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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 conditionsevents 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 designoriented 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 designoriented 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 bargepier 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 thirtyfive 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 fourbarge 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 140 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 multibarge 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 MeirDomberg (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 codespecified equivalent static loads. However, this is a new and emerging area of bridgerelated structural research. Recent analytical studies focusing on barge impact [5,6] have involved the development and use of highresolution 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 mechanicsthey are not generally appropriate for use in routine bridge design due to the considerable computational resources often required. In this thesis, designoriented medium and low resolution dynamic analysis techniques are developed that approximate the behavior of more refined high resolution bargepier 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. LSDYNA [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. FBPier [8,9]). * Low resolution Simple, loworder 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, designoriented 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 loaddeformation 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 loaddeformation 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 pseudostatic. 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 nonimpact 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 worstcase 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 worstcase scenario of a headon 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 MeirDornberg. 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 semianalytical approach was used based on data from twentysix 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 twentyfour 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 massidealization of the shippier 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 MeirDomberg [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 fullscale 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 bargetobarge interaction in a flotilla during a collision, and to investigate a new energyabsorbing 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 bargewall interaction. Following the prototype tests, fullscale experiments were conducted on the Robert C. Byrd Lock and Dam in Gallipolis, West Virginia in 1999 [16]. These experiments consisted of a fifteenbarge flotilla of jumbo openhopper barges impacting the lock wall with and without a fendering system. There were fortyfour 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 fullscale plastic range experiments that will be used to determine actual crushing strength and performance of inland waterway barges. Currently, an ongoing 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 designoriented impactload 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 fullscale barge impact tests on the older structure before it is demolished. After the new bridge has been opened to traffic, a fullscale 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 generatedsustained versus highly oscillatorywas 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 contactimpact finite element analysis software and refined barge, pier, and soilstructure 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 LSDYNA [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 headon, perpendicular manner (Figure 2). The jumbo class hopper barge59.5 m long, 10.7 m wide, 3.7 m deep, 1.8 MN empty weight, 6.9 MN fully loaded weightmakes 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 stressstrain 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 Pier3 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, pieri and pier3 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. Pier1 (adjacent to the navigation channel) and pier3 (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. Pier1 is considered a typical example of an impact resistant pier because of its size and stiffness, while pier3 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. Soilstructure interaction effects between the piles and the surrounding soil including nonlinear soil response, plastic deformation, gap formation, and pilegroup effectswere modeled using thousands of nonlinear lateral and vertical soil spring elements. Unique loaddeformation 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 multiDOF nonlinear dynamic pier analysis program (FBPier [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 forcedeformation 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 precomputed barge crush curve is employed to model the stiffness of the barge bow the medium resolution impact analyses (Figure 3). Barge forcedeformation  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 precomputed crush data exist (experimental testing, finite element analysis, or specificationprescribed 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 expensivefor this purpose is the idea that these analyses are onetime 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 pier1 and pier3 (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 forcedeformation data (crush data) were smoothed and resampled (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 elasticplastic loaddeformation 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 pier1 200 Static crush data for pier3  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 pier3  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 pier1 & pier3. 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 thub are the barge displacements at times t+h, t, and th 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 thub (8) which uses data at times t and th 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 elasticplastic 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 timestepping 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 FBPier 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 bargedynamics module to the FBPier 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 crosssectional 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 elasticplastic 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. Flowchart 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 thub + = +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 thub Pb =Pb()Pb Pb 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. Flowchart 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 uptodate 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 thub) and barge crush data (tabp and thabax) 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 "backandforth" 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(tl 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 FBPier 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 twoDOF model to represent dynamic bargepier 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 loaddeformation 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 twoDOF analysis, evaluation of this simplified technique is justified. 6.2 Implementation Implementation of the twoDOF 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 twoDOF 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 (FBPier, LSDYNA, etc.). In this thesis, pushover data for pier1 and pier3 were obtained from high resolution analyses conducted on combined pier/soil models using LSDYNA. The data obtained were then validated against analyses conducted on the same piers using FBPier. 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 12 on the loaddisplacement curve shown in Figure 8a represents the soil/pier loading curve obtained from a pushover analysis. Path 23 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 wI b) Figure 8. Depiction of pier/soil loading and unloading behavior, a) Loaddisplacement curve from pushover including various unloading paths; b) Single pile and soil model describing points on loaddisplacement curve 0 In the barge impact cases simulated in this thesis, plastic soil deformation is considered in the soil response but pierstructure 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 23 of Figure 8. In the low resolution analysis technique implemented here, path 23 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 reloading 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 twoDOF 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 bargepiersoil interaction. A variety of cases were simulated for both pier1 and pier3 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 crossvalidated 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 bargepier 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 (4knots) (190 to) (2825 kipft) (1900 tons) Half loaded 1.03 m/s 0.53 MJ B Pier1 9.34 MN B Pie(2knots) 0 t) (390 kipft) (1050 tons) 2.06 m/s Empty 0.40 MJ C 1.80 MN _C (4knots) 1.0 (297 kipft) (200 tons) Half loaded 0.51 m/s 0.13 MJ D 9.34 MN (1knot) 9.34 MN (98 kipft) (1050 tons) Half loaded 0.26 m/s 0.03 MJ E Pier3 9.34 MN E P 3 (0.5knots) 9.05 to) (24 kipft) (1050 tons) 2.06 m/s Empty 0.40 MJ F 1.80 MN (4knots) (20 (297 kipft) (200 tons) 77.1 Comparison of Dynamic Simulation Results Case A, a 4knot fully loaded barge impact on pier1, 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 forcedeformation 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 forcedeformation 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 : 4knot fully loaded impact on pier1. a) Time history of impact force; b) Time history of pier displacement; c) Barge forcedeformation .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 lesssustained, more transient loading conditions. Predicted force histories for Case B and C are plotted in Figure 10. Case B, a 2knot half loaded impact condition (Figure 10a), produces an oscillatory loading history, while case C, a 4knot empty condition (Figure 10b), has an even more transient shortterm force history. For both of these cases, all three analyses peak at approximately the same time and load level and exhibit timevariation 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 pier1. a) Case B : a 2knot half loaded barge impact; b) Case C : a 4knot empty loaded barge impact Case D, E, and F correspond to impacts on pier3, the less impact resistant pier. Due to the increased flexibility of pier3, 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 pier3. 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 : 1knot half loaded impact on pier3. 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 codebased 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 AF previously cited plus an additional intermediate impactenergy condition (Case G : pier1, 4knots, 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 (kipft) 0 1000 2000 3000 S ............................ .............. .............................................. .... .................... ............. 1 6 0 0 8...7. 1800 7 C ii_~..........~......r 1600 1400 1200 5~~~lo   1000 S.________ 800 S.  AASHTO Relationship S..............................................Pier1,4 knots fully loaded 600 B Pier1, 2 knots, half loaded 2D .......... ... ........... ... C Pier1, 4 knots, empty D Pier3, 1 knots, half loaded 400 E Pier3, 0.5 knots, half loaded ..................................................................... F Pier3, 4 knots, em pty 200 G Pier1, 4 knots, half loaded 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Kinetic energy (MNm) 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 LSDYNA, the static load application was achieved by applying the loads slowly enough so as to eliminate all inertial (dynamic) effects. To crosscheck 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 FBPier in static analysis mode). Results obtained from the pseudostatic analyses and the true static analyses were virtually identical. Maximum sustained pier displacement data for impact conditions on the relatively stiff pier1 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 pier3 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 (kipft) 0 1000 2000 3000 175 15 0 ............... .. ............. ... 6 Displacement produced by HD .. application of AASHTO 2 150    equivalent static load A Pier 1, 4knots, fully loaded 25 B Pier1, 2knots, half loaded C Pier1, 4knots, empty G Pier1, 4knots, 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) Pieri results; b) Pier3 results Kinetic energy (kipft) 0 100 200 300 600 Displacement produced by 500 application of AASHTO 20 equivalent static load D Pier3, 1knot, half loaded E Pier3, 0.5knots, half loaded S 400 ...... F Pier3, 4knots, 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 pier3 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 loaddeformation 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 nonimpact 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 bargepier 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 acceptablealthough slightly conservativein cases where the impact event produces sustained bargepier 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 LSDYNA 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) LSDYNA 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) LSDYNA 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 FBPier 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 AcSimplecSimpl1+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 xintercept 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 FBPier FBPier crush APPENDIX B LOW RESOLUTION ANALYSIS PROGRAM In this appendix, a Mathcad program is presented that documents the implementation of the loworder analysis technique described earlier in this thesis. The program was codeveloped with Mr. David Cowan and his contributions are gratefully acknowledged. The specific example presented here is for impact Case A, the 4knot fully loaded barge impact on the impact resistant pier, pier1. Low Resolution BargePier Interaction Model Pier1 : 4knot, 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) LoadDisplacement Data... Eur:= 1340kips id ...barge crush plastic unloading slope Import barge_crush data (loaddeformation) : 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 (loaddisplacement) 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 : Loaddisplacement 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) MassBarge... dwt := 1900tons mb:= dwt g ...dead weight tonnage 2 mb = 9.8423 ips's ...mass of barge in MassPier.. 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:= 66.16in77.61ir 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) + (nPierA + p .s (nPile.A.pile L.pil) mp = 3.918kipss m Damping... 0 0 kipss 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 kipss 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. < iAt 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 xdisplacement xp:= outT v ( +2) ( ) ( <+4) () T +5) ab :=out T) ...barge xacceleration ap := outT + Fb :=( S+6> ( <)S+7, Fb:= out ) ...barge xforce 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 xdisplacement ...pier xvelocity ...pier xacceleration ...pier xforce ...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 C1 . ) 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 Forcedeformation 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 loaddeflection 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 Cvo K.xo) M "Contact Spring" PcY Pc < +1 AcY < Ac +1 PcY kcv <  AcY for ie (E + l)..last(Pc) Pc. Pc. 1 11 kcv.  1 Aci Aci1 "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 voAt +  "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 2M" 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 1kc 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,i2 X 4 J+2,i1 2. At x. 2 x. + x. J,i1 Ji2 X 4 j+4,i1 2 At "Force" ftemp < kc (Xl, i1 I x0, iI1) ks xl, i1 for j 6.. 7 x. ftemp. x.,i1 + ftempj6 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 (Agap) 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 .AAy) 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 pier1 and pier3 are validated against corresponding high resolution analysis results. Both a pseudostatic 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 C1. Pier1 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 C2. Pier3 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 C3. Pier1 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 4knot 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 D1. 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 D2. 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 D3. Impact force vs. barge crush 6 5 1 7 0 00 00 )o JO L 8( S6( B 1 2knot Half loaded 0.25 0.5 0.75 1 Time (sec) High resolution analysis Medium resolution analysis Low resolution analysis Figure D4. 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 D514. 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 D6. Impact force vs. barge crush 1000 800 600 * ,1 400 200 0 r1n m \ 0.25 High resolution Medium resolution Low resolution Figure D7. Time history of impact force Case : Pier: Velocity : Payload : C 1 4knot 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 D8. Time history of pier displacement Barge crush depth (m) 2 3 .L.......................HB.I. rI  25 50 75 100 Barge crush depth (mm) High resolution analysis Medium resolution analysis Low resolution analysis Figure D9. 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 1knot 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 D10. 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 D11. 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 D12. Impact force vs. barge crush Case : Pier: Velocity : Payload : 2.5 2 E 3 0.5knot 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 D13. 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 D14. 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 D15. 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 D16. Time history of impact force F 3 4knot Empty 700 600 500 z 400 300  200 100 0 6 4 0 a2 0 2 0 0.25 0.5 0.75 Time (sec) High resolution Medium resolution Low resolution Figure D17. 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 S400 8 300 i i e 200 a ^'  mo ICI El ii 0 70 80 90 100 High resolution Medium resolution Low resolution Figure D18. 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 #R293, 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 : KTC946. 5. 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Ph.D. Dissertation, University of Florida, Department of Civil Engineering, 1999. 21. Mathsoft Engineering & Educuation, Inc. MathCad 2001i User's Guide i/ il 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. 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. 