SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN

Transcription

1 SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN By MICHAEL THOMAS DAVIDSON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 27 1

2 27 Michael Thomas Davidson 2

3 To my wife, Kiristen 3

4 ACKNOWLEDGMENTS This material is based on work supported under a National Science Foundation Graduate Research Fellowship. However, this thesis would not have been completed without the support of several individuals. First, the insight and guidance of Dr. Gary Consolazio has proven invaluable. His willingness to invest time in helping graduate students become effective analysts and independent researchers will undoubtedly garner countless and vast returns. The author also wishes to thank Dr. Marc Hoit, Dr. Petros Christou, and Dr. Jae Chung for their assistance with extending the capabilities of FB-MultiPier. A graduate student deserving of many thanks and much future success is David Cowan, whose brilliance seems to be limitless. Finally, the author wishes to thank his wife Kiristen, his family, and his friends for their enduring love and fellowship. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF TABLES...7 LIST OF FIGURES...8 ABSTRACT...12 CHAPTER 1 INTRODUCTION LITERATURE REVIEW Experimental Research Analytical Research COUPLED BARGE COLLISION ANALYSIS Introduction Barge Loading and Unloading Behavior Coupled Analysis Algorithm Use of Experimental Data for Coupled Analysis Validation Barge Impact Test Cases Selected for Validation: Case 1 and Case Software Selection and Model Development Coupled Analysis Module Parameters Accounting for Payload Sliding During Impact Testing Comparison of Analytical and Experimental Data Case Case SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS Overview Linearized Barge Force-Crush Relationship Reduction of the Bridge Model Uncoupled Condensed Stiffness Matrix Lumped Mass Approximation Multiple-Pier Coupled Analysis Simplification Algorithm

6 5 SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES Introduction Geographical Information, Structural Configuration, and Impact Conditions Case Case Case Comparison of Simplified and Full-Resolution Results Conclusions from Simplified-Coupled Analysis Demonstrations Dynamic Amplification of the Impacted Pier Column Internal Forces CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH...62 APPENDIX 6.1 Conclusions Recommendations for Future Research...63 A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA...64 B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS...72 C D SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT...76 ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS...95 REFERENCES...1 BIOGRAPHICAL SKETCH

7 LIST OF TABLES Table page 1-1 Case descriptions: use, configuration, and impact data

8 LIST OF FIGURES Figure page 3-1 Coupling between barge and bridge (after Consolazio and Cowan 25) Stages of barge crush (after Consolazio and Cowan 25) Structural configurations analyzed (not to relative scale) SDF barge force-crush relationship derived from experimental and analytical data Sliding criterion between payload and barge Comparison of Case 1 coupled analysis output and P1T4 experimental data Comparison of Case 2 coupled analysis output and B3T4 experimental data Comparison of Case 2 coupled analysis output and B3T4 experimental data: Impulse Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown) Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model Structural configuration analyzed in Case Plan view of multiple pier numerical model and location of lumped masses in two-span single-pier model Structural configuration analyzed in Case Structural configuration analyzed in Case Comparison of Case 3 simplified and full-resolution coupled analyses Comparison of Case 4 simplified and full-resolution coupled analyses Comparison of Case 5 simplified and full-resolution coupled analyses Time computation comparison of coupled analyses Comparison of demonstration case simplified, full-resolution, and static analyses...61 A-1 Analytical output comparison to experimental P1T4 barge impact data...65 A-2 Analytical output comparison to experimental P1T5 barge impact data

9 A-3 Analytical output comparison to experimental P1T6 barge impact data...67 A-4 Analytical output comparison to experimental P1T7 barge impact data...68 A-5 Analytical output comparison to experimental B3T2 barge impact data...69 A-6 Analytical output comparison to experimental B3T3 barge impact data...7 A-7 Analytical output comparison to experimental B3T4 barge impact data...71 C-1 Case 3 AASHTO curve coupled analysis output comparison at impact location...77 C-2 Case 3 AASHTO curve coupled analysis output comparison at pier column top...78 C-3 Case 3 AASHTO curve coupled analysis output comparison at pile head...79 C-4 Case 4 AASHTO curve coupled analysis output comparison at impact location...8 C-5 Case 4 AASHTO curve coupled analysis output comparison at pier column top...81 C-6 Case 4 AASHTO curve coupled analysis output comparison at pile head...82 C-7 Case 5 AASHTO curve coupled analysis output comparison at impact location...83 C-8 Case 5 AASHTO curve coupled analysis output comparison at pier column top...84 C-9 Case 5 AASHTO curve coupled analysis output comparison at pile head...85 C-1 Case 3 bilinear curve coupled analysis output comparison at impact location...86 C-11 Case 3 bilinear curve coupled analysis output comparison at pier column top...87 C-12 Case 3 bilinear curve coupled analysis output comparison at pile head...88 C-13 Case 4 bilinear curve coupled analysis output comparison at impact location...89 C-14 Case 4 bilinear curve coupled analysis output comparison at pier column top...9 C-15 Case 4 bilinear curve coupled analysis output comparison at pile head...91 C-16 Case 5 bilinear curve coupled analysis output comparison at impact location...92 C-17 Case 5 bilinear curve coupled analysis output comparison at pier column top...93 C-18 Case 5 bilinear curve coupled analysis output comparison at pile head

10 LIST OF ABBREVIATIONS L or L R or R [ F ] appended to symbol, indicates symbol exclusivity to left-flanking structure appended to symbol, indicates symbol exclusivity to right-flanking structure flexibility matrix [ K condensed ] condensed stiffness matrix K coupling off-diagonal (coupling) stiffness term K Δ translational stiffness term K θ plan-view rotational stiffness term unit M unit moment m H mass of half-span of superstructure m b mass of barge m p mass of payload u initial sliding velocity of payload unit V unit shear force coupling V θ shear due to coupled stiffness and plan-view rotation V Δ shear due to translational stiffness and translation W p weight of payload Δ translation Δ M translation due to unit moment Δ V translation due to unit shear μ static coefficient of friction between payload and barge 1

11 θ plan-view rotation θ M plan-view rotation due to unit moment θ V plan-view rotation due to unit shear 11

12 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN Chair: Gary R. Consolazio Cochair: Marc I. Hoit Major: Civil Engineering By Michael Thomas Davidson August 27 The American Association of State and Highway Transportation Officials barge impact provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to determine structural demand on bridge piers. However, conclusions drawn from experimental full-scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand with consideration of additional forces generated from dynamic effects. Static quantification of bridge pier demand due to barge impact ignores mass related inertial forces generated by the superstructure which can amplify restraint of underlying pier columns. An algorithm for efficiently performing coupled nonlinear dynamic barge impact analysis on simplified bridge structure-soil finite element models is presented in this thesis. The term coupled indicates the impact of a finite element bridge model and a respective single degree-of-freedom barge model traveling at a specified initial velocity with a specified force-deformation relationship. Coupled analysis is validated using experimental data. Also, results from simplified and full-resolution analyses are compared for several cases to illustrate robustness of the algorithm for various barge impact energies and pier types. Simplified coupled dynamic analysis is shown to accurately capture dynamic forces and amplification effects. 12

13 CHAPTER 1 INTRODUCTION Potential loss of life and detrimental economic consequences due to bridge failure from waterway vessel collision have been realized numerous times throughout modern history. Catastrophic bridge failure events due to vessel collision, which occur approximately once a year worldwide (Larsen 1993), led to the development of bridge design specifications for vessel collision. The American Association of State and Highway Transportation Officials (AASHTO) Guide Specification and Commentary for Vessel Collision Design of Highway Bridges is used along with characteristics of a given waterway and the accompanying waterway traffic to determine static design loads, which are applied to respective piers for impact design purposes (AASHTO 1991). Even though the AASHTO specifications are used for bridge pier design due to ship and barge collision, limited barge impact data was available for use in their development. In April 24, Consolazio et al. (26) conducted full-scale experimental barge impact testing on bridge piers of the Old St. George Island Causeway Bridge located in Apalachicola, Florida. Key findings from the experiments that are pertinent to the research presented in this thesis include: Inertial forces due to acceleration of bridge component masses can contribute significantly to overall pier response during a collision event; Significant portions of the impact load can transfer (or shed ) into the superstructure; and, Superstructure resistance is comprised of displacement-dependent and mass-dependent (inertial) forces. Inertial forces can produce a momentary increase in pier restraint during initial impact stages, and amplify structural demand on pier columns. Restraint of a bridge pier due to acceleration of the mass of the overlying superstructure, and the corresponding amplification of forces developed in the pier columns during initial stages of barge collision events, are not addressed in current static design procedures. In contrast, dynamic time-history analysis of bridges inherently accounts for such amplification effects. 13

14 However, due to the unique characteristics of each bridge, impact load time-histories vary from bridge to bridge. Coupled dynamic analysis addresses this issue by employing a single degree-of-freedom (SDF) barge mass, impact velocity, and vessel force-crush relationship to simulate barge impact at a specified bridge pier location. This method enables efficient time-history analysis that yields time-varying barge collision load and bridge response data specific to each bridge structure. To validate the procedure, coupled analysis is performed and compared with experimental data for single-pier and multiple-pier cases. A summary of all analysis cases presented in this thesis is given in Table 1-1. However, coupled full-resolution bridge finite element (FE) models are cumbersome to analyze dynamically and time-history analysis of models of such size is not common in current practice. To facilitate use of coupled analysis in design settings, simplifying modifications are made to the barge and bridge structural models subject to impact. Specifically, to alleviate the onus of developing an appropriate barge force-crush relationship for each of the possible barge types, a simplified crush curve that is in accordance with current AASHTO design standards is employed. Second, an algorithm is presented which incorporates coupled analysis but reduces a multiple-pier model to essentially a pseudo-single pier model (with adjacent spans, springs, and lumped masses) thereby significantly reducing required analysis time. Simplified-coupled dynamic barge impact analysis is performed and compared to results from full-resolution models for a range of bridge and collision configurations. In comparison to full bridge model coupled time-history analysis, results from respective simplified models are sufficiently accurate for design purposes. Comparisons are also made between static and dynamic analysis predictions of bridge pier structural demand for each case. By employing coupled analysis with a simplified crush curve and simplified bridge structural model, design-oriented software is produced that can 14

15 efficiently quantify collision induced bridge pier demand, including capture of dynamic amplification effects. 15

16 Table Case descriptions: use, configuration, and impact data. Barge impact parameters Case Use a No. Piers Spans Weight Velocity Energy 1 V MN (64 T) 1.33 m/s (2.59 knots).484 MN-m (357 kip-ft) 2 V MN (344 T).787 m/s (1.53 knots).97 MN-m (71.3 kip-ft) 3 U/D MN (2 T) 1.3 m/s (2. knots).96 MN-m (7.9 kip-ft) 4 D MN (22 T) 1.54 m/s (3. knots) 2.18 MN-m (161 kip-ft) 5 D MN (772 T) 3.63 m/s (7. knots) 46. MN-m (34 kip-ft) a V = Validation; U = Uncoupled Condensed Stiffness Calculation; D = Demonstration 16

17 CHAPTER 2 LITERATURE REVIEW 2.1 Experimental Research In 1983, Meier-Dörnberg conducted reduced scale impact tests on barge bows using a pendulum impact hammer. Static crush tests were also performed on reduced scale barge bows. Results from this study were used to develop relationships between kinetic energy, barge bow crush depth, and static impact force. These relationships comprise a major portion of the collision-force calculation procedure adopted in the AASHTO specifications (1991). However, this research did not address phenomena such as bridge superstructure effects and dynamic amplification, nor did the tests involve pier or bridge response. During this same time and afterward, full-scale experimental barge collision tests were conducted in connection with the U.S. Army Corps of Engineers (USACE). In 1989, lock gate impact tests were performed with a nine-barge flotilla traveling at low velocities at Lock and Dam 26 near Alton, Illinois (Goble et al. 199). In 1997, four-barge flotilla impact tests were conducted on concrete lock walls at Old Lock and Dam 2, near Pittsburgh, Pennsylvania (Patev et al. 23). Additional lock wall tests were conducted with a fifteen-barge flotilla in 1998 at the Robert C. Byrd Lock and Dam in West Virginia (Arroyo et al. 23). All of these tests were performed on lock walls and lock gates, which produce fundamentally different structural responses to collision loading in comparison to that of bridge piers. The impact testing (Consolazio et al. 26) of the old St. George Island Bridge, constructed in the 196s, constitutes the only experimental research that explicitly measured barge impact forces on bridge piers using full-scale tests. The experiments were divided into three series of impact tests using a single barge and various pier/bridge structural configurations. The first series (termed the P1 series) consisted of eight impacts on a single, stiff channel pier 17

18 (termed Pier 1-S) by a loaded barge with an impact weight of 5.37 MN (64 T) and impact velocities approaching 1.8 m/s (3.5 knots). The second series of tests (termed the B3 series) consisted of four impacts on a multi-span, multi-pier partial bridge structure by an empty barge with an impact weight of 3.6 MN (344 T) and impact velocities approaching.78 m/s (1.5 knots). The third series (termed the P3 series) consisted of three impacts on a single, flexible pier (termed Pier 3-S) by an empty barge with an impact weight of 3.6 MN (344 T) and impact velocities approaching.95 m/s (1.8 knots). These tests form an important dataset for validating barge collision analysis methods. 2.2 Analytical Research Development and analysis of very high-resolution contact-impact FE models (those with tens to hundreds of thousands of elements) that simulate nonlinear dynamic barge impact on bridge piers have been feasible as a research tool for approximately a decade. In preparation for the full-scale St. George Island experimental barge impact testing, high-resolution FE pier models were developed to determine appropriate experimental conditions with respect to barge impact velocity and safety (Consolazio et al. 22). Reanalysis of the models using experimental data complimented the research findings from the experimental program (Consolazio et al. 26). High-resolution FE models of single-barges and multi-barge flotillas were analyzed when pier columns of various shape and dimension were subject to a variety of barge impact simulations (Yuan 25). These analytical results were used to develop a set of empirical formulas for barge impact force quantification as an improvement to the current static design procedures. Also, high-resolution FE single-barge models were developed and subjected to quasi-static loading by various stiff impactors in an effort to better quantify barge force-crush relationships (Consolazio and Cowan 23). 18

19 As an alternative to very high resolution contact-impact FE analysis, coupled barge-pier analysis was developed (Consolazio et al. 24a, Consolazio et al. 24b). Coupled analysis simulates a SDF barge model (with specified mass, velocity, and force-crush relationship) colliding with a multiple degree-of-freedom (MDF) bridge-pier-soil model. The coupled analysis required the use of a barge force-crush relationship, which was developed for a common barge type using high-resolution FE models. The force-crush curves encompass loading and unloading behavior derived from quasi-static cyclic loading (Consolazio and Cowan 25). 19

20 CHAPTER 3 COUPLED BARGE COLLISION ANALYSIS 3.1 Introduction Within the context of coupled analysis, the term coupled refers to the use of a shared contact force between the barge and impacted bridge structure (Fig. 3-1). The impacting barge is assigned a mass, initial velocity, and bow force-crush relationship. Traveling at the prescribed initial velocity, the barge impacts a specified location on the bridge structure and generates a time-varying impact force in accordance with the force-crush relationship of the barge and the relative displacements of the barge and bridge model at the impact location. The barge is represented by a SDF model, and the pier structural configurations and soil parameters of the impacted bridge structure constitute a MDF model. The MDF pier-soil model, subject to the shared time-varying impact force, displaces, develops internal forces, and interacts with the SDF barge model through the shared impact force during the analysis. Hence, coupled analysis automatically generates the barge impact load time-history specific to each bridge structural configuration and impacting barge type. This overcomes the challenge of pre-quantifying the time-varying barge impact load as a necessary component of time-history analysis Barge Loading and Unloading Behavior Barge behavior is represented by a force-crush relationship, consisting of a loading curve, unloading curves, and a specified yield point (Fig. 3-2). The yield point represents the crush depth beyond which plastic deformations occur. Any subsequent unloading beyond this point is determined according to the specified unloading curves. Until the crush depth corresponding to yield is reached, loading and unloading occurs elastically along the specified curve (Fig. 3-2A). A series of unloading curves represent the unloading behavior at various attained maximum crush depths (Fig. 3-2B). After unloading, if the barge is no longer in contact with the pier, no 2

21 impact force is generated (Fig. 3-2C). Alternatively, if reloading occurs (Fig. 3-2D), it is assumed to occur along the previously traveled unloading curve. Plastic deformation subsequent to complete reloading occurs along the originally specified loading curve (Fig. 3-2D). Additional details of this model are given in Consolazio and Cowan (25) Coupled Analysis Algorithm Algorithmically, the coupled analysis procedure involves a SDF barge code interacting with a separate nonlinear dynamic pier-soil analysis code at a specified node of the MDF pier-soil model. Specifically, coupled analysis utilizes an explicit time-step barge impact force determination procedure and links the output, the resulting impact force, with a respective numerical MDF pier-soil model analysis code (Hendrix 23). The pier-soil analysis code then responds to the impact force by generating iterative displacements and forces throughout the MDF model Use of Experimental Data for Coupled Analysis Validation Coupled analysis was previously developed and demonstrated as a proof-of-concept using analytical data (Consolazio and Cowan 25). Output from very high-resolution FE models consisting of a MDF impacting barge and a MDF impacted pier were compared to output obtained from coupled analysis of a SDF barge and MDF pier model. At present, experimental data is now available for validation of the coupled analysis procedure. Using data from the full-scale barge impact experiments (Consolazio et al. 26), validation of the coupled analysis procedure is carried out in four stages: select appropriate pier structures from the experimental dataset; develop respective models in a nonlinear dynamic finite element analysis (NDFEA) code capable of conducting coupled analysis; analyze the models using respective barge impact conditions and coupled analysis; and, compare time-history results from the coupled analysis to those obtained experimentally. 21

22 3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2 Data was collected more extensively from Pier 1-S than from any other pier in the 24 full-scale experimental test set (Consolazio et al. 26). Furthermore, a single pier is representative of the type of structure often used in static design procedures for barge collision analysis (Knott and Prucz 2). Hence, a single pier (Pier 1-S) was selected for coupled analysis validation using experimental data (Fig. 3-3A). Of the eight experimental tests conducted on Pier 1-S, the fourth test (termed P1T4) consisted of a head-on impact at an undamaged portion of the barge bow, as would be assumed in bridge design. Test P1T4, with velocity and impact weight as specified in Table 1-1, was selected for Case 1. In addition to validating the coupled analysis procedure for a single-pier, data from the partial bridge (B3 series) tests were employed for validation purposes. Regarding impact conditions used for validation, the fourth test (termed B3T4) generated the largest pier response among the B3 test series. Hence, test B3T4, with velocity and impact weight as specified in Table 1-1, was selected for Case 2 (Fig. 3-3B). 3.3 Software Selection and Model Development Coupled analysis was previously implemented in the commercial pier analysis software, FB-Pier (23), and was shown to produce force and displacement time-histories in agreement with those obtained from high-resolution contact-impact FE pier-soil model simulations. Subsequent to implementation of the coupled analysis procedure in FB-Pier, an enhanced package called FB-MultiPier (27) was released. FB-MultiPier possesses the same analysis capabilities as FB-Pier (including coupled analysis) but also has the ability to analyze bridge structures containing superstructure elements. Therefore, FB-MultiPier was selected for all model development and analysis conducted in this study. 22

23 FB-MultiPier employs fiber-based frame elements for piles, pier columns, and pier caps; flat shell elements for pile caps; beam elements, based on gross section properties, for superstructure spans; and, distributed nonlinear springs to represent soil stiffness. Transfer beams transmit load from bearings, for which the stiffness and location are user-specified, to the superstructure elements. FB-MultiPier permits Rayleigh damping, which was applied to all structural elements in the models used for this study such that approximately 5% of critical damping was achieved over the first five natural modes of vibration. FB-MultiPier allows either linear elastic or material-nonlinear analysis of structural elements. Linear elastic analysis was selected for all structural (non-soil) element components of models used in this study. This approach was taken because the 24 full-scale barge impact experiments were non-destructive (Consolazio et al. 26) and post-test inspection of the pier structures subjected to collision loading indicated that the structural components had remained largely in the elastic range. Structural models of Case 1 and Case 2 (Fig. 3-3A and Fig. 3-3B, respectively) were developed from original construction drawings and direct site investigation measurements. The Case 2 structural model was limited to four piers, with springs representing the stiffness contributions of piers beyond Pier 5-S (Fig. 3-3B), as contribution to structural response from these piers was expectedly small (Consolazio et al. 26). The soil model spring system for Case 1 was developed based on boring logs and dynamic soil properties obtained from a geotechnical investigation conducted in parallel with the 24 full-scale barge impact testing (McVay et al. 25). For the development of the Case 2 soil-spring system, boring logs formed the sole data source available. 23

24 For each model, a preliminary analysis was conducted in which the experimentally measured time-history load was directly applied at the impact point for the specified test case. The resulting displacement time-history of the structure was then compared to the experimentally measured displacement time-history at the impact point. Output from the direct analysis and comparison to experimental data aided in calibration of each model. Consequently, because analytical application of the experimentally measured load time-history was shown to produce pier response in agreement with that of the experimental data, the direct analysis comparison provided a baseline means of judging the efficacy of the coupled analysis procedure Coupled Analysis Module Parameters Within the coupled analysis procedure, the barge is modeled by a SDF point mass and nonlinear compression spring. Barge impact conditions for the validation cases (P1T4 and B3T4) were directly measured during the experimental tests. Thus, the experimental impact weights and velocities were directly input into analytical Case 1 as 5.37 MN (64 T) traveling at 1.33 m/s (2.59 knots) and Case 2 as 3.6 MN (344 T) traveling at.79 m/s (1.53 knots), respectively. The loading portion of the barge force-crush relationship used for Case 1 and Case 2 (Fig. 3-4) was developed from impact-point force and displacement time-history data measured during the P1T4 test; P1T4 was selected because of the undamaged bow impact location and head-on nature of the collision event. The portion of the barge force-crush relationship up to the peak force was obtained by performing coupled analysis using P1T4 impact conditions, and an initially arbitrary force-crush relationship. After analysis completion, the coupled analysis prediction of impact force was compared to that experimentally measured during the P1T4 test. The analytical force-crush relationship was then adjusted to more closely match that measured experimentally. After several iterations of this calibration process, a force-crush loading 24

25 relationship was obtained that produced force time-history data in agreement with the experimental measurements of impact force. The experimentally derived loading portion of the force-crush curve (Fig. 3-4) has a peak impact force value of 5.74 MN (165 kips) at a crush depth of 12.7 cm (4.75 in). Explicit derivation of forces beyond this point, pertaining to the barge-bow impact force-crush relationship, was not possible using the experimental dataset. However, barge bow force-crush data are available in the literature that apply to the shape of the impacted pier in the P1T4 test; specifically, a rectangular (flat) surface impactor. This data was obtained by subjecting a high-resolution FE barge model to quasi-static crushing by square (flat) 1.8 m (6 ft) and 2.4 m (8 ft) impactors (Consolazio and Cowan 23). In the present study, barge force-crush parameters pertaining to crush depths beyond that corresponding to the peak force were proportioned from high-resolution FE force-crush data. Specifically, these parameters are: the yield point, structural softening beyond the peak force, and the force plateau level beyond softening (Fig. 3-4). The unloading curves (Fig. 3-4) chosen for Case 1 and Case 2 exhibit steeper unloading paths at smaller crush-depths and shallower unloading paths at larger crush depths. The unloading curves are consistent, with respect to qualitative shape, with those employed in a prior study for a common barge type subject to quasi-static crush by square piers (Consolazio and Cowan 25) Accounting for Payload Sliding During Impact Testing During the Pier 1-S test series, payload in the form of m (55 ft) reinforced concrete bridge superstructure span segments was placed on the barge to simulate a loaded impact condition. However, the payload was observed to slide during the collision events, implying the development of frictional forces and dissipation of energy (Consolazio et al. 26). In general bridge design, the payload would not be assumed to slide. However, for the purpose of 25

26 validating the coupled analysis procedure as accurately as possible, enhancements were made to the pre-existing coupled analysis procedure to numerically account for payload sliding (Fig. 3-5). At each time-step and iteration, the ratio of barge acceleration (which, before sliding occurs, is equal to the payload acceleration) to gravitational acceleration was computed and compared to the static coefficient of friction ( μ ) between the barge and the payload. When the acceleration ratio exceeded the static coefficient of friction, sliding was initiated (Fig. 3-5B). At sliding initiation, the barge payload was assigned an initial velocity ( ) relative to the underlying barge, equal to the corresponding current velocity of the barge-payload system. The payload was assumed to continue sliding until the initial payload kinetic energy was completely dissipated through friction. At all points in time during which sliding occurred, a constant frictional force, equal to the product of the static coefficient of friction and the weight of the payload ( applied to the barge. When the sliding kinetic energy of the payload barge was dissipated, the payload mass ( m ) and barge mass ( m ) were assumed to rejoin as a single loaded p b ), was barge-payload system, as before sliding (Fig. 3-5A). For the P1T4 test, a sliding distance of.376 m (14.8 in) was predicted from the module modifications, which agreed very well with the observed payload slide of approximately.38 m (15 in). 3.4 Comparison of Analytical and Experimental Data Case 1 The Case 1 impact load time-history (Fig. 3-6A) is nearly identical to the respective experimental curve up to the peak load, and expectedly so, because the portion of the barge force-crush relationship (Fig. 3-4), up to the peak impact load, was derived from the impact force and displacement data acquired during the Case 1 (P1T4) collision event. Additionally, the analytical and experimental agreement for portions of the Case 1 force time-history curve u W p 26

27 beyond the peak justifies the assumptions made during the development of the load softening, load plateau, and unloading components of the force-crush curve (Fig. 3-4). The analytically determined peak value of pier displacement exceeds the experimental value by 16% (Fig. 3-6B). Supplementary coupled analyses of the Pier 1-S model were conducted with impact velocities measured during similar and higher impact-energy P1 series tests. Comparisons of displacement output from these analyses (Appendix A) to respective experimental data show discrepancies of comparable or lesser magnitude to those of Case Case 2 Case 2, in direct contrast to Case 1, consists of a low-energy barge collision event on a flexible pier with superstructure restraint. Case 1 and Case 2 share only the barge force-crush relationship derived from the P1T4 experimental data. The Case 2 experimental and analytical force time-histories (Fig. 3-7A) embody similar qualitative shapes; however, the analytical peak force magnitude is larger than the experimental counterpart. Despite the disparity in magnitude, numerical integration of the curves indicates that the shape and magnitude of the impulse, as a function of time, agree well between the experimental and analytical results (Fig. 3-8). This implies that the change in momentum of the barge was accurately predicted by the coupled analysis and produced a pier response similar to that measured experimentally. The concord of the analytical and experimental time-history of displacement (Fig. 3-7B) demonstrates the proficiency of the coupled analysis procedure in adequately predicting barge collision response for piers of varying stiffness. Accurate pier response predictions are maintained while incorporating superstructure effects. Agreement of pier response is the most important outcome of the coupled analysis procedure, as the accompanying internal forces generated throughout the MDF pier-soil model ultimately govern the pier structural member design. The coupled analysis procedure effectively shifts the analytical focus away from 27

28 determination of the barge impact force, and centers the emphasis on determining pier structural demand. Coupled analysis also inherently captures dynamic phenomena exhibited during barge-bridge collisions. As evidenced by the time-history plots of Case 1 and Case 2 (Fig. 3-6 and Fig. 3-7), the peak impact force and displacement do not occur simultaneously for individual experimental test cases involving appreciable impact-energies (Consolazio et al. 26). Static procedures do not account for peak load-displacement time disparity or the potential amplification effects intrinsic to the early stages of collision events for bridge structures. Coupled analysis automatically accounts for these effects. 28

29 Barge motion Barge and bridge models are coupled together through a common contact force Bridge motion superstructure Pier structure F F Soil stiffness Barge Crushable bow section of barge SDF barge model MDF bridge model Figure 3-1. Coupling between barge and bridge (after Consolazio and Cowan 25). 29

30 Impact Force Impact Force Elastic loading/unloading Initiation of unloading Yield point Loading curve Crush Depth A Unloading curve Crush Depth B Impact Force Impact Force Plastic loading occurs along loading curve Barge and bridge not in contact Crush Depth C Reloading occurs along same path as unloading Crush Depth D Figure 3-2. Stages of barge crush (after Consolazio and Cowan 25). A) Loading. B) Unloading. C) Barge not in contact with pier. D) Reloading and continued plastic deformation. 3

31 Pier 1-S Impact Pier 2-S Pier 3-S Pier 4-S Pier 5-S A Impact Springs modeling additional spans beyond Pier 5-S Figure 3-3. Structural configurations analyzed (not to relative scale). A) Case 1: Single pier. B) Case 2: Four piers with superstructure. B 31

32 Impact Force (MN) Crush Distance (in) Crushable barge bow SDF Barge Loading Curve Unloading Curves MDF Pier Crush Distance (mm) Impact Force (kips) Figure 3-4. SDF barge force-crush relationship derived from experimental and analytical data. 32

33 Payload m p W p No relative motion m b F μ Barge Barge acceleration < Gravitational acceleration Total barge-payload mass contributes to impact force Static coefficient of friction between barge and payload A m p Payload W p u m b μw p F Barge Barge mass and constant payload frictional force contribute to impact force Barge acceleration > Gravitational acceleration Static coefficient of friction between barge and payload B Figure 3-5. Sliding criterion between payload and barge. A) No sliding. B) Sliding. 33

34 5 Impact Force (MN) Coupled Analysis Output Experimental Data Impact Force (kips) Time (s) A 15 Pier Displacement (mm) Coupled Analysis Output Experimental Data Pier Displacement (in) Time (s) B Figure 3-6. Comparison of Case 1 coupled analysis output and P1T4 experimental data. A) Impact force. B) Pier displacement. 34

35 2 Impact Force (MN) Coupled Analysis Output Experimental Data Impact Force (kips) Time (s) A 5 Pier Displacement (mm) Coupled Analysis Output Experimental Data Pier Displacement (in) Time (s) B Figure 3-7. Comparison of Case 2 coupled analysis output and B3T4 experimental data. A) Impact force. B) Pier displacement. 35

36 Impulse (MN-sec).2.1 Coupled Analysis Output Experimental Data Impulse (kip-sec) Time (s) Figure 3-8. Comparison of Case 2 coupled analysis output and B3T4 experimental data: Impulse. 36

37 CHAPTER 4 SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS 4.1 Overview At current computer processing speeds, barge impact time-history analysis of bridge models can require between tens of minutes to several hours of processing time. Two simplifications may be applied to the coupled analysis of bridge structural models to reduce analysis time and facilitate its use in design settings. First, a simplified alternative to the experimentally and analytically derived crush curve may be used in design when more detailed barge force-crush behavior is not available. The bilinear curve found in the current static AASHTO design specifications (Fig. 4-1) may be used for general barge-bridge collision design applications. Second, multiple-pier models may be reduced to a pseudo-single pier model (with two attached superstructure spans) and analyzed to produce results that match to a satisfactory degree of accuracy, those obtained from corresponding full-resolution (multi-span, multi-pier) models. 4.2 Linearized Barge Force-Crush Relationship The nonlinear loading portion of the barge force-crush curve, developed from P1T4 experimental data (Fig. 4-1), is specific to the barge used in the 24 impact experiments. Phenomena such as structural-softening beyond the peak force level for each combination of barge type and impactor shape are not well documented in the literature and further study is warranted before these components of barge bow crushing behavior may be quantified for general application. Hence, the use of a simple bilinear force-crush relationship, such as that found in the AASHTO barge-collision specifications, is desirable at present as long as such a curve produces reasonable results. 37

38 The AASHTO force-crush relationship is in reasonable agreement with the P1T4 experimentally determined based force-crush curve. The crush depth at which the AASHTO and experimental curves shift from the initial linear portion to the subsequent linear portion occur at mm (4.8 in) and mm (4.75 in), respectively. For convenience, these locations are termed the shift points. Note that the AASHTO force corresponding to the shift point, 6.17 MN (1386 kips), is significantly greater than that found in the experimentally based curve, 4.74 MN (165 kips). Additionally, the AASHTO curve exhibits positive stiffness regardless of crush depth, whereas the curve employed in the validation of the coupled analysis method is assumed to exhibit perfectly plastic behavior at high crush depths (Fig. 4-1). Consequently, the AASHTO curve yields higher impact forces than the experimental data for all barge crush depths, and is therefore conservative. 4.3 Reduction of the Bridge Model Barges impart predominantly horizontal forces to impacted bridge piers during collision events. Displacement and acceleration based superstructure restraint (due to superstructure stiffness and mass, respectively) can attract a significant portion of the horizontal forces and cause the impact load to shed to the superstructure (Consolazio et al. 26). The horizontal force shed to the superstructure then propagates (initially) away from the impacted pier. Consequently, lateral translational and plan-view rotational stiffnesses influence the structural response as the force propagates through the superstructure from the impacted pier to adjacent piers. Simultaneously, the distributed mass of the superstructure alternates between a source of inertial resistance to a source of inertial load that respectively restrains or must be absorbed by other portions of the bridge structure. Simplification of the multiple-pier structural model, therefore, must adequately retain the influence of adjacent non-impacted (the lateral translational 38

39 and plan-view rotational stiffnesses of the adjacent piers; and, the dynamically participating mass of the superstructure) Uncoupled Condensed Stiffness Matrix The stiffness DOF of a bridge model, beyond the superstructure spans that extend from the impacted pier (Fig. 4-2), may be approximated by equivalent lateral translational springs and plan-view rotational springs. These springs are linear elastic and represent the predominant DOF of the linear elastic structural elements in the full-resolution model at piers adjacent to the impacted pier. Soil nonlinearities at piers other than the impacted pier are ignored during formation of the translational and rotational springs. Replacement of numerous DOF from the flanking portions of a full bridge model (Fig. 4-2) by two uncoupled springs at each end of a simplified two-span single-pier model may be described in terms of a condensed stiffness matrix: K K K Δ coupling [ K ] = condensed coupling K (4.1) θ where [ is the condensed stiffness matrix of the flanking bridge portion eliminated at K condensed ] each side of the impacted pier; KΔ is the condensed lateral translational stiffness term; K coupling is an off-diagonal stiffness term that couples the translational DOF to the rotational DOF; and K θ is the condensed stiffness plan-view rotational stiffness term. In the simplified model, the diagonal terms K and K are represented by translational and rotational springs, respectively, Δ θ and the K terms are neglected. The exclusion of K in the simplified model is coupling coupling justified by examining the forces generated by the condensed stiffness terms on one side of an example five-pier model. 39

40 A channel pier was added to the previously discussed four-pier Case 2 model, using bridge plans of the old St. George Island Bridge. This new five-pier model (Fig. 4-3) is referred to as Case 3, as defined in Table 1-1. Through flexibility inversion (Fig. 4-2), the left-flanking bridge structure in Case 3 (consisting of Pier 1-S to Pier 2-S), is reduced to the 2-DOF linear elastic condensed stiffness matrix in Eq. (4.1), where K Δ = 97. MN/m (554 kip/in); K = 2.58E+5 MN-m/rad (2.28E+9 kip-in/rad); and, K = 398 MN/rad θ [ ] coupling ( kip/rad). In row one of K, the K term may be interpreted as a horizontal condensed coupling shear force generated when a unit rotation (1 rad) is induced at the right-most node of the left-flanking structure. Static application of a load of 1.84 MN (414 kips) to the central pier of the Case 3 five-pier model induces a plan-view rotation of θ = 6.35E-6 rad at the location of the condensed stiffness. The horizontal shear produced as a result of this rotation is: V = K (4.2) coupling θ couplingθ where coupling V θ is the shear produced from the coupling of rotational and translational DOF. In this instance, coupling V θ = 2.53E-3 MN (.568 kips). In comparison, the horizontal shear produced as a result of diagonal lateral stiffness K = 97. MN/m (554 kip/in) and lateral displacement Δ Δ = 4.62 mm (.182 in) is: V = K Δ Δ (4.3) Δ where V Δ is the shear produced directly from lateral translation. For this loading, V Δ =.448 MN (11 kips). The amount of horizontal shear generated at the location of the condensed stiffness matrix, due to the coupling stiffness term, is very small relative to the amount of horizontal shear 4

41 coupling generated due to the diagonal stiffness term ( is only.6% of V ). A similar examination of the K and K terms yields ratios of comparable values ( Appendix B). The large θ coupling V θ difference in magnitude between the two shear forces demonstrates that the off-diagonal stiffness Δ terms of [ K condensed ] generate negligible forces relative to those generated by the diagonal stiffness terms. Uncoupling the condensed stiffness terms by applying two independent springs is therefore warranted for design applications, as the uncoupled springs form a reasonable static approximation of the stiffness of the excluded portions of the model. As a further simplification to the full-bridge model, the diagonal stiffness terms K Δ and K θ may be approximated by direct inversion of the individual diagonal flexibility coefficients. Specifically, this entails directly inverting the translational Δ V and rotational θ M displacements, respectively, induced by the application of a unit shear force unit V and unit force-couple unit M on the applicable flanking structure (Fig. 4-2). This approximation produces only nominally different magnitudes of stiffness with respect to that obtained by a flexibility matrix inversion (Appendix B) and is simpler to carry out. If significant nonlinear behavior is expected at non-impacted piers, then loads representative of the forces that will be shed to the superstructure, and subsequently transmitted into these piers, should be used to compute displacements (flexibility coefficients). Inversion of flexibility coefficients formed in this manner yields a condensed secant stiffness that may then be employed in the simplified model as described previously Lumped Mass Approximation Mass is attributed to each node of the NDFEA models in this study, which consequently, approximate a distributed mass system under dynamic loading. Therefore, a portion of mass of the excluded structural components is assumed to contribute to the structural response of the 41

42 simplified models. This mass is assumed to fall within the tributary area (Fig. 4-4) extending along the spans beyond the piers adjacent to the impacted pier of a given full-resolution model. The mass is lumped and placed at respective ends of the simplified model. The lumped mass simplification is combined with the stiffness approximation (Fig. 4-2) to complete the simplified two-span single-pier model. 4.4 Multiple-Pier Coupled Analysis Simplification Algorithm Simplified coupled analysis occurs in two stages. First, the two-span single-pier model is assembled by replacing extraneous portions of the multiple-pier model with uncoupled linear elastic springs and half-span lumped tributary masses. Coupled analysis is then performed, as previously discussed, with the AASHTO bilinear crush-curve being employed for the barge. The simplification algorithm automatically retains the ability to capture dynamic effects, such as amplification, not addressed in static procedures. Furthermore, hundreds to thousands of DOF are eliminated because the non-impacted piers and respective superstructure spans from the full-resolution model are omitted from the model. 42

43 Impact Force (MN) Crush Distance (in) Derived from experimental data AASHTO Impact Force (kips) Crush Distance (mm) Figure 4-1. Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown). 43

44 P-1 P-2 P-3 P-4 P-5 P-6 P-7 Impact location on full bridge model Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans Left-flanking structure Right-flanking structure P-1 P-2 P-3 P-5 P-6 P-7 Apply unit shear force at center of P-3 pile cap and center of P-5 pile cap P-1 P-2 P-3 P-5 P-6 P-7 V unit Record shear-induced translations and rotations at center of P-3 pile cap and center of P-5 pile cap V unit P-1 P-2 P-3 Δ VL θ VL θ VR Δ VR P-5 P-6 P-7 Apply unit moment at center of P-3 pile cap and center of P-5 pile cap P-1 P-2 P-3 P-5 P-6 P-7 M unit M unit Record moment-induced rotations and translations at center of P-3 pile cap and center of P-5 pile cap P-5 θ MR P-6 P-1 P-2 P-3 P-7 Δ MR Δ ML -1 L F = Form 2x2 left-flanking and right-flanking flexibility matrices using displacements and invert to form condensed stiffness Δ VL θ VL Δ ML θ ML -1 L K = condensed L K Δ L K coupling K L coupling K θ L θ ML -1 R F = Δ VR θ VR Δ MR θ MR -1 R K = condensed LR K Δ R K coupling K R coupling K θ R Neglect off-diagonal stiffness and replace flanking-structures in full bridge model with diagonal stiffness as uncoupled springs L K Δ L K θ L K Δ L K θ L P-4 LR K Δ K θ R LR K Δ K θ R Impact location on two-span single-pier model Figure 4-2. Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model. 44

45 Pier 1-S Pier 2-S Pier 3-S Pier 4-S Pier 5-S Impact Figure 4-3. Structural configuration analyzed in Case 3. 45

46 P-1 P-2 P-3 P-4 P-5 P-6 P-7 Impact location on full bridge model Form left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans Left-flanking structure Right-flanking structure P-1 P-2 P-3 P-5 P-6 P-7 Calculate mass of half-span beyond P-3 Calculate mass of half-span beyond P-5 P-1 P-2 P-3 P-5 P-6 P-7 m HL m HR Form lumped mass equal to m HL Form lumped mass equal to m HR m HL m HR Apply lumped masses in place of flanking-structure masses in full bridge model P-4 m HL m HR Impact location on two-span single-pier model Figure 4-4. Plan view of multiple pier numerical model and location of lumped masses in twospan single-pier mode. 46

47 CHAPTER SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES 5.1 Introduction To illustrate the efficacy of the simplification algorithm, three demonstration cases (FB-MultiPier bridge models) are presented. Each model was developed using methods representative of those employed by bridge designers. Impact conditions prescribed for the models are such that the range of scenarios encountered in practical bridge design for barge impact loading is well represented. The cases employ the AASHTO bilinear barge crush-curve, consist of impacted pier models of increasing impact resistance, and are subjected to impacts with corresponding increases in impact energy. Time-history output of internal pier structural member forces obtained from both full-resolution and simplified models are subsequently compared for each case. Each full-resolution model contains five piers: a centrally located impact pier and additional structural components (soil, non-impacted piers, and superstructure spans) for a length of two spans to either side of the central pier. A five-pier model contains a sufficient number of piers and spans such that inclusion of additional piers would increase analytical computation costs without appreciably improving the computed structural response. The appropriateness of the decision to limit the full-resolution models to five piers is substantiated by the consistently negligible acceleration response exhibited by the outer-most piers included in the five-pier models. Alternatively stated, the added restraint provided by including additional piers is not necessary, as the outer-most piers of the five-pier models are only nominally active throughout the barge impact analysis. 47

48 A single time-step increment,.25 sec, was employed for all demonstration analyses. Each model also utilized Rayleigh damping, which is configured such that the first five vibration modes undergo damping at approximately 5% of critical damping. 5.2 Geographical Information, Structural Configuration, and Impact Conditions Case 3 The first demonstration case consists of analysis of the previously described Case 3 model (Fig. 4-3). This model was based on the old St. George Island Bridge from the Apalachicola Bay area, linking St. George Island to mainland Florida, in the southeastern United States. Apalachicola Bay is located approximately 8.5 km (5 mi) southwest of Tallahassee, Florida in the panhandle portion of the state. The structure of the old St. George Island Bridge, constructed in the 196s, was detailed in a prior report (Consolazio et al. 26). Pertinent to demonstration Case 3, the superstructure spanning from Pier 2-S to Pier 5-S (Fig. 4-3) consisted of 23 m (75.5 ft) concrete girder-and-slab segments overlying concrete piers with waterline footings. Spanning the navigation channel and one additional pier to either side, a 189 m (619.5 ft) continuous three-span steel girder and concrete slab segment rested on Pier 1-S and Pier 2-S, each containing a mudline footing and steel H-piles. The central pier in Case 3, Pier 3-S, contained two tapered rectangular pier columns, with a 1.5 m (5 ft) wide impact face at approximately the same elevation as the top of a small shear strut that spanned between the two 1.2 m (4 ft) thick waterline pile-cap segments. The pier rested on eight battered.5 m (2 in) square prestressed concrete piles, each containing a free length of approximately 3.7 m (12 ft). The Case 3 FE model includes the southern channel pier and extends southward from the centerline of barge traffic. The impacted pier, Pier 3-S, was constructed before the AASHTO provisions were written (1991), and was flexible as it was not a channel pier. The pier was 48

49 located m (38 ft) from the channel centerline, which was significantly closer to a distance of three times the impacting vessel length, 138 m (45 ft), than the distance to the edge of the navigation channel, m (124 ft). Per the AASHTO specifications, the pier would be subject to a reduced impact velocity, approaching that of the yearly mean current velocity (Consolazio et al. 22). The kinetic energy (Table 1-1) associated with an empty jumbo-hopper barge drifting at the yearly mean current velocity for the Apalachicola Bay is representative of a low-energy impact condition Case 4 Escambia Bay abuts Pensacola, Florida, in the southeastern United States. Case 4 (Fig. 5-1) consists of impact analysis of a model based on the Escambia Bay Bridge. Structural components of this bridge model were derived from bridge plans developed in the 196s. The superstructure spanning from Pier 2-W to Pier 2-E consists of a 125 m (41 ft) continuous three-span steel girder and concrete slab. A 28 m (92 ft) concrete girder-and-slab segment spans the underlying concrete piers beyond Pier 2-E. All piers, except for the channel piers denoted as Pier 1-E and Pier 1-W, contain two pier columns, a shear wall, pile cap, and waterline footing foundation. The channel piers in Case 4 each contain two tapered rectangular pier columns, with a 2.6 m (8.5 ft) wide head-on impact face at approximately the mid-height elevation of a 5.3 m (17.5 ft) shear wall. The pier columns and shear wall overlie a 1.5 m (5 ft) thick mudline footing and 1.8 m (6 ft) tremie seal. The channel pier foundations consist of eighteen battered and nine plumb.6 m (24 in) square prestressed concrete piles. The Case 4 FE model includes both of the channel piers and three auxiliary piers. The impacted pier, Pier 1-E, was constructed before the AASHTO provisions were written (1991), but contains large impact resistance relative to the impacted pier from Case 3, as Pier 1-E is a channel pier. Impact on a channel pier with a relatively high impact resistance was chosen to 49

50 demonstrate the accuracy of the simplification algorithm for the medium-energy impact of a fully-loaded jumbo-hopper barge and towboat, traveling at a higher speed than the mean waterway velocity (Table 1-1) Case 5 Case 5 (Fig. 5-2) consists of impact analysis of piers from the new St. George Island Bridge, which replaced the old St. George Island Bridge in 24. The structural model of the new St. George Island Bridge was derived from construction drawings. Per these drawings, Pier 46 through Pier 49 support five cantilever-constructed Florida Bulb-T girder-and-slab segments at span lengths of m (27.5 ft) for the channel and 78.5 m (257.5 ft) for the flanking spans. Due to haunching, the depth of the post-tensioned girders vary from 2 m (6.5 ft) at drop-in locations to 3.7 m (12 ft) at respective pier cap beam bearing locations. Simply supported Florida Bulb-T beams with a depth equal to that of the haunched beams at the drop-in locations span either side of Pier 5. All piers included in this model contain two pier columns, a shear strut centered near a respective pier column mid-height, a pile cap, and a waterline footing system. The central pier in Case 5, Pier 48, contains two round 1.8 m (6 ft) pier columns, a (6.5 ft) thick pile cap, and fourteen battered and one plumb 1.4 m (4.5 ft) diameter prestressed cylinder piles with a 3 m (1 ft) concrete plug extending earthward from the pile cap. The new St. George Island Bridge was designed in accordance with current AASHTO barge collision design standards and provided a means of validating the simplification algorithm for barge impact energies similar to those used in present day design. The Case 5 FE model includes both of the channel piers and three auxiliary piers. The impacted pier, Pier 48 was designed for a static impact load of MN (3255 kips). With respect to the static AASHTO design impact load, an energy equivalent impact condition (Appendix D) is employed in Case 5. The prescribed vessel mass and velocity yields an impact kinetic energy equivalent to four 5

51 fully-loaded jumbo class hopper barges and a towboat traveling slightly above typical waterway vessel speeds for the Apalachicola Bay waterway (Table 1-1). 5.3 Comparison of Simplified and Full-Resolution Results In bridge design applications related to waterway vessel collision, the analytically quantified internal forces in a given pier structure govern subsequent structural component sizing. Hence, accurate determination of internal forces is a necessary outcome of a bridge structural analysis method. To highlight the ability of simplified analysis to accurately quantify design forces over the full range of impacted pier structures, time-histories of internal shear force induced by the impact loading are shown for the top of the impacted pier column and an underlying pile-head node for Case 3 through Case 5 shown in Fig. 5-3 through Fig. 5-5, respectively (additional comparisons of the impact force, displacements, and internal moments are documented in Appendix C). The predictions of load duration (the time during which the barge and pier are in contact), common to both simplified and full-resolution analyses, are.26 sec,.78 sec, and 2.9 sec, respectively, for Case 3, Case 4, and Case 5. At points in time greater than the respective load durations, each bridge is in an unloaded condition and undergoes damped free-vibration. Accordingly, pier response to time-history barge collision analysis may be divided into two phases: first a load-phase then a free-vibration phase. In all three demonstration cases, peak internal pier forces occur during the load-phase (.13 sec,.17 sec, and 2.1 sec for Case 3, Case 4, and Case 5, respectively). Therefore, agreement between the simplified and full-resolution models is most critical during the load-phase, as forces obtained during this phase ultimately govern bridge pier member design. Simplified analysis retains the ability to accurately capture forces during the load-phase of response (Fig. 5-3 through Fig. 5-5 for each case, respectively). Peak shear forces generated by full-resolution and simplified analysis during 51

52 the load-phase for each case differ by less than 2%. Reduced, yet still reasonable, agreement with respect to period of response and subsequent peak values of shear force occur during the free-phase of response for each case, however, such agreement is less critical and typically irrelevant for design purposes. Case 3 through Case 5 were analyzed on a Dell Latitude D61 notebook computer using a single 2.13 GHz Intel PentiumM CPU and FB-MultiPier. The computation times necessary for analysis completion of the simplified models were only 8%, 7.5%, and 8.4% of those required for the full-resolution models of Case 3 through Case 5, respectively (Fig. 5-6). All cases required significantly less than an hour to complete 8, 8, and 16 time-steps of analysis, respectively. Engineering judgment is required to determine the appropriate amount of analysis time specified. However, analysis generally need not be conducted beyond the end of load-phase, as evidenced by forces during the load-phase for Case 3 through Case Conclusions from Simplified-Coupled Analysis Demonstrations Excellent agreement is observed during the load-phase response of the full-resolution and simplified test cases, especially with respect to peak internal forces generated at various locations of the impacted piers. From a design perspective, reasonable agreement between full and simplified analytical results is also observed during the free-phase portions of respective time-history responses. Time-histories of internal shear force, moment, and displacement are adequately captured by the simplification algorithm, despite the simplifying stiffness and mass assumptions that are made. The time necessary to analyze the simplified models is significantly less than one hour in each case, which is in contrast to the several hours necessary to analyze respective full-resolution models. It should be noted that all FB-MultiPier analyses were conducted in compilation debug 52

53 mode. Considerable additional reduction in analysis time is expected if the same analyses were to be conducted in a release compilation or commercial version of FB-MultiPier. 5.5 Dynamic Amplification of the Impacted Pier Column Internal Forces Application of the simplification algorithm to each of the demonstration cases inherently incorporates mass and acceleration based inertial forces that emerge from integration of the dynamic system equations of motion. The simplification algorithm accurately captures dynamic amplification of forces generated in the pier columns that would be absent from static analysis results. Dynamic amplification in each case may be quantified by considering the maximum pier column shears developed in models subjected to static application of the peak impact load predicted through the coupled analysis. The peak shear and moments developed in the pier due to static loading are then compared to those from the simplified and full-resolution dynamic analyses (Fig. 5-7) With respect to peak pier column structural demand, the dynamic analyses are in excellent agreement with each other for all cases. However, the peak magnitudes of the statically generated shears and moments, respectively, correspond to 59% and 64% of the magnitude of the dynamically obtained counterparts for Case 3; and, 38% and 37%, respectively, for Case 4 (Fig. 5-7). In each of these cases, a static analysis employing a dynamically obtained peak impact load leads to un-conservative predictions of peak pier column demand, as static analysis only encompasses stiffness considerations. In contrast, dynamic analyses incorporate both stiffness and inertial effects associated with the superstructure and therefore capture dynamic amplification of pier column forces due to the mass of the superstructure. Furthermore, the simplified procedure retains the ability to capture pier column force amplification as evidenced by the agreement between the simplified and full-resolution output pertaining to peak pier column demand. 53

54 The impact energy specified in Case 5 is of sufficient magnitude to cause the barge and impacted pier to remain in contact for a time greater than several periods of the fundamental pier vibration mode. Consequently, the inertial forces in the impacted pier begin to dissipate due to damping effects. This is evidenced by attenuation of oscillation exhibited in the pile head shear force time-history for Case 5 from.1 sec to 2.5 sec (Fig. 5-5B). Despite the continued dynamic activity in the top of the Pier 48 pier columns throughout the analysis (Fig. 5-5A), the overall pier behavior approaches that of a static response as the impact load approaches a maximum value. Additionally, because the AASHTO barge bow force-crush relationship (Fig. 4-1) maintains a positive stiffness regardless of crush depth, the Case 5 peak impact force occurs at a time in which the dynamic component of behavior of Pier 48 has substantially diminished. Therefore, the peak pier column demands are driven by a static response in this case. As a result, there is not a great difference between dynamic and static response (Fig. 5-7). 54

55 Pier 2-W Pier 1-W Pier 1-E Pier 2-E Pier 3-E Impact Figure 5-1. Structural configuration analyzed in Case 4. 55

56 Pier 46 Pier 47 Pier 48 Pier 49 Pier 5 Impact Figure 5-2. Structural configuration analyzed in Case 5. 56