACSCHAH ACSCHAS BLTCHAM BLTCHAH EGBCHAM MBCCHAM NSGCHAM NSGCHAH NSGCHAS NSGOFFL OSGCHAM OSGOFFM PNCCHAM RNGCHAM RNGCHAH RNGOFFL RNGOFFM SBZCHAM SBZCHAH OPTS superstructure Spring superstructure (Mean: 1.36) (Mean: 1.46) Figure 420. SBIA pier column shear demands relative to CVIA. 87 PAGE 88 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Foundation moment relative to CVIA ACSCHAM ACSCHAH ACSCHAS BLTCHAM BLTCHAH EGBCHAM MBCCHAM NSGCHAM NSGCHAH NSGCHAS NSGOFFL OSGCHAM OSGOFFM PNCCHAM RNGCHAM RNGCHAH RNGOFFL RNGOFFM SBZCHAM SBZCHAH OPTS superstructure Spring superstructure (Mean: 1.47) (Mean: 1.45) Figure 421. SBIA foundation mo ment demands relative to CVIA. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Foundation shear relative to CVIA ACSCHAM ACSCHAH ACSCHAS BLTCHAM BLTCHAH EGBCHAM MBCCHAM NSGCHAM NSGCHAH NSGCHAS NSGOFFL OSGCHAM OSGOFFM PNCCHAM RNGCHAM RNGCHAH RNGOFFL RNGOFFM SBZCHAM SBZCHAH OPTS superstructure Spring superstructure (Mean: 1.34) (Mean: 1.32) Figure 422. SBIA foundation sh ear demands relative to CVIA. 88 PAGE 89 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Total bearing shear relative to CVIA ACSCHAM ACSCHAH ACSCHAS BLTCHAM BLTCHAH EGBCHAM MBCCHAM NSGCHAM NSGCHAH NSGCHAS NSGOFFL OSGCHAM OSGOFFM PNCCHAM RNGCHAM RNGCHAH RNGOFFL RNGOFFM SBZCHAM SBZCHAH OPTS superstructure Spring superstructure (Mean: 1.45) (Mean: 1.56) Figure 423. SBIA total bearing shear demands relative to CVIA. 89 PAGE 90 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS Summary and Conclusions Previously conducted experiment al and analytical research has highlighted the importance of dynamic effects that are present during barg ebridge collision even ts. Specifically, these studies identified superstructure inertia as a critical component of dynamic bridge pier behavior. Furthermore, current static analysis methods do not account for dynamic amplification of pier forces generated by inertial effects. Thus, dynamic analytical methods were developed, as part of prior research, which directly consider the effects of bridge ma ss distribution and bridge motions developed during barge impact. While timedomain dynamic analysis proce duresnotably the c oupled vessel impact analysis (CVIA) procedureprovide accurate predic tions of bridge response to barge collision loads, the computationa l requirements of such methods can be prohibitive in some design situations. In addition, the struct ural characteristics of a bridge in preliminary design may not be sufficiently welldefined to permit a detailed dynam ic analysis. Thus, a need was identified to develop an equivalent static an alysis procedure that simply and conservatively accounts for dynamic amplifications present duri ng barge impact events. The research presented in this thesis has been carried out to address this need. An equivalent static analys is procedure was developed that statically emulates superstructure inertial restraint. The proposed static bracketed impact analysis (SBIA) method consists of two static load cases. The first load case addresses inertial resistance by means of a static inertial load applied at the superstructure elevation, in addition to the barge impact load. This inertial load is quantified using empirical expressions that relate inertial force to readily available structural parameters. The second load case excludes the inerti al load, and provides a 90 PAGE 91 reliable means of quantifying maximum foundation forces. By bracketing pier forces between both load cases, reasonably conservative estimat es of pier demands are assessed, including dynamic amplifications. Recommendations Consideration of superstructure inertial effects: It is recommended that superstructure inertia be considered as part of the bridge design process for barge collision. Supe rstructure inertia can greatly amplify pier forces, and such amplifications are not considered in existi ng static analysis pr ocedures. Dynamically amplified design forces should be quantified using either the time domain CVIA method or the newly developed equiva lent static SBIA method. Use of static bracketed impact analysis (SBIA) procedure: For preliminary design, or in cases where tim edomain dynamic analysis is not warranted, the SBIA procedure is recommended to sta tically approximate dynamic amplification effects. This empirical method provides a conservative but simple static approach to impactresistant bridge design. This thesis de monstrates that SBIA predicts conservative estimates of dynamic pier forces for a wide ra nge of structural configurations and impact conditions. Use of maximum member de sign forces from SBIA: To maximize the simplicity of the SBIA method, certain lim itations in terms of accuracy had to be imposed. While the static lo ad cases employed by SBIA are able to conservatively bracket maximum design forces arising in major structural bridge components (pier columns, piles, drilled shafts), the detailed ve rtical profiles of structural demand (moment, shear) produced by the SBIA load cases are not n ecessarily conservative at every elevation. Hence, it is recomme nded that each major bridge component be designed with uniform capacity along its entire length such that every section of the component possesses sufficient capacity to re sist the maximum forces predicted by the SBIA method. 91 PAGE 92 92 APPENDIX A COMPARISON OF SBIA AND CVIA RESULTS In this appendix, SBIA (equiva lent static) analysis results are compared to corresponding CVIA (dynamic) simulation results. For each bridge in the parametric study, three analyses were conducted: 1) Fully dynamic CVIA, 2) SBIA usi ng onepier, twospan (OPTS) models, and 3) SBIA using springbased superstructure models. The data presented in this appendix form the basis for the relative demand ratios pres ented in Figures 419 through 423. Figures presented in this appendix show th e maximum magnitudes of displacement, shear force, and moment at each vertical elevation, fo r all three analyses. Elevation data have been adjusted such that the elevation datum (0 ft.) corresponds to the midplane elevation of the pile cap. In each figure, a gray rectangle is used to represent the pile cap vertical thickness. Each figure represents one impact en ergy condition applied to a respec tive bridgepier configuration. PAGE 93 Peak Pier Demands Column moment: 62823 kipft 97201 kipft 83425 kipft Column shear: 1127 kips 2146 kips 2202 kips Pile Moment: 2157 kipft 4110 kipft 4018 kipft Pile Shear: 248 kips 385 kips 369 kips Total Bearing Shear: 1196 kips 2841 kips 2925 kips CVIA SBIA OPTS SBIA SPRING 0.1 0 0.1 0.2 0.3 0.4 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 40000 80000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft)0 1000 2000 3000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip) SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) PierP i l e PierP i l e PierP i l e Impact Figure A1. Comparison of CV IA, SBIAOPTS, and SBIAspring results. Br idge: Acosta Bridge (ACS) channel pier. Impact condition: 2030 tons at 2.5 knots. 93 PAGE 94 0.2 0 0.2 0.4 0.6 0.8 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) 0 40000 80000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft) Peak Pier Demands Column moment: 63873 kipft 97201 kipft 83425 kipft Column shear: 1145 kips 2146 kips 2202 kips Pile Moment: 2797 kipft 4110 kipft 4018 kipft Pile Shear: 272 kips 385 kips 369 kips Total Bearing Shear: 1223 kips 2841 kips 292 5 kips CVIA SBIA OPTS SBIA SPRING 0 1000 2000 3000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip) Pier P i l e Pier P i l e Impact Pier P i l e SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A2. Comparison of CV IA, SBIAOPTS, and SBIAspring results. Br idge: Acosta Bridge (ACS) channel pier. Impact condition: 5920 tons at 5.0 knots. 94 PAGE 95 0.2 0 0.2 0.4 0.6 0.8 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 40000 80000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft) 0 1000 2000 3000 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip) Peak Pier Demands Column moment: 64036 kipft 97201 kipft 83425 kipft Column shear: 1148 kips 2146 kips 2202 kips Pile Moment: 2798 kipft 4110 kipft 4018 kipft Pile Shear: 272 kips 385 kips 369 kips Total Bearing Shear: 1227 kips 2842 kips 292 5 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A3. Comparison of CV IA, SBIAOPTS, and SBIAspring results. Br idge: Acosta Bridge (ACS) channel pier. Impact condition: 7820 tons at 5.0 knots. 95 PAGE 96 Peak Pier Demands Column moment: 7165 kipft 7826 kipft 7934 kipft Column shear: 394 kips 411 kips 413 kips Pile Moment: 14124 kipft 22324 kipft 22753 kipft Pile Shear: 1123 kips 1638 kips 1644 kips Total Bearing Shear: 849 kips 891 kips 922 kips CVIA SBIA OPTS SBIA SPRING 0 10000 20000 30000 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 Moment (kipft)0 500 1000 1500 2000 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 Shear (kip)1 0 1 2 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A4. Comparison of CV IA, SBIAOPTS, and SBIAspri ng results. Bridge: SR20 at Bl ountstown (BLT) channel pier. Impact condition: 2030 tons at 2.5 knots. 96 PAGE 97 Peak Pier Demands Column moment: 7281 kipft 7826 kipft 7934 kipft Column shear: 404 kips 411 kips 413 kips Pile Moment: 17569 kipft 22324 kipft 22753 kipft Pile Shear: 1216 kips 1638 kips 1644 kips Total Bearing Shear: 883 kips 892 kips 922 kips CVIA SBIA OPTS SBIA SPRING 0 10000 20000 30000 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 Moment (kipft)1 0 1 2 3 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 500 1000 1500 2000 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 Shear (kip) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A5. Comparison of CV IA, SBIAOPTS, and SBIAspri ng results. Bridge: SR20 at Bl ountstown (BLT) channel pier. Impact condition: 5920 tons at 5.0 knots. 97 PAGE 98 0 1000 2000 3000 4000 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Moment (kipft) Peak Pier Demands Column moment: 2589 kipft 2940 kipft 3057 kipft Column shear: 106 kips 113 kips 117 kips Pile Moment: 106 kipft 158 kipft 156 kipft Pile Shear: 79 kips 91 kips 90 kips Total Bearing Shear: 43 8 kips 461 kips 480 kips CVIA SBIA OPTS SBIA SPRING 0 50 100 150 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Shear (kip) 1 0.5 0 0.5 1 1.5 50 40 30 20 10 0 10 20 30 40 50 60 70 80 CVIA SBIAOPTS SBIASPRING Displacement (in)* Nodal height (ft) (Pile cap midplane elevation = 0 ft) Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A6. Comparison of CV IA, SBIAOPTS, and SBIAspring results. Bri dge: Eau Gallie Bridge (EGB) channel pier. Impact condition: 2030 tons at 2.5 knots. 98 PAGE 99 Peak Pier Demands Column moment: 3331 kipft 3722 kipft 3825 kipft Column shear: 203 kips 266 kips 293 kips Pile Moment: 123 kipft 158 kipft 159 kipft Pile Shear: 75 kips 90 kips 90 kips Total Bearing Shear: 410 kips 459 kips 505 kips CVIA SBIA OPTS SBIA SPRING 0 100 200 300 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Shear (kip)0 1000 2000 3000 4000 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Moment (kipft)1 0 1 2 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A7. Comparison of CV IA, SBIAOPTS, and SB IAspring results. Bridge: Melbourne Causeway (MBC) channel pier. Impact condition: 2030 tons at 2.5 knots. 99 PAGE 100 Peak Pier Demands Column moment: 6954kipft 8312kipft 9308kipft Column shear: 603kips 742kips 821kips Pile Moment: 3023kipft 3775kipft 3724kipft Pile Shear: 377kips 459kips 454kips Total Bearing Shear: 1114kips 1376kips 1400kips CVIA SBIA OPTS SBIA SPRING 2 1 0 1 2 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRINGDisplacementsDisplacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 200 400 600 800 1000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 ShearsShear (kip)0 2000 4000 6000 8000 10000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 MomentsMoment (kipft) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A8. Comparison of CV IA, SBIAOPTS, and SBIAspri ng results. Bridge: new St. Geor ge Island (NSG) channel pier. Impact condition: 2030 tons at 2.5 knots. 100 PAGE 101 2 1 0 1 2 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 200 400 600 800 1000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip)0 2000 4000 6000 8000 10000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft) Peak Pier Demands Column moment: 7194 kipft 8312 kipft 9308 kipft Column shear: 638 kips 742 kips 821 kips Pile Moment: 3070 kipft 3775 kipft 3724 kipft Pile Shear: 383 kips 459 kips 454 kips Total Bearing Shear: 1171 kips 1376 kips 1400 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A9. Comparison of CV IA, SBIAOPTS, and SBIAspri ng results. Bridge: new St. Geor ge Island (NSG) channel pier. Impact condition: 5920 tons at 5.0 knots. 101 PAGE 102 0 2000 4000 6000 8000 10000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft)0 200 400 600 800 1000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip)2 1 0 1 2 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) Peak Pier Demands Column moment: 7233 kipft 8312 kipft 9308 kipft Column shear: 644 kips 742 kips 821 kips Pile Moment: 3076 kipft 3775 kipft 3724 kipft Pile Shear: 384 kips 459 kips 454 kips Total Bearing Shear: 1180 kips 1376 kips 1400 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A10. Comparison of CVIA SBIAOPTS, and SBIAspring results. Bridge: new St. Geor ge Island (NSG) channel pier. Impact condition: 7820 tons at 7.5 knots. 102 PAGE 103 0 1000 2000 3000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft)0 50 100 150 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip)0.6 0.4 0.2 0 0.2 0.4 70 60 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) Peak Pier Demands Column moment: 1518 kipft 2671 kipft 2898 kipft Column shear: 61 kips 95 kips 103 kips Pile Moment: 521 kipft 1060 kipft 1039 kipft Pile Shear: 71 kips 141 kips 140 kips Total Bearing Shear: 12 8 kips 218 kips 231 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A11. Comparison of CVIA SBIAOPTS, and SBIAspring results. Bridge: new St. George Island (NSG) offchannel pier. Impact condition: 200 tons at 1.0 knots. 103 PAGE 104 Peak Pier Demands Column moment: 6710 kipft 8025 kipft 7689 kipft Column shear: 405 kips 490 kips 500 kips Pile Moment: 107 kipft 189 kipft 184 kipft Pile Shear: 73 kips 88 kips 87 kips Total Bearing Shear: 721 kips 860 kips 973 kips CVIA SBIA OPTS SBIA SPRING 1 0 1 2 3 4 50 40 30 20 10 0 10 20 30 40 50 60 CVIA SBIAOPTS SBIASPRING Displacement (in)Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 200 400 600 50 40 30 20 10 0 10 20 30 40 50 60 Shear (kip)0 2000 4000 6000 8000 10000 50 40 30 20 10 0 10 20 30 40 50 60 Moment (kipft) PierP i l e Impact PierP i l e PierP i l e SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A12. Comparison of CVIA SBIAOPTS, and SBIAspring results. Bridge: old St. Geor ge Island (OSG) channel pier. Impact condition: 2030 tons at 2.5 knots. 104 PAGE 105 Peak Pier Demands Column moment: 989 kipft 1217 kipft 1217 kipft Column shear: 64 kips 76 kips 76 kips Pile Moment: 220 kipft 406 kipft 343 kipft Pile Shear: 36 kips 57 kips 52 kips Total Bearing Shear: 109 kips 161 kips 165 kips CVIA SBIA OPTS SBIA SPRING 0.5 0 0.5 1 1.5 70 60 50 40 30 20 10 0 10 20 30 40 50 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 20 40 60 80 70 60 50 40 30 20 10 0 10 20 30 40 50 Shear (kip)0 500 1000 1500 70 60 50 40 30 20 10 0 10 20 30 40 50 Moment (kipft) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A13. Comparison of CVIA SBIAOPTS, and SBIAspring results. Bridge: old St. George Island (OSG) offchannel pier. Impact condition: 200 tons at 1.0 knots. 105 PAGE 106 0 1000 2000 3000 4000 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Moment (kipft)0 100 200 300 400 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Shear (kip)2 1 0 1 2 3 4 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) Peak Pier Demands Column moment: 2334 kipft 3165 kipft 3177 kipft Column shear: 175 kips 338 kips 282 kips Pile Moment: 201 kipft 295 kipft 305 kipft Pile Shear: 61 kips 84 kips 84 kips Total Bearing Shear: 31 8 kips 537 kips 400 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A14. Comparison of CV IA, SBIAOPTS, and SBIAspring results. Br idge: Pineda Causeway (PNC) channel pier. Impact condition: 2030 tons at 2.5 knots. 106 PAGE 107 0 20000 40000 60000 70 60 50 40 30 20 10 0 10 20 30 40 50 Moment (kipft)0 500 1000 1500 2000 70 60 50 40 30 20 10 0 10 20 30 40 50 Shear (kip)0.4 0.2 0 0.2 0.4 0.6 70 60 50 40 30 20 10 0 10 20 30 40 50 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) Peak Pier Demands Column moment: 35960 kipft 53783 kipft 51966 kipft Column shear: 743 kips 881 kips 1160 kips Pile Moment: 13291 kipft 17229 kipft 18760 kipft Pile Shear: 1451 kips 1710 kips 1789 kips Total Bearing Shear: 1028 kips 1219 kips 1667 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A15. Comparison of CVIA, SB IAOPTS, and SBIAspring results. Bridge: Ringling (RNG) channel pier. Impact condition: 2030 tons at 2.5 knots. 107 PAGE 108 0 20000 40000 60000 70 60 50 40 30 20 10 0 10 20 30 40 50 Moment (kipft)0 500 1000 1500 2000 70 60 50 40 30 20 10 0 10 20 30 40 50 Shear (kip)0.4 0.2 0 0.2 0.4 0.6 70 60 50 40 30 20 10 0 10 20 30 40 50 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) Peak Pier Demands Column moment: 37180 kipft 53783 kipft 51966 kipft Column shear: 788 kips 881 kips 1160 kips Pile Moment: 13818 kipft 17229 kipft 18760 kipft Pile Shear: 1494 kips 1710 kips 1789 kips Total Bearing Shear: 1199 kips 1219 kips 1667 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A16. Comparison of CVIA, SB IAOPTS, and SBIAspring results. Bridge: Ringling (RNG) channel pier. Impact condition: 5920 tons at 5.0 knots. 108 PAGE 109 Peak Pier Demands Column moment: 12678 kipft 19346 kipft 16233 kipft Column shear: 282 kips 412 kips 528 kips Pile Moment: 7134 kipft 8476 kipft 7766 kipft Pile Shear: 410 kips 526 kips 500 kips Total Bearing Shear: 428 kips 573 kips 759 kips CVIA SBIA OPTS SBIA SPRING 0 100 200 300 400 500 600 70 60 50 40 30 20 10 0 10 20 30 40 Shear (kip)0.1 0 0.1 0.2 70 60 50 40 30 20 10 0 10 20 30 40 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft) PierP i l e PierP i l e PierP i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A17. Comparison of CVIA, SBIA OPTS, and SBIAspring results. Bridge: Ringling (RNG) offchannel pier. Impact condition: 200 tons at 1.0 knots. 109 PAGE 110 0.5 0 0.5 1 70 60 50 40 30 20 10 0 10 20 30 40 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 20000 40000 60000 80000 70 60 50 40 30 20 10 0 10 20 30 40 Moment (kipft)0 500 1000 1500 2000 70 60 50 40 30 20 10 0 10 20 30 40 Shear (kip) Peak Pier Demands Column moment: 52823 kipft 65276 kipft 55154 kipft Column shear: 1214 kips 1388 kips 1793 kips Pile Moment: 16536 kipft 20116 kipft 19967 kipft Pile Shear: 1156 kips 1361 kips 1322 kips Total Bearing Shear: 1483 kips 1923 kips 2562 kips CVIA SBIA OPTS SBIA SPRING Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A18. Comparison of CVIA, SBIA OPTS, and SBIAspring results. Bridge: Ringling (RNG) offchannel pier. Impact condition: 2030 tons at 2.5 knots 110 PAGE 111 0.5 0 0.5 1 1.5 2 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 CVIA SBIAOPTS SBIASPRING Displacement (in)*Nodal height (ft) (Pile cap midplane elevation = 0 ft)0 10000 20000 30000 40000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 Moment (kipft) Peak Pier Demands Column moment: 27487 kipft 35046 kipft 35467 kipft Column shear: 654 kips 881 kips 918 kips Pile Moment: 354 kipft 570 kipft 560 kipft Pile Shear: 86 kips 114 kips 113 kips Total Bearing Shear: 643 kips 1040 kips 1082 kips CVIA SBIA OPTS SBIA SPRING 0 200 400 600 800 1000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 Shear (kip) Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A19. Comparison of CVIA, SBIA OPTS, and SBIAspring results. Br idge: Seabreeze (SBZ) channel pier. Impact condition: 2030 tons at 2.5 knots. 111 PAGE 112 112 0 10000 20000 30000 40000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 Moment (kipft) Peak Pier Demands Column moment: 27807 kipft 35046 kipft 35467 kipft Column shear: 661 kips 881 kips 918 kips Pile Moment: 362 kipft 570 kipft 560 kipft Pile Shear: 88 kips 114 kips 113 kips Total Bearing Shear: 650 kips 1040 kips 108 2 kips CVIA SBIA OPTS SBIA SPRING 0 200 400 600 800 1000 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 Shear (kip) 0.5 0 0.5 1 1.5 2 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 CVIA SBIAOPTS SBIASPRING Displacement (in) Nodal height (ft) (Pile cap midplane elevation = 0 ft) Pier P i l e Pier P i l e Pier P i l e Impact SBIA displacement profiles shown were obtained using SBIA Load Case 1 with the moment IRF equation (Equation 5.15) Figure A20. Comparison of CVIA, SBIA OPTS, and SBIAspring results. Br idge: Seabreeze (SBZ) channel pier. Impact condition: 5920 tons at 5.0 knots PAGE 113 APPENDIX B DEMONSTRATION OF SBIA METHOD Introduction In this appendix, the SBIA method is dem onstrated for the New Trammel Bridge, in northwestern Florida. For this example, a threebarge flotilla (5920 tons w ith tug) collides with the channel pier at 5.0 knots. Barge impact occu rs near the top of a 30.5 ft tall shear wall that connects two 9 ft diameter drilled shafts ( Figure B1 ). Two circular pier columns (5.5 ft diameter), which are axially collinear with each foundation shafts, span from the foundation elements to the top of the pier. Impact location Uncoupled springs Uncoupled springs Figure B 1. Structural configura tion for New Trammel Bridge. Demonstration of SBIA Procedure Prior to constructing the SBIA load cases, the vessel impact force must be computed. For this bridge, impact occurs near the top of the sh ear wall, which has a 9ft diameter round impact 113 PAGE 114 surface (). Thus, the barge yield force is determined in accordance with pw9f t Figure B2 and Eqn. B1 Note that this yield force occurs at a crush depth (a BY ) of 2 in. 0 6 12 18 24 3036 0 2,000 4,000 6,000 8,000 10 9 ft 1,770 kipsPBY : Barge yield load (kips) F l a t 1 5 0 0 + 6 0 wP F l a t 3 0 0 + 1 8 0 wP R o u n d 1 5 0 0 + 3 0 wPft 10 wP ft 10 wP Figure B 2. Barge yield load determinati on for 9ft round impact surface. BY pP150030w150030(9)1770kips ( B 1) With the yield force quantifie d, the impact force correspond ing to the highenergy barge collision (5920ton flotilla, traveling at 5.0 knots) is computed. First, the series stiffness of the barge and pier/soil system (k S ) is calculated per Eqn. B3 Note that the stiffness of the pier/soil system for this bridge (k P ) must be quantified as shown in Figure B3 and Eqn. B2 For this example, P BY is applied to the pier to quantify k P ; however, if the calculated impact load (P B) is found to be less than P B BY then this process should be repeated to obtain a more accurate estimate of k P 114 PAGE 115 BYP 1770kips p 1.83in Figure B 3. Determination of pier stiffness (k P ). BY P pP 1770 k 963kip/in 1.83 (B 2) 1 1 BY S BYPa 121 k 461kip/in Pk1770963 ( B 3) Thus, the highenergy crush force is com puted given the barge tow velocity (v Bi ) of 5.0 knots (101 in/s) and mass (m B) of 5920 tons (30.7 kip/in/s ): B 2 BBiSB BY BBYPvkm10146130.712,015kips 12,015kipsP PP1,770kips (B 4) This calculation illustrates that the incoming ki netic energy of the barge tow is sufficient to yield the barge bow, generating the maximum crush force for this pier (1770 kips). Load Case 1 With the barge impact load (P B) quantified, the SBIA load ca ses are constructed. For Load Case 1, the amplified static impact load is computed: B B1.45P1.45(1770)2567kips (B 5) 115 PAGE 116 This amplified impact load is used for each pa rt of Load Case 1, regardless of the demand type of interest. However, unique piertop loads are computed, corresponding to pier moment, pier shear, and total bearing shear. To quantify these loads, correspondi ng IRFs are calculated, based on the bridge structural parameters illustrated in Figure B4 and Figure B5 Weight of shaded regionp1815kipsW Clear height of pier columnp37fth Figure B 4. Determination of pier weight (W P ) and pier height (h p ). Bridge deck Girders Impacted pier Flanking pier Remove impact pier and apply load equal to 0.25PB at impact pier location Measure superstructure deflection ( sup) at impact pier location sup 0.25 PB 116 PAGE 117 Figure B 5. Determination of superstructure stiffness (k sup ). As illustrated in Figure B4 the total weight of this pier (W p ) is 1815 kips, and the height of the pier (h p ) is 37 ft. The lateral superstructure stiffness (k sup ) is 199 kip/in, as determ ined using the process shown in Figure B5 Due to the empirical na ture of the IRF equations, the units shown above must be used. Thus, sup m p pk 4.5 4.5(199) IRF0.22 0.22 0.48 h( 3 7 ) W (1815) ( B 6) sup v p pk 3.0 3.0(199) IRF0.36 0.36 0.54 h( 3 7 ) W (1815) ( B 7) sup b p pk 7.0 7.0(199) IRF0.37 0.37 0.78 h( 3 7 ) W (1815) (B 8) The amplified impact load (as computed in Eqn. B5 ) is applied at the impact location for all three analyses. For each of the three analyses, the corr esponding IRF (as calculated in Eqns. B6 B7 and B8 ) is multiplied by the impact force (P B), and this load is applied at the pier cap beam center of gravity, in the opposite direction of impact ( B Figure B6 ). 117 PAGE 118 Pier cap C.G. B1.45P 2567kips col max,1V922kipB1380kips0.78P Pier cap C.G. B1.45P 2567kipscol max,1M7,934kipftfnd. max,1M11,596kipft B850kips0.48P Pier cap C.G. B1.45P 2567kips col max,1V413kipfnd. max,1V886kip B956kips0.54P Figure B 6. Loading conditions and maximum de mand predictions for Load Case 1. With the loading conditions for Load Case 1 deve loped, the structure is statically analyzed. Predictions of pier (column a nd foundation) moment, pier (c olumn and foundation) shear, and total bearing shear demands are quantif ied using the respective analyses ( Figure B6 ). These design forces are additionally summarized in Table B1 Load Case 2 SBIA Load Case 2 is also analyzed as shown in Figure B7 From this single analysis, all pertinent member forces are qua ntifiedpier moments, pier shears, and total bearing shear. These demands are compared to those obtained from Load Case 1 in Table B1 The amplified impact load for Load Case 2 is calculated as: B1.85P1.85(1770)3275kips (B 9) 118 PAGE 119 B1.85P 3275kipscol max,2M6,688kipft fnd. max,2M22,753kipft col max,2V343kipfnd. max,2V1,644kipbrg tot,1V657kip Figure B 7. Loading conditions and maximum de mand predictions for Load Case 2. Results summary Design forces predicted by Load Ca ses 1 and 2 are summarized in Table B1. For each demand type, the maximum is selected for design. In this example, Load Case 1 controlled pier column and bearing design forces, while Load Ca se 2 controlled foundation design forces. This pattern is typical of the SBIA procedure; however, it is possible, given specific pier configurations and loading conditi ons, for either load case to dominate a given demand. Thus, the maximum demand predicted between both load cases must be considered for design. Table B 1. SBIA demand prediction summary. Load Case 1 Load Case 2 Maximum Calibrated to pier moment Calibrated to pier shear Calibrated to bearing shear Column moment (kipft) 7,934 6,688 7,934 Column shear (kips) 413 343 413 Foundation moment (kipft) 11,596 22,753 22,753 Foundation shear (kips) 886 1,644 1,644 Total bearing shear (kips) 922 657 922 119 PAGE 120 LIST OF REFERENCES AASHTO (1991). Guide Specification and Commentary for Vessel Collision Design of Highway Bridges, AASHTO, Washington D.C. AASHTO (2009). Guide Specification and Commentary for Vessel Collision Design of Highway Bridges, 2 nd Edition, AASHTO, Washington D.C. AASHTO (2008). LRFD Bridge Design Specifications, 4 th Edition, AASHTO, Washington D.C. Cameron, J., Nadeau, J., and LoSciuto, J. (2007), Ultimate Strength Analysis of Inland Tank Barges, Marine Safety Center, United States Coast Guard, Washington D.C. Consolazio, G. R., Cook, R. A., McVay, M. C., Cowan, D. R., Bi ggs, A. E., and Bui, L. (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. Consolazio, G. R. and Cowan, D. R. (2005). Num erically Efficient Dynamic Analysis of Barge Collisions with Bridge Piers. ASCE Journal of Structural Engineering, 131(8), 12561266. Consolazio, G. R. and Davidson, M. T. (2008). Simplified Dynamic Barge Collision Analysis for Bridge Design. Journal of the Transportation Research Board, 2050, 1325. Consolazio, G. R., Davidson, M. T., and Cowan, D. R. (2009a). Barge Bow ForceDeformation Relationships for BargeBridge Collision Analysis. Journal of the Transportation Research Board, (In press). Consolazio, G. R., Getter, D. J., and Davidson, M. T. (2009b). A Static Analysis Method for BargeImpact Design of Bridges with Consideration of Dynamic Amplification, Structures Research Report No. 2009/68901, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida. Consolazio, G. R., McVay, M. C., Cowan D. R ., Davidson, M. T., and Getter, D. J. (2008). Development of Improved Bridge Desi gn Provisions for Barge Impact Loading. Structures Research Report No. 2008/51117, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida. Davidson, M. T., Consolazio, G. R., and Getter, D. J., Dynamic Amplification of Pier Column Internal Forces Due to BargeBridge Collision. Journal of the Transportation Research Board, (In review), 2010. FBMultiPier (2009). FBMultiPier Users Manual. Florida Bridge Software Institute, University of Florida, Gainesville, Florida. LSTC (2008). LSDYNA Keyword Users Manual: Version 971, Livermore Software Technology Corporation, Livermore, CA. 120 PAGE 121 Liu, C. and Wang, T. L. (2001). State wide Vessel Collision Design for Bridges. ASCE Journal of Bridge Engineering, 6(3), 213219. MeierDrnberg, K. E. (1983). Ship Collisions, Safety Zone s, and Loading Assumptions for Structures in Inland Waterways, Verein Deutscher Ingenieure (Association of German Engineers) Report No. 496, 19. Pearson, E. S. and Hartley, H. O. (1958). Biometrika Tables for Statisticians (2 nd Edition), Cambridge, NY. Precast/Prestressed Concrete Institute (PCI) (2004). PCI Design Handbook (5 th Edition), PCI, Chicago, IL. Yuan, P., Harik, I. E., and Davidson, M. T. (2005). MultiBarge Flotilla Impact Forces on Bridges, Research report, Kentucky Transporta tion Center, College of Engineering, University of Kent ucky, Lexington, KY. 121 PAGE 122 BIOGRAPHICAL SKETCH The author was born in Ogden, Utah, in 1981. In September 2002, he began attending Daytona Beach Community College where he later earned an Associate of Arts degree in August 2005. Subsequently, he began attend ing the University of Florida, where he received the degree of Bachelor of Science in Civil Engineering in May 2008. The author enrolled in graduate school at the University of Florida in September 2008, where he anticipates receiving the degree of Master of Engineering. The aut hor plans to continue work towa rd a Doctor of Philosophy degree at the University of Florida upon graduation. 122 