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Dynamic Analysis Techniques for Quantifying Bridge Pier Response to Barge Impact Loads


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DYNAMIC ANALYSIS TECHNIQUES FOR QUANTIFYING BRIDGE PIER RESPONSE TO BARGE IMPACT LOADS By JESSICA LAINE HENDRIX A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2003

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ii ACKNOWLEDGMENTS The successful completion of this thesis and the research discussed herein would not have been possible without the guidance and support of my graduate advisor, Dr. Gary R. Consolazio. The knowledge and estimable work ethic he imparted to me will continue to serve me in all of my future endeavors, and for that I am very grateful. I would also like to acknowledge the following people who were instrumental in the successful completion of my graduate work: Dr. Mark Williams with the Bridge Software Institute, David Cowan, and William Yanko. Their help was truly invaluable and they are living proof that patience truly is a virtue. I would also like to thank my family and friends for their unyielding faith and confidence, without which none of this would have been possible.

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iii TABLE OF CONTENTS page ACKNOWLEDGMENTS...................................................................................................ii TABLE OF CONTENTS...................................................................................................iii CHAPTER 1. INTRODUCTION.........................................................................................................1 2. AASHTO BARGE IMPACT PROVISIONS...............................................................6 3. CHARACTERIZATION OF BARGE IMPACT LOADS...........................................9 3.1 Ship Collision Studies.............................................................................................9 3.2 Barge Collision Studies.........................................................................................10 4. HIGH RESOLUTION BARGE IMPACT ANALYSIS.............................................13 5. MEDIUM RESOLUTION BARGE IMPACT ANALYSIS.......................................16 5.1 Modeling Barge Crush Behavior..........................................................................17 5.2 Integration of Dynamic Barge Behavior...............................................................21 5.3 Coupling Between the Barge and Pier..................................................................23 6. LOW RESOLUTION BARGE IMPACT ANALYSIS..............................................30 6.1 Description............................................................................................................30 6.2 Implementation......................................................................................................31 7. DISCUSSION OF RESULTS.....................................................................................35 7.1 Comparison of Dynamic Simulation Results........................................................36 7.2 Comparison of Dynamic and Equivalent Static Impact Loads.............................42 8. CONCLUSIONS AND RECOMMENDATIONS.....................................................47 APPENDIX A SMOOTH HIGH RESOLUTION CRUSH DATA....................................................49

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iv B LOW RESOLUTION ANALYSIS PROGRAM........................................................54 C MEDIUM RESOLUTION VALIDATION CASES...................................................68 D COMPLETE DYNAMIC ANALYSIS RESULTS.....................................................71 REFERENCES..................................................................................................................84 BIOGRAPHICAL SKETCH............................................................................................86

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v 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 Engineering DYNAMIC ANALYSIS TECHNIQUES FOR QUANTIFYING BRIDGE PIER RESPONSE TO BARGE IMPACT LOADS By Jessica Laine Hendrix August 2003 Chair: Gary R. Consolazio Major Department: Civil and Coastal Engineering Current bridge design provisions specify the use of an equivalent static load approach to represent barge impacts loading conditions—events that are, in fact, often highly dynamic in character. This thesis presents three different dynamic analysis techniques as alternative procedures for quantifying barge impact loads on bridge piers. The first technique utilizes high resolution finite element impact simulation to quantify impact loads. The remaining two techniques are both design-oriented in nature but differ in their level of sophistication: one is capable of modeling complex structural pier behavior while the other is limited to inclusion of only certain types of pier nonlinearity. A variety of barge impact analyses are conducted using the two design-oriented techniques and the results are then compared to results obtained from high resolution nonlinear finite element impact simulations. These comparisons show that designoriented, computationally efficient analysis techniques are capable of modeling a wide

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vi range of dynamic barge-pier interaction and can reasonably predict the dynamic loads generated by during such events. Additional comparisons are made between the equivalent static load method prescribed by current bridge design specifications and the dynamic analysis techniques presented here. Impact loads and structural deformations predicted by the static method and the dynamic analysis techniques are compared for barge collisions of varying kinetic energies. Results from such comparisons indicate that, for barge collision events associated with relatively low kinetic energy levels, dynamic analysis techniques are preferable. For more severe impact conditions, the use of equivalent static loads for design purposes is acceptable.

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1 CHAPTER 1 INTRODUCTION The use of vessel impact loads in bridge design is a concept that has gained mainstream attention only within the past two decades. It was previously a common belief among the engineering community that the probability of such an event was so unlikely that it could be disregarded [1]. It was not until 1980, with the collapse of the Sunshine Skyway Bridge in Tampa, Florida that designers began to realize the significance of vessel collisions. As a result of an errant cargo ship, The Summit Venture the Sunshine Skyway Bridge was destroyed and thirty-five people lost their lives [2]. Although the collapse of the Sunshine Skyway was just one of several catastrophic failures in recent history, it raised the world’s consciousness regarding the need to design bridges for the possibility of vessel impact. Since then, important strides have been made in an effort to develop modern principles and standards for the design of bridges against such events. Unfortunately, barge collisions continue to be a very real problem in the United States and worldwide. On September 15, 2001, a four-barge tow struck the Queen Isabella Causeway, the only bridge leading to South Padre Island, Texas, killing eight people. Substantial damage was not only imparted to the bridge but to the local economy as well [3]. Less than a year later, another fatal bridge collapse occurred. On May 26, 2002 an Interstate 40 bridge near Webber’s Falls, Oklahoma collapsed into the Arkansas River killing fourteen people, after being hit by an errant barge. The I-40 Bridge, built in 1967, passed all required safety checks just months before the accident [3]. Thus, there is

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2 a clear need for research focusing on the development of improved methods for predicting barge impact loads and evaluating the structural response to such loads. Compared to the data available for ship collisions, very little experimental data exists with respect to barge collision impact forces. This is due not only to the expense involved in carrying out experimental studies, but also to the inherent complexity of measuring such a phenomenon. One such complication is the variety of barge configurations that transit inland waterways. The U.S. Army Corps of Engineers reports [4] that there are over two thousand different types and sizes of barges using the U.S. inland waterway system. From such a variety of barge types, almost any combination of size and quantity may be configured into a multi-barge flotilla. As a result of the limited availability of barge impact data, the American Association of State Highway and Transportation Officials (AASHTO) Guide Specification for Vessel Collision Design of Highway Bridges [2] relies on data produced by a single study conducted by Meir-Dornberg (discussed in more detail later) in 1983. Furthermore, the AASHTO provisions stipulate the use of equivalent static loads to represent impact events that often have significant dynamic components of load and bridge response. Because barge impact loading conditions often control the design of new bridges, it is important that alternative load prediction procedures be explored. Dynamic structural analysis techniques offer one possible alternative to the use of code-specified equivalent static loads. However, this is a new and emerging area of bridge-related structural research. Recent analytical studies focusing on barge impact [5,6] have involved the development and use of high-resolution nonlinear dynamic finite element models. While such models are capable of quantifying dynamic impact loads—

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3 as well as providing valuable insight into the relevant collision mechanics—they are not generally appropriate for use in routine bridge design due to the considerable computational resources often required. In this thesis, design-oriented medium and low resolution dynamic analysis techniques are developed that approximate the behavior of more refined high resolution barge-pier interaction models (Figure 1). Throughout this document, reference will made to high, medium, and low resolution analysis techniques. In this context, these terms are defined as follows : High resolution Highly refined finite element models, generally consisting of tens of thousands of elements [5,6], which are analyzed using dynamic finite element analysis codes (e.g. LS-DYNA [7]) capable of representing contact and localized nonlinear buckling. Medium resolution Finite element models generally consisting of hundreds to thousands of elements that are analyzed using commercial dynamic finite element analysis codes (e.g. FB-Pier [8,9]). Low resolution Simple, low-order numerical models utilizing a very small number of lumped masses and nonlinear springs to represent barge and pier behavior. In regard to the medium resolution analysis technique presented here, focus is placed on the development of a procedure that can be used to couple nonlinear dynamic barge behavior to existing commercially available, design-oriented dynamic structural analysis codes. The low resolution analysis technique presented focuses instead on a simpler spreadsheet based prediction of dynamic impact loads using direct integration techniques of a set of coupled differential equations. In a design environment, structural response to such loads would then need to be evaluated using a dynamic structural analysis package.

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4 m b m p Direction of barge motion Zone of contact between barge and pier Pier column Pier of total mass m p Pile cap Steel H-piles supporting pier Barge of total mass m g b Soil springs Reinforced concrete pier a) k b m b m p Gap element permitting transfer of compressive loads only (zero tension) Nonlinear spring element representing the barge load-deformation crush relationship Single dof model of barge as a point mass b) k b k p m b m p Single dof model of pier as a point mass Nonlinear soil spring representing lateral load-deformation behavior of pier c) Figure 1. Dynamic analysis techniques used to simulate barge impact events. a) High resolution model; b) Medium resolution model: simplified barge model; c) Low resolution model: simplified barge and pier models

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5 In all cases treated here, particular focus is given to evaluating the accuracy of the dynamic loads predicted by the medium and low resolution models. Previous studies [5,6] have demonstrated that barge impact loading conditions can vary from highly dynamic and oscillatory in nature to sustained, nearly pseudo-static. The nature of the loads produced depends on the specific characteristics of the collision event (barge mass, impact speed, pier stiffness, etc.). The ability of the medium and low resolution analysis techniques to predict load and pier response will be evaluated by comparing results produced by these procedures to results obtained from high resolution analyses. Also of interest, is the extent to which impact loads predicted by dynamic analysis in general compare to the equivalent static loads specified by the AASHTO procedures. In the following chapters, an overview of the existing AASHTO procedures for barge impact design will be given, followed by descriptions of the newly proposed dynamic analysis methods. Following that, an impact resistant pier and a non-impact resistant pier are analyzed using high, medium, and low resolution models and the results are compared.

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6 CHAPTER 2 AASHTO BARGE IMPACT PROVISIONS The AASHTO barge impact provisions [2,10] apply to the design of all bridges spanning shallow draft inland waterways carrying barge traffic. The AASHTO Guide Specification for Vessel Collision Design and the AASHTO LRFD Bridge Design Specification differ in their risk analysis method, but follow the same procedure for predicting expected barge impact loads. These provisions are based on empirical equations, rational analysis based on theory, and model testing supported by analysis [2]. Such an approach was taken due to the lack of available experimental data involving barges striking bridge piers. In addition, at the time the specifications were written, computer analysis programs available to design engineers were incapable of handling dynamic effects and the presence of nonlinearities in structural systems. The AASHTO provisions are intended to provide a simplified procedure for computing an equivalent static load for a barge impact in lieu of a full dynamic impact analysis. The calculations begin with the collection of required data for the bridge design (vessel traffic, vessel transit speed, loading characteristics, bridge geometry, waterway and navigable channel geometry, water depths, and environmental conditions). Barge impact loads are then evaluated on the basis of energy considerations. The translational kinetic energy for a moving vessel is given by AASHTO [10] as : 2500 MV C KE H (1)

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7 where K Eis the vessel kinetic energy (joule), H C is the hydrodynamic mass coefficient, M is the vessel displacement tonnage (metric ton), and V is the vessel impact speed (m/sec). It should be noted that Eq. 1 is an empirical equation based on the standard relationship for translational kinetic energy of a moving body commonly expressed as : 22 1 MV KE (2) where M is the mass of the vessel (kg). The hydrodynamic mass coefficient, H C, included in the AASHTO equation for kinetic energy, is present to account for additional inertia forces caused by the mass of the water moving with the vessel. Several variables are accounted for in the determination of H C including water depth, underkeel clearances, shape of the vessel, speed, currents, position and location of the vessel in relation to the pier, direction of travel, stiffness of the barge, and the cleanliness of the hull underwater. Based on a previous study [11], a simplified expression has been adopted by AASHTO in the case of a vessel moving in a forward direction at a high velocity (the worst-case scenario). Under such conditions, the procedure recommended by AASHTO depends only on the underkeel clearances [10] : For large underkeel clearances ) 5 0 ( Draft : 05 1 H C For small underkeel clearances ) 1 0 ( Draft : 25 1 H C where the draft is the distance between the bottom of the vessel and the floor of the waterway. For underkeel clearances between these two limits, H C is estimated by interpolation. The next step in the AASHTO provisions is to determine the barge bow (front section of the barge) damage depth. During an impact event, energy can be absorbed or

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8 dissipated in a variety of ways, including displacement and/or plastic deformation of the barge and bridge pier (including any fendering system), friction, and also rotation in the event of an eccentric impact [12]. Considering AASHTO’s assumption that the impact loads developed represent the worst-case scenario of a head-on collision, one of the primary ways energy is dissipated is through crushing of the bow. AASHTO’s relationship between kinetic energy and barge crush is represented by the following equation [10] : ) 1 ) 10 3 1 ( 1 ( 31007 KE R aB B (3) In this equation B a is the barge bow damage depth (mm) and B R given by 67 10 B B is the barge width modification factor, where B B is the barge width (m). The barge width modification factor is included to account for a barge width other than the standard value of 10.67 meters (the average barge width used in U.S. inland waterways) [2]. Once B a is determined, the barge impact force is calculated by [10] as : mm a mm a R a R a PB B B B B B B100 100 1600 ) 10 0 6 ( ) 10 0 6 (6 4 (4) where B P is the equivalent static barge impact force (N). Note that Eqs. 3 and 4 are based on the AASHTO LRFD Bridge Design provisions [10] but are modified to include the barge width modification factor ( B R ) presented in the AASHTO Vessel Collision Specifications [2], in accordance with results from Meir-Dornberg.

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9 CHAPTER 3 CHARACTERIZATION OF BARGE IMPACT LOADS As stated in the previous chapter, very few analytical or experimental studies have been conducted to characterize the response of bridge piers to impact loads generated during vessel collision events. In this chapter, a review of relevant previous studies is provided. 3.1 Ship Collision Studies Ship collision events have been studied to a much greater extent than have barge collisions. Two primary ship collision studies form the basis for most current theories relating to ship impact loading. The first study is that of V.U. Minorsky [13]. This study was conducted in 1959 to analyze collisions with reference to protection of nuclear powered ships, and focused on predicting the extent of vessel damage during a collision. A semi-analytical approach was used based on data from twenty-six actual collisions. From this data, Minorsky determined a linear relationship between the deformed steel volume and the absorbed impact energy. The second key study of the time was conducted by Woisin in West Germany [14]. Woisin was also interested in the deformation of nuclear powered ships in the event of a collision. Data were collected from twenty-four collision tests of scaled ship models colliding with each other. Results from this study were used to develop the current AASHTO equation for calculating equivalent static ship impact force [2].

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10 In 1990, Prucz and Conway published a paper [15] discussing analytical research conducted on the dynamic effects associated with ship collisions with bridge piers. Their paper presented a simplified numerical procedure, based on a lumped mass-idealization of the ship-pier system, to investigate the dynamic effects associated with ship collision events. 3.2 Barge Collision Studies With regard to barge impact loads, the most significant experimental study conducted to date is that of Meir-Dornberg [2]. This research included both static and dynamic loading on scaled models of the European Barge, Type IIa, similar in dimension to the U.S. standard jumbo hopper barge. No significant differences were found between the static and dynamic impact force. However, the tests did not involve interaction between vessels and bridge piers. From this research, equations were developed that related equivalent static barge impact force, barge deformation, and impact deformation energy. These equations were modified slightly and adopted by AASHTO for use in computing barge impact loads. More recently, the U.S. Army Corps of Engineers (USACE) completed the first full-scale barge impact experiments on a concrete lock wall [16]. The USACE Waterway Experiment Station performed the experiments in order to verify the current analytical model used to design inland waterway navigation structures and also to support the development of more innovative structures. Structures such as lock walls are subject to frequent barge flotilla impacts and must be designed accordingly. The current single degree of freedom model used by the Army Corps to quantify design loads is believed to be overly conservative and therefore produces very costly structures. The goals of this research were to determine a baseline response of a barge impact with a lock wall,

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11 measure impact forces, quantify barge-to-barge interaction in a flotilla during a collision, and to investigate a new energy-absorbing fendering system. Since this type of experiment had never been done in the past, prototype barge impact experiments were first conducted on Allegheny River Lock and Dam 2 in Pittsburgh, Pennsylvania in 1997 [17]. The primary purposes for conducting the prototype tests were to determine how to quantify and measure barge impact forces and to gain a better understanding of the nature of the barge-wall interaction. Following the prototype tests, full-scale experiments were conducted on the Robert C. Byrd Lock and Dam in Gallipolis, West Virginia in 1999 [16]. These experiments consisted of a fifteen-barge flotilla of jumbo open-hopper barges impacting the lock wall with and without a fendering system. There were forty-four impact experiments in all. The experiments ranged in impact angles from 5 to 25 degrees and in impact velocities from 0.15 to 1.2 m/sec. All of the impacts were within the elastic deformation range of the barge flotilla. Results from this study are being used to develop full-scale plastic range experiments that will be used to determine actual crushing strength and performance of inland waterway barges. Currently, an on-going investigation is being conducted by the University of Florida [18,19] and the Florida Department of Transportation (FDOT) to quantify impact loads generated on bridge piers during barge collision events. A combination of experimental testing and analytical modeling are being used to characterize the dynamic nature of impact loads that arise during barge collisions, to compare these loads to the equivalent static loads prescribed by the AASHTO provisions, and to develop appropriate design-oriented impact-load prediction procedures.

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12 Replacement of the existing St. George Island Causeway Bridge (near Apalachicola, Florida) with a newly constructed bridge has afforded the opportunity to conduct full-scale barge impact tests on the older structure before it is demolished. After the new bridge has been opened to traffic, a full-scale hopper barge will be driven into selected piers of the older structure at several different impact speeds while the dynamic impact loads imparted to the piers are monitored. In addition, deformations of the piers, surrounding soil, and barge bow will be monitored throughout each impact event. Complimenting the experimental components of this investigation, a variety of finite element based analytical studies have also been conducted to quantify the impact loads generated during barge collisions and to aid in planning the physical impact tests (selection of piers to be tested, selection of impact speeds, etc.). Consolazio et al. [5,18, 19] have developed high resolution finite element models of a barge and two piers of the existing St. George Island Causeway Bridge (including representation of both the structural properties and soil properties). Nonlinear finite element analysis codes have been applied to these models to study the crush characteristics of hopper barges [6], and to quantify the dynamic loads that are imparted to bridge piers during collision events [5]. In the latter study, the character of the impact loads generated—sustained versus highly oscillatory—was found to be a function primarily of barge impact speed and pier stiffness. In addition, this study also revealed that during severe collision events, more than half of the kinetic energy of impact may be dissipated through plastic deformation of the barge bow.

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13 CHAPTER 4 HIGH RESOLUTION BARGE IMPACT ANALYSIS Numerical prediction of lateral impact loads imparted to bridge piers during barge collision events is most accurately achieved through the use of contact-impact finite element analysis software and refined barge, pier, and soil-structure interaction models. In this thesis, the high resolution models previously developed by Consolazio et al. [5,18, 19], and the nonlinear dynamic finite element code LS-DYNA [7] have been used to predict baseline impact data to which results obtained from the medium and low resolution analysis techniques, presented in the following chapters, can be compared. All impact conditions considered herein involve a jumbo class hopper barge striking a pier bridge in a head-on, perpendicular manner (Figure 2). The jumbo class hopper barge—59.5 m long, 10.7 m wide, 3.7 m deep, 1.8 MN empty weight, 6.9 MN fully loaded weight—makes up more than 50% of the entire barge fleet operating in the U.S. inland waterway system. Furthermore, this type of barge is the baseline vessel upon which the AASHTO barge impact provisions are established. For these reasons, the finite element barge model used in this study matched the mass, geometry, and structural configuration of a typical jumbo hopper barge. A key difference between the high resolution analysis and the lower resolution methods presented later in this document lies in the modeling of the barge. High resolution analysis, as defined here, involves the use of a very detailed finite element barge model. The barge model developed by Consolazio et al. is based on detailed

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14 structural plans obtained from a leading U.S. barge manufacturer, rather than on the crush relationship assumed by AASHTO, i.e. Eqs. 3 and 4. Intended for use in frontal impact simulations, the model uses more than 25,000 shell elements to represent internal structural members (plates, channels, angles, etc.) in the bow section of the barge. By combining this level of geometric discretization with nonlinear steel stress-strain relationships, bow crushing and energy dissipation during collisions with relatively rigid concrete piers can be accurately simulated. Frictional effects, internal buckling and contact, and buoyancy effects are also included in the model. Hopper barge Steel piles and soil springs Concrete piles and soil springs Pier-1 Pier-3 Buoyancy springs Figure 2. High resolution barge, pier-1 and pier-3 finite element models The high resolution structural pier models used in this study (Figure 2) represent two of the support piers of the previously cited St. George Island Causeway Bridge. Pier-1 (adjacent to the navigation channel) and pier-3 (third from the channel) were chosen for this study because they represent a significant range in both structural stiffness

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15 and impact resistance, and therefore yielded impact force results representative of a wide variety of pier types. Pier-1 is considered a typical example of an impact resistant pier because of its size and stiffness, while pier-3 is representative of a more flexible, nonimpact resistant pier. The pier models, developed using construction plans for the bridge, include pier bents, pile caps, and piles. The models do not take into account any structural contribution or inertial effects of the superstructure. Soil-structure interaction effects between the piles and the surrounding soil— including nonlinear soil response, plastic deformation, gap formation, and pile-group effects—were modeled using thousands of nonlinear lateral and vertical soil spring elements. Unique load-deformation curves were specified for each spring based on soil boring data obtained for the bridge site. For a more detailed descriptions of the barge, pier, or soil models, the reader is referred to Refs. [18, 19]. Impact data predicted by nonlinear dynamic analysis of these models are used to compare results predicted by medium and low resolution analysis techniques later in this thesis.

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16 CHAPTER 5 MEDIUM RESOLUTION BARGE IMPACT ANALYSIS Although high resolution dynamic analysis techniques are capable of yielding accurate impact load data, such methods are also computationally intensive and thus, not always practical for use by design engineers. In this chapter, a medium resolution analysis technique is described in which a single degree of freedom (DOF) nonlinear barge is coupled to an existing multi-DOF nonlinear dynamic pier analysis program (FB-Pier [8]). The coupling is implemented in a way that necessitates only minimal modifications to the dynamic pier analysis code. Conducting a barge impact analysis requires consideration of both dynamic behavior and nonlinear structural behavior. Properly modeling structural behavior (the inelastic force-deformation response of the barge) is particularly important as it affects both force development and energy dissipation during impact. The approach taken here is to approximate dynamic nonlinear barge behavior by independently representing dynamic behavior (mass related inertial resistance) and nonlinear structural behavior (barge crushing). The total mass of the barge is represented as a single degree of freedom (SDOF) point mass while inelastic structural response of the barge bow is modeled using a nonlinear crush relationship (a b P vs. ba curve). A pre-computed barge crush curve is employed to model the stiffness of the barge bow the medium resolution impact analyses (Figure 3).

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17 m b a b P b a b u b K ur Barge force-deformation relationship modeled using data from high resolution finite element simulation a) m p m b P b P b u b u p b) Figure 3. Dynamic barge and pier/soil modules. a) Barge module showing origin of contact force; b) Contact force linking the two modules 5.1 Modeling Barge Crush Behavior While several sources of pre-computed crush data exist (experimental testing, finite element analysis, or specification-prescribed expressions such as Eq. 4) in this study, high resolution finite element simulations have been used to generate the barge crush data needed. Key to the concept of using high resolution finite element analysis— which was cited above as being computationally expensive —for this purpose is the idea that these analyses are one-time events conducted in advance. In the present study, static crush simulations have been conducted [19] using the high resolution barge model described in the previous chapter to generate data relating barge force, b P to crush deformation, ba The crush characteristics of a barge are functions only of the barge itself and the geometry (shape and size) of the object imposing the crush damage (i.e., the

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18 pier). Thus, crush data can be computed in advance using high resolution analysis and then stored for subsequent use in a medium resolution analysis model. It is by this strategy that complex crushing behavior can be efficiently modeled using a SDOF barge model. Because static barge crush behavior is dependent on the size and shape of the pier [19], separate crush curves were generated for pier-1 and pier-3 (since they have different column widths). However, data generated from finite element crush analyses often exhibit small scale dynamic deviations (Figure 4a) from the primary crush curve. Deviations of this type complicate the process of applying direct dynamic integration techniques to the SDOF barge model. Therefore, in this study, the raw high resolution force-deformation data (crush data) were smoothed and re-sampled (Figure 4b) before being incoporated into the medium resolution model. For additional information on the smoothing technique employed, refer to Appendix A. Smoothed crush curves, along with an unloading and reloading stiffness (ur K ), have been used to define an elastic-plastic load-deformation relationship for the barge. All possible load paths have been included in the model (initial loading, unloading, reloading, etc.). Figure 5 illustrates the various stages that the model can exhibit during an impact event. While the barge is in contact with the pier, the crush is computed as p b bu u a max, where bu and pu are the barge and pier displacement, respectively. Figure 5a represents the virgin loading stage of the barge (bya is the yield crush deformation). Once ba exceeds bya, the barge will continue to load plastically, causing permanent inelastic deformation.

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19 0 1 2 3 4 5 6 0 100 200 300 400 500 600 700 800 900 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 35 Force (MN) Force (kip)Barge crush depth (mm) Barge crush depth (in) Static crush data for pier-1 Static crush data for pier-3 a) 0 1 2 3 4 5 6 0 100 200 300 400 500 600 700 800 900 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 35 Force (MN) Force (kip)Barge crush depth (mm) Barge crush depth (in) Smoothed crush data for pier-1 Smoothed crush data for pier-3 b) Figure 4. Comparison of static and smoothed crush curves for pier-1 & pier-3. a) Static crush curves; b) Smoothed crush curves

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20 Unloading (Figure 5b) occurs at the specified slope (ur K ). As unloading progresses, the barge may eventually cease to contact the pier. At this stage, 0bP and the barge retains a permanent plastic deformation bpa From this point in time forward, any computed ) (p b bu u a for which bbpaa represents a condition in which a finite size gap has formed between the barge and the pier face, and thus 0bP (Figure 5c). Closure of this gap must first occur before subsequent reloading (Figure 5d) along the slope,ur K can take place. P b a b a by a bmax a) K ur a b a by P b b) P b a b a bp K ur c) P b a b a bp K ur d) Figure 5. Stages of barge crush. a) Loading stage; b) Elastic unloading stage; c) Gap formed; d) Reloading stage

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21 5.2 Integration of Dynamic Barge Behavior Given the nonlinearities inherent in barge crushing, integration of the barge equation of motion is only practical using numerical methods. In this study, viscous damping effects associated with the water surrounding the barge are neglected (1 HC ). Energy dissipation (damping) in the barge model occurs only through plastic crushing. Examining Figure 3, the equation of motion is then written as bbbmuP (5) where bm is the barge mass, bu is the barge acceleration, and b P is the contact force acting between the barge and pier. Assuming that the mass of the barge remains constant, then evaluating Eq. 5 at time t we have tt bbbmuP (6) where t bu and t b P are the barge acceleration and force at time t The acceleration of the barge at time t can be estimated using the central difference equation : ) 2 ( 12 b h t b t b h t b tu u u h u (7) where h is the time step size; and th bu, t bu and th bu are the barge displacements at times th t and th respectively. Substituting Eq. 7 into Eq. 6 yields the explicit integration central difference method (CDM ) dynamic update equation 2(/)2thttth bbbbbuPhmuu (8)

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22 which uses data at times t and th to predict the displacement b h tu of the barge at time th Since the barge crush behavior is nonlinear, the force t b P is computed using the elastic-plastic loading and unloading model described in the previous section. Choice of the CDM for this application was based primarily on the relative ease with which nonlinear behavior can be taken into account. However, since the method is known to be conditionally stable, consideration had to be given to the choice of time step size (h) and the appropriateness of the method for this particular application. For CDM integration of SDOF systems, a reasonable choice of time step size is 10crhhT where T is the natural period of the system. If crh for the barge (integrated using the explicit central difference method) is much smaller than crh for the pier/soil system (integrated using the implicit Newmark’s method; described in the next section), then the single DOF barge would control time-stepping for the entire coupled barge/pier/soil system. If computational efficiency is a consideration, such a condition would clearly not be desirable. In fact, this condition does not occur and CDM is therefore, a suitable choice of integration technique. For the SDOF barge, 2bb bTmk, therefore conditions leading to the minimum crh would be minimum mass bm and maximum stiffness bk The minimum mass of a barge is its mass when empty (no payload). The maximum barge stiffness (slope of the crush curve) occurs during the initial loading stage (before plastic deformation softens the system and reduces bk ; see Figure 4). Considering these conditions, the computed value of crh for the barge is found to be

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23 10crbhT 0.01 sec. Prior experiences in conducting dynamic pier analyses using the FB-Pier code have consistently shown that use of time steps larger than 0.01 sec. often leads to convergence failure. Thus, the CDM represents a suitable and efficient choice for the application of barge impact analysis. 5.3 Coupling Between the Barge and Pier Implementation of the medium resolution dynamic analysis technique involved adding a barge-dynamics module to the FB-Pier program [8] (hereafter referred to as the “pier analysis program” or the “pier/soil module” depending on the context) and then making appropriate modifications so as to couple the barge and pier together. As Figure 3 suggests, this coupling has been accomplished through the use of a common impact force b P that acts on both the barge and the pier (at the point of contact between the two). Pier columns in the pier program are modeled using materially nonlinear frame elements (based on a cross-sectional fiber model). As such, the barge impact force b P is applied to the pier as a time varying nodal force. Conceptually, the overall coupled barge/pier/soil system (i.e., the medium resolution analysis model) can be thought of as two separate modules : a pier/soil control module (Figure 6), and a barge module (Figure 7). The dynamic time integration process is primarily controlled by the pier/soil module with the barge module determining the magnitude of the impact force b P based on the dynamic barge and pier motions, and based on the elastic-plastic barge crush model. Coupling is achieved through the insertion of three minimally intrusive links into the pier/soil module (Figure 6).

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24 form stiffness, mass, damping,, KMC form effective stiffnessˆ []([],[],[]) KfnKMC Initialize pier/soil module For each time step i = 1, ... For each iteration j = 1, ... Check for convergence of pier and soil ... Form external load vector {} F000{},{},{}0 uuu 0 t 1ˆˆ {}[]{} uKF {}{}{}thtuuu[]({})thKfnuˆ []([],[],[]) KfnKMC {}({})thRfnu ˆ {}{}{}([],[]) FFRfnMC 1ˆˆ {}[]{} uKF{}{}{} uuu no no ˆ max({||}) FTOL max({||}) uTOL yes yes Check for convergence of coupled barge/pier/soil system by checking convergence of barge force predictions. ( see barge module for details) {}{}bFFP {}{}bFFP bPTOL no mode = CONV updated is returned yes Form new external load vector with updated .... bP Record converged pier, soil, barge data and advance to next time step form stiffness, mass, damping,, KMC {}0 R initialize internal force vector ... insert barge force into external load vectorbF ˆ {}{}{}([],[]) FFRfnMC ... form effective force vector p u Extract displacement of pier at impact point from Barge module returns computed barge impact forcebP computed is returnedbP mode = CALC; is sent p u Compute barge impact force based on displacements of the pier and barge ( see barge module for details) p ubPbu mode = INIT Instruct barge module to perform initialization Initialize state of barge ( see barge module for details) Pier/soil module Barge module{}thu 4 3 2 1 Barge module Barge module Time for which a solution is sought is denoted as t+hbu converged are returned and bP ... update estimate of displacements at time t+hbP Figure 6. Flow-chart for nonlinear dynamic pier/soil control module

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25 0000 max0bbbpbuaaa 0()th bbuuh ()()0bbPP 0 intialize internal barge module cycle counter For each iteration k = 1, ... () 2()2l thtth bbbbbuPhmuu ()ththth bbpauu() max(,,)thtt bbbpbPfnaaa() 2(/)2thtth bbbbbuPhmuu no bPTOL ()(1) bbbPPP Return current barge force to pier module. () bP()()(1) bbbPPP If mode=CONV, then the pier/soil module has converged. Now, determine if the overall coupled barge/pier/soil system has converged by examing the difference between barge forces for current cycle and previous cycle ... () bPTOL yestht bbuutth bbuumax,,ttt bbpbaaa ... and ... Coupled barge/pier/soil system has converged, thus th bu is now a converged value. Update displacement data for next time step : save converged barge crush parameters Return to pier/soil module for next time step.()()(1)1bbbPPP()(1)() bbbPPP Return barge force for current cycle to pier module() bP Convergence not achieved. ... compute incremental barge force using a weighted average ... compute updated barge force1 increment internal cycle counter mode=INIT no yes Return to pier/soil module mode=CALC yes mode=CONV yes no yes no 2 4 1 3 Barge module Figure 7. Flow-chart for nonlinear dynamic barge module

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26 Each link instructs the barge module to perform a particular task : mode=INIT : Perform initialization of the barge module mode=CALC : Calculate an initial estimate of barge force at the start of a time step mode=CONV : Determine if convergence of the overall coupled system has occurred In this study, implicit direct time step integration in the pier/soil system was accomplished using Newmark’s method [20]. As Figure 6 indicates, the overall flow of the process involves an outer loop that controls time stepping, and an inner loop that controls iteration to convergence (satisfaction of dynamic equilibrium at each time step). For brevity, only the key aspects of the algorithm that are relevant to the discussion here are presented in the flowchart; for complete details the reader is referred to Ref. [20]. At the beginning of each time step, the barge module is invoked (mode=CALC) to calculate an estimation of impact force b P for that time step. Determination of b P first requires that the crush be computed as max,bbpbpauua In performing this computation, the pier/soil module has supplied the most up-to-date estimation of the pier displacement p u An iterative variation on the CDM is then used (Figure 7) to satisfy the dynamic equation of motion for the barge. At each pass through this iteration loop, the barge impact force estimation is refined until convergence is achieved and the calculated b P value is returned to the pier/soil module. After assembling the b P value into the appropriate location in the pier/soil external load vector (which may contain other loads such as gravity), the pier/soil module iterates until it has reached dynamic equilibrium (i.e., convergence). During this process, the value of b P that has been merged into the load vector is not altered (attempting to

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27 update the barge force within each pier/soil equilibrium iteration results in unreliable convergence behavior). Once the pier/soil module has converged, the displacement of the pier at the impact point ( p u ) is extracted from the pier/soil displacement vector {} u. While the pier/soil system has converged at this stage, the overall coupled barge/pier/soil system may still not have converged. Determination as to overall convergence is accomplished by once again invoking the barge module (now as mode=CONV). Using the newly updated p u value provided by the pier/soil module, the barge module once again carries out iterative CDM integration on the SDOF barge. It is very important to note that each time the barge module is invoked, time integration is performed using displacement data (t bu and th bu) and barge crush data (t bpa and maxth ba) that correspond to previous points in time at which the entire coupled barge/pier/soil system achieved convergence. Once a new estimation of the barge force has been computed, the difference in value between the current invocation (“cycle ”) and the previous invocation (“cycle 1 ”) is computed (see the calculation ()()(1)bbbPPP in Figure 7). If ()b P is sufficiently small, then not only has the pier/soil system converged, but the prediction of barge force for this time step has also converged and therefore the entire coupled system is in dynamic equilibrium. Thus, if ()b P TOL is satisfied, the barge module instructs the pier/soil module to advance to the next time step. If instead, () b P TOL then an updated estimate of barge force b P must be computed and returned to the pier/soil module. While the simplest choice would be to

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28 return the value of ()b P just used in the ()()(1)bbbPPP calculation, this is in fact a rather poor choice. Because the pier/soil module and the barge module each iterate to convergence independently in an alternating “back-and-forth” fashion (which has been found to be necessary in order to ensure robust convergence to dynamic equilibrium), the barge forces ()b P predicted during sequential barge module invocations (i.e., cycles = 0,1,2, etc.) tend to oscillate as the two systems seek to achieve coupled dynamic equilibrium. This oscillation in ()b P values can result in slow coupled convergence and typically involves sequentially computed values of ()b P that alternate in sign. To diminish these oscillations and accelerate convergence, a damping (relaxation) technique was implemented in the barge module. Rather than returning the raw computed ()b P value from the iterative CDM process to the pier/soil module, an exponentially decaying historical averaging process is used to compute a damped increment of barge impact force : c b b b a bP P P) 1 ( ) () 1 ( (9) In this expression, part a is the damped force increment; part b is the “raw” force increment computed for the current cycle; part c is the damped force increment from the previous cycle; and is a factor that determines the relative weighting of current versus previous data in determining the damped incremental change (0 indicates that previous data should be disregarded altogether). Note that by the recursive nature of this process, part c contains not only data for the previous cycle but also data for all

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29 previous cycles. The influence of older cycles diminishes in an exponentially decaying fashion that is controlled by the choice of With the damped increment of force ()b P determined (using Eq. 9), the actual barge force is computed by the barge module (Figure 7) as : ) ( ) 1 ( ) ( b b bP P P (10) and returned to the pier module (the force is denoted simply as b P in Figure 6 since the cycle number concept is local to the barge module). After assembling the new b P value into a clean copy of the external load vector {} F to form {} F the pier/soil module resumes the process of iterating toward convergence using the newly formed load vector. Advancement to a new time step only occurs when the overall coupled barge/pier/soil system has converged. While the barge module described here has been implemented specifically within the FB-Pier dynamic pier analysis program, the techniques presented are sufficiently general that they can be implemented within most nonlinear dynamic finite element structural analysis codes. Results obtained using this medium resolution dynamic analysis technique are presented later in this thesis.

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30 CHAPTER 6 LOW RESOLUTION BARGE IMPACT ANALYSIS 6.1 Description In the previous chapter, the goal was to demonstrate a methodology by which a nonlinear dynamic pier model could be coupled to a very simple barge model to enable prediction of impact loads and structural responses during barge collisions. Given sufficient nonlinear sophistication in the pier model, localized structural failures (e.g., plastic hinging of columns or piles) could be taken into consideration with the medium resolution analysis technique. In this chapter the goal is instead to present a very low order dynamic analysis technique capable of generating approximate time histories of dynamic impact load. In theory, pier response to such time histories of impact load could then be analyzed using a separate dynamic structural analysis package of the designer’s choice. A key assumption in this technique, however, is that the both the pier and the barge can be adequately represented using simple nonlinear SDOF models (Figure 1c). Individual point masses are used to represent the barge ( mb) and the pier ( mp). A nonlinear spring/gap element identical to that used in the medium resolution analysis is used to represent barge behavior. In regard to the pier, the total structure mass is concentrated at the theoretical barge impact point and a nonlinear spring (anchored to the reference frame of the dynamic system) is used to represent combined structure/soil resistance to the imparted impact load.

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31 Clearly, there are limitations to the accuracy of such a simplified analytical model. If the vertical distribution of mass in the pier is such that the location of the actual center of mass is significantly different than that assumed in the SDOF model, then the ability of the two-DOF model to represent dynamic barge-pier interaction will be limited. Furthermore, with only a single pier DOF, modes of vibration at frequencies higher than the fundamental sway mode will not be included in the dynamic analysis. Similarly, certain types of localized failures (e.g., hinging of a pier column) cannot be represented because such behavior would necessitate the inclusion of response modes other than the fundamental sway mode. However, by using a nonlinear load-deformation relationship (described below) to represent combined structure/soil resistance to applied loading, some types of plastic deformation can be reasonably included in the model. For example, plastic hinging of a pile, while constituting a localized form of failure, will tend to result in an overall softening of the structure in the sway mode and thus can be approximately represented even with a SDOF model. Given such capabilities and given the minimal computational resources needed to perform the two-DOF analysis, evaluation of this simplified technique is justified. 6.2 Implementation Implementation of the two-DOF dynamic analysis technique has been accomplished through the development of a Mathcad [21] based program that implements the central difference method [22] of direct time step integration (a copy of the complete program is included in Appendix B of this thesis). Data obtained from high resolution static barge crush analyses (described earlier) are used to represent the nonlinear response of the barge in the two-DOF model. Similarly, the resistance of the

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32 pier/soil system to applied lateral loading is represented using a nonlinear loading curve and a linear unloading curve. The loading curve is generated by conducting a static nonlinear lateral load analysis of the pier using any pier analysis tool available to the engineer (FB-Pier, LS-DYNA, etc.). In this thesis, pushover data for pier-1 and pier-3 were obtained from high resolution analyses conducted on combined pier/soil models using LS-DYNA. The data obtained were then validated against analyses conducted on the same piers using FB-Pier. Unlike the SDOF barge model, the SDOF pier/soil model does not utilize a gap element because unloading of the pier does not necessarily result in residual deformation (i.e., the presence of a “gap”). Consider the single pile/soil model depicted in Figure 8. Loading path 1-2 on the load-displacement curve shown in Figure 8a represents the soil/pier loading curve obtained from a pushover analysis. Path 2-3 represents an unloading path for a condition in which plastic deformation has not occurred in the pier structure (e.g., in the piles) but for which some soil springs have sustained plastic deformation. Upon removal of the external load, elastic strain energy (stored primarily in the springs that have remained elastic during the loading) restore the pile to its initial position and gaps form in the soil springs that have sustained plastic deformation. Path 24 describes an alternate unloading path from a condition in which both the soil springs and the structure itself have sustained permanent plastic deformation. In this case, despite the restorative capacity of the elastic soil springs, a residual displacement (4 ) results from the permanent structural deformation.

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33 +P + -P 2 1 3 2 4 4 Virgin load curve from static pushover analysis Approximate unloading/re-loading path a) P 2 1 2 3 4 4 Formation of soil gap Residual displacement due to permanent structural deformation b) Figure 8. Depiction of pier/soil loading and unloading behavior. a) Load-displacement curve from pushover including various unloading paths; b) Single pile and soil model describing points on load-displacement curve

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34 In the barge impact cases simulated in this thesis, plastic soil deformation is considered in the soil response but pier-structure response is set to remain in the elastic range. For such cases, residual structural deformations will not occur and unloading will always occur along a path similar to path 2-3 of Figure 8. In the low resolution analysis technique implemented here, path 2-3 has been simplified and approximated by a simple linear secant line extending between the point of maximum sustained deformation and the origin (Figure 8). Unloading and re-loading of the combined pier/soil system occur along this secant line. If, after unloading, substantial reloading occurs, the pier/soil system will reload along the current secant stiffness until the previously sustained maximum displacement state is reached. At this point, the system will once again begin to load along the virgin loading curve. Subsequent unloading would occur along an updated secant line extending from the origin to the new point of maximum sustained deformation. The same behavior is true if the structure travels in the reverse direction. The pier/soil behavior model described above is assumed to hold true in both the positive and negative directions. However, separate secant unloading lines are maintained in the positive and negative directions using the maximum pier/soil displacements sustained in each of these directions. While a more sophisticated model of unloading and reloading could certainly be implemented (e.g., using a nonlinear unloading curve) the simplified secant model was deemed reasonable for initial evaluation of the two-DOF analysis technique.

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35 CHAPTER 7 DISCUSSION OF RESULTS Using the three levels of dynamic analyses described in the previous chapters, barge impact simulations were conducted to evaluate the ability of the medium and low resolution models to predict dynamic barge-pier-soil interaction. A variety of cases were simulated for both pier-1 and pier-3 of the St. George Island Causeway Bridge in order to cover a representative range of realistic single barge impact conditions. All of the simulations consisted of jumbo hopper barges (of varying payloads and initial velocities) impacting one of the pier models. In each case, the hydrodynamic mass coefficient has been set to a unit value (1 H C ) to simplify comparisons to AASHTO predicted loads. Table 1 lists the six cases presented herein along with their corresponding impact parameters. It must be noted before presenting impact simulation results that, prior to conducting the barge impact simulations, the high and medium resolution analysis techniques were cross-validated against each other using static lateral load analyses and a dynamic triangular pulse load analysis. Results from these cases (shown in Appendix C) confirmed that for controlled static and dynamic loading conditions, the two models predicted very similar responses. Any differences in predicted barge impact data are thus related primarily to barge-pier interaction effects.

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36 Table 1. Barge impact simulation cases. Case Pier Model Initial velocity Payload condition Kinetic energy A 2.06 m/s (4-knots) Fully loaded 16.90 MN (1900 tons) 3.83 MJ (2825 kip-ft) B 1.03 m/s (2-knots) Half loaded 9.34 MN (1050 tons) 0.53 MJ (390 kip-ft) C Pier-1 2.06 m/s (4-knots) Empty 1.80 MN (200 tons) 0.40 MJ (297 kip-ft) D 0.51 m/s (1-knot) Half loaded 9.34 MN (1050 tons) 0.13 MJ (98 kip-ft) E 0.26 m/s (0.5-knots) Half loaded 9.34 MN (1050 tons) 0.03 MJ (24 kip-ft) F Pier-3 2.06 m/s (4-knots) Empty 1.80 MN (200 tons) 0.40 MJ (297 kip-ft) 77.1 Comparison of Dynamic Simulation Results Case A, a 4-knot fully loaded barge impact on pier-1, is the most severe of all the impact cases considered here. Simulation results computed for this case using each of the three dynamic impact analysis techniques, are shown in Figure 9. The dynamic impact forces, shown in Figure 9a, all achieve approximately the same peak value and exhibit substantial load for a sustained duration of time (equal to or greater than the period of vibration, approximately T= 0.73 seconds). In order to compare the level of structural response predicted by each analysis, the time varying pier displacements measured at the point of barge impact, predicted by each method are compared in Figure 9b. Both the peak displacements and the period of vibration predicted by all three methods compare well. Barge force-deformation results (shown in Figure 9c) indicate nearly identical predictions of maximum sustained dynamic barge crush. In addition, the energy dissipated through plastic deformation of the barge bow (approximately equal to the area under the force-deformation curve) is also nearly identical in all three cases.

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37 0 1 2 3 4 5 6 00.250.50.7511.251.51.752 0 200 400 600 800 1000 1200Impact force (MN) Impact force (kip)Time (sec) High resolution analysis Medium resolution analysis Low resolution analysis a) -25 0 25 50 75 100 125 150 175 00.250.50.7511.251.51.75 2 0 1 2 3 4 5 6 Pier displacement (mm) Pier displacement (in)Time (sec) High resolution analysis Medium resolution analysis Low resolution analysis b) Figure 9. Results for Case A : 4-knot fully loaded impact on pier-1. a) Time history of impact force; b) Time history of pier displacement; c) Barge force-deformation

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38 0 1 2 3 4 5 6 0200400600800 1000 1200 0 200 400 600 800 1000 1200 010203040 Impact force (MN) Impact force (kip)Barge crush depth (mm) Barge crush depth (in) High resolution analysis Medium resolution analysis Low resolution analysis c) Figure 9. Continued Whereas Case A is representative of a high energy impact condition, Case B and C represent less severe impact conditions. The intent in conducting the simulations for Case B and C is to evaluate the abilities of the various models to predict less-sustained, more transient loading conditions. Predicted force histories for Case B and C are plotted in Figure 10. Case B, a 2-knot half loaded impact condition (Figure 10a), produces an oscillatory loading history, while case C, a 4-knot empty condition (Figure 10b), has an even more transient short-term force history. For both of these cases, all three analyses peak at approximately the same time and load level and exhibit time-variation of loading, indicating good agreement in terms of predicted dynamic interaction between the barge and the pier.

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39 High resolution analysis Medium resolution analysis Low resolution analysis 0 1 2 3 4 00.250.50.7511.251.5 1.75 2 0 200 400 600 800 Impact force (MN) Impact force (kip)Time (sec) a) 0 1 2 3 4 0 100 200 300 400 500 600 700 800 900 Impact force (MN) Impact force (kip) High resolution analysis Medium resolution analysis Low resolution analysis 00.250.50.7511.251.5 1.752 Time (sec) b) Figure 10. Time histories of impact force on pier-1. a) Case B : a 2-knot half loaded barge impact; b) Case C : a 4-knot empty loaded barge impact

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40 Case D, E, and F correspond to impacts on pier-3, the less impact resistant pier. Due to the increased flexibility of pier-3, substantial dynamic interaction occurs between the barge and the pier and, as a result, the dynamic loads imparted to the structure are highly oscillatory in nature. In Figure 11a, predicted impact loads for Case D are presented. All three simulations agree with respect to the initial peak load predicted and the subsequent decrease to zero load (at approximately t= 0.11 seconds). Following this point, the predicted impact loads differ but all exhibit a similar highly oscillatory characteristic. However, of equal importance to the nature of the dynamic load is the structural response of the pier to that applied loading. Despite the differences in the predicted dynamic loading, the pier displacements predicted by each analysis technique (shown in Figure 11b) are in reasonable correlation. Thus, at least based on this measure of structural response, the structural severities of impact predicted by all three methods tend to be in agreement. Results obtained for Cases E and F exhibited similar characteristics to those shown for Case D. Detailed simulation results for Cases A through F are shown in Appendix D. In Cases D, E, and F, very little barge deformation occurred during impact Due to the flexible nature of pier-3. As a result, negligible enery dissipation occurred through barge crushing for these cases. Results for Cases A through F are summarized in Table 2 where peak (maximum magnitude) dynamic forces, pier displacements, and barge crush deformations are reported. For the six cases listed, the medium and low resolution analysis techniques predict peak data that are in sufficient agreement with the values predicted by the high resolution analysis.

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41 0 0.5 1 1.5 2 2.5 3 00.250.5 0.75 1 0 100 200 300 400 500 600 Impact force (MN) Impact force (kip)Time (sec) High resolution analysis Medium resolution analysis Low resolution analysis a) High resolution analysis Medium resolution analysis Low resolution analysis -50 -25 0 25 50 75 100 125 00.250.50.7511.251.51.752 -1 0 1 2 3 4 Pier displacement (mm) Pier displacement (in)Time (sec) b) Figure 11. Results for Case D : 1-knot half loaded impact on pier-3. a) Time history of impact force; b) Time history of pier displacement

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42 Table 2. Peak impact loads, pier displacements, and barge crushes High resolution analysis Medium resolution analysis Low resolution analysis Case Peak impact load MN (kip) Peak pier disp. mm (in) Peak barge crush mm (in) Peak impact load MN (kip) Peak pier disp. mm (in) Peak barge crush mm (in) Peak impact load MN (kip) Peak pier disp. mm (in) Peak barge crush mm (in) A 4.67 (1050) 143.04 (5.64) 1024.75 (40.34) 5.12 (1150) 156.46 (6.16) 929.64 (36.60) 5.08 (1142) 154.05 (6.07) 1015.21 (39.97) B 3.38 (761) 84.31 (3.32) 101.28 (3.99) 3.75 (844) 91.70 (3.61) 72.64 (2.86) 3.50 (786) 92.28 (3.64) 80.52 (3.17) C 3.71 (824) 51.54 (2.03) 106.42 (4.19) 3.74 (841) 49.28 (1.94) 84.58 (3.33) 3.54 (796) 50.32 (1.98) 94.82 (3.73) D 2.37 (534) 109.26 (4.30) 15.13 (0.60) 2.37 (532) 101.35 (4.00) 11.28 (0.44) 2.53 (570) 98.22 (3.87) 20.77 (0.82) E 1.56 (352) 41.60 (1.64) 4.90 (0.19) 1.65 (372) 47.75 (1.88) 5.49 (0.22) 2.20 (494) 49.07 (1.93) 9.26 (0.36) F 3.07 (690) 170.98 (6.74) 65.62 (2.58) 2.85 (640) 136.65 (5.38) 66.55 (2.62) 2.78 (625) 132.00 (5.20) 95.43 (3.76) 7.2 Comparison of Dynamic and Equivalent Static Impact Loads Based on the comparisons presented above, it may be concluded that designoriented dynamic analysis techniques offer the promise of alternative approaches to designing bridge piers for barge impact conditions. While additional research and validation efforts are needed before dynamic analysis procedures can serve in lieu of code-based load determination methodologies (e.g., the AASHTO provisions), in concept, at least, such approaches seem viable and even desirable. In this section, impact loads and structural response parameters computed using dynamic analysis techniques are compared to data predicted by implementing the AASHTO equivalent static load computation approach. For purposes of comparing dynamic and equivalent static methods, it is preferable to use the most accurate dynamic data available. In this study, the high resolution dynamic analysis technique discussed

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43 earlier yields the most accurate data (although similar results have been obtained from medium and low resolution analyses as was demonstrated above). Results for a total of seven high resolution simulations are considered : Cases A-F previously cited plus an additional intermediate impact-energy condition (Case G : pier-1, 4-knots, half loaded). The first comparison presented here focuses on predicted impact loads. In Figure 12, peak dynamic loads from high resolution analyses are compared to equivalent static loads computed using the AASHTO provisions (Eqs. 1, 3, 4). It is important to note that the AASHTO equivalent static loads are unfactored in the sense that they are associated with only the barge event, not with the probability that the event will occur. Results shown in the figure indicate that for the higher energy impact conditions, AASHTO predicts loads that are greater than the peak dynamic loads predicted by the simulations. 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 200 400 600 800 1000 1200 1400 1600 1800 0 1000 2000 3000 Impact force (MN) Impact force (kip)Kinetic energy (MN-m) Kinetic energy (kip-ft) AASHTO Relationship Pier-3, 1 knots, half loaded Pier-3, 0.5 knots, half loaded Pier-1, 4 knots, empty A B C D E F G Pier-3, 4 knots, empty Pier-1, 2 knots, half loaded Pier-1, 4 knots, half loaded Pier-1, 4 knots, fully loaded D C E F B A G Figure 12. Relationship between barge impact force and kinetic energy

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44 For lower energy impact conditions, Figure 12 indicates peak dynamically predicted loads that are greater than those predicted by AASHTO. However, these loads tend to be transient or highly oscillatory in nature, as the time histories of impact force presented in the previous section revealed. Therefore, comparison to equivalent static loads is unsuitable. Instead, an alternate approach is required in which structural responses to impact loading (dynamic or static) are compared. Since pile forces (moments, shears, etc.) are closely linked to lateral pier displacements, comparing maximum sustained pier displacements provides a means by which static and dynamic analysis predictions can be duly compared. To enable such a comparison, maximum sustained pier displacements were determined for each impact case from the high resolution dynamic analyses. Next, a large number of hypothetical impact conditions were chosen with a distribution of impact energies that covered the range indicated in Figure 12. For each hypothetical impact condition, an AASHTO equivalent static load was computed using Eqs. 3 and 4. These loads were then applied in a static sense to each of the high resolution pier models so that the resulting static pier displacements could be determined. Since the high resolution models are dynamic models intended for analysis using LS-DYNA, the static load application was achieved by applying the loads slowly enough so as to eliminate all inertial (dynamic) effects. To cross-check the validity of the results obtained by this procedure, pier displacement data were also computed by conducting true static analyses on the medium resolution pier models (using FB-Pier in static analysis mode). Results obtained from the pseudo-static analyses and the true static analyses were virtually identical.

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45 Maximum sustained pier displacement data for impact conditions on the relatively stiff pier-1 are shown in Figure 13a. In general, data predicted by both the dynamic and static analysis procedures agree well. In all cases, the AASHTO equivalent static loads produce conservative predictions of structural response. In Figure 13b, results for several impact conditions on the more flexible pier-3 are presented. Several discrepancies between the dynamic and static analysis results are noted. For the two lowest energy impact conditions (cases D and E), dynamic analysis predicts pier displacements significantly in excess of those predicted by application of the AASHTO static loads. Conversely, the static AASHTO load application results in very conservative predictions of pier response for the higher energy impact conditions (e.g., case F). Figure 13. Relationship between pier displacement and kinetic energy. a) Pier-1 results; b) Pier-3 results 0 25 50 75 100 125 150 175 00.51 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 0100020003000 Pier displacement (mm) Pier displacement (in)Kinetic energy (MJ) Kinetic energy (kip-ft) C B G Displacement produced by application of AASHTO equivalent static load Pier-1, 4-knots, fully loaded Pier-1, 2-knots, half loaded Pier-1, 4-knots, empty Pier-1, 4-knots, half loaded A B C G A a)

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46 0 100 200 300 400 500 600 0 0.075 0.15 0.225 0.3 0.375 0.45 0 5 10 15 20 0 100 200 300 Pier displacement (mm) Pier displacement (in)Kinetic energy (MJ) Kinetic energy (kip-ft) Pier-3, 1-knot, half loaded Pier-3, 0.5-knots, half loaded Pier-3, 4-knots, empty D E F Displacement produced by application of AASHTO equivalent static load E F D b) Figure 13. Continued The variable nature of the discrepancies observed in the pier-3 comparisons derives from the presence of significant dynamic interaction between the barge and pier and the resulting oscillatory nature of the imparted impact loads. In such circumstances, equivalent static loads are not able to accurately predict the severity of the structural response and the use of rational dynamic analysis techniques is recommended.

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47 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS Three levels of dynamic analysis techniques have been presented as alternatives to the current AASHTO equivalent static barge impact load computation procedures. A high resolution analysis technique utilizing a nonlinear finite element impact simulation has been presented as a means of achieving accurate prediction of barge and pier response during barge collision events. However, this technique is generally not appropriate for routine use in bridge design due to the substantial computational resources needed to conduct such simulations. A medium resolution analysis technique has been developed by modifying a commercially available dynamic pier analysis program, while a very low resolution analysis technique has been implemented as a spreadsheet program. Both of these techniques utilize barge load-deformation relationships that have been obtained from high resolution nonlinear contact finite element simulations. Barge impact simulations have been presented for six different impact conditions on models of an impact resistant pier and a non-impact resistant pier. The accuracy of the medium and low resolution analysis techniques has been evaluated by comparing results produced by these procedures to results obtained from high resolution analyses. Results presented in the comparisons reveal that the medium and low resolution models are capable of simulating dynamic barge-pier interaction during a collision event. In addition, both methods have been shown to produce impact force time histories that are

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48 sufficiently accurate for bridge design purposes. In the future, improvements may be made to the low resolution pier/soil model and to the single DOF barge model to improve their representation of unloading behavior. Comparisons have also been made between dynamic and static load and displacement prediction methods. Impact force and pier displacement data have been computed using the AASHTO barge impact provisions and compared to dynamically predicted data for several different impact scenarios. These comparisons demonstrate that a static load approach is acceptable—although slightly conservative—in cases where the impact event produces sustained barge-pier interaction. The comparisons also demonstrate that there are circumstances in which a dynamic analysis should be strongly considered as a design alternative.

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49 APPENDIX A SMOOTH HIGH RESOLUTION CRUSH DATA In this appendix, a Mathcad program is presented that documents the technique used to smooth the high resolution crush data. The smoothed crush data is then applied to the medium resolution model to represent the nonlinear behavior of the barge. The specific example presented here is a 6 ft. square pier statically crushing the bow of a hopper barge.

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50 Smoothed Crush DataORIGIN0 ORIGIN 6 ft. Square Crush Test Data LS-DYNA crushData 01 0 1 2 3 4 5 6 7 8 9 00 0.11264.2 0.17437.34 0.32564.9 0.41628.18 0.44691.02 0.53748.34 0.61796.52 0.68823.41 0.77828.45 ...import 6ft_crush_fast_static.txt ccrushData ...deflection (in) PccrushData 1 ...load (kips) c .max max c c .max 34.226 (in) nlast c 05101520253035 0 500 1000 1500 LS-DYNA 6ft Square Crush Test DataDeflection (in)Load (kips)

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51 Smoothed Crush Data bw1.2 bw22 PcSmooth1ksmooth cPc bw1 PcSmooth2ksmooth cPc bw2 05101520253035 0 500 1000 1500 LS-DYNA 6ft Square Crush Test DataDeflection (in)Load (kips)

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52 Simplify number of data points : simpCrushncSimp18 ncSimp242 .critical .571 "set initial starting point to 0" cSimple0 PcSimple0 cSimplei .critical ncSimp11 i PcSimpleilinterp cPcSmooth1 cSimplei i 1 ncSimp11 for cSimplei c .max .critical ncSimp2 () i1 ncSimp1 () .critical PcSimpleilinterp cPcSmooth2 cSimplei incSimp1 ncSimp2ncSimp1 1 for Eur_x0 cSimplencSimp11 T Eur_y0PcSimplencSimp11 T EurEur_y 1 Eur_x 1 cSimplePcSimpleEurEur_xEur_yreturn ...number of desired crush data points cannot exceed 200 for FB-Pier cSimplePcSimpleEurEur_xEur_ysimpCrush 05101520253035 0 500 1000 1500 Simplified Crush DataDeflection (in)Load (kips)

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53 EurceilEur () Eur1.34103 ...(kips/in) approximate unloading slope abyEur_x1() aby0.571 ...(in) approximate yield point (pick from plot and output) *Note: Make sure unloading curve does not have a negative or 0 x-intercept FBPier_crushaugment cSimplePcSimple ...export into "xport_smth_6ft_sq_crush.txt" and then copy to FB-Pier FBPier_crush 01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 00 0.082232.88 0.163357.686 0.245478.629 0.326593.256 0.408652.247 0.489708.631 0.571764.615 1.372792.731 2.174806.775 2.975801.661 3.776811.33 4.578825.896 5.379844.532 6.18873.079 6.981910.025

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54 APPENDIX B LOW RESOLUTION ANALYSIS PROGRAM In this appendix, a Mathcad program is presented that documents the implementation of the low-order analysis technique described earlier in this thesis. The program was co-developed with Mr. David Cowan and his contributions are gratefully acknowledged. The specific example presented here is for impact Case A, the 4-knot fully loaded barge impact on the impact resistant pier, pier-1.

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55 Ps p ier 1 ki p s Pc b ar g e 1 ki p s spier in cbarge in Soil Spring : Contact Spring : pier 01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 00 0.0247.5 0.0572.5 0.1197.5 0.19122.5 0.26147.5 0.32172.5 0.37197.5 0.41222.5 0.44247.5 0.47272.5 0.5297.5 0.55322.5 0.62347.5 0.69372.5 0.78397.5 barge 01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 00 0.08232.88 0.16357.69 0.24478.63 0.33593.26 0.41652.25 0.49708.63 0.57764.62 1.37792.73 2.17806.77 2.97801.66 3.78811.33 4.58825.9 5.38844.53 6.18873.08 6.98910.02 Import pier_pushover data (load-displacement) Import barge_crush data (load-deformation) : ...barge crush plastic unloading slope Eur1340kips in Load-Displacement Data... ...applied load (not time dependent) F 0 0 kips Force... Input Parameters : rad1 knot1.687809857ft s tons2000lbf kips1000lbf ORIGIN ORIGIN0 Define Units : Pier-1 : 4-knot, fully loadedLow Resolution Barge-Pier Interaction Model

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56 Load-displacement data plots... 051015202530354045 0 200 400 600 800 1000 1200 Barge CrushCrush (in)Load (kips) 0123456789 0 500 1000 1500 2000 Soil SpringDisplacement (in)Load (kips) (used to define nonlinearity of pier/soil spring)

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57 L .pier 44.1146ft ...height of pier from top of pile cap to bottom of pier cap A .pileCap 822.5ft2 ...area of pile cap t .cap 5ft ...thickness of pile cap A .pierCap 31.354ft2 ...area of pier cap L .pierCap 29.83ft ...length of pier cap A .shr 60ft2 ...area of shear wall L .shr 16.9ft ...length of shear wall (between inside of columns) mp .c A .pileCap t .cap nPierA .pier L .pier A .pierCap L .pierCap A .shr L .shr .s nPileA .pile L .pile mp3.918 kipss2 in ...mass of the Pier Damping... C 0 0 0 0 kipss in ...system damping matrix Mass-Barge... dwt1900tons ...dead weight tonnage mbdwtg mb9.8423 kipss2 in ...mass of barge Mass-Pier... .c 2.251007 kipss2 in4 .s 7.3451007 kipss2 in4 ...mass density of concrete and steel nPile36 ...number of piles A .pile 21.09in2 ...area of HP14x73 L .pile 50.5ft ...length of pile nPier2 ...number of piers A .pier 66.16in 77.61 in ...average area of pier

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58 mp mp masu Pc Pc foru c c disu Ps Ps foru s s disu CC timu masu vovo timu disu Eur Eur stifu Build Matricies : Mass... M mb 0 0 mp ...system mass matrix Stiffness... kco Pc 1 c 1 kso Ps 1 s 1 ...(kip/in) initial barge crush stiffness and soil spring stiffness kcEurstifu ...(kip/in) barge crush unloading curve Ko kco kco kco kcokso ...initial system stiffness matrix Timing Considerations... t0.001s ...time step n2000 ...number of points Initial Conditions... vo4knot ...initial barge velocity vo81.0149ins Normalize Input : Normalizing Parameters... masu kipss2 in forukips timusec disuin stifukipsin Normalize Variables... t t timu F F foru mb mb masu

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59 ...barge x-displacement xpoutT 1 ...pier x-displacemen t vboutT 2 ...barge x-velocity vpoutT 3 ...pier x-velocity aboutT 4 ...barge x-acceleration apoutT 5 ...pier x-acceleration FboutT 6 ...barge x-force FpoutT 7 ...pier x-force Calculate barge crush... cxbxp () crushixbixpi xbixpi0 if crushi0 otherwise i rowsxb ()1 for crush return crushcxbxp () Time... t tii t i n1 for t Initial Conditions... xo 0 0 ...displacement vo vo 0 ...velocity Timing Considerations... genvalsKoM () 40.1299 11.401 ...natural frequency ...check time step size check t t t_maxmin 10 check"Time Step OK" t t_max if check"Error : Reduce Time Step" otherwise check return check t t"Time Step OK" Central Difference Method : outCDEurPc cPs sxovo tnMCF00 xboutT

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60 Plot Results : 00.20.40.60.811.21.41.61.82 0 20 40 Barge Displacement Pier Displacement Displacement HistoryTime (sec)Displacement (in) 00.20.40.60.811.21.41.61.82 0 200 400 600 800 1000 1200 Force HistoryTime (sec)Force (kips)

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61 0510152025303540 0 500 1000 Force-deformationCrush (in)Force (kips) 101234567 500 0 500 1000 1500 Soil SpringDisplacement (in)Force (kips)Fp xp Write Results : dataaugmenttFb crush xb xp () WRITEPRN"OUT_p1_4kn_full_2dof.prn" ()data

PAGE 68

62 CDM Central Difference Function :CDPccPssxovotnMCFgapcgaps*Pc and Dc, and Ps and Ds represent the load-deflection values for the contact and soil springs respectively *xo and vo are the initial displacement (typically 0) and initial velocity of the two masses *Dt is the time step and n is the number of time steps in the simulation *M and C are the mass and damping matricies *F is the force (static) applied to each mass *gapc and gaps are the sizes of the initial gaps (typically 0) for the contact and soil springs respectively CDargIn ()"Central Difference Algorithm" "Initial Stiffness" EurPc cPs sxovo tnMCFgapcgapsargIn ORIGIN kc Pc 1 c 1 ks Ps 1 s 1 K kc kc kc kcks "Initial Acceleration" ao 1 M FCvo Kxo () "Contact Spring" PcYPc 1 cY c 1 kcvPcY cY kcviPciPci1 ci ci1 i 1 lastPc () for

PAGE 69

63 "Soil Spring Positive Loading" gapsfgaps PsYfPs 1 sYf s 1 ksvfPsYf sYf ksvfiPsiPsi1 si si1 i 1 lastPs () for "Soil Spring Negative Loading" gapsbgaps PsYbPs 1 sYb s 1 ksvbPsYb sYb ksvbiPsiPsi1 si si1 i 1 lastPs () for "Displacement at Back Time Step" xpxovo t ao t2 2 "Effective Mass" m_ M t2 C 2 t "Displacement for the First Time Step" xj xoj j 1 for x_tempm_1 FK 2M t2 xo M t2 C 2 t xp xj 1 x_tempj j 1 for x_ppxo x_px_temp i 2 n for

PAGE 70

64 "Displacement at the ith Time Step" x_tempm_1 FK 2M t2 x_p M t2 C 2 t x_pp "Contact Spring" _cx_tempx_temp 1 kc0 K kc kc kc kcks _cgapc if P_ckcv _cgapc gapc0 if P_cEur _cgapc otherwise kc P_c _c K kc kc kc kcks _cgapc _c cY if PcYPcYkcvy _c cY cY _c kc PcY cY gapc cY PcY Eur K kc kc kc kcks break _c cy if y 1 last c for _c cY if

PAGE 71

65 "Soil Spring Positive Loading" _sx_temp 1 ks0 K kc kc kc kcks _sgapsf _s0 if P_sfksvf _sgapsf ks P_sf _s K kc kc kc kcks _sgapsf _s sYf if PsYfPsYfksvfy _s sYf sYf _s ks PsYf sYf ksvfks gapsf sYf PsYf ksvf K kc kc kc kcks break _s sy if y 1 last s for _s sYf if "Soil Spring Negative Loading" ks0 K kc kc kc kcks _sgapsb _s0 if P_sbksvb _sgapsb ks P_sb _s K kc kc kc kcks _sgapsb _s sYb if

PAGE 72

66 PsYbPsYbksvby _s sYb sYb _s ks PsYb sYb ksvbks gapsb sYb PsYb ksvb K kc kc kc kcks break _s sy if y 1 last s for _s sYb if "Derivatives" xji x_tempj xj2 i1 xji xji2 2 t xj4 i1 xji 2xji1 xji2 t2 j 1 for "Force" f_temp kcx1i1 x0i1 ksx1i1 xji1 f_tempj6 j67 for x_ppx_p x_px_temp outij xij j n1 for i 7 for out

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67 Formation of gap element : Initially: 0 gap Loop: if < gap 0 k if gap < < Y P k gap k PE The stiffness used in the elastic region cannot be the elastic stiffness due to the gap (Point 1). if Y < E k Y P Y gap Y Y P k Y Y P k Y P Y P The yield point and the gap are updated each time (basically the elastic region is shifted). The stiffness used in the plastic region cannot be the plastic stiffness due to the stiffness change from elastic to plastic (Point 2). Stiffness for Point 1 if Elastic Stiffness is used Gap updated Point 1 Point 2 Stiffness for Point 2 if Plastic Stiffness is used Initial Yield Point Updated Yield Point P

PAGE 74

68 APPENDIX C MEDIUM RESOLUTION VALIDATION CASES In this appendix, validation simulations are presented for the medium resolution analysis models discussed earlier in this thesis. Medium resolution analysis results for both pier-1 and pier-3 are validated against corresponding high resolution analysis results. Both a pseudo-static lateral load analysis and a triangular pulse load analysis are conducted to confirm that similar pier responses are predicted by the medium and high resolution analysis techniques.

PAGE 75

69 Validation Case: Static lateral load analysis Figure C-1. Pier-1 static lateral load analysis Figure C-2. Pier-3 static lateral load analysis

PAGE 76

70 Validation Case: Triangular pulse load analysis Figure C-3. Pier-1 pulse load analysis

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71 APPENDIX D COMPLETE DYNAMIC ANALYSIS RESULTS In this appendix, complete dynamic analysis results are presented for each the six simulation cases discussed briefly within the body of this thesis. For each included here, the time history of impact force, time history of pier displacement, and barge forcedeformation relationships are presented in full.

PAGE 78

72 Case : A Pier : 1 Velocity : 4-knot Payload : Fully loaded Figure D-1. Time history of impact force

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73 Figure D-2. Time history of pier displacement Figure D-3. Impact force vs. barge crush

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74 Case : B Pier : 1 Velocity : 2-knot Payload : Half loaded Figure D-4. Time history of impact force

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75 Figure D-514. Time history of pier displacement Figure D-6. Impact force vs. barge crush

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76 Case : C Pier : 1 Velocity : 4-knot Payload : Empty Figure D-7. Time history of impact force

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77 Figure D-8. Time history of pier displacement Figure D-9. Impact force vs. barge crush

PAGE 84

78 Case : D Pier : 3 Velocity : 1-knot Payload : Half loaded Figure D-10. Time history of impact force

PAGE 85

79 Figure D-11. Time history of pier displacement Figure D-12. Impact force vs. barge crush

PAGE 86

80 Case : E Pier : 3 Velocity : 0.5-knot Payload : Half loaded Figure D-13. Time history of impact force

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81 Figure D-14. Time history of pier displacement Figure D-15. Impact force vs. barge crush

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82 Case : F Pier : 3 Velocity : 4-knot Payload : Empty Figure D-16. Time history of impact force

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83 Figure D-17. Time history of pier displacement Figure D-18. Impact force vs. barge crush

PAGE 90

84 REFERENCES 1. Frandsen, A.G. and H. Langso. “Ship Collision Problems, Great Belt Bridge, International Enquiry.” IABSE Proceedings. Zurich, Switzerland: International Association of Bridge and Structural Engineering, 1980 : P31/80. 2. AASHTO. Guide Specification and Commentary for Vessel Collision Design of Highway Bridges Washington, DC : American Association of State Highway and Transportation Officials, 1991. 3. “Towboats and Bridges : A Dangerous Mix.” GCMA Report #R-293, Revision 2. 2002. Gulf Coast Mariners Association . 4. Whitney, M.W., Harik, I., Griffin, J., and Allen, D. “Barge Impact Loads for the Maysville Bridge.” Interim Research Report Kentucky Transportation Center, 1994 : KTC-94-6. 5. Consolazio, G.R., Lehr, G.B., and McVay, M.C. “Dynamic Finite Element Analysis of Vessel-Pier-Soil Interaction During Barge Impact Events.” Journal of the Transportation Research Board Washington, D.C., 2003 (In press). 6. Consolazio, G. R. and Cowan, D.R. “Nonlinear Analysis of Barge Crush Behavior and its Relationship to Impact Resistant Bridge Design.” Computers & Structures 81 :547-557 (2003). 7. Livermore Software Technology Corporation (LSTC). LS-DYNA Theoretical Manual Livermore, CA, 1998. 8. Florida Bridge Software Institute. FB-PIER Users Manual University of Florida, Department of Civil & Coastal Engineering, 2000. 9. Hoit, M.I., McVay, M., Hays, C., and Andrade, W. “Nonlinear Pile Foundation Analysis Using Florida-Pier,” ASCE Journal of Bridge Engineering, Volume 1 Number 4, pp 135-142, 1996. 10. AASHTO. LRFD Bridge Design Specifications and Commentary Washington, DC : American Association of State Highway and Transportation Officials, 1994. 11. Permanent International Association of Navigation Congresses (PIANC). Report of the International Commission for Improving the Design of Fender Systems Brussels, Belgium, 1984.

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85 12. Larsen, Ole D. “Ship Collision with Bridges : The Interaction between Vessel Traffic and Bridge Structures.” IABSE Structural Engineering Document 4. Zurich, Switzerland : International Association of Bridge and Structural Engineering, 1993. 13. Minorsky, V.U. “An Analysis of Ship Collisons with Reference to Protection of Nuclear Power Plants.” Journal of Ship Research 4.2 (1959) : 1-4. 14. Woisin, G. “The Collision Tests of the GKSS.” Jahrbuch der Schiffbautechnischen Gesellschaft 70 (1976) : 465-487. 15. Prucz, Zolan and William, C.B. “Ship Collision with Bridge Piers – Dynamic Effects.” 69th Annual Meeting of the Transportation Research Board Washington, DC : Transportation Research Board (TRB), 1990 : 890712. 16. Patev, R.C. “Full-scale Barge Impact Experiments.” Transportation Research Circular 491 (1999) : 36-42. 17. Patev, R.C and Barker, B.C. “Prototype Barge Impact Experiments, Allegheny Lock and Dam 2, Pittsburgh, Pennsylvania.” US Army Corps of Engineers ERDC/ITL TR-03-2, 2003. 18. Consolazio, G.R., Cook, R.A., Lehr, G.B., and Bollmann, H.T., Barge Impact Testing of the St. George Island Causeway Bridge Phase I : Feasibility Study, Structures Research Report No. 783, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida, January 2002. 19. Consolazio, G.R., Cook, R.A., Biggs, D.R., Cowan, D.R., and Bollmann, H.T., Barge Impact Testing of the St. George Island Causeway Bridge Phase II : Design of Instrumentation Systems, Structures Research Report No. 883, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida, April 2003. 20. Fernandez, C., Jr. Nonlinear Dynamic Analysis of Bridge Piers Ph.D. Dissertation, University of Florida, Department of Civil Engineering, 1999. 21. Mathsoft Engineering & Educuation, Inc. MathCad 2001i User’s Guide with Reference Manual Cambridge, MA, 2001. 22. Tedesco, Joseph W., William G. McDougal, and C. Allen Ross. Structural Dynamics Theory and Application Melo Park, CA : Addison Wesley Longman, Inc., 1999.

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86 BIOGRAPHICAL SKETCH The author was born on October 27, 1979, in Tampa, Florida. After graduating as valedictorian of Hillsborough High School’s class of 1997 in Tampa, she attended Florida State University where she received her Bachelor of Science degree in Civil Engineering (graduating magna cum laude) in August 2001. She then began graduate school at the University of Florida in the College of Engineering, Department of Civil and Coastal Engineering. The author plans to receive her Master of Engineering degree in August 2003, with a concentration in structural engineering. She will begin her professional career as a bridge designer with Figg Bridge Engineers of Tallahassee, Florida in May 2003 after successful completion of her graduate work.


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Title: Dynamic Analysis Techniques for Quantifying Bridge Pier Response to Barge Impact Loads
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Material Information

Title: Dynamic Analysis Techniques for Quantifying Bridge Pier Response to Barge Impact Loads
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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DYNAMIC ANALYSIS TECHNIQUES FOR QUANTIFYING BRIDGE PIER
RESPONSE TO BARGE IMPACT LOADS















By

JESSICA LAINE HENDRIX


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





UNIVERSITY OF FLORIDA


2003















ACKNOWLEDGMENTS

The successful completion of this thesis and the research discussed herein would

not have been possible without the guidance and support of my graduate advisor, Dr.

Gary R. Consolazio. The knowledge and estimable work ethic he imparted to me will

continue to serve me in all of my future endeavors, and for that I am very grateful.

I would also like to acknowledge the following people who were instrumental in

the successful completion of my graduate work: Dr. Mark Williams with the Bridge

Software Institute, David Cowan, and William Yanko. Their help was truly invaluable

and they are living proof that patience truly is a virtue. I would also like to thank my

family and friends for their unyielding faith and confidence, without which none of this

would have been possible.















TABLE OF CONTENTS
page

A C K N O W L E D G M E N T S ......... .. ............. ...................................................................ii

T A B L E O F C O N T E N T S ......... ................. ............................................... ....................iii

CHAPTER

1. IN T R O D U C T IO N ............................ ............................................................ .............. 1

2. AASHTO BARGE IMPACT PROVISIONS ...................................................... 6

3. CHARACTERIZATION OF BARGE IMPACT LOADS ....................... ..............9

3 .1 S h ip C ollision Stu dies .................................................. .................. ........ .. .. .. 9
3.2 B arge C ollision Studies ......................................... .. .. .................. .............. 10

4. HIGH RESOLUTION BARGE IMPACT ANALYSIS .......................................... 13

5. MEDIUM RESOLUTION BARGE IMPACT ANALYSIS............... ................. 16

5.1 M odeling B arge C rush B behavior .......................................................................... 17
5.2 Integration of Dynamic Barge Behavior .......................... ............................. 21
5.3 Coupling Between the Barge and Pier ........ ............................. ..............23

6. LOW RESOLUTION BARGE IMPACT ANALYSIS ............... .... .............. 30

6 .1 D e scrip tio n ............................................................................. 3 0
6 .2 Im plem entation ...................................... .............. .................. .. ................ 3 1

7. D ISCU SSION O F RE SU LTS ......................................... ................... .............. 35

7.1 Comparison of Dynamic Simulation Results................................ .................. 36
7.2 Comparison of Dynamic and Equivalent Static Impact Loads ........................... 42

8. CONCLUSIONS AND RECOMMENDATIONS .................................................47

APPENDIX

A SMOOTH HIGH RESOLUTION CRUSH DATA ........................................... 49









B LOW RESOLUTION ANALYSIS PROGRAM ................................................. 54

C MEDIUM RESOLUTION VALIDATION CASES .............. .............. 68

D COMPLETE DYNAMIC ANALYSIS RESULTS...................... .................. 71

R E F E R E N C E S .................. .................................................................. ............ .. .. 8 4

BIOGRAPHICAL SKETCH ........................................................................ 86














































iv















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 Engineering


DYNAMIC ANALYSIS TECHNIQUES FOR QUANTIFYING BRIDGE PIER
RESPONSE TO BARGE IMPACT LOADS

By

Jessica Laine Hendrix

August 2003

Chair: Gary R. Consolazio

Major Department: Civil and Coastal Engineering

Current bridge design provisions specify the use of an equivalent static load

approach to represent barge impacts loading conditions-events that are, in fact, often

highly dynamic in character. This thesis presents three different dynamic analysis

techniques as alternative procedures for quantifying barge impact loads on bridge piers.

The first technique utilizes high resolution finite element impact simulation to quantify

impact loads. The remaining two techniques are both design-oriented in nature but differ

in their level of sophistication: one is capable of modeling complex structural pier

behavior while the other is limited to inclusion of only certain types of pier nonlinearity.

A variety of barge impact analyses are conducted using the two design-oriented

techniques and the results are then compared to results obtained from high resolution

nonlinear finite element impact simulations. These comparisons show that design-

oriented, computationally efficient analysis techniques are capable of modeling a wide









range of dynamic barge-pier interaction and can reasonably predict the dynamic loads

generated by during such events.

Additional comparisons are made between the equivalent static load method

prescribed by current bridge design specifications and the dynamic analysis techniques

presented here. Impact loads and structural deformations predicted by the static method

and the dynamic analysis techniques are compared for barge collisions of varying kinetic

energies. Results from such comparisons indicate that, for barge collision events

associated with relatively low kinetic energy levels, dynamic analysis techniques are

preferable. For more severe impact conditions, the use of equivalent static loads for

design purposes is acceptable.














CHAPTER 1
INTRODUCTION

The use of vessel impact loads in bridge design is a concept that has gained

mainstream attention only within the past two decades. It was previously a common

belief among the engineering community that the probability of such an event was so

unlikely that it could be disregarded [1]. It was not until 1980, with the collapse of the

Sunshine Skyway Bridge in Tampa, Florida that designers began to realize the

significance of vessel collisions. As a result of an errant cargo ship, The Summit Venture,

the Sunshine Skyway Bridge was destroyed and thirty-five people lost their lives [2].

Although the collapse of the Sunshine Skyway was just one of several catastrophic

failures in recent history, it raised the world's consciousness regarding the need to design

bridges for the possibility of vessel impact. Since then, important strides have been made

in an effort to develop modern principles and standards for the design of bridges against

such events.

Unfortunately, barge collisions continue to be a very real problem in the United

States and worldwide. On September 15, 2001, a four-barge tow struck the Queen

Isabella Causeway, the only bridge leading to South Padre Island, Texas, killing eight

people. Substantial damage was not only imparted to the bridge but to the local economy

as well [3]. Less than a year later, another fatal bridge collapse occurred. On May 26,

2002 an Interstate 40 bridge near Webber's Falls, Oklahoma collapsed into the Arkansas

River killing fourteen people, after being hit by an errant barge. The 1-40 Bridge, built in

1967, passed all required safety checks just months before the accident [3]. Thus, there is






2


a clear need for research focusing on the development of improved methods for

predicting barge impact loads and evaluating the structural response to such loads.

Compared to the data available for ship collisions, very little experimental data

exists with respect to barge collision impact forces. This is due not only to the expense

involved in carrying out experimental studies, but also to the inherent complexity of

measuring such a phenomenon. One such complication is the variety of barge

configurations that transit inland waterways. The U.S. Army Corps of Engineers reports

[4] that there are over two thousand different types and sizes of barges using the U.S.

inland waterway system. From such a variety of barge types, almost any combination of

size and quantity may be configured into a multi-barge flotilla.

As a result of the limited availability of barge impact data, the American

Association of State Highway and Transportation Officials (AASHTO) Guide

Specification for Vessel Collision Design of Highway Bridges [2] relies on data produced

by a single study conducted by Meir-Domberg (discussed in more detail later) in 1983.

Furthermore, the AASHTO provisions stipulate the use of equivalent static loads to

represent impact events that often have significant dynamic components of load and

bridge response. Because barge impact loading conditions often control the design of

new bridges, it is important that alternative load prediction procedures be explored.

Dynamic structural analysis techniques offer one possible alternative to the use of

code-specified equivalent static loads. However, this is a new and emerging area of

bridge-related structural research. Recent analytical studies focusing on barge impact

[5,6] have involved the development and use of high-resolution nonlinear dynamic finite

element models. While such models are capable of quantifying dynamic impact loads-









as well as providing valuable insight into the relevant collision mechanics-they are not

generally appropriate for use in routine bridge design due to the considerable

computational resources often required.

In this thesis, design-oriented medium and low resolution dynamic analysis

techniques are developed that approximate the behavior of more refined high resolution

barge-pier interaction models (Figure 1). Throughout this document, reference will made

to high, medium, and low resolution analysis techniques. In this context, these terms are

defined as follows :

* High resolution
Highly refined finite element models, generally consisting of tens of thousands of
elements [5,6], which are analyzed using dynamic finite element analysis codes (e.g.
LS-DYNA [7]) capable of representing contact and localized nonlinear buckling.
* Medium resolution
Finite element models generally consisting of hundreds to thousands of elements that
are analyzed using commercial dynamic finite element analysis codes (e.g.
FB-Pier [8,9]).
* Low resolution
Simple, low-order numerical models utilizing a very small number of lumped masses
and nonlinear springs to represent barge and pier behavior.


In regard to the medium resolution analysis technique presented here, focus is placed on

the development of a procedure that can be used to couple nonlinear dynamic barge

behavior to existing commercially available, design-oriented dynamic structural analysis

codes. The low resolution analysis technique presented focuses instead on a simpler

spreadsheet based prediction of dynamic impact loads using direct integration techniques

of a set of coupled differential equations. In a design environment, structural response to

such loads would then need to be evaluated using a dynamic structural analysis package.













Zone of contact between
barge and pier
Direction of
barge motion

total
smb


Gap element permitting
transfer of compressive -
loads only (zero tension)

Nonlinear spring element
representing the barge
load-deformation crush
relationship


Single dof model of
barge as a point mass


mb mp
kbk
Single dof model of
pier as a point mass

Nonlinear soil spring representing
lateral load-deformation behavior of pier

c)

Figure 1. Dynamic analysis techniques used to simulate barge impact events, a) High
resolution model; b) Medium resolution model: simplified barge model; c) Low
resolution model: simplified barge and pier models


Reinforced
concrete pier


Soil springs









In all cases treated here, particular focus is given to evaluating the accuracy of the

dynamic loads predicted by the medium and low resolution models. Previous studies

[5,6] have demonstrated that barge impact loading conditions can vary from highly

dynamic and oscillatory in nature to sustained, nearly pseudo-static. The nature of the

loads produced depends on the specific characteristics of the collision event (barge mass,

impact speed, pier stiffness, etc.). The ability of the medium and low resolution analysis

techniques to predict load and pier response will be evaluated by comparing results

produced by these procedures to results obtained from high resolution analyses. Also of

interest, is the extent to which impact loads predicted by dynamic analysis in general

compare to the equivalent static loads specified by the AASHTO procedures.

In the following chapters, an overview of the existing AASHTO procedures for

barge impact design will be given, followed by descriptions of the newly proposed

dynamic analysis methods. Following that, an impact resistant pier and a non-impact

resistant pier are analyzed using high, medium, and low resolution models and the results

are compared.














CHAPTER 2
AASHTO BARGE IMPACT PROVISIONS

The AASHTO barge impact provisions [2,10] apply to the design of all bridges

spanning shallow draft inland waterways carrying barge traffic. The AASHTO Guide

Specification for Vessel Collision Design and the AASHTO LRFD Bridge Design

Specification differ in their risk analysis method, but follow the same procedure for

predicting expected barge impact loads. These provisions are based on empirical

equations, rational analysis based on theory, and model testing supported by analysis [2].

Such an approach was taken due to the lack of available experimental data involving

barges striking bridge piers. In addition, at the time the specifications were written,

computer analysis programs available to design engineers were incapable of handling

dynamic effects and the presence of nonlinearities in structural systems.

The AASHTO provisions are intended to provide a simplified procedure for

computing an equivalent static load for a barge impact in lieu of a full dynamic impact

analysis. The calculations begin with the collection of required data for the bridge design

(vessel traffic, vessel transit speed, loading characteristics, bridge geometry, waterway

and navigable channel geometry, water depths, and environmental conditions). Barge

impact loads are then evaluated on the basis of energy considerations. The translational

kinetic energy for a moving vessel is given by AASHTO [10] as :

KE = 500CHAV2 (1)









where KE is the vessel kinetic energy (joule), CH is the hydrodynamic mass coefficient,

M is the vessel displacement tonnage (metric ton), and V is the vessel impact speed

(m/sec). It should be noted that Eq. 1 is an empirical equation based on the standard

relationship for translational kinetic energy of a moving body commonly expressed as :

KE =I 2
2 (2)


where M is the mass of the vessel (kg). The hydrodynamic mass coefficient, CH,

included in the AASHTO equation for kinetic energy, is present to account for additional

inertia forces caused by the mass of the water moving with the vessel. Several variables

are accounted for in the determination of CH including water depth, underkeel

clearances, shape of the vessel, speed, currents, position and location of the vessel in

relation to the pier, direction of travel, stiffness of the barge, and the cleanliness of the

hull underwater. Based on a previous study [11], a simplified expression has been

adopted by AASHTO in the case of a vessel moving in a forward direction at a high

velocity (the worst-case scenario). Under such conditions, the procedure recommended

by AASHTO depends only on the underkeel clearances [10] :

* For large underkeel clearances (> 0.5 Draft) : CH =1.05

* For small underkeel clearances ( 0.1 Draft) : CH = 1.25

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

waterway. For underkeel clearances between these two limits, CH is estimated by

interpolation.

The next step in the AASHTO provisions is to determine the barge bow (front

section of the barge) damage depth. During an impact event, energy can be absorbed or









dissipated in a variety of ways, including displacement and/or plastic deformation of the

barge and bridge pier (including any fendering system), friction, and also rotation in the

event of an eccentric impact [12]. Considering AASHTO's assumption that the impact

loads developed represent the worst-case scenario of a head-on collision, one of the

primary ways energy is dissipated is through crushing of the bow. AASHTO's

relationship between kinetic energy and barge crush is represented by the following

equation [10] :

3100 7
aB 0( 1+(1.3.107)KE- 1) (3)
RB

In this equation aB is the barge bow damage depth (mm) and RB, given by By /10.67,

is the barge width modification factor, where BB is the barge width (m). The barge

width modification factor is included to account for a barge width other than the standard

value of 10.67 meters (the average barge width used in U.S. inland waterways) [2]. Once

aB is determined, the barge impact force is calculated by [10] as :


_(6.0. 104)aB RB aB <100mm (4)
B (6.0.106) +1600.a -RB a lO100mm
where Pg is the equivalent static barge impact force (N). Note that Eqs. 3 and 4 are

based on the AASHTO LRFD Bridge Design provisions [10] but are modified to include

the barge width modification factor (RB) presented in the AASHTO Vessel Collision

Specifications [2], in accordance with results from Meir-Dornberg.















CHAPTER 3
CHARACTERIZATION OF BARGE IMPACT LOADS

As stated in the previous chapter, very few analytical or experimental studies have

been conducted to characterize the response of bridge piers to impact loads generated

during vessel collision events. In this chapter, a review of relevant previous studies is

provided.

3.1 Ship Collision Studies

Ship collision events have been studied to a much greater extent than have barge

collisions. Two primary ship collision studies form the basis for most current theories

relating to ship impact loading. The first study is that ofV.U. Minorsky [13]. This study

was conducted in 1959 to analyze collisions with reference to protection of nuclear

powered ships, and focused on predicting the extent of vessel damage during a collision.

A semi-analytical approach was used based on data from twenty-six actual collisions.

From this data, Minorsky determined a linear relationship between the deformed steel

volume and the absorbed impact energy. The second key study of the time was

conducted by Woisin in West Germany [14]. Woisin was also interested in the

deformation of nuclear powered ships in the event of a collision. Data were collected

from twenty-four collision tests of scaled ship models colliding with each other. Results

from this study were used to develop the current AASHTO equation for calculating

equivalent static ship impact force [2].









In 1990, Prucz and Conway published a paper [15] discussing analytical research

conducted on the dynamic effects associated with ship collisions with bridge piers. Their

paper presented a simplified numerical procedure, based on a lumped mass-idealization

of the ship-pier system, to investigate the dynamic effects associated with ship collision

events.

3.2 Barge Collision Studies

With regard to barge impact loads, the most significant experimental study

conducted to date is that of Meir-Domberg [2]. This research included both static and

dynamic loading on scaled models of the European Barge, Type IIa, similar in dimension

to the U.S. standard jumbo hopper barge. No significant differences were found between

the static and dynamic impact force. However, the tests did not involve interaction

between vessels and bridge piers. From this research, equations were developed that

related equivalent static barge impact force, barge deformation, and impact deformation

energy. These equations were modified slightly and adopted by AASHTO for use in

computing barge impact loads.

More recently, the U.S. Army Corps of Engineers (USACE) completed the first

full-scale barge impact experiments on a concrete lock wall [16]. The USACE Waterway

Experiment Station performed the experiments in order to verify the current analytical

model used to design inland waterway navigation structures and also to support the

development of more innovative structures. Structures such as lock walls are subject to

frequent barge flotilla impacts and must be designed accordingly. The current single

degree of freedom model used by the Army Corps to quantify design loads is believed to

be overly conservative and therefore produces very costly structures. The goals of this

research were to determine a baseline response of a barge impact with a lock wall,









measure impact forces, quantify barge-to-barge interaction in a flotilla during a collision,

and to investigate a new energy-absorbing fendering system. Since this type of

experiment had never been done in the past, prototype barge impact experiments were

first conducted on Allegheny River Lock and Dam 2 in Pittsburgh, Pennsylvania in 1997

[17]. The primary purposes for conducting the prototype tests were to determine how to

quantify and measure barge impact forces and to gain a better understanding of the nature

of the barge-wall interaction.

Following the prototype tests, full-scale experiments were conducted on the

Robert C. Byrd Lock and Dam in Gallipolis, West Virginia in 1999 [16]. These

experiments consisted of a fifteen-barge flotilla of jumbo open-hopper barges impacting

the lock wall with and without a fendering system. There were forty-four impact

experiments in all. The experiments ranged in impact angles from 5 to 25 degrees and in

impact velocities from 0.15 to 1.2 m/sec. All of the impacts were within the elastic

deformation range of the barge flotilla. Results from this study are being used to develop

full-scale plastic range experiments that will be used to determine actual crushing

strength and performance of inland waterway barges.

Currently, an on-going investigation is being conducted by the University of

Florida [18,19] and the Florida Department of Transportation (FDOT) to quantify impact

loads generated on bridge piers during barge collision events. A combination of

experimental testing and analytical modeling are being used to characterize the dynamic

nature of impact loads that arise during barge collisions, to compare these loads to the

equivalent static loads prescribed by the AASHTO provisions, and to develop appropriate

design-oriented impact-load prediction procedures.









Replacement of the existing St. George Island Causeway Bridge (near

Apalachicola, Florida) with a newly constructed bridge has afforded the opportunity to

conduct full-scale barge impact tests on the older structure before it is demolished. After

the new bridge has been opened to traffic, a full-scale hopper barge will be driven into

selected piers of the older structure at several different impact speeds while the dynamic

impact loads imparted to the piers are monitored. In addition, deformations of the piers,

surrounding soil, and barge bow will be monitored throughout each impact event.

Complimenting the experimental components of this investigation, a variety of

finite element based analytical studies have also been conducted to quantify the impact

loads generated during barge collisions and to aid in planning the physical impact tests

(selection of piers to be tested, selection of impact speeds, etc.). Consolazio et al. [5,18,

19] have developed high resolution finite element models of a barge and two piers of the

existing St. George Island Causeway Bridge (including representation of both the

structural properties and soil properties). Nonlinear finite element analysis codes have

been applied to these models to study the crush characteristics of hopper barges [6], and

to quantify the dynamic loads that are imparted to bridge piers during collision events [5].

In the latter study, the character of the impact loads generated-sustained versus highly

oscillatory-was found to be a function primarily of barge impact speed and pier

stiffness. In addition, this study also revealed that during severe collision events, more

than half of the kinetic energy of impact may be dissipated through plastic deformation of

the barge bow.















CHAPTER 4
HIGH RESOLUTION BARGE IMPACT ANALYSIS

Numerical prediction of lateral impact loads imparted to bridge piers during barge

collision events is most accurately achieved through the use of contact-impact finite

element analysis software and refined barge, pier, and soil-structure interaction models.

In this thesis, the high resolution models previously developed by Consolazio et al. [5,18,

19], and the nonlinear dynamic finite element code LS-DYNA [7] have been used to

predict baseline impact data to which results obtained from the medium and low

resolution analysis techniques, presented in the following chapters, can be compared.

All impact conditions considered herein involve a jumbo class hopper barge

striking a pier bridge in a head-on, perpendicular manner (Figure 2). The jumbo class

hopper barge-59.5 m long, 10.7 m wide, 3.7 m deep, 1.8 MN empty weight, 6.9 MN

fully loaded weight-makes up more than 50% of the entire barge fleet operating in the

U.S. inland waterway system. Furthermore, this type of barge is the baseline vessel upon

which the AASHTO barge impact provisions are established. For these reasons, the

finite element barge model used in this study matched the mass, geometry, and structural

configuration of a typical jumbo hopper barge.

A key difference between the high resolution analysis and the lower resolution

methods presented later in this document lies in the modeling of the barge. High

resolution analysis, as defined here, involves the use of a very detailed finite element

barge model. The barge model developed by Consolazio et al. is based on detailed










structural plans obtained from a leading U.S. barge manufacturer, rather than on the crush

relationship assumed by AASHTO, i.e. Eqs. 3 and 4. Intended for use in frontal impact

simulations, the model uses more than 25,000 shell elements to represent internal

structural members (plates, channels, angles, etc.) in the bow section of the barge. By

combining this level of geometric discretization with nonlinear steel stress-strain

relationships, bow crushing and energy dissipation during collisions with relatively rigid

concrete piers can be accurately simulated. Frictional effects, internal buckling and

contact, and buoyancy effects are also included in the model.



Hopper barge
S 1 ier- Pier-3





SBuoy1 I Concrete piles
Buoyancy f II stand soil springs
springs I_ I'II

IIi I ll
I I I
Steelpiles ,,.I II
soil spri,
I I ,





I I



Figure 2. High resolution barge, pier-i and pier-3 finite element models

The high resolution structural pier models used in this study (Figure 2) represent

two of the support piers of the previously cited St. George Island Causeway Bridge.

Pier-1 (adjacent to the navigation channel) and pier-3 (third from the channel) were

chosen for this study because they represent a significant range in both structural stiffness









and impact resistance, and therefore yielded impact force results representative of a wide

variety of pier types. Pier-1 is considered a typical example of an impact resistant pier

because of its size and stiffness, while pier-3 is representative of a more flexible, non-

impact resistant pier. The pier models, developed using construction plans for the bridge,

include pier bents, pile caps, and piles. The models do not take into account any

structural contribution or inertial effects of the superstructure.

Soil-structure interaction effects between the piles and the surrounding soil-

including nonlinear soil response, plastic deformation, gap formation, and pile-group

effects-were modeled using thousands of nonlinear lateral and vertical soil spring

elements. Unique load-deformation curves were specified for each spring based on soil

boring data obtained for the bridge site.

For a more detailed descriptions of the barge, pier, or soil models, the reader is

referred to Refs. [18, 19]. Impact data predicted by nonlinear dynamic analysis of these

models are used to compare results predicted by medium and low resolution analysis

techniques later in this thesis.














CHAPTER 5
MEDIUM RESOLUTION BARGE IMPACT ANALYSIS

Although high resolution dynamic analysis techniques are capable of yielding

accurate impact load data, such methods are also computationally intensive and thus, not

always practical for use by design engineers. In this chapter, a medium resolution

analysis technique is described in which a single degree of freedom (DOF) nonlinear

barge is coupled to an existing multi-DOF nonlinear dynamic pier analysis program

(FB-Pier [8]). The coupling is implemented in a way that necessitates only minimal

modifications to the dynamic pier analysis code.

Conducting a barge impact analysis requires consideration of both dynamic

behavior and nonlinear structural behavior. Properly modeling structural behavior (the

inelastic force-deformation response of the barge) is particularly important as it affects

both force development and energy dissipation during impact. The approach taken here

is to approximate dynamic nonlinear barge behavior by independently representing

dynamic behavior (mass related inertial resistance) and nonlinear structural behavior

(barge crushing). The total mass of the barge is represented as a single degree of freedom

(SDOF) point mass while inelastic structural response of the barge bow is modeled using

a nonlinear crush relationship (a Pb vs. ab curve). A pre-computed barge crush curve is

employed to model the stiffness of the barge bow the medium resolution impact analyses

(Figure 3).









Barge force-deformation ---------------------
relationship modeled using data from Pb
high resolution finite element simulation

rub ab

................" ...... K u r

^. ^. ^ '*--"-'-- ---- '--------------- b

a)

-ub -Up

mb [--Pb Pb --u
.9. ,.S2 mn


b)

Figure 3. Dynamic barge and pier/soil modules. a) Barge module showing origin of
contact force; b) Contact force linking the two modules

5.1 Modeling Barge Crush Behavior

While several sources of pre-computed crush data exist (experimental testing,

finite element analysis, or specification-prescribed expressions such as Eq. 4) in this

study, high resolution finite element simulations have been used to generate the barge

crush data needed. Key to the concept of using high resolution finite element analysis-

which was cited above as being computationally expensive-for this purpose is the idea

that these analyses are one-time events conducted in advance. In the present study, static

crush simulations have been conducted [19] using the high resolution barge model

described in the previous chapter to generate data relating barge force, Pb, to crush

deformation, ab. The crush characteristics of a barge are functions only of the barge itself

and the geometry (shape and size) of the object imposing the crush damage (i.e., the









pier). Thus, crush data can be computed in advance using high resolution analysis and

then stored for subsequent use in a medium resolution analysis model. It is by this

strategy that complex crushing behavior can be efficiently modeled using a SDOF barge

model.

Because static barge crush behavior is dependent on the size and shape of the pier

[19], separate crush curves were generated for pier-1 and pier-3 (since they have different

column widths). However, data generated from finite element crush analyses often

exhibit small scale dynamic deviations (Figure 4a) from the primary crush curve.

Deviations of this type complicate the process of applying direct dynamic integration

techniques to the SDOF barge model. Therefore, in this study, the raw high resolution

force-deformation data (crush data) were smoothed and re-sampled (Figure 4b) before

being incorporated into the medium resolution model. For additional information on the

smoothing technique employed, refer to Appendix A.

Smoothed crush curves, along with an unloading and reloading stiffness (Ku,),

have been used to define an elastic-plastic load-deformation relationship for the barge.

All possible load paths have been included in the model (initial loading, unloading,

reloading, etc.). Figure 5 illustrates the various stages that the model can exhibit during

an impact event. While the barge is in contact with the pier, the crush is computed as

ab = max(ub -Up), where ub and up are the barge and pier displacement, respectively.

Figure 5a represents the virgin loading stage of the barge (aby is the yield crush

deformation). Once ab exceeds aby, the barge will continue to load plastically, causing

permanent inelastic deformation.










Barge crush depth (in)
0 5 10 15 20 25 30 35
6

1200


1000





S600

. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
400


SStatic crush data for pier-1 200
Static crush data for pier-3 -

0 100 200 300 400 500 600 700 800 900
Barge crush depth (mm)

a)

Barge crush depth (in)
0 5 10 15 20 25 30 35


1200


1000


800 .-
o3
.. . ---- ------------- .. ... ... ... .. ... ... ...
600


400


Smoothed crush data for pier- 200
Smoothed crush data for pier-3 --
0- 0
0 100 200 300 400 500 600 700 800 900
Barge crush depth (mm)

b)

Figure 4. Comparison of static and smoothed crush curves for pier-1 & pier-3. a) Static
crush curves; b) Smoothed crush curves










Unloading (Figure 5b) occurs at the specified slope (Ku,). As unloading

progresses, the barge may eventually cease to contact the pier. At this stage, Pb = 0 and

the barge retains a permanent plastic deformation abp. From this point in time forward,


any computed ab = (ub -up) for which ab < abp represents a condition in which a finite


size gap has formed between the barge and the pier face, and thus Pb = 0 (Figure 5c).

Closure of this gap must first occur before subsequent reloading (Figure 5d) along the

slope, Kur can take place.

Pb Pb








Kur


I ab ab
aby abmax aby

a) b)

Pb Pb






Kur Kur


-- ab ab
abp abp

c) d)

Figure 5. Stages of barge crush. a) Loading stage; b) Elastic unloading stage; c) Gap
formed; d) Reloading stage









5.2 Integration of Dynamic Barge Behavior

Given the nonlinearities inherent in barge crushing, integration of the barge

equation of motion is only practical using numerical methods. In this study, viscous

damping effects associated with the water surrounding the barge are neglected (CH =1).

Energy dissipation (damping) in the barge model occurs only through plastic crushing.

Examining Figure 3, the equation of motion is then written as

mb Ub = Pb (5)

where mb is the barge mass, iib is the barge acceleration, and Pb is the contact force

acting between the barge and pier. Assuming that the mass of the barge remains

constant, then evaluating Eq. 5 at time t, we have

mb tb = tPb (6)


where tib and tPb are the barge acceleration and force at time t. The acceleration of the

barge at time t can be estimated using the central difference equation :

tb (t+hub -2tub +-ub) (7)
h


where h is the time step size; and t+hub, tub, and t-hub are the barge displacements at

times t+h, t, and t-h respectively. Substituting Eq. 7 into Eq. 6 yields the explicit

integration central difference method (CDM) dynamic update equation

t+hub = -(tPb h2 / mb) + 2t ub t-hub (8)









which uses data at times t and t-h to predict the displacement t+hub of the barge at time

t+h. Since the barge crush behavior is nonlinear, the force tPb is computed using the

elastic-plastic loading and unloading model described in the previous section.

Choice of the CDM for this application was based primarily on the relative ease

with which nonlinear behavior can be taken into account. However, since the method is

known to be conditionally stable, consideration had to be given to the choice of time step

size (h) and the appropriateness of the method for this particular application. For CDM

integration of SDOF systems, a reasonable choice of time step size is h < h, = (T/10)

where T is the natural period of the system. If hc, for the barge (integrated using the

explicit central difference method) is much smaller than hc, for the pier/soil system

(integrated using the implicit Newmark's method; described in the next section), then the

single DOF barge would control time-stepping for the entire coupled barge/pier/soil

system. If computational efficiency is a consideration, such a condition would clearly not

be desirable.

In fact, this condition does not occur and CDM is therefore, a suitable choice of

integration technique. For the SDOF barge, Tb = 27rm k therefore conditions

leading to the minimum hr, would be minimum mass mb and maximum stiffness kb.

The minimum mass of a barge is its mass when empty (no payload). The maximum

barge stiffness (slope of the crush curve) occurs during the initial loading stage (before

plastic deformation softens the system and reduces kb; see Figure 4). Considering these

conditions, the computed value of hr, for the barge is found to be









h, =_ (Tb/10) 0.01 sec. Prior experiences in conducting dynamic pier analyses using

the FB-Pier code have consistently shown that use of time steps larger than 0.01 sec.

often leads to convergence failure. Thus, the CDM represents a suitable and efficient

choice for the application of barge impact analysis.

5.3 Coupling Between the Barge and Pier

Implementation of the medium resolution dynamic analysis technique involved

adding a barge-dynamics module to the FB-Pier program [8] (hereafter referred to as the

"pier analysis program" or the "pier/soil module" depending on the context) and then

making appropriate modifications so as to couple the barge and pier together. As

Figure 3 suggests, this coupling has been accomplished through the use of a common

impact force Pb that acts on both the barge and the pier (at the point of contact between

the two). Pier columns in the pier program are modeled using materially nonlinear frame

elements (based on a cross-sectional fiber model). As such, the barge impact force Pb is

applied to the pier as a time varying nodal force.

Conceptually, the overall coupled barge/pier/soil system (i.e., the medium

resolution analysis model) can be thought of as two separate modules : a pier/soil control

module (Figure 6), and a barge module (Figure 7). The dynamic time integration process

is primarily controlled by the pier/soil module with the barge module determining the

magnitude of the impact force Pb based on the dynamic barge and pier motions, and

based on the elastic-plastic barge crush model. Coupling is achieved through the

insertion of three minimally intrusive links into the pier/soil module (Figure 6).













--------------------- Pier/soil module --


Initialize pier/soil module

T -nttmrt laroe module tn nerfnrm initiali7azti


mode ------------------

mode = INIT
o S


t=o

0fu}, 0u, O = 0
{R} ={0} initialize internal force vector

[K], [M], [C] form stiffness, mass, damping

[K]= fn([K],[M],[C]) form effective stiffness
For each time step i = 1,
STime for which a solution is sought is denoted as t+h
Form external load vector {F}

Extract displacement of pier at mode = CALC; Up is sent
impact point from t+h{u}

Barge module returns computed computed Pb is returned
barge impact force Pb

{F} = {F} P ... insert barge force Fb into external load vector

{F}={F}-{R}+fn([M],[C]) ... form effective force vector

{Au}= [K] {F}

For each iterations = 1, ..
t+h{u}= t{u}+{Au} ...update estimate of
displacements at time t+h
[K]= f('+h u})

[K]= fn([K],[M],[C])

{R}= fn(+h{u})
{F}= {F}-{R}+fn([M],[C])

S{u} =[K] 1{F}
{Au} = {Au} +{Su}

Check for convergence
of pier and soil ...
no
max(: TOL


no
max({\F\}) TOL

yes
........ ..
mode= CONV


Form new external
load vector with updated Pb is returned
S updated Pb ....
{F}= {F} Pb


Record converged
Spier, soil, barge data
and advance to next convened ub and P
time step are returned

-------------------------------------------------------/


Barge module
......... Barge module .......
Initialize state of barge
(see barge module for details)


........ Barge module ............

Compute barge impact force
Pb based on displacements of
the pier up and barge ub.
(see barge module for details)






























......... Barge module ..........

Check for convergence of
coupled barge/pier/soil system
by checking convergence of
barge force predictions.
(see barge module for details)


---n o < T O L
yes


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













..------------------............... Barge mdule ------------------


o- mode=INIT
no


: o o
ub= ab= a= abmax = 0
S .-(" 'h
S= 0 ntialize internal barge module cycle counter

I p = 0
( Return to pier/soil module


St+hub = (-Pb(') h2/b)+2 tub t-hub

+ = +1 increment internal cycle counter

For each iteration k= 1,
t+hab = (t+hub t+hup)

Pb() fn +hab tabp, tbmax)

t+hb =(-Pb() h2/mb)+2 tub t-hub

Pb =Pb()-Pb


|Pb no I


mode=CALC
no
yes

2 Return current barge
S force I pier module
S----------------------------------------- ---

mode=CONV

yes
If mode=CONV, then the pier/soil module has converged
Now, determine if the overall coupled barge/pier/soil
system has converged by examing the difference between
barge forces for current cycle and previous cycle

=pb(0)_pb(-1)

S TOL
yes
no
.. .. .. .. -- -- -- -- - -- -- -- -- -- -- -- - -- -- -- -- --


Convergence not achieved compute incremental
Sbarge force using a
weighted average

P) () + compute updated barge
b =Pb + force

Return barge force Pb) for current cycle to pier module


Coupled barge/pier/soil system has converged, thus
is now a converged value Update displacement data for
next time step

hub= ub and = t+hub

Stab bp, bmax save converged barge crush parameters

r = Return to pier/soil module for next time step






Figure 7. Flow-chart for nonlinear dynamic barge module









Each link instructs the barge module to perform a particular task :

* mode=INIT : Perform initialization of the barge module
* mode=CALC : Calculate an initial estimate of barge force at the start of a time step
* mode=CONV : Determine if convergence of the overall coupled system has occurred

In this study, implicit direct time step integration in the pier/soil system was

accomplished using Newmark's method [20]. As Figure 6 indicates, the overall flow of

the process involves an outer loop that controls time stepping, and an inner loop that

controls iteration to convergence (satisfaction of dynamic equilibrium at each time step).

For brevity, only the key aspects of the algorithm that are relevant to the discussion here

are presented in the flowchart; for complete details the reader is referred to Ref. [20].

At the beginning of each time step, the barge module is invoked (mode=CALC)

to calculate an estimation of impact force Pb for that time step. Determination of Pb first

requires that the crush be computed as ab =max((ub -up),abp). In performing this

computation, the pier/soil module has supplied the most up-to-date estimation of the pier

displacement up An iterative variation on the CDM is then used (Figure 7) to satisfy

the dynamic equation of motion for the barge. At each pass through this iteration loop,

the barge impact force estimation is refined until convergence is achieved and the

calculated Pb value is returned to the pier/soil module.

After assembling the Pb value into the appropriate location in the pier/soil

external load vector (which may contain other loads such as gravity), the pier/soil module

iterates until it has reached dynamic equilibrium (i.e., convergence). During this process,

the value of Pb that has been merged into the load vector is not altered (attempting to









update the barge force within each pier/soil equilibrium iteration results in unreliable

convergence behavior). Once the pier/soil module has converged, the displacement of

the pier at the impact point (up) is extracted from the pier/soil displacement vector {u}.

While the pier/soil system has converged at this stage, the overall coupled barge/pier/soil

system may still not have converged.

Determination as to overall convergence is accomplished by once again invoking

the barge module (now as mode=CONV). Using the newly updated up value provided

by the pier/soil module, the barge module once again carries out iterative CDM

integration on the SDOF barge. It is very important to note that each time the barge

module is invoked, time integration is performed using displacement data (tub and

t-hub) and barge crush data (tabp and t-habax) that correspond to previous points in

time at which the entire coupled barge/pier/soil system achieved convergence. Once a

new estimation of the barge force has been computed, the difference in value between the

current invocation ("cycle ") and the previous invocation ("cycle -1") is computed

(see the calculation AP() =Pb ) -Pb(-1) in Figure 7). If AP(C) is sufficiently small,

then not only has the pier/soil system converged, but the prediction of barge force for this

time step has also converged and therefore the entire coupled system is in dynamic

equilibrium. Thus, if AP(C) < TOL is satisfied, the barge module instructs the pier/soil

module to advance to the next time step.

If instead, AP(t) > TOL, then an updated estimate of barge force Pb must be

computed and returned to the pier/soil module. While the simplest choice would be to









return the value of Pb(') just used in the AP) = Pb ) -Pb() calculation, this is in fact

a rather poor choice. Because the pier/soil module and the barge module each iterate to

convergence independently in an alternating "back-and-forth" fashion (which has been

found to be necessary in order to ensure robust convergence to dynamic equilibrium), the

barge forces Pb(t) predicted during sequential barge module invocations (i.e., cycles r

= 0,1,2, etc.) tend to oscillate as the two systems seek to achieve coupled dynamic

equilibrium. This oscillation in Pb() values can result in slow coupled convergence and

typically involves sequentially computed values of APIt) that alternate in sign.

To diminish these oscillations and accelerate convergence, a damping (relaxation)

technique was implemented in the barge module. Rather than returning the raw

computed Pb(t) value from the iterative CDM process to the pier/soil module, an

exponentially decaying historical averaging process is used to compute a damped

increment of barge impact force :

APb=(1-/u)APb(t) +/uAPb(t-l
1 (9)
a b c

In this expression, part a is the damped force increment; part b is the "raw" force

increment computed for the current cycle; part c is the damped force increment from the

previous cycle; and / is a factor that determines the relative weighting of current versus

previous data in determining the damped incremental change (u = 0 indicates that

previous data should be disregarded altogether). Note that by the recursive nature of this

process, part c contains not only data for the previous cycle f but also data for all









previous cycles. The influence of older cycles diminishes in an exponentially decaying

fashion that is controlled by the choice of ,u.

With the damped increment of force APIb() determined (using Eq. 9), the actual

barge force is computed by the barge module (Figure 7) as :

Pb() = Pb (l +) (10)

and returned to the pier module (the force is denoted simply as Pb in Figure 6 since the

cycle number concept is local to the barge module). After assembling the new Pb value

into a clean copy of the external load vector {F} to form {F}, the pier/soil module

resumes the process of iterating toward convergence using the newly formed load vector.

Advancement to a new time step only occurs when the overall coupled barge/pier/soil

system has converged.

While the barge module described here has been implemented specifically within

the FB-Pier dynamic pier analysis program, the techniques presented are sufficiently

general that they can be implemented within most nonlinear dynamic finite element

structural analysis codes. Results obtained using this medium resolution dynamic

analysis technique are presented later in this thesis.















CHAPTER 6
LOW RESOLUTION BARGE IMPACT ANALYSIS

6.1 Description

In the previous chapter, the goal was to demonstrate a methodology by which a

nonlinear dynamic pier model could be coupled to a very simple barge model to enable

prediction of impact loads and structural responses during barge collisions. Given

sufficient nonlinear sophistication in the pier model, localized structural failures (e.g.,

plastic hinging of columns or piles) could be taken into consideration with the medium

resolution analysis technique.

In this chapter the goal is instead to present a very low order dynamic analysis

technique capable of generating approximate time histories of dynamic impact load. In

theory, pier response to such time histories of impact load could then be analyzed using a

separate dynamic structural analysis package of the designer's choice. A key assumption

in this technique, however, is that the both the pier and the barge can be adequately

represented using simple nonlinear SDOF models (Figure Ic). Individual point masses

are used to represent the barge (mb) and the pier (mp). A nonlinear spring/gap element

identical to that used in the medium resolution analysis is used to represent barge

behavior. In regard to the pier, the total structure mass is concentrated at the theoretical

barge impact point and a nonlinear spring (anchored to the reference frame of the

dynamic system) is used to represent combined structure/soil resistance to the imparted

impact load.









Clearly, there are limitations to the accuracy of such a simplified analytical

model. If the vertical distribution of mass in the pier is such that the location of the

actual center of mass is significantly different than that assumed in the SDOF model, then

the ability of the two-DOF model to represent dynamic barge-pier interaction will be

limited. Furthermore, with only a single pier DOF, modes of vibration at frequencies

higher than the fundamental sway mode will not be included in the dynamic analysis.

Similarly, certain types of localized failures (e.g., hinging of a pier column) cannot be

represented because such behavior would necessitate the inclusion of response modes

other than the fundamental sway mode.

However, by using a nonlinear load-deformation relationship (described below) to

represent combined structure/soil resistance to applied loading, some types of plastic

deformation can be reasonably included in the model. For example, plastic hinging of a

pile, while constituting a localized form of failure, will tend to result in an overall

softening of the structure in the sway mode and thus can be approximately represented

even with a SDOF model. Given such capabilities and given the minimal computational

resources needed to perform the two-DOF analysis, evaluation of this simplified

technique is justified.

6.2 Implementation

Implementation of the two-DOF dynamic analysis technique has been

accomplished through the development of a Mathcad [21] based program that

implements the central difference method [22] of direct time step integration (a copy of

the complete program is included in Appendix B of this thesis). Data obtained from high

resolution static barge crush analyses (described earlier) are used to represent the

nonlinear response of the barge in the two-DOF model. Similarly, the resistance of the









pier/soil system to applied lateral loading is represented using a nonlinear loading curve

and a linear unloading curve. The loading curve is generated by conducting a static

nonlinear lateral load analysis of the pier using any pier analysis tool available to the

engineer (FB-Pier, LS-DYNA, etc.). In this thesis, pushover data for pier-1 and pier-3

were obtained from high resolution analyses conducted on combined pier/soil models

using LS-DYNA. The data obtained were then validated against analyses conducted on

the same piers using FB-Pier.

Unlike the SDOF barge model, the SDOF pier/soil model does not utilize a gap

element because unloading of the pier does not necessarily result in residual deformation

(i.e., the presence of a "gap"). Consider the single pile/soil model depicted in Figure 8.

Loading path 1-2 on the load-displacement curve shown in Figure 8a represents the

soil/pier loading curve obtained from a pushover analysis. Path 2-3 represents an

unloading path for a condition in which plastic deformation has not occurred in the pier

structure (e.g., in the piles) but for which some soil springs have sustained plastic

deformation. Upon removal of the external load, elastic strain energy (stored primarily in

the springs that have remained elastic during the loading) restore the pile to its initial

position and gaps form in the soil springs that have sustained plastic deformation. Path 2-

4 describes an alternate unloading path from a condition in which both the soil springs

and the structure itself have sustained permanent plastic deformation. In this case,

despite the restorative capacity of the elastic soil springs, a residual displacement (A4)

results from the permanent structural deformation.












Virgin load curve from
static pushover analysis


A2

twl(-ll


O
"r
(D


Formation of
soil gap


A4 Residual displacement
due to permanent
structural deformation



t-,w-- -w-I


b)
Figure 8. Depiction of pier/soil loading and unloading behavior, a) Load-displacement
curve from pushover including various unloading paths; b) Single pile and soil model
describing points on load-displacement curve


0









In the barge impact cases simulated in this thesis, plastic soil deformation is

considered in the soil response but pier-structure response is set to remain in the elastic

range. For such cases, residual structural deformations will not occur and unloading will

always occur along a path similar to path 2-3 of Figure 8. In the low resolution analysis

technique implemented here, path 2-3 has been simplified and approximated by a simple

linear secant line extending between the point of maximum sustained deformation and the

origin (Figure 8). Unloading and re-loading of the combined pier/soil system occur along

this secant line. If, after unloading, substantial reloading occurs, the pier/soil system will

reload along the current secant stiffness until the previously sustained maximum

displacement state is reached. At this point, the system will once again begin to load

along the virgin loading curve. Subsequent unloading would occur along an updated

secant line extending from the origin to the new point of maximum sustained

deformation. The same behavior is true if the structure travels in the reverse direction.

The pier/soil behavior model described above is assumed to hold true in both the

positive and negative directions. However, separate secant unloading lines are maintained

in the positive and negative directions using the maximum pier/soil displacements

sustained in each of these directions. While a more sophisticated model of unloading and

reloading could certainly be implemented (e.g., using a nonlinear unloading curve) the

simplified secant model was deemed reasonable for initial evaluation of the two-DOF

analysis technique.















CHAPTER 7
DISCUSSION OF RESULTS

Using the three levels of dynamic analyses described in the previous chapters,

barge impact simulations were conducted to evaluate the ability of the medium and low

resolution models to predict dynamic barge-pier-soil interaction. A variety of cases were

simulated for both pier-1 and pier-3 of the St. George Island Causeway Bridge in order to

cover a representative range of realistic single barge impact conditions. All of the

simulations consisted of jumbo hopper barges (of varying payloads and initial velocities)

impacting one of the pier models. In each case, the hydrodynamic mass coefficient has

been set to a unit value (CH =1) to simplify comparisons to AASHTO predicted loads.

Table 1 lists the six cases presented herein along with their corresponding impact

parameters.

It must be noted before presenting impact simulation results that, prior to

conducting the barge impact simulations, the high and medium resolution analysis

techniques were cross-validated against each other using static lateral load analyses and a

dynamic triangular pulse load analysis. Results from these cases (shown in Appendix C)

confirmed that for controlled static and dynamic loading conditions, the two models

predicted very similar responses. Any differences in predicted barge impact data are thus

related primarily to barge-pier interaction effects.










Table 1. Barge impact simulation cases.

Case Pier Model Initial velocity Payload condition Kinetic energy
Fully loaded
2.06 m/s Fully loaded 3.83 MJ
A 16.90 MN
_A (4-knots) (190 to) (2825 kip-ft)
(1900 tons)
Half loaded
1.03 m/s 0.53 MJ
B Pier-1 9.34 MN
B Pie(2-knots) 0 t) (390 kip-ft)
(1050 tons)
2.06 m/s Empty 0.40 MJ
C 1.80 MN
_C (4-knots) 1.0 (297 kip-ft)
(200 tons)
Half loaded
0.51 m/s 0.13 MJ
D 9.34 MN
(1-knot) 9.34 MN (98 kip-ft)
(1050 tons)
Half loaded
0.26 m/s 0.03 MJ
E Pier-3 9.34 MN
E P 3 (0.5-knots) 9.05 to) (24 kip-ft)
(1050 tons)
2.06 m/s Empty 0.40 MJ
F 1.80 MN
(4-knots) (20 (297 kip-ft)
(200 tons)

77.1 Comparison of Dynamic Simulation Results

Case A, a 4-knot fully loaded barge impact on pier-1, is the most severe of all the

impact cases considered here. Simulation results computed for this case using each of the

three dynamic impact analysis techniques, are shown in Figure 9. The dynamic impact

forces, shown in Figure 9a, all achieve approximately the same peak value and exhibit

substantial load for a sustained duration of time (equal to or greater than the period of

vibration, approximately T= 0.73 seconds). In order to compare the level of structural

response predicted by each analysis, the time varying pier displacements measured at the

point of barge impact, predicted by each method are compared in Figure 9b. Both the

peak displacements and the period of vibration predicted by all three methods compare

well. Barge force-deformation results (shown in Figure 9c) indicate nearly identical

predictions of maximum sustained dynamic barge crush. In addition, the energy

dissipated through plastic deformation of the barge bow (approximately equal to the area

under the force-deformation curve) is also nearly identical in all three cases.

















.. . .................. .................








.....^................................................
..
' j*


************** **^ v


0 0.25 0.5 0.75 1
Time (sec)


0 0.25 0.5 0.75


High resolution analysis
Medium resolution analysis -----+
Low resolution analysis --











................. .. ....... ......... ...... ..... ..... i .................. ..................
.......... .................
d.3







.

-i
I-


1
Time (sec)


1.25 1.5 1.75 2


1.25 1.5 1.75


1200



1000



800



600 Z



400



200



0


Figure 9. Results for Case A : 4-knot fully loaded impact on pier-1. a) Time history of
impact force; b) Time history of pier displacement; c) Barge force-deformation


.t .High resolution analysis
.................. ...... fi .. ............................. M medium resolution analysis .....+.....
Medium resolution analysis .....+.....
'i // '} 'Low resolution analysis -







t
!i \- ------






............ ................ ................... ...... ............ ..... .................. ..................



................. ..... .................. ..................
---- -- -









: :+....
Pf"~


----~ ""'


m










Barge crush depth (in)
0 10 20 30 40
6 .

High resolution analysis -- 1200
5 Medium resolution analysis ..... .....
:/ Low resolution analysis a -
1000


800




1 .--! ------ ---- -------- --- -]-An^ ----------- 0
3 ................ ... ........................ ..... ........ .. ..... ...... ....... ........................









0 0
0 200 400 600 800 1000 1200
Barge crush depth (mm)

c)

Figure 9. Continued

Whereas Case A is representative of a high energy impact condition, Case B and

C represent less severe impact conditions. The intent in conducting the simulations for

Case B and C is to evaluate the abilities of the various models to predict less-sustained,

more transient loading conditions. Predicted force histories for Case B and C are plotted

in Figure 10. Case B, a 2-knot half loaded impact condition (Figure 10a), produces an

oscillatory loading history, while case C, a 4-knot empty condition (Figure 10b), has an

even more transient short-term force history. For both of these cases, all three analyses

peak at approximately the same time and load level and exhibit time-variation of loading,

indicating good agreement in terms of predicted dynamic interaction between the barge

and the pier.














800




600




400




200


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


0 0.25 0.5 0.75


1
Time (sec)


ouu

500

400

300

200

100


1.25 1.5 1.75 2


Figure 10. Time histories of impact force on pier-1. a) Case B : a 2-knot half loaded
barge impact; b) Case C : a 4-knot empty loaded barge impact









Case D, E, and F correspond to impacts on pier-3, the less impact resistant pier.

Due to the increased flexibility of pier-3, substantial dynamic interaction occurs between

the barge and the pier and, as a result, the dynamic loads imparted to the structure are

highly oscillatory in nature. In Figure la, predicted impact loads for Case D are

presented. All three simulations agree with respect to the initial peak load predicted and

the subsequent decrease to zero load (at approximately t= 0.11 seconds). Following this

point, the predicted impact loads differ but all exhibit a similar highly oscillatory

characteristic. However, of equal importance to the nature of the dynamic load is the

structural response of the pier to that applied loading. Despite the differences in the

predicted dynamic loading, the pier displacements predicted by each analysis technique

(shown in Figure 1 Ib) are in reasonable correlation. Thus, at least based on this measure

of structural response, the structural severities of impact predicted by all three methods

tend to be in agreement.

Results obtained for Cases E and F exhibited similar characteristics to those

shown for Case D. Detailed simulation results for Cases A through F are shown in

Appendix D. In Cases D, E, and F, very little barge deformation occurred during impact

Due to the flexible nature of pier-3. As a result, negligible enery dissipation occurred

through barge crushing for these cases.

Results for Cases A through F are summarized in Table 2 where peak (maximum

magnitude) dynamic forces, pier displacements, and barge crush deformations are

reported. For the six cases listed, the medium and low resolution analysis techniques

predict peak data that are in sufficient agreement with the values predicted by the high

resolution analysis.











3

High resolution analysis -
Medium resolution analysis +..
2.5 .................................... ................... ...... Low resolution analysis -

2 .
2 ... .. ............ ... .. ..... ..... .. ...... ...... ... .... .@ ...... ..... .... ....................... .....................................
\ I I i ; A i : i ,


1 .5 ..... ........... .... .. ..... ... ................ .................................

A%


0.25 0.5 0.75
Time (sec)


0 0.25 0.5 0.75


1 1.25 1.5 1.75 2
Time (sec)

b)


Figure 11. Results for Case D : 1-knot half loaded impact on pier-3. a) Time history of
impact force; b) Time history of pier displacement


600


500


400


300


200 E


100


0


High resolution analysis
.................. ....... M edium resolution analysis ....-+..... ..
Low resolution analysis -




..... ../. .
.............. ...................... ..... ................. .................. .................. ................. .................




............. .. .... .......: ....... .
... ............... ................... ..... ........ ....... .....




------------------------------ -- ---------********--- ************ ** **** --- ----



I^ I II


50

.- 25


4


3


2


a



0


-1










Table 2. Peak impact loads, pier displacements, and barge crushes


High resolution analysis Medium resolution analysis Low resolution analysis

Peak Peak Peak Peak Peak Peak Peak Peak Peak
Case
impact pier barge impact pier barge impact pier barge
load disp. crush load disp. crush load disp. crush
MN mm mm MN mm mm MN mm mm
(kip) (in) (in) (kip) (in) (in) (kip) (in) (in)
4.67 143.04 1024.75 5.12 156.46 929.64 5.08 154.05 1015.21
A
(1050) (5.64) (40.34) (1150) (6.16) (36.60) (1142) (6.07) (39.97)
3.38 84.31 101.28 3.75 91.70 72.64 3.50 92.28 80.52
B
(761) (3.32) (3.99) (844) (3.61) (2.86) (786) (3.64) (3.17)
3.71 51.54 106.42 3.74 49.28 84.58 3.54 50.32 94.82
C
(824) (2.03) (4.19) (841) (1.94) (3.33) (796) (1.98) (3.73)
2.37 109.26 15.13 2.37 101.35 11.28 2.53 98.22 20.77
D
(534) (4.30) (0.60) (532) (4.00) (0.44) (570) (3.87) (0.82)
1.56 41.60 4.90 1.65 47.75 5.49 2.20 49.07 9.26
E
(352) (1.64) (0.19) (372) (1.88) (0.22) (494) (1.93) (0.36)
3.07 170.98 65.62 2.85 136.65 66.55 2.78 132.00 95.43
F
(690) (6.74) (2.58) (640) (5.38) (2.62) (625) (5.20) (3.76)


7.2 Comparison of Dynamic and Equivalent Static Impact Loads

Based on the comparisons presented above, it may be concluded that design-

oriented dynamic analysis techniques offer the promise of alternative approaches to

designing bridge piers for barge impact conditions. While additional research and

validation efforts are needed before dynamic analysis procedures can serve in lieu of

code-based load determination methodologies (e.g., the AASHTO provisions), in

concept, at least, such approaches seem viable and even desirable.

In this section, impact loads and structural response parameters computed using

dynamic analysis techniques are compared to data predicted by implementing the

AASHTO equivalent static load computation approach. For purposes of comparing

dynamic and equivalent static methods, it is preferable to use the most accurate dynamic

data available. In this study, the high resolution dynamic analysis technique discussed










earlier yields the most accurate data (although similar results have been obtained from

medium and low resolution analyses as was demonstrated above). Results for a total of

seven high resolution simulations are considered : Cases A-F previously cited plus an

additional intermediate impact-energy condition (Case G : pier-1, 4-knots, half loaded).

The first comparison presented here focuses on predicted impact loads. In

Figure 12, peak dynamic loads from high resolution analyses are compared to equivalent

static loads computed using the AASHTO provisions (Eqs. 1, 3, 4). It is important to

note that the AASHTO equivalent static loads are unfactored in the sense that they are

associated with only the barge event, not with the probability that the event will occur.

Results shown in the figure indicate that for the higher energy impact conditions,

AASHTO predicts loads that are greater than the peak dynamic loads predicted by the

simulations.

Kinetic energy (kip-ft)
0 1000 2000 3000


S ............................ .............. .............................................. .... .................... ............. 1 6 0 0
8...--7---. 1800
7 C------------- i-------i-----_~..........~......r 1600

1400

1200
5~~~lo -- ----
1000

S.________ 800
S. -- AASHTO Relationship
S..............................................Pier-1,4 knots fully loaded 600
B Pier-1, 2 knots, half loaded
2D .......... ... ........... ... C Pier-1, 4 knots, empty
D Pier-3, 1 knots, half loaded 400
E Pier-3, 0.5 knots, half loaded
..................................................................... F Pier-3, 4 knots, em pty 200
G Pier-1, 4 knots, half loaded

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Kinetic energy (MN-m)

Figure 12. Relationship between barge impact force and kinetic energy









For lower energy impact conditions, Figure 12 indicates peak dynamically

predicted loads that are greater than those predicted by AASHTO. However, these loads

tend to be transient or highly oscillatory in nature, as the time histories of impact force

presented in the previous section revealed. Therefore, comparison to equivalent static

loads is unsuitable. Instead, an alternate approach is required in which structural

responses to impact loading (dynamic or static) are compared. Since pile forces

(moments, shears, etc.) are closely linked to lateral pier displacements, comparing

maximum sustained pier displacements provides a means by which static and dynamic

analysis predictions can be duly compared.

To enable such a comparison, maximum sustained pier displacements were

determined for each impact case from the high resolution dynamic analyses. Next, a

large number of hypothetical impact conditions were chosen with a distribution of impact

energies that covered the range indicated in Figure 12. For each hypothetical impact

condition, an AASHTO equivalent static load was computed using Eqs. 3 and 4. These

loads were then applied in a static sense to each of the high resolution pier models so that

the resulting static pier displacements could be determined. Since the high resolution

models are dynamic models intended for analysis using LS-DYNA, the static load

application was achieved by applying the loads slowly enough so as to eliminate all

inertial (dynamic) effects. To cross-check the validity of the results obtained by this

procedure, pier displacement data were also computed by conducting true static analyses

on the medium resolution pier models (using FB-Pier in static analysis mode). Results

obtained from the pseudo-static analyses and the true static analyses were virtually

identical.











Maximum sustained pier displacement data for impact conditions on the relatively

stiff pier-1 are shown in Figure 13a. In general, data predicted by both the dynamic and

static analysis procedures agree well. In all cases, the AASHTO equivalent static loads

produce conservative predictions of structural response. In Figure 13b, results for several

impact conditions on the more flexible pier-3 are presented. Several discrepancies

between the dynamic and static analysis results are noted. For the two lowest energy

impact conditions (cases D and E), dynamic analysis predicts pier displacements

significantly in excess of those predicted by application of the AASHTO static loads.

Conversely, the static AASHTO load application results in very conservative predictions

of pier response for the higher energy impact conditions (e.g., case F).


Kinetic energy (kip-ft)
0 1000 2000 3000
175

15 0 ............... .. ............. ... 6









Displacement produced by
HD .. application of AASHTO 2
150 ------- ------------------------- ---






equivalent static load
A Pier- 1, 4-knots, fully loaded
25 B Pier-1, 2-knots, half loaded
C Pier-1, 4-knots, empty
G Pier-1, 4-knots, half loaded
0 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Kinetic energy (MJ)

a)

Figure 13. Relationship between pier displacement and kinetic energy. a) Pier-i results;
b) Pier-3 results










Kinetic energy (kip-ft)
0 100 200 300
600

Displacement produced by
500 application of AASHTO 20
equivalent static load
D Pier-3, 1-knot, half loaded
E Pier-3, 0.5-knots, half loaded
S 400 ...... F Pier-3, 4-knots, empty................
g 15

300 5


s 200


100


0 0i i
0 0.075 0.15 0.225 0.3 0.375 0.45
Kinetic energy (MJ)

b)

Figure 13. Continued

The variable nature of the discrepancies observed in the pier-3 comparisons

derives from the presence of significant dynamic interaction between the barge and pier

and the resulting oscillatory nature of the imparted impact loads. In such circumstances,

equivalent static loads are not able to accurately predict the severity of the structural

response and the use of rational dynamic analysis techniques is recommended.















CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS

Three levels of dynamic analysis techniques have been presented as alternatives

to the current AASHTO equivalent static barge impact load computation procedures. A

high resolution analysis technique utilizing a nonlinear finite element impact simulation

has been presented as a means of achieving accurate prediction of barge and pier

response during barge collision events. However, this technique is generally not

appropriate for routine use in bridge design due to the substantial computational

resources needed to conduct such simulations.

A medium resolution analysis technique has been developed by modifying a

commercially available dynamic pier analysis program, while a very low resolution

analysis technique has been implemented as a spreadsheet program. Both of these

techniques utilize barge load-deformation relationships that have been obtained from

high resolution nonlinear contact finite element simulations.

Barge impact simulations have been presented for six different impact conditions

on models of an impact resistant pier and a non-impact resistant pier. The accuracy of

the medium and low resolution analysis techniques has been evaluated by comparing

results produced by these procedures to results obtained from high resolution analyses.

Results presented in the comparisons reveal that the medium and low resolution models

are capable of simulating dynamic barge-pier interaction during a collision event. In

addition, both methods have been shown to produce impact force time histories that are









sufficiently accurate for bridge design purposes. In the future, improvements may be

made to the low resolution pier/soil model and to the single DOF barge model to improve

their representation of unloading behavior.

Comparisons have also been made between dynamic and static load and

displacement prediction methods. Impact force and pier displacement data have been

computed using the AASHTO barge impact provisions and compared to dynamically

predicted data for several different impact scenarios. These comparisons demonstrate

that a static load approach is acceptable-although slightly conservative-in cases where

the impact event produces sustained barge-pier interaction. The comparisons also

demonstrate that there are circumstances in which a dynamic analysis should be strongly

considered as a design alternative.















APPENDIX A
SMOOTH HIGH RESOLUTION CRUSH DATA

In this appendix, a Mathcad program is presented that documents the technique

used to smooth the high resolution crush data. The smoothed crush data is then applied

to the medium resolution model to represent the nonlinear behavior of the barge. The

specific example presented here is a 6 ft. square pier statically crushing the bow of a

hopper barge.











Smoothed Crush Data


7 := ORIGIN


6 ft. Square Crush Test Data LS-DYNA


0 1
0 0 0
1 0.11 264.2
2 0.17 437.34
3 0.32 564.9
4 0.41 628.18
5 0.44 691.02
6 0.53 748.34
7 0.61 796.52
8 0.68 823.41
9 0.77 828.45


...import 6ft_crush_fast_static.txt


Ac := crushData

Pc:= crushData +1"

Ac.max:= maAc)


...deflection (in)

...load (kips)


Ac.max= 34.226


n := last(Ac)


LS-DYNA 6ft Square Crush Test Data


1500


1000


5 10 15 20
Deflection (in)


25 30 35


ORIGIN-- 0


crushData :













Smoothed Crush Data
bwl := .2

PcSmooth := ksmooth(Ac,Pc,bwl)


1500


bw2:= 2

PcSmooth2:= ksmooth(Ac,Pc,bw2)


LS-DYNA 6ft Square Crush Test Data


1000





500
500


5 10 15 20 25 30
Deflection (in)


'I '.


'A
-N
'--1 -


I.- -











Simplify number of data points :

simpCrush:= ncSimpl 8 ...number of desired crush data points
Simp2 42 cannot exceed 200 for FB-Pier
ncSimp2 <- 42
A.critical .571
"set initial starting point to 0"
AcSimple -- 0
PcSimple -- 0

for ie (E + 1).. (ncSimpl- 1 +)

A.critical +
AcSimplq e- *(ti+ J
A(ncSimpl 1 + E)
PcSimple *- linterp(Ac, PcSmoothl,AcSimplq)

for i e (ncSimpl + E).. (ncSimp2 + ncSimpl 1 + E)

(Ac.max- Acritical)
AcSimplq -- .(i + 1 ncSimpl) + A.critical
(ncSimp2)
PcSimple .- linterp(Ac, PcSmooth2, AcSimplq)

Eur_x- (0 AcSimplecSimpl-1+E )T

Eury -(0 PcSimple cSipl+E )T

Eur -- Eur_y+ Eur x +-

return (AcSimple PcSimple Eur Eur x Eury)


(AcSimple PcSimple Eur Eurx Eur_y):= simpCrush


5 10 15 20
Deflection (in)


Simplified Crush Data
1500 i I I I


25 30 35











Eur:= ceil(Eur)

aby :=Eur_x(+1)


Eur= 1.34x 103 ...(kips/in) approximate unloading slope


|aby =0.571


...(in) approximate yield point (pick from plot and output)


*Note: Make sure unloading curve does not have a negative or 0 x-intercept

FBPiercrush := augment (AcSimple, PcSimple)


0 1
0 0


0.082
0.163
0.245
0.326
0.408
0.489
0.571
1.372
2.174
2.975
3.776
4.578
5.379
6.18
6.981


232.88
357.686
478.629
593.256
652.247
708.631
764.615
792.731
806.775
801.661
811.33
825.896
844.532
873.079
910.025


...export into "xportsmth_6ft sq_crush.txt"
and then copy to FB-Pier


FBPier crush















APPENDIX B
LOW RESOLUTION ANALYSIS PROGRAM

In this appendix, a Mathcad program is presented that documents the

implementation of the low-order analysis technique described earlier in this thesis. The

program was co-developed with Mr. David Cowan and his contributions are gratefully

acknowledged.

The specific example presented here is for impact Case A, the 4-knot fully loaded

barge impact on the impact resistant pier, pier-1.












Low Resolution Barge-Pier Interaction Model

Pier-1 : 4-knot, fully loaded


Define Units:

ORIGIN:= 0

kips 1000. lbf

Input Parameters :

Force...


E := ORIGIN

tons 20001bf


knot 1.687809857 ft + s


F:= )kips


...applied load (not time dependent)


Load-Displacement Data...

|Eur:= 1340kips id ...barge crush plastic unloading slope


Import barge_crush data (load-deformation) :


0 1
0 0 0
1 0.08 232.88
2 0.16 357.69
3 0.24 478.63
4 0.33 593.26
5 0.41 652.25
6 0.49 708.63
7 0.57 764.62
8 1.37 792.73
9 2.17 806.77
10 2.97 801.66
11 3.78 811.33
12 4.58 825.9
13 5.38 844.53
14 6.18 873.08
15 6.98 910.02


Import pier_pushover data (load-displacement)


pier :


0 1
0 0 0
1 0.02 47.5
2 0.05 72.5
3 0.11 97.5
4 0.19 122.5
5 0.26 147.5
6 0.32 172.5
7 0.37 197.5
8 0.41 222.5
9 0.44 247.5
10 0.47 272.5
11 0.5 297.5
12 0.55 322.5
13 0.62 347.5
14 0.69 372.5
15 0.78 397.5


Contact Spring :


Ac := barge in

Pc := barge kips


Soil Spring :

As := pier -in

Ps pie +1 .kips
Ps :=pier -kips


rad 1


barge :















Load-displacement data plots...



1200



1000



800

C- A
600



400



200



0
0 5 1






2000





1500





1000











0
0 1


0 15 20 25 30 35 40 45
Crush (in)


2 3 4 5 6 7 8 9
Displacement (in)


(used to define nonlinearity of pier/soil spring)











Mass-Barge...

|dwt := 1900tons|


mb:= dwt g


...dead weight tonnage

2
mb = 9.8423 ips's ...mass of barge
in


Mass-Pier..


2
07 kips. s
p c := 2.25 10
.4
m


2
7.34510 07 kips s
.4
m


...mass density of concrete and steel


nPile:= 3

.pile:= 21.09in

L.pile:= 50.5



A.pier:= 66.16in-77.61ir
Lr:=44.1146fl
L.pier := 44.1146fF

A.pileCap:= 822.5ft1

Itcap 1 := 5

A.pierap:= 31.354fti
LpierCap j
L.pierCap := 29.83 1

A.sh:= 60 ft2

L.sh=16.9f

mp:=p .c[(A.pileCap.t.cap) + (nPier-A
+ p .s (nPile.A.pile L.pil)


mp = 3.918kipss
m


Damping...

0 0 kips-s
C: 0) in


...number of piles

...area of HP14x73

...length of pile

...number of piers
...average area of pier

...height of pier from top of pile cap to bottom of pier cap


...area of pile cap

...thickness of pile cap


...area of pier cap

...length of pier cap


...area of shear wall

...length of shear wall (between inside of columns)


.pierL.pier) + (A.pierCapL.pierCap) + (A.shr L.shr) -.



...mass of the Pier


...system damping matrix


P.s :












Timing Considerations...

At:= 0.001s ...time step


Initial Conditions...

vo := 4. knot ...initial barge


vo = 81.0149in s

Normalize Input:

Normalizing Parameters...

2
kips-s
masu:= foru := kips
in

Normalize Variables...

At F
At:= F:=
timu foru


Pc Ac
Pc := Ac := -
foru disu


timu tin
C := C-- vo := vo --
masu dis


Build Matricies :

Mass...


(mb 0 .s
M := mp...sy
M 0 mprn

Stiffness...

Pc Ps_
+1 +1
kco := kso :=-
AcE +1 AsE +


kc := Eur + stifu


kco -kco
Ko:=
.-kco kco + kso)


|n := 200(


...number of points


velocity


timu:= sec disu := in


mb
mb:=-
masu


Ps
Ps:.
foru


Eur
Eur:=
stifu


stifu := kips + in


mp
mp :=
masu


As
As:-
disu


stem mass matrix


...(kip/in) initial barge crush stiffness and soil spring stiffness



...(kip/in) barge crush unloading curve



...initial system stiffness matrix


U
Uu










Time...

t:= for i~ ..(n +)
t. <- i-At

t


Initial Conditions...


xo:= ) ... disi


Timing Considerations...


o := genvals(Ko, M)



checkAt(At) := Atma

check
check
return


placement


vo := "




= 40.1299)
C 11.401)


.x <- mi -)
100)
e- "Time Step OK" if At < Atmax
e- "Error : Reduce Time Step" otherwise
check


Central Difference Method :

out:= CD((Eur Pc Ac Ps As xo vo At n M C F 0 0))

xb:= outT) ...barge x-displacement xp:= outT
v ( +2) ( ) vb := outT ) ...barge x-velocity vp := outT +
( <+4) () T +5)
ab :=out T) ...barge x-acceleration ap := outT +

Fb :=( S+6> ( <)S+7,
Fb:= out ) ...barge x-force Fp:= outT

Calculate barge crush...

c(xb, xp) := for ie ..(rows(xb) 1 +)
crush (xb xp) if xbi xp) > 0

crush. <- 0 otherwise

return crush

crush := c(xb, xp)


...pier x-displacement

...pier x-velocity

...pier x-acceleration

...pier x-force


...velocity





...natural frequency


...check time step size












Plot Results :


40






I 20






0


0 0.2 0.4 0.6 0.8

Barge Displacement
..... Pier Displacement


1 1.2 1.4 1.6 1.8
Time (sec)


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time (sec)


Displacement History


Force History


1200



1000



800

C-1
.-
) 600



400



200



0
0



















1000





C-



S500









0
0 5 10 15 20
Crush (in)


C--
0
s
Y, Ep
C)
C)
C4


-1 0 1 2 3

xp
Displacement (in)


25 30 35 40


4 5 6 7


Write Results :

data := augment (t, -Fb, crush, xb, xp)


WRITEPRI("OUT_pl_4kn_full_2dofprn" ) := data


Force-deformation


Soil Spring











Central Difference Function:


CD(Pc,Ac,Ps, As,xo, vo, At,n,M, C,F, gapc, gaps)

*Pc and Dc, and Ps and Ds represent the load-deflection values for
the contact and soil springs respectively
*xo and vo are the initial displacement (typically 0) and initial
velocity of the two masses
*Dt is the time step and n is the number of time steps in the
simulation
*M and C are the mass and damping matricies
*F is the force (static) applied to each mass
*gapc and gaps are the sizes of the initial gaps (typically 0) for the
contact and soil springs respectively

2D(argIn) "Central Difference Algorithm"
"Initial Stiffness"

(Eur Pc Ac Ps As xo vo At n M C F gapc gaps) <- argIn
S-- ORIGIN
Pc+
kc <- -



ks --P
AsE+1
(kc -kc "
-kc kc + ks)
"Initial Acceleration"
1
ao <- --(F C-vo K.xo)
M
"Contact Spring"
PcY Pc
< +1
AcY <- Ac +1

PcY
kcv <- -
AcY

for ie (E + l)..last(Pc)
Pc. Pc.
1 1-1
kcv. --
1 Aci Aci-1












"Soil Spring Positive Loading"

gapsf <- gaps

PsYf Ps

AsYf <- AsE +1

PsYf
ksvf <- ---
S AsYf

for i E + 1.. last(Ps)

Ps. Psi-
1- 1
ksvf. --
1 Asi Asi_1

"Soil Spring Negative Loading"

gapsb <- gaps

PsYb ---Ps
+1

AsYb <- -As +1

PsYb
ksvb ---
AsYb

for i E + 1.. last(Ps)

Ps.- Ps.
1- 1
ksvb. ---
1 Asi Asi_1

"Displacement at Back Time Step"

2
ao At
xp <- xo vo-At + -

"Effective Mass"

M C
m <--+-
At2 2.At

"Displacement for the First Time Step"

for je .. + 1

x. -- xo.
), = J

x temp <- m F K -xo ( -xp


for j e .. + 1

x. ,E- x temp.

xpp <- xo
x_p <- xtemp

for ie (E + 2)..n











"Displacement at the ith Time Step"
xtemp 1 2-M" M C
x temp <- m_ F K '-x_p -- xpp
At2) At2 2 At
"Contact Spring"
A_c <- xtemp, xtemp +1

if A_c < gapc
kc 0
kc -kc "
-kc kc + ks)
if A_c gapc A Ac < AcY
P_c <- kcv, -(A_c gapc) if gapc = 0

P_c <- Eur (A_c gapc) otherwise
Pc
kc <- -
Ac
kc -kc "
K-
-kc kc+ ks)
for y (E + ).. last(Ac) if A_c > AcY
if A_c < Acy

PcY PcY + kcv (A_c AcY)

AcY <- A c
PcY
kc <- -
AcY
PcY
gapc <- AcY -
Eur
kc -kc
K-
K -kc kc + ks)
break











"Soil Spring Positive Loading"
A_s -- xtemp,+

if A_s < gapsf A A_s > 0
ks 0
( kc -kc
-kc kc + ks)
if A_s > gapsf A A_s < AsYf
P_sf -- ksvf (A_s gapsf)

P sf
ks <- -
As
Skc -kc >
K -
-kc kc + ks)
for ye (E + 1)..last(As)
if A_s < Asy

PsYf PsYf + ksvf (As AsYf)
y
AsYf +- s
PsYf
ks <-- Psf
AsYf
ksvf, <- ks

PsYf
gapsf <- AsYf -
ksvf,

kc -kc )
K -
-kc kc + ks)
break
"Soil Spring Negative Loading"
if A_s > gapsb A A_s < 0
ks <- 0
Skc -kc
\-kc kc + ks)

if A_s < gapsb A A_s > AsYb

P_sb <- ksvb *(As gapsb)


P sb
ks <- -
As
(kc
K -- -kc
1-kc


if A s > AsYf


-kc )
kc + ks)












for y e (E + 1).. last(As) if As < AsYb

if A_s > -Asy

PsYb PsYb + ksvb (A_s -AsYb)

AsYb -- A s
PsYb
ks <- P--
AsYb

ksvb, <- ks

PsYb
gapsb <- AsYb
ksvb ,

kc -kc )
K -
-kc kc + ks)
break
"Derivatives"

for je .. (E + 1)
x. x temp.
J,i J
x. x. .-
j,i j,i-2
X 4-
J+2,i-1 2. At

x. 2 x. + x.
J,i-1 Ji-2
X 4-
j+4,i-1 2
At
"Force"

f-temp <-- kc (Xl, i-1 I- x0, i-I1)

ks xl, i-1

for j 6.. 7
x. ftemp.
x.,i-1 +- f-tempj-6

xpp -- xp
x_p -- xtemp

for ie E..7

for je S ..(n- 1 + E)
out. x .
1,J 1,J










Formation of gap element:

Initially:
gap = 0
Loop:
if A< gap
k=0
if gap A < Ay
P= kE (A-gap)

k=
A
The stiffness used in the elastic region cannot be the elastic stiffness due to
the gap (Point 1).
if Ay < A
Py =Py +kp .A-Ay)
Ay =A

PY
Ay


gap= A kE
y E
The yield point and the gap are updated each time (basically the elastic region is
shifted).
The stiffness used in the plastic region cannot be the plastic stiffness due to the
stiffness change from elastic to plastic (Point 2).



Stiffness for Point 1 if Elastic Stiffness is used

I Updated Yield Point
Initial Yield Point Point 2

P
/Point 1




/ Stiffness for Point 2 if Plastic Stiffness is used


Gap updated















APPENDIX C
MEDIUM RESOLUTION VALIDATION CASES

In this appendix, validation simulations are presented for the medium resolution

analysis models discussed earlier in this thesis. Medium resolution analysis results for

both pier-1 and pier-3 are validated against corresponding high resolution analysis

results. Both a pseudo-static lateral load analysis and a triangular pulse load analysis are

conducted to confirm that similar pier responses are predicted by the medium and high

resolution analysis techniques.












Validation Case: Static lateral load analysis

Displacement (in)
0 1 2 3 4 5 6
9

8

7-

6 I



4-

/
------ ----- ---------/-----


5 --- --- -- --
0 .4 ------------------. ........


7 8 9


50 100 150 200
Displacement (mm)


High Resolution Medium Resolution



Figure C-1. Pier-1 static lateral load analysis


Displacement (in)
0 1 2 3 4 5 6






--------------- -- -------- ----- ---------------- ------------ -- ------ ---- -------------



--------------- i---- ----------~ --- /------ - - - - - - - - - - -
- - - - - - - --.. . - - - - - - - - - - - - - - - - - - . . . . .


- /
.. . .. . .. / / .. . .. . .. . .. . .. . .. . .. . .. . .


. . L . . .L . . .L . . . . . . . /. . .

( "


) I


0 25 50 75 100
Displacement (mm)


High Resolution -


Figure C-2. Pier-3 static lateral


125 150



Medium Resolution


load analysis


2000




1500




1000


500
0


500


S0
250















500


400


300 :


200


100


0
175











Validation Case: Triangular pulse load analysis


0.5 1 1.5
Time (sec)


High resolution analysis


Medium resolution analysis


Figure C-3. Pier-1 pulse load analysis


-50


-75


-100


3


2


1 -


0

1 -

-2


-3
-3















APPENDIX D
COMPLETE DYNAMIC ANALYSIS RESULTS

In this appendix, complete dynamic analysis results are presented for each the six

simulation cases discussed briefly within the body of this thesis. For each included here,

the time history of impact force, time history of pier displacement, and barge force-

deformation relationships are presented in full.













A
1
4-knot
Fully loaded


i I,, i


0.25 0.5 0.75 1
Time (sec)


High resolution analysis
Medium resolution analysis


1.25 1.5 1.75-
1.25 1.5 1.75


Low resolution analysis


Figure D-1. Time history of impact force


Case :
Pier:
Velocity :
Payload :


1200


1000


800
0
600


400


200



2






























--........- ---.. .. .........-- ---

EL El
Uj.n- 7


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


High resolution analysis Low resolution analysis -----
Medium resolution analysis


Figure D-2. Time history of pier displacement


Barge crush depth (in)
0 10 20 30 40


1200


1000


t ------------ --
V ^ _A^
Vl" Vt fl.I
; ; ^ a


0 200


400 600 800
Barge crush depth (mm)


High resolution analysis
Medium resolution analysis


4



-- -- na~-------- 0
2(



1000 1200



Low resolution analysis ---


Figure D-3. Impact force vs. barge crush


6


5



1



7




-0


00


00
)o

JO L


8(


S6(










B
1
2-knot
Half loaded


0.25 0.5 0.75 1
Time (sec)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-4. Time history of impact force


Case :
Pier:
Velocity :
Payload :


-z
500

400

300

200

100

0















3



2






0
-1
Co

-Q


0.5 1 1.5 2
Time (sec)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-514. Time history of pier displacement


Barge crush depth (in)
2


25 50 75 100
Barge crush depth (mm)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-6. Impact force vs. barge crush


1000



800



600 *

,1
400



200



0































r1n


m \


0.25


High resolution
Medium resolution


Low resolution


Figure D-7. Time history of impact force


Case :
Pier:
Velocity :
Payload :


C
1
4-knot
Empty


i -


-z
500

400

300


0.5 0.75 1
Time (sec)






















25



.0 -'----- -


-25 -


0 0.25 0.5
Time (sec)


High resolution analysis
Medium resolution analysis


-------------- --------------------------





0.75



Low resolution analysis


Figure D-8. Time history of pier displacement


Barge crush depth (m)
2 3


----.L................-.......HB.I.-- ------------r---I----- -------
25 50 75 100
Barge crush depth (mm)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-9. Impact force vs. barge crush


.a] "
*3 -if


2













-1
a,
a,




ca

o i


-1


1000



800



600

I-
400



200


"~











Case :
Pier:
Velocity :
Payload :


3


2.5


D
3
1-knot
Half loaded


600


500


400
t0
O
300


200


0 0.25 0.5 0.75 1
Time (sec)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-10. Time history of impact force















4


3


2


0

-T



-1


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


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-11. Time history of pier displacement


Barge crush depth (in)
0 0.1 0.2 0.3 0.4 0.5 0.6


0.8 0.9


I//


400 1
0
300
200

200 e


10 15
Barge crush depth (mm)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-12. Impact force vs. barge crush










Case :
Pier:
Velocity :
Payload :


2.5



2


E
3
0.5-knot
Half loaded


500
------------------ ;------------- 7 ---- - - - - -- - - - - - - - - -

S- 400


11- 300 o


7- 200



'! u 1 0 0



0 0.5 1 1.5 2
Time (sec)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-13. Time history of impact force















1.5



1



0.5



-0.5


-0.5


0.5 1 1.5 2
Time (sec)


High resolution analysis
Medium resolution analysis


Low resolution analysis


Figure D-14. Time history of pier displacement


Barge crush depth (in)
0.2


500


400


1 300

200


3'4-. 6,,'


2 3 4 5 6
Barge crush depth (mm)


High resolution analysis
Medium resolution analysis


7 8 9 10



Low resolution analysis ---


Figure D-15. Impact force vs. barge crush


7
-


0 1











Case :
Pier:
Velocity :
Payload :


3.5


3


2.5


- .. ., ,' ..... .... .. .. .......... ... ... .
0 0.25 0.5 0.75
Time (sec)


High resolution
Medium resolution


Low resolution


Figure D-16. Time history of impact force


F
3
4-knot
Empty


700

600

500
-z
400

300 |

200

100

0















6



4



0

a-2

0


-2


0 0.25 0.5 0.75
Time (sec)


High resolution
Medium resolution


Low resolution


Figure D-17. Time history of pier displacement


Barge crush depth (in)
2


0 10 20 30


------- --- a-----














40 50 60
Barge crush depth (mm)


700

u^-J -: 600
t -- ----.. o

S500

S-400 8

300
i i e

200


-a------------ -^-----'-------- ----------
mo ICI El ii 0
70 80 90 100


High resolution
Medium resolution


Low resolution


Figure D-18. Impact force vs. barge crush















REFERENCES


1. Frandsen, A.G. and H. Langso. "Ship Collision Problems, Great Belt Bridge,
International Enquiry." IABSE Proceedings. Zurich, Switzerland: International
Association of Bridge and Structural Engineering, 1980 : P31/80.

2. AASHTO. Guide Specification and Commentary for Vessel Collision Design of
Highway Bridges. Washington, DC : American Association of State Highway and
Transportation Officials, 1991.

3. "Towboats and Bridges : A Dangerous Mix." GCMA Report #R-293, Revision 2.
2002. Gulf Coast Mariners Association. 293r2.htm>.

4. Whitney, M.W., Harik, I., Griffin, J., and Allen, D. "Barge Impact Loads for the
Maysville Bridge." Interim Research Report. Kentucky Transportation Center, 1994
: KTC-94-6.

5. Consolazio, G.R., Lehr, G.B., and McVay, M.C. "Dynamic Finite Element Analysis
of Vessel-Pier-Soil Interaction During Barge Impact Events." Journal of the
Transportation Research Board, Washington, D.C., 2003 (In press).

6. Consolazio, G. R. and Cowan, D.R. "Nonlinear Analysis of Barge Crush Behavior
and its Relationship to Impact Resistant Bridge Design." Computers & Structures
81 :547-557 (2003).

7. Livermore Software Technology Corporation (LSTC). LS-DYNA Theoretical
Manual. Livermore, CA, 1998.

8. Florida Bridge Software Institute. FB-PIER Users Manual. University of Florida,
Department of Civil & Coastal Engineering, 2000.

9. Hoit, M.I., McVay, M., Hays, C., and Andrade, W. "Nonlinear Pile Foundation
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BIOGRAPHICAL SKETCH


The author was born on October 27, 1979, in Tampa, Florida. After graduating as

valedictorian of Hillsborough High School's class of 1997 in Tampa, she attended

Florida State University where she received her Bachelor of Science degree in Civil

Engineering (graduating magna cum laude) in August 2001. She then began graduate

school at the University of Florida in the College of Engineering, Department of Civil

and Coastal Engineering. The author plans to receive her Master of Engineering degree

in August 2003, with a concentration in structural engineering. She will begin her

professional career as a bridge designer with Figg Bridge Engineers of Tallahassee,

Florida in May 2003 after successful completion of her graduate work.