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PAGE 1 1 DETERMINATION OF BARGE FLOTILLA IMPACT FORCES ON FLEXIBLE GUIDE WALLS By ROBERT A LAN WALTERS 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 2012 PAGE 2 2 2012 Robert A lan Walters PAGE 3 3 ACKNOWLEDGMENTS This thesis work could not have been completed without the support and guidance of a number of individuals. The author would like to extend appreciation and thanks to Dr. Gary Consolazio, who continuously provided motivation and expertise to enable the aut hor to further develop his engineering knowledge and judgment. Thanks are also extended to Dr. H.R. Hamilton and Dr. Ronald A. Cook for their input and assistance through the development and execution of this research. Finally, the author would like to tha nk his girlfriend, Jessica Carter, along with his family and friends for providing unwavering support throughout this journey. PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................3 LIST OF FIGURES .........................................................................................................................7 LIST OF ABBREVIATIONS ........................................................................................................11 ABSTRACT ...................................................................................................................................13 CHAPTER 1 INTRODUCTION ..................................................................................................................15 2 BARGE FLOTILLA FINITE ELEMENT MODEL ..............................................................17 Flotilla Finite Element Model Overview ................................................................................17 Structural Modeling ................................................................................................................18 Impacting Barge ..............................................................................................................18 Non Impactin g Barge ......................................................................................................20 Barge Interactions Within Flotilla Model ...............................................................................20 Additional Modeling Considerations ......................................................................................23 3 FINITE ELEMENT MODELING OF FLEXIBLE GUIDANCE STRUCTURE .................36 Introduction .............................................................................................................................36 Development of Rigid Cell and Thrust Block Finite Element Models ..................................36 Development of Flexible Wall Finite Element Model ...........................................................39 Overview of Modeling Approach ....................................................................................39 Reinforcement .................................................................................................................40 Concrete Solid Elements .................................................................................................41 Concrete Beam Elements ................................................................................................42 Contact Shells ..................................................................................................................44 Match Cast Seg ments ......................................................................................................45 Flexible Wall Damping ...................................................................................................47 4 VERIFICATION AND VALIDATION OF FLEXIBLE WALL MODEL ...........................70 Model Verification Using Independent C ross Sectional Analysis .........................................70 Simplified Sectional Analysis .........................................................................................70 Sectional Analysis with Foundation Supports .................................................................71 Model Validation Using Winfield Lock and Dam Experimental Test Data ..........................72 Finite Element Modeling of Load Measurement System ...............................................73 Simulations of Instrumented Barge Im pacts on the Winfield Flexible Wall ..................75 5 IMPACT SIMULATIONS AND RESULTS .........................................................................85 PAGE 5 5 Overview .................................................................................................................................85 Force Time Histories of Impact Simulations .........................................................................85 Sensitivity of Barge Flotilla Impact Forces to Flotilla Size ...................................................87 6 CONCLUSIONS ..................................................................................................................101 REFERENCES ............................................................................................................................104 BIOGRAPHICAL SKETCH .......................................................................................................106 PAGE 6 6 LIST OF TABLES Table page 31 Bearing pad stiffness values ...............................................................................................53 32 Post tensioned tendon parameters ......................................................................................53 51 Barge flotilla impact conditions .........................................................................................90 52 Summary of results from 3x3 barge flotilla impact simulations ........................................90 53 Summary of results from 3x5 barge flotilla impact simulations ........................................90 54 Comparison of results between 3x3 and 3x5 flotilla impact simulations ..........................91 55 Effect of flotilla length on peak impact forces ...................................................................91 56 Effect of floti lla width on peak impact forces ...................................................................91 PAGE 7 7 LIST OF FIGURES Figure page 21 Typical 3x5 barge flotilla in transit ....................................................................................25 22 Jumbo hopper barge schematics ........................................................................................26 23 Jumbo hopper barge flotilla schematics .............................................................................26 24 Jumbo hopper barge flotilla FE model ...............................................................................27 25 Jumbo hopper barge finite element model .........................................................................27 26 Barge bow zone ..................................................................................................................28 27 Barge structural steel stress strain relationship ..................................................................28 28 Strain rate model used for mild steel barge components ...................................................29 29 Non impacting barge FE model .........................................................................................29 210 Lashing configuration on barge flotilla ..............................................................................30 211 Exterior fore/aft wires ........................................................................................................30 212 Interior fore/aft wires .........................................................................................................31 213 Breast wires ........................................................................................................................32 214 Scissor wires ......................................................................................................................33 215 Lashi ng configuration combinations on a 3x3 flotilla .......................................................34 216 Bow to stern FE crushing simulation for determining contact stiffness of rigid barges ...35 217 Force deformation relationship for bow to stern contact stiffness of rigid, nonimpacting barges ................................................................................................................35 218 Barge buoyancy spring schematic .....................................................................................35 31 Rendering of flexible wall guidance system ......................................................................54 32 Flexible wall finite element model ....................................................................................54 33 Schematic of foundations and flexible wall .......................................................................55 34 Schematic of foundation assemblies with connection details ............................................56 PAGE 8 8 35 Rendering of foundation finite element model ..................................................................57 36 Gravity load bearing pad footprints superimposed on rigid cell FE mesh ........................57 37 Hybrid modeling approach using both solid elements and beam elements .......................58 38 Pattern of steel reinforcement within concrete cross section .............................................58 39 Post tensioned tendon locations .........................................................................................59 310 Post tensioned tendon material model ...............................................................................59 311 Schematic of solid cross section ........................................................................................60 312 Schematic of transition cross section .................................................................................60 313 Rendering of solid element modeling ................................................................................61 314 Division of hollow concrete cross section .........................................................................62 315 Cross section integration scheme .......................................................................................62 316 Nonlinear concrete material model fit ...............................................................................63 317 Contact ring schematic .......................................................................................................64 318 Interaction between barge and contact rings ......................................................................65 319 Segment contact face on outer segment of flexible wall ...................................................66 320 Grout cracking upon joint opening ....................................................................................66 321 Sensitivity of debonded length of post tensioning at segment joints ................................67 322 Mid span deflection of flexible wall for 3x3 flotilla impact at 17 with initial velocity of 2.9 fps ............................................................................................................................67 323 Frequency analysis of displacement timehistory of flexible wall ....................................68 324 Deflection time history for damped pluck test of flexible wall at mid span .....................68 325 Overview of numerical process used to model post tensioning ........................................69 41 Schematic of moment curvature FE simulation ................................................................77 42 Moment curvature relationships obtained from cross sectional analysis and fini te element analysis .................................................................................................................77 43 Mechanism of axial load increase within flexible wall .....................................................78 PAGE 9 9 44 Point sampled moment curvature procedure .....................................................................79 45 Comparison of moment curvature relationships from cross sectional analysis and finite element analysis of the wall and foundation model .................................................80 46 Definition of impact conditions at Winfield Lock and Dam .............................................80 47 Load measurement s ystem (beam, shear pins, and clevis mounts) attached to corner of barge ..............................................................................................................................81 48 Overview of finite element model of load measurement system ......................................81 49 Use of nodal rigid bodies in finite element model of loadmeasurement system ..............82 410 Rendering of loadmeasuring beam constrained to barge ..................................................82 411 Force time history for Experiment 10 and finite element simulations with and without the loadmeasurement beam .................................................................................83 412 Deflection time history for Experiment 10 and finite element simulations with and without the loadmeasurement beam .................................................................................83 413 Force time history for Experiment 20 and finite element simulations with and without the loadmeasurement beam .................................................................................84 414 Deflection time history for Experiment 20 and finite element simulations with and without the loadmeasurement beam .................................................................................84 51 Schematic of impact condition for flotilla impact simulations ..........................................92 52 Force time histories for 3 x3 barge flotillas impacting at 10 ............................................92 53 Force time histories for 3x3 barge flotillas impacting at 20 ............................................93 54 Force time histories for 3x3 barge flotillas impacting at 30 ............................................93 55 Force time histories for 3x5 barge flotillas impacting at 10 ............................................94 56 Force time histories for 3x5 barge flotillas impacting at 20 ............................................94 57 Force time histories for 3x5 barge flotillas impacting at 30 ............................................95 58 Simplified illustration of flotilla deformation during impact ............................................95 59 Barge flotilla deformation for impact condition 20 and 3 fps ..........................................96 510 Force time histories for 3x3 and 3x5 barge flotillas impacting at 10 with 2 fps initial velocity ...............................................................................................................................96 PAGE 10 10 511 Force time histories for 3x3 and 3x5 barge flotillas impacting at 10 with 3 fps initial velocity ...............................................................................................................................97 512 Force time histories for 3x3 and 3x5 barge flotillas impacting at 20 with 3 fps initial velocity ...............................................................................................................................97 513 Force time histories for 3x3 and 3x5 barge flotillas impacting at 20 with 4 fps initial velocity ...............................................................................................................................98 514 Force time histories for 3x3 and 3x5 barge flotillas impacting at 30 with 5 fps initial velocity ...............................................................................................................................98 515 Peak impact forces plotted as a function of momentum ....................................................99 516 Barge flotilla configurations considered in sensitivity study .............................................99 517 Effect of flotilla length on impact force time histories ....................................................100 518 Effect of flotilla width on impact force time histories .....................................................100 PAGE 11 11 LIST OF ABBREVIATION S A cross sectional area (in2) Ar material specific shape factor for Ramberg Osgood model Br material specific shape factor for Ramberg Osgood model C material based constant for Cowper Symonds strain rate model (sec1) Cr material specific shape factor for Ramberg Osgood model db diameter of strand (in) E modulus of elasticity (kips/in2) Ec modul us of elasticity for concrete in compression (kips/in2) Ect modulus of elasticity for concrete in tension (kips/in2) Ep modulus of elasticity for 7wire strand (kips/in2) cf peak compressive stress of concrete (kips/in2) fcr peak tensile stress of concrete (kips/in2) fpe effective prestress stress (kips/in2) fps strand stress at nominal member resistance (kips/in2) I moment of inertia (in4) k shape factor for Collins and Mitchell material model L span length (in) L1 L7 lashing ty pes in barge flotilla ld development length (in) M internal section moment (kip in) My moment about Y axis of cross section (kipin) Mz moment about Z axis of cross section (kipin) n shape factor for Collins and Mitchell material model nm mode shape PAGE 12 12 P sum of forces on cross section (kip) Q material based constant for Cowper Symonds strain rate model V0 initial velocity of barge flotilla (feet/sec) change in axial load as a result of loading (kips) logarithmic decrement strain c strain at peak compressive stress cr strain at peak tensile stress p eff effective plastic strain rate (in/in sec) oblique barge impact angle (deg) FHWA mandated multiplier of 1.6 damping ratio mass density (kipsec2/in4) stress (kips/in2) c stress of concrete in compress ion (kips/in2) y dynamic dynamic yield stress (kips/in2) pu ultimate tensile stress of 7 wire strand (kips/in2) y static static yield stress (kips/in2) t stress of concrete in tension (kips/in2) curvature (in1) 0 zero curvature (in1) u curvature at ultimate section moment (in1) natural frequency (rad/sec) PAGE 13 13 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 DETERMINATION OF BARGE FLOTILLA IMPACT FORCES ON FLEXIBLE GUIDE WALLS By Robert A lan Walters May 2012 Chair: Gary Consolazio Major: Civil Engineering Current design practices for waterway guidance and protection structures employed by the U.S. Army Corps of Engineers are based on data developed from full scale impact experiments conducted on rigid guidance structures. Given tha t the inventory of navigational structures designed by the USACE also includes flexible systems, it is important to determine whether application of the current impact load determination procedures to such structures produces appropriate, overly conservati ve, or unconservative designs. This study addresses this issue by developing, verifying, and validating a flexible guide wall finite element model that accounts for nonlinear concrete and mild reinforcing steel behavior, the effects of post tensioned segme ntal concrete construction, and a variable cross sectional shape. Subsequently, impact simulations are conducted with the wall model and a highresolution nonlinear multi barge flotilla model at various impact angles and impact velocities to quantify flotilla impact forces on the wall. Results reveal that peak impact forces are primarily a function of mass contained within the first row of barges in the flotilla. Because cable lashings are relatively flexible in comparison to the stiffnesses of individual barges, during a flotilla wall collision event, only barges in the first row of the flotilla are redirected immediately by the wall. Consequently, flotilla width is found to strongly influence peak impact force while barge flotilla length is found to have minimal, almost PAGE 14 14 negligible, influence. These results also suggest that the common design practice of using only flotilla momentum, which depends on both the width and length of a flotilla, to predict peak impact force is not likely to lead to the most efficient design standards when applied to multi barge flotilla impacts. PAGE 15 15 CHAPTER 1 INTRODUCTION Navigational guidance structures serve an important role in helping to ensure safe operations at locks and dams throughout the United States. These structures provide dual functionality by aiding vessel operators in aligning their vessels with locks, and b y protecting nearby waterway facilities such as hydroelectric dams. With the high volume of lockages that occur on the navigable waterways of the United States, guidance structures can be subjected to daily impacts from vessels; most commonly barge flotillas. Procedures currently employed by the U.S. Army Corps of Engineers (USACE) for the calculation of barge impact forces on waterway guidance and protection structures are based primarily on data obtained from investigations of barge impacts on rigid structural systems. However, given that the inventory of navigational structures designed by the USACE also includes flexible systems, it is important to determine whether application of the current impact load determination procedures to such structures produces appropriate, overly conservative, or unconservative designs. Typical approaches used to quantify impact loads involve relatively expensive impact experiments, which are often limited for safety reasons to low energy impact conditions. However, wit h ever increasing ability to generate and analyze complex finite element models, impact forces imparted on structures by barge flotillas can be quantified using finite element analysis with accuracy suitable for design. Furthermore, finite element simulation provides the ability to quantify impact forces for highenergy impact events that push materials well into their nonlinear ranges. Coupled together with experimental validation data, finite element impact simulation can provide significant advantage in the development of accurate and efficient design guidelines for the USACE. PAGE 16 16 The work described in this report documents the development of a flexible guide wall finite element (FE) model and dynamic FE impact simulations conducted on the guide wall. A high resolution, nonlinear FE barge flotilla model has been used as the impacting vessel for the simulations in this study. Impact forces have been quantified for a variety of impact conditions by varying angle of obliquity and impact velocity. Force time histo ry results from these simulations have been compared to those from the Winfield Lock and Dam experiments (Barker et al. in press). In addition, the influence of a load measurement beam that was attached to the corner of the test barges in both the Robert C Byrd Lock and Dam experiments (Patev et al. 2003) and the Winfield Lock and Dam experiments, has been investigated to determine the effect this system has on measured impact force. PAGE 17 17 CHAPTER 2 BARGE FLOTILLA FINIT E E LEMENT MODEL Flotilla Finite Element Model Overview Each impact simulation conducted for this study utilizes a high resolution finite element model of a barge flotilla. A barge flotilla, such as the one shown in Figure 2 1, is an assembly of individual barges, typically of similar size and configuration, which are connected together by a series of wire ropes (also known as lashings) For this study, two specific barge flotilla configurations are modeled: a 3x5 barge flotilla (consisting of three strings, or columns, of five barges per string 15 barges total) and a 3x3 barge flotilla (consisting of three strings of three barges per string 9 barges total). The barge flotilla FE models used in this study are created using the same methodology as that documented in the report Development of Finite Element Models for Studying Multibarge Flotilla Impacts (Consolazio et al. 2010). Nevertheless, a brief overview of the barge flotilla FE model is described in the present report to provide self contained documentation of the capabilities of the model. Each barge flotilla FE model is comprised of a series of jumbo hopper river barges measuring 195 ft long by 35 ft wide and weighing 2000 tons each (where 1 ton = 2000 lbs). Two configurations of this jumbo hopper river barge are employed in each flotilla m odel: single raked and double raked. Single raked barges are raked (tapered through the depth) at the bow only, whereas double raked barges are raked at both the bow and stern. Single raked barges are present at the foremost and aft most barge positions of each string of barges while double raked barges are only present in between the fore and aft most barges in each string. A schematic of each jumbo hopper barge is shown in Figure 22. PAGE 18 18 The two barge flotilla configurations used in this study are shown in Figure 23. The 3x5 barge flotilla has overall dimensions of 975 ft by 105 ft (length by width) with a total weight of 30,000 tons and the 3x3 barge flotilla has overall dimensions of 585 ft by 105 ft (length by width) with a total weight of 18,000 tons. A rendering of the FE model of each flotilla is shown in Figure 24. Structural Modeling Two types of individual barge FE models are used within the flotilla models for this study. A single highly discretized barge within each flotilla, referred to as the impacting barge, is the only barge to make physical contact with the target structure (structure to be impacted). The high level of discretization associated with the impacting barge is necessary to enable accurate representation of the contact interaction between the target structure and impacting barge. The remaining barges within a given flotilla are referred to as nonimpacting barges. The primary role of nonimpacting barges is to facilitate modeling of both the dynamic response resu lting from barge to barge contact and the lashing interactions of adjacent barges during impact. Note that nonimpacting barges never make contact with the target structure. Impacting Barge The high resolution impacting barge FE model is composed of more than 900,000 nonlinear shell elements modeled in LS DYNA (LSTC 2009); is consistent with available detailed structural plans; and is made up of three barge zones: the bow zone, the stern zone, and the hopper zone. Each zone is discretely modeled with inter nal structural members and external plate surfaces. Internal structural members consist of angle, channel, or plate sections. Internal member thicknesses and external plate thicknesses vary between 5/16 in. and 5/8 in., as determined by the structural plan s. Figure 25 shows a rendering of the impacting barge. An PAGE 19 19 example of the high level of mesh discretization is demonstr ated in Figure 26, which shows a rendering of the bow zone of the impacting barge FE model. The impacting barge is given a nonlinear constitutive relationship (effective true stress vs. effective plastic strain) representing A36 structural steel, which is shown in Figure 27. This relationship is employed for all structural and plate members within the barge FE model, which are meshed using 4 node fully integrated shell elements. Importantly, each structural member is modeled with sufficient mesh density to allow for local bu ckling and local material failure. Material failure is represented by element deletion within LS DYNA and occurs when the effective plastic strain of the steel reaches 0.2 in./in. In addition to the base constitutive relationship, a model to account for st rain rate effects is also included. Strain rate effects occur at the material level and act to increase the overall effective stiffness of structural members by raising the yield stress of the material as the plastic strain rate increases. In this study, t hese effects are modeled using the Cowper Symonds strain rate model, which is given by Equation 21. 11yp Q dynamic eff y staticC ( 21) In this relationship, dynamic yield stress ( y dynamic ) is obtained by modifying the static yield stress ( y static ) based on the effective plastic strain rate ( p eff ) and two material dependent parameters, C and Q. For mild steel, C and Q are taken as 40.5 sec1 and 5, respectively (Jones 1997). In Figure 28, the increase of dynamic yield stress (relative to the static yield stress) predicted by the Cowper Symonds model is illustrated for mild steel. PAGE 20 20 Non Impacting Barge The pr imary role of the nonimpacting barge FE model is to provide a means of efficiently representing the mass related inertial properties of a barge as well as the dynamic interactions between barges (through contact and lashings) during and after impact. In p ractical terms, it is not computationally feasible to utilize a high resolution deformable barge model at each position within a flotilla to model these effects. For this reason, the nonimpacting barge is modeled in a way that retains the external geometr y of a high resolution barge, as well as the inertial and mass properties, but has a lower mesh resolution than the highresolution barge FE model. The low resolution nonimpacting barge has approximately 4,000 shell elements compared to the 900,000 shell elements of the highresolution impacting barge. Shell elements defining the external geometry of the nonimpacting barge are modeled as rigid elements, and no internal structural elements are modeled. Inertial and mass properties measured using the highr esolution barge model are assigned to (imposed upon) the rigid nonimpacting barge. This procedure allows the nonimpacting barge to mimic the rigid body dynamics of a high resolution barge model when external forces are imposed upon it. However, being a r igid model, the appropriate structural stiffness of the nonimpacting barge model, which affects bargeto barge contact forces, is not automatically generated by the mesh. Instead, the structural stiffness of the high resolution barge is introduced into the low resolution nonimpacting rigid barge model through contact definitions (discussed later) in a way that produces the appropriate stiffness relationship. A rendering of the nonimpacting barge FE model is shown in Figure 29. Barge Interactions Within Flotilla Model Individual barges within a barge flotilla are joined together using wire ropes (also referred to as lashings) that wrap aro und bitts (cylindrical posts) which are affixed to the barge deck. A PAGE 21 21 photograph of a typical lashing configuration is provided in Figure 2 10. Lashi ng configurations used within this study are consistent with those used in the full scale barge impact tests conducted by the USACE at Gallipolis Locks (Patev et al. 2003). Each wire rope within the FE model is assigned an appropriate geometric configurati on; a set of material properties that represent the nonlinear stiffness of the lashing; and a failure criterion based on ultimate capacity. Depending upon the location of the wire rope within the overall flotilla, an appropriate ultimate tensile of either 90 kips (for 1 in. diameter wire rope) or 120 kips (for 1.25 in. diameter wire rope) is assigned. By including a failure criterion, the flotilla model has the ability to experience either full or partial break up wherein the individual barges are free to s eparate from one another and move independently. Each pair of adjacent barges within a flotilla is lashed together by wrapping the barge bitts in a specific pattern, referred to as a lashing configuration. Different configurations are used to lash differen t types of barge pairs (endto end, side to side, or diagonal) and to resist different loads imposed by common flotilla maneuvers. Lashings are layered on top of each other when more than one configuration is required at the same location. In total, seven lashing configurations are used in the 3x3 and 3x5 barge flotilla models. Fore/aft wires (Figure 211 to 212) are used to secure end to end barge pairs together at shared corners, providing longitudinal rigidity to each barge string. The configuration used along the exterior edges of the flo tilla (L1) is unique in that it employs a 1 in. diameter wire rope rated at a 90 kip break strength. Every other configuration in the flotilla, including the fore/aft wires along the interior edges (L2 L3), uses a 1.25 in. diameter wire rope rated at a 120 kip break strength. PAGE 22 22 Breast wires (Figure 213 ) are used to connect side to side barge pairs together at shared corners, and act to keep the port and starboard st rings from lagging behind the center string during travel. Because of their different orientations, they are sometimes referred to separately as towing wires (L4) that are engaged when the center string is pushed forward, and backing wires (L5) that are en gaged when the center string is pulled backward. Scissor wires (L6 L7, Figure 214) connect diagonal barge pairs together at every four corner interface. They se rve to increase flexural rigidity, allowing the flotilla to be steered more easily by the rear tug. Figure 215 shows each lashing configuration combination for a 3x3 barge flotilla. In addition to barges interacting with one another through lashing forces, they also interact by generating contact forces. In the barge flotilla FE model, contact definitions are defined between each set of adjacent barges. For the purposes of numerical efficiency, barges not initially in contact with each other do not have contact pairings defined between them. For instance, for a 3x3 flotilla, a barge in the lead (fore) row has no contact defined between itself and a barge in the trailing (aft) row since they are not initially adjacent to each other. Since all barges except the impacting barge are rigidized, the contact stiffness cannot be obtained directly from the structural stiffness of the barges themselves (since the non impacting bar ges are structurally rigid). Contact stiffness (and therefore structural stiffness) for a non impacting barge is thus accounted for through the use of a prescribed contact stiffness relationship, which is representative of the stiffness of impacting barges To obtain such relationships, highresolution FE models of deformable barges are quasi statically crushed together. Nonlinear force deformation relationships (stiffnesses) can then be extracted from the results of the crushing simulation. PAGE 23 23 An example of t his procedure for a bow to stern contact interface is shown in Figure 216. The force deformation relationship obtained from this simulation (plotted in Figure 217 ) is prescribed for each bow to stern contact interface defined within the flotilla FE model. During an impact, as the rigid barges contact each other, the contact impact algorithm in LS DYNA will compute the contact penalty forces from this prescribed relationship, thereby mimicking the stiffnesses of two highresolution deformable barges. For each barge within the flotilla, th ree types of contact may be defined: sideto side contact; bow to bow contact; and bow to stern contact. For each such case, a forcedeformation relationship is generated from a high resolution crushing simulation, and then applied to the appropriate conta ct definitions within the flotilla FE model. Additional Modeling Considerations In each impact simulation conducted in this study, the effects of both gravitational forces and buoyancy forces acting on the barge flotilla are included. Buoyant uplift forc es underneath each barge are modeled by introducing individual buoyancy springs over the bottom surface of the barge model. For the highresolution barge model, approximately 26,400 discrete springs are attached to the barge bottom nodes, whereas each nonimpacting barge employs approximately 900 buoyancy springs. The stiffness of each buoyancy spring is computed by determining the tributary area of the barge bottom surface supported by the spring, and then multiplying this value by the density of water (62 .4 lb/ft3). By using a large number of springs with relatively small tributary areas, the resulting stiffness values are small, ranging from 0.001 kip/in to 0.004 kip/in, thereby precluding the development of unrealistically concentrated buoyant forces dur ing barge motions. Each buoyancy spring is 200 in. in length and connects to a support node (above the barge) that is freely able to translate in the horizontal plane (Figure 218) but restrained against vertical PAGE 24 24 motion. As such, the barge model hangs from the collection of buoyancy springs and is able to translate arbitrarily large distances in the horizontal plane (plan view) without r esistance. Vertical motions of the barge, however, cause appropriate changes in the distribution of vertical uplift forces, which are based on changes in the submerged depth of the barge. Because the buoyancy springs are always in tension, the vertical support node of each spring tracks (in plan view) with the corresponding node at the bottom surface of the barge. Consequently, the buoyancy springs remain vertical at all points during the simulation, regardless of the horizontal motions that the flotilla may undergo. This is particularly beneficial should a partial or full flotilla breakup occur during a simulation. Additional aspects of buoyancy modeling, such as calibration of the buoyancy springs and gapping of buoyant springs at the raked barge bow, ar e described in Consolazio et al. (2010). PAGE 25 25 Figure 21. Typical 3x5 barge flotilla in transit (after USACE 2007) PAGE 26 26 A B Figure 22. Jumbo hopper barge schematics A) Single raked barge. B) Double raked barge. A B C D Figure 23. Jumbo hopper barge flotil la schematics. A) 3x5 pl an view. B) 3x5 elevation view. C) 3x3 plan view. D) 3x3 elevation view B owStarboard PortS t e r n1950 35 0Hopper region Stern 12 0 2 0 12 6350 1620 276 56 Stern Bow B owStarboard PortS t e r n195035 0Hopper region Bow Stern 12 0 2 0350 1400 270 276 Stern 12 0 2 0 9750 1050 9750 140 1050 5850 140 5850 PAGE 27 27 A B Figure 24. Jumbo hopper barge flotilla FE model (mesh not shown for clarity). A) 3x5 flotilla B) 3x3 flotilla A B Figure 25. Jumbo hopper barge finite element model (mesh not shown for clarity). A) Perspective view B) Exploded view PAGE 28 28 A B Figure 26. Barge bow zone. A) Structural configuration. B) Finite element mesh Figure 27. Barge structural steel stress strain r elationship. Internal rake truss Top hull plate Bottom hull plate Headlog plate Hopper zo ne Bow zone Effective plastic strain (in./in.)E f fe c t i ve t rue s t re s s (ks i ) 0 0.05 0.1 0.15 0.2 0 20 40 60 80 Failure strain = 0.2 PAGE 29 29 Figure 28. Strain rate model used for mild steel barge components Figure 29. Nonimpacting barge FE model (mesh shown) PAGE 30 30 Figure 210. Lashing configuration on barge flotilla Figure 211. Exterior fore/aft wires (rated for 90 ki p break strength) Wire rope lashing Bitt Direction of flotilla motionP or t ba r ge s t r i ngL1 PAGE 31 31 Figure 212. Interior fore/aft wires (rated for 120 kip break strength) C enter bar ge s tr ing Por t bar ge s t r ingL2 L3 Direction of flotilla motion PAGE 32 32 Figure 213. Breast wires (rat ed for 120 kip break strength) Direction of flotilla motion C e nt e r ba r ge s t r i ng P o r t ba r g e s t r i ngL4 L5 PAGE 33 33 Figure 214. Scissor wires (rated for 120 kip break strength) Direction of flotilla motionC e nt e r ba r ge s t r i ng P or t ba r ge s t r i ngL6 L7 PAGE 34 34 Figure 215. Lashing configuration combinations on a 3x3 flotilla A B C D B C D A C Starboard lashings mirrored about centerline L1 L4 L5 L5 L7 L4 L3 L6 L2 Port lashings shown A Direction of flotilla motion PAGE 35 35 A B Figure 216. Bow to stern FE crushing simulation for determining contact stiffness of rigid barges. A) At beginning of simulation. B) During simulation Figure 217. Force deformation relationship for bow to stern contact stiffness of rigid, nonimpacting barges Figure 218. Barge buoyancy spring schematic. Deformation (in)Force (kip) 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000 Vertical restraint Buoyancy spring 200Foremost buoyancy spring Spring attached to bottom surface of barge 49 8 6 PAGE 36 36 CHAPTER 3 FINITE ELEMENT MODELING OF FLEXIBLE GUID ANCE STRUCTURE Introduction In this study, a portion of the lock approach wall at the Winfield Lock and Dam (Figure 31) in Red House, WV is modeled with finite elements to conduct barge flotilla impact simulations. The lock approach wall is a segmental post tensioned concrete wall (referred to as a flexible wall) that is simply supported by a foundation at each end. Each foundation is comprised of a concrete cell and concrete thrust block, which connect to the flexible wall using bearing pads and tieback anchors. The specific portion of the Winfield Lock and Dam considered in this study consists of a single span flexible wall unit and two foundation structures. The flexible wall is modeled using a combination of solid, shell, and beam elements that represent the concrete cross section and internal reinforcement. Foundation structures are modeled with solid elements, and bearing pads and tieback anchors are modeled with beam elements. Nonlinear materials are utilized where necessary to ensure accurate representation of the flexible wall response during impact. A rendering of the FE model of the flexible wall and support system is provided in Figure 3 2. Development of Rigid Cell and Thrust Block Finite Element Models Each flexible wall span is supported by two foundation assemblies that consist of a concrete cell footing an d a concrete thrust block. A schematic of the concrete cell and concrete thrust block is shown in Figure 33. Each cell is a massive cylindrical reinforced concrete structure with a diameter of 35 ft 6 in. and a height of 39 ft 2 in, approximating the depth to the top of rock in the channel. Cast integrally with each cell is a thrust block, also made of reinforced concrete, that measures approximately 10 ft by 14 ft with a height of 8 ft. Each thrust block and PAGE 37 37 rigid cell connect to the flexible wall span using a series of bearing pads and tieback anchors as shown in Figure 34. Since the concrete cell is considerably stiffer and more massive than the flexible wall, the deformations of the cell footing during impact will have a negligible contribution to the response of the overall sy stem. For this reason, the concrete cell is modeled as rigid, and is therefore referred to as a rigid cell. Each rigid cell is modeled with solid elements (Figure 35A ) having a rigid material model and a unit weight of 150 lb/ft3 to be representative of concrete. Because each cell is rigid, soil properties are not considered and therefore a fixed boundary condition is applied to the bottom of each cell. The assumption of negligible soil response and rigid cell displacement is based on observations at the Winfield Lock and Dam experimental impact study (Barker et al. in press). Relative to the rigid cell, the thrust block is substantia lly less massive and has at least minimal potential to affect the response of the flexible wall system. Consequently, each thrust block is modeled with deformable solid elements (Figure 35 B ) that have an elastic concrete material model with a modulus of elasticity of 4030 ksi and a unit weight of 150 lb/ft3. Each thrust block is merged with the corresponding rigid cell at coincident nodal locations between the bottom face of the thrust block and the top face of the rigid cell. Nodal merging provides a perfect bond between the rigid cell and thrust block, reflecting the cast in place construction of the foundation. At each foundation, the flexible wall is connected to a thrust block and rigid cell by four bearing pads and two tieback anchors (recall Figure 34). Of the four bearing pads, two are located on the rigid cell, which resist gravity loads from the flexible wall; the two remaining bearing pads are positioned on the thrust block and transfer impact loads from the flexible wall to PAGE 38 38 the thrust block. Furthermore, two tieback anchors extend from the thrust block into the flexible wall, preventing the flexible wall from walking off of the foundation. Each bearing pad is modeled as a grid of discrete spring elements distributed over the footprint of the pad in the fini te element mesh. Each discrete spring is assigned a stiffness corresponding to the tributary area of the bearing pad it represents (in both shear and axial modes) such that the summed stiffness of all tributary discrete springs represents the stiffness of the overall bearing pad. An illustration of this process is shown in Figure 3 6, in which the footprint of the two gravity load bearing pads is superimposed on the FE mesh of the rigid cell. Gravity load bearing pads on the rigid cell each have a footprint of 1 ft 6 in. by 2 ft 4 in. The thrust block bearing pad has a footprint of 1 ft 4 in. by 3 ft 6 in. and the stop block bearing pad has a f ootprint of 10 in. by 2 ft 2 in. All bearing pads have a 2 in. thickness with a total of three internal 14 gauge steel layers suspended in an elastomer with a durometer hardness of 50. Stiffness values are calculated for each bearing pad using the method d escribed by Gent (2001). Computed values of overall bearing pad stiffness are given in Table 3 1. Note that these are not the values applied t o individual discrete springs. Since the springs work in parallel, the stiffness values shown in Table 31 are divided by the number of spring s in the bearing pad to calculate tributary stiffness values. In addition, axial bearing pad springs are defined such that they act in compression only and any displacements resulting in tension instead produce zero force between connected elements (ther eby modeling gap formation between the connected elements). Tieback anchors are modeled using a cable beam element within LS DYNA. Cable elements are unique beam elements that can be prescribed a specified internal force level over a given time period A cable element formulation is necessary for dealing with issues related to the staged construction of the wall system, which will be discussed later in the chapter. Each tieback PAGE 39 39 anchor is a 1.375 in. diameter thread bar and is modeled with an elastic st eel material model having a modulus of elasticity of 29000 ksi. Tieback anchors are debonded through their entire length; therefore, each is modeled such that only the nodes at either end of the tieback are connected to the thrust block and flexible wall. Development of F lexible Wall F inite E lement M odel Overview of Modeling A pproach The flexible wall is a posttensioned segmental concrete system spanning 118 ft between two foundation supports. Over the total span, the wall is divided into three equal length match cast segments within which three unique cross sectional shapes are used. A solid cross section, hollow cross section, and transition cross section (from solid to hollow) are all present in the two outer segments of the flexible wall. Within th e single inner segment, only the hollow cross section is used. To accurately capture the nonlinear behavior of the flexible wall, but to remain computationally efficient, it becomes infeasible to model the entire flexible guidance structure using three dim ensional solid continuum elements. Therefore, a hybrid approach (Figure 37) is instead taken with modeling the flexible wall. A combination of solid elements (used near the foundation supports) and nonlinear beam elements (used in the interior portion of the span) are used to represent the structural properties of the concrete within the flexible wall. In addition to modeling the concrete portions of the wall, two types of steel reinforcement are modeled: mild steel bars and high strength multistrand post tensioned tendons. Each steel bar and tendon is modeled using a series of discrete beam elements located throughout the entire len gth of the flexible wall system (in both the solid element and beam element model zones). All reinforcement details and the shape of the concrete cross section are both modeled according to plan sets acquired from the USACE for the Winfield Lock and Dam. PAGE 40 40 Reinforcement Mild steel and post tensioned multi strand tendons are located within the wall unit as indicated in Figure 3 8. Mild steel c onsists of #5 and #9 reinforcing bars (rebar). Post tensioned multistrand tendons are comprised of varying arrangements of in. diameter 270 ksi low relaxation 7 wire strands. Each piece of mild steel reinforcement in the flexible wall is modeled separat ely as an individual discrete beam element, and each multi strand tendon is represented by a single beam element having an area equal to the summation of all strand areas present within the tendon. Note that only longitudinal steel within the wall is model ed; shear reinforcement is not included in the model. Each steel bar and tendon within the cross section is discretized into 6 in. long elements along the entire length (118 ft) of the flexible wall. At each of the two joints where the precast concrete se gments meet, the post tensioned tendons are modeled as continuous across the joint, but the rebar elements are discontinuous (i.e., rebar elements terminate at each end of the precast segments). For mild steel rebar, the pattern of the reinforcement remain s the same throughout the entire span length. However, the post tensioned tendon profile varies within the solid and transition cross section zones. A schematic showing the tendon layout within the solid and hollow cross sections is provided in Figure 3 9. Within the transition zone between the solid and hollow cross sections, the tendon profile is assumed to change linearly between the layouts shown in Figure 39. Multistrand tendon cross sectional areas and post tensioning forces, as noted from the structural wall plans, are provided in Table 32. Mild steel reinforcement bars are modeled with an elastic, perfectly plastic material model, with a modulus of elasticity of 29000 ksi and a yield stress of 60 ksi. Post tensioning strands are given a nonlinear material model corresponding to the Ramberg Osgood function (Equation 31). PAGE 41 41 11 1rr pr pu C rA EA B ( 31) This function is suggested by Collins and Mitchell (1991) for 270 ksi low relaxation 7 wire strands. This function is illustrated in Figure 310. For the Ramberg Osgood function, Ep is the strand modulus (28500 ksi) and Ar, Br, and Cr are material based constants (0.025, 118, and 10, respecti vely for 270 ksi low relaxation 7 wire strands). Once the function reaches the pu = 270 ksi), the response becomes perfectly plastic for all larger strains. Concrete Solid Elements Each end segment of the flexible wall system is co mprised of a beam element region and a solid element region. The solid element region is modeled nearest the cell and thrust block foundation supports for the purpose of facilitating connections to the tieback anchors and to the bearing pads. Two unique cr oss sections, both a solid cross section and a transition cross section (from solid to hollow), are modeled in this zone with the solid elements. Dimensions of the solid cross section are shown in Figure 311. Dimensions of the ends of the transition cross section are shown in Figure 312 A and 312B with a linear variation assumed between the two schematics shown. For solid elements within the solid end block and transition sections of the wall, a linear elastic concrete material model is employed with a modulus of elasticity of 4030 ksi, and a unit weight of 153 pcf. Throughout most of the finite element mesh (Figure 313 ), 8 node hexahedral solid elements are used, however, 4 node tetrahedral elements are also employed in areas where the transition cross section tapers in thickness. All solid elements in the finite element mesh are discretized so as to produce an element length of 6 in. along the length of the span. Nodal locations within the flexible wall mesh align with nodal locations in the thrust block and rigid cell meshes so that these components can be PAGE 42 42 connected together using bearing pad beam elements (i.e., beam elements representing the stiffness properties of the bearing pads). Similarly, tieback anchor elements connect nodes in the thrust block solid mesh to nodes in the flexible wall solid mesh. The 6 in. element discretiz ation in the longitudinal direction is also important in that it permits enforcement of strain compatibility across the various components of the composite wall cross section. As is evident in Figure 313 B the reinforcement beam element nodes are not coincident with those of the solid elements. Without nodal coincidence (collocation of nodes), strain compatibility cannot be obta ined using nodal merging. However, the nodes are all coplanar since both the reinforcement beam elements and the concrete solid elements are discretized at a 6 in. element length along the span direction of the wall. Therefore, to enforce strain compatibil ity in the composite wall model, a constraint set is applied to all coplanar nodes at each 6 in. cross sectional slice of the wall. Consequently, nodes that are initially coplanar will remain coplanar (plane sections remain plane) as the crosssection of th e wall translates and rotates during loading. Concrete B eam E lements As noted earlier, the hollow portion of the flexible wall (recall Figure 37) is modeled with beam elements rather than with hexahedral or tetrahedral solid elements. In this mesh zone, individual beam elements, with a nonlinear material model, represent the structural behavior of each of the four faces of the concret e cross section (Figure 3 14). Because each face is modeled with cross section integrated beam elements, as opposed to resultant be am elements, an arbitrary number of integration points can be specified for each rectangular face beam in the cross section. By then specifying a large number of integration points for each face beam, a highresolution profile of stress can be calculated for each beam element, which results in an accurate depiction of the nonlinear flexural response of the overall hollow cross section (Figure 315). PAGE 43 43 Consistent with other parts of the model, each of the four face beam elements is discretized at a 6 in. element length along the span of the flexible wall. Since this element discretization coincides with that of the steel reinforcement (reb ar and tendons), constraints are applied to all coplanar nodes, at every 6 in. cross sectional slice, such that all coplanar nodes translate and rotate together (same constraint scheme as solids). This method provides strain compatibility between steel rei nforcement and the concrete face beam elements and enforces the condition that plane sections remain plane. Each face beam uses a concrete specific material model built into LS DYNA (LSTC 2007) in which nonlinear compressive behavior and tensionstiffenin g are taken into account. The LS DYNA material model is limited, however, in that the compression curve must use the Hognestad parabola, and the tensionstiffening curve is required to be linear. Material model parameters are defined so as to target the co mpression model suggested by Collins and Mitchell (1991) and the tensionstiffening model suggested by Hsu (1993). The compression portion of the target concrete material model, as suggested by Collins an d Mitchell (1991) is defined by Equations 32 through 3 6. Units for the compression model are defined in MPa for stress and modulus of elasticity. The compressive strength of concrete ( cf ) is taken as 5 ksi (34.47 MPa), as noted on the structural wall plans provide by the USACE. '33206900ccEf ( 32) '0.8 17 cf n ( 33) '(1)c c c fn En ( 34) '1 if 0.67(/62) if c c ck f ( 35) '(/) 1(/)c cc nk cn f n ( 36) PAGE 44 44 The tension portion of the target concrete material model, as sugg ested by Hsu (1993) is given by Equations 37 through 3 9. cr, is assumed to be 0.00008 in/in in the present study. Units for the tension model are defined in ksi for stress and modulus of elasticity. Figure 316 shows the resulting fit between the LS DYNA material model and the suggested models by Hsu, Collins, and Mitchell. '471000ct cEf ( 37) '0.003751000cr cff ( 38) 0.4for for /ct cr t cr crcrE f ( 39) Contact S hells Since the materially nonlinear portion of the flexible wall is modeled using beam elements (as opposed to solid elements), providing the ability of the barge flotilla FE model to make contact with the beam elements (during impact) requires special modeling procedures. To create a contactable surface for the beam elements, a separate entity referred to as a contact ring is employed. A contact ring defines the perimeter surface geometry of the flexible wall, as shown in Figure 317, and provides a surface for the barge to contact during impact simulation. Each contact ring is composed of a series of rigid shell elements, which together have the same physical dimensions as the perimeter of the flexible wall. Contact rings are spaced at 6 in. intervals along the axis of the flexible wall, and are 6.1 in. long (Figure 3 17 ) such that a small amount of overlap is provided from one contact ring to the next. Furthermore, the contact rings are spaced at 6 in. intervals so that they can be included in the constraint set between the reinforcement and conc rete beam elements. Consequently, the contact rings rotate and translate with the beam elements as the wall undergoes flexural deformation. PAGE 45 45 As a barge flotilla FE model impacts the flexible wall model, contact is made directly between the surface of barge and the contact rings. Contact forces imposed on the rings are then transmitted, through the planar constraints, to the nonlinear beam elements representing both the concrete and the reinforcement. Figure 3 18 shows a schematic of this contact scheme. Note that, in the figure, only a single beam element (representative of all concrete and reinforcement beam elements) is shown for clarity of pre sentation. While contact is defined between the impacting barge and the contact rings, contact is not defined between adjacent rings. Hence, the individual contact rings can overlap each other freely if necessary. Additionally, while the contact rings use a structural rigid material model, they are also assigned a contact stiffness equal to a modulus of elasticity of 4030 ksi (representative of concrete). Finally, since the contact rings are purely a computational mechanism used to transmit impact load, the y are defined as being approximately massless so as not to adversely alter the dynamic response of the composite wall model during impact. Match C ast Segments As previously discussed, the flexible wall is a segmental post tensioned system divided into three equal length match cast segments: two outer segments and a single inner segment. Within the outer segments, both beam elements and solid elements are employed to represent the composite concrete cross section. For the inner segment, only beam element s represent the cross section. Moreover, each of the three segments is modeled as an independent structure. Between the segments, at each segment joint, no continuity of concrete or mild steel exists. However, post tensioning is continuous across these joi nts. To model the effect of applying post tensioning to independent segments, not only must the post tensioning elements be modeled, but a surface at each joint for the post tensioning to react upon must also be modeled. A rigid shell surface, referred to as the segment contact face PAGE 46 46 (Figure 3 19), is modeled as the reaction surface for the post tensioning. For each segment joint, a segment conta ct face is modeled on each side of the joint and has the approximate shape of the hollow wall cross section. Each segment contact face is modeled with a structural rigid material model but is given a contact stiffness (modulus of elasticity) of 4030 ksi. C ontact defined between each set of adjacent segment contact faces is given a coefficient of friction of 0.6 per ACI 11.6.4.3 (ACI 2008). An initial gap of 0.001 in. is introduced between each set of adjacent segment contact faces to avoid initial contact s urface penetrations at the beginning of the numerical simulation. During extreme flexural deformation, the segment joints have the potential to form a gap on the tension side of the flexible wall, thereby inducing tension into the grout surrounding the post tensioning at the joints. Consequently, a sensitivity study is conducted to determine the effect of debonding the post tensioning (due to grout cracking) at the segment joints (Figure 320). Due to a lack of data in the literature regarding bond strength of multi strand tendons, it is assumed that the development length of individual prestressing strands will serve as a reasonable estimate of debonded length that might result under severe cracking. The American Association of State Highway and Transportation Officials (AASHTO) suggests a deve lopment length (AASHTO 2007) calculated using Equation 310. Development length is computed using the effective prestress stress (fpe), the diameter of the strand (db), and the expected stress in the steel at nominal member resistance (fps). In ad that is applied for additional conservatism. 2 3d pspeblffd ( 310) Trial debonded lengths on each side of the segment joint, equal to the AASHTO specified development length and one half of the AASHTO development length, are considered in the PAGE 47 47 sensitivity study. Tendon debonding is modeled by removing the rotational and translational co nstraints between the post tensioning elements and the concrete beam elements, thus removing strain compatibility between the tendons and the concrete. For each debondedlength case, a mid span point load is applied quasi statically and increased in magnit ude until flexural failure of the wall occurs. Load deflection data at mid span, extracted from the sensitivity analysis results, are plotted in Figure 321. This comparison indicates that there is very little change in the response of the flexible wall for the debonding lengths tested. Furthermore, the stiffest response is provided by the non debonded case, which would result in conservative calculation of impact forces under barge impact loading. Therefore, the flexible wall model used in this study employs no debonding of the post tensioning at the segment joints. Flexible Wall D amping Structural damping is included in the flexible wall model and as is typical for reinforced concrete structures, a damping level of 5% of critical damping is targeted. To determine the appropriate damping parameters for use in the LS DYNA finite element model, the concept of Rayleigh damping is considered. In the Ra yleigh damping model, several modes of vibration, typically those that are most active under the loading condition of interest, are targeted to a specific level of damping. Establishing damping parameters therefore requires the identification of modes of vibration that are most active during barge impacts on the flexible wall. To facilitate this identification, a flotilla impact simulation is conducted using the flexible wall model in order to obtain deflection time history data. The impact condition analyz ed to obtain this data involves a 3x3 barge flotilla impacting the flexible wall at mid span with an angle of obliquity of 17 and an initial velocity of 2.9 fps. This impact condition corresponds to the highest energy test conducted during the Winfield Lo ck and Dam barge impact study. For PAGE 48 48 this condition, the computed midspan deflection time history of the flexible wall is shown in Figure 322 (sampled at a frequency of 2000 Hz). The deflection data presented in Figure 322 suggest the presence o f at least two (but possibly more) active modes of vibration: an overall low frequency mode (with an approximate period of 1 sec. and frequency of 1 Hz) and a second, higher frequency mode superimposed on the lower frequency mode. To establish suitable Ray leigh coefficients for damping in the concrete wall, it is necessary to determine which modal frequencies correspond to vibrations in the wall and which correspond to unrelated oscillatory deformations in the barge flotilla. This modal separation process i s carried out as follows. 1) The wall is approximated as behaving in a manner similar to a beam of uniformly distributed mass. 2) Closedform analytical expressions are used to compute the natural frequencies of the equivalent beam representation. 3) A num erical frequency analysis is performed, using a Fast Fourier Transform (FFT), on the response data shown in Figure 3 22 to determine which of the previously analytically identified modal frequencies participate most strongly in the response of the wall to barge impact loading. 4) Once the frequencies of the key mode (or modes) of the wall that are active during impact load ing have been identified, appropriate Rayleigh damping coefficients to be computed to produce the desired damping level of 5% in the wall. Approximating the flexible wall as a simply supported beam with uniformly distributed mass, the natural frequencies can be calculated as documented by Tedesco et al. (1999) in Equation 311. m) for span length (L) modulus of elasticity (E), moment of inertia about the desired axis of bending sectional area (A). Using this equation, the natural frequencies of the flexible wall, for the first three modes of vibration, are 5 Hz, 21 Hz, and 47 Hz. PAGE 49 49 2 m nn EI LA ( 311) Figure 323 shows the results of an FFT analysis of the deflection data shown in Figure 322. Superimposed on the FFT data are the first, second, and third natural frequencies obtained by approximating the flexible wall as a simply supported beam. It is evident f rom Figure 323 that the lowest frequency content (~1Hz) in the FFT spectrum corresponds not to the wall, but to oscillations in the barge flotilla. Furthermore for the impact condition analyzed, by far the most active wall vibration mode is the first mode: the Fourier coefficient for this mode is more than an order of magnitude larger than the coefficients for modes two or three. For this reason, only the wall mode 1 frequency is considered in calculating the concrete wall Rayleigh damping parameters for LS DYNA. Upon calculating the damping parameters based on achieving approximately 5% of critical damping for the first natural mode, and applying these paramete rs to the flexible wall finite element model, a damped free vibration pluck test is numerically simulated, and the resulting deflection time history is analyzed to verify that 5% damping exists in the model. This deflection time history obtained from the pluck test is shown in Figure 324. Th e logarithmic decrement method (shown by Equations 312 and 313) is utilized to compute the damping ratio. To calculate the logarith measured (where x0 is the first peak and xn is the second peak that is n oscillations away). Applying this calculation to the data shown in Figure 324 reveals that the damping is 4.9%, which is sufficiently close to the target value of 5%. 01 lnnx nx ( 312) PAGE 50 50 21 2 1 ( 313) Staged Construction Issues and Ancillary Loading Prior to subjecting the flexible wall to barge impact loading, gravity buoyancy, and post tensioning loading conditions must first be applied. At the construction stage where the structure has achieved static equilibrium under these load conditions, there is zero force in the tieback anchors and all bearing pads are in cont act with the wall. Satisfying these prerequisites prior to conducting an impact simulation is complicated by the development of substantial elastic shortening during application of post tensioning forces. Additionally, this elastic shortening will naturall y induce forces into the tieback anchors before impact and will cause contact to be lost between the flexible wall and the stop block bearing pads. Therefore, these aspects of staged construction must be addressed to ensure that the flexible wall, under st atic equilibrium, will respond accurately to impact loading. The initialization period required to apply the static loading conditions (gravity, buoyancy, and post tensioning) and bring the beam into static equilibrium must be as short as possible to minimize computational cost. The entire initialization period, therefore, takes place over the course of one second of simulation time. For the first 0.5 sec. of this initialization period, the gravity, buoyancy, and post tensioning loads are ramped up from 0% to 100% of their respectively defined loads. During the next 0.5 sec., the static loads are held constant and an overdamped condition is imposed on the structure to help achieve static equilibrium as rapidly as possible. As described previously, the post tensioned multi strand tendon beam elements are modeled using a special formulation called cable beam elements. A cable element is a tension  PAGE 51 51 only beam element formulation that induces a prescribed force level on the element by contracting element length through time. This behavior is particularly useful for applying post tensioning loads to the flexible wall. Acco r ding to the USACE plan sets for the Winfield flexible wall, a posttension force of 6217 kips is distributed through the 10 multi strand tendons (recall Figure 39 and Table 3 2). In the FE model, this post tensioning load is ramped up from t = 0 sec to t = 0.5 sec and then held constant for another 0.5 seconds (from t = 0.5 to t = 1.00 sec) while the flexi ble wall achieves static equilibrium (Figure 325). After equilibrium is reached, the cable elements stop contracting in length, which locks i n the post tension force. After this point, the elements have the ability to deform and to interact as normal beam elements, but no longer impose changes in length to introduce additional axial loads. During application of the post tensioning load, signif icant elastic shortening occurs in the flexible wall, which poses two problems. First, the shortening will introduce artificial forces into the tieback anchors; forces which would not normally occur in the actual physical structure. Second, at each stop bl ock bearing pad, a gap will develop that is equal to half the length of elastic shortening of the wall. If uncorrected, upon impact by a barge, the wall will undergo a relatively large longitudinal shift (displacement) until contact between the wall and stop block bearing pad is abruptly reengaged. Again, this behavior does not correspond to the physical structure. To solve these issues, alternative modeling techniques are utilized for each component. Each stop block bearing pad element, while present in the model, is denoted inactive at the beginning of the static initialization period. A time based LS DYNA sensor is used to activate each stop block bearing pad element just prior the end of the initialization period (at t = 0.99 sec.; PAGE 52 52 just before impa ct begins). By activating the bearing pad elements in this way, each such element is considered undeformed at the length that results after elastic shortening of the wall has occurred. Hence, axial compressive loads applied after element activation are i mmediately resisted by the stiffness of the bearing pad (i.e., no gap exists). Each tieback anchor element is treated similarly. As previously noted, the tieback anchors are modeled using cable beam elements. Cable beam elements the same elements used to model post tensioning have the ability to contract in element length through time to achieve a specified tensile force level within the element. Since the tieback anchor elements should have zero force after static initialization (and the corresponding el astic wall shortening), a negligibly small target force level is specified during the initialization phase (from t = 0 sec to t = 0.99 sec). Consequently, the tieback anchor elements will contract or lengthen, as necessary, to achieve the specified nearly zero target axial force level. At the end of the initialization period, the element resumes behavior as a beam element. PAGE 53 53 Table 31. Bearing pad stiffness values Axial stiffness (kip/in) Shear stiffness (kip/in) Gravity bearing 13339 27 Thrust block bearing 18920 36 Stop block bearing 3807 14 Table 32. Post tensioned tendon parameters Tendon location No. of strands Tendon area (in 2 ) Post tension force (kip) A 31 4.743 747.0 B 31 4.743 747.0 C 31 4.743 747.0 D 31 4.743 747.0 E 24 3.672 578.3 F 31 4.743 747.0 G 31 4.743 747.0 H 24 3.672 578.3 I 12 1.836 289.2 J 12 1.836 289.2 258 39.474 6217 PAGE 54 54 Figure 31. Rendering of flexible wall guidance system ( S ource: http://www.stingraysurfgraphics.com/garage/projects/dams/win04.jpg Last accessed October, 2011) Figure 32. Flexible wall finite element model. Rigid cell Thrust block Single flexible wall span Cell Thrust block Flexible wall PAGE 55 55 A B Figure 33. Schematic of foundations and flexible wall. A) Plan view B) Elevation view 100 Flexible wall Cell Thrust block 140 124 129 129 Cell Flexible wall Top of rock 392 Thrust block 129 100 129 80 PAGE 56 56 A B Figure 34. Schematic of foundation assemblies with connection details. A) Plan view B) Section view 2 90 Cell Tieback anchor (typ.) Stop block Flexible wall Gravity load bearing pad 9 20 19 26 26 20 Thrust block 26 Thrust block bearing pad Stop block bearing pad 2 50 20 70 50 50 210 42 12 Thrust block Tieback anchor Cell Gravity load bearing pads Thrust block bearing pad PAGE 57 57 A B Figure 35. Rendering of foundation finite element model. A) Rigid cell B) Thrust block attached to rigid cell Figure 36. Gravity load bearing pad footprints superimposed on rigid cell FE mesh. Thrust block 26 Bearing footprint covers 20 nodes Bearing footprint covers 16 nodes 26 50 Rigid cell mesh (6 x 6) 9 Tributary bearing pad discrete spring Rigid cell outline Flexible wall outline PAGE 58 58 Figure 37. Hybrid modeling approach using both solid elements and beam elements Figure 38. Pattern of steel rein forcement within concrete cross section Solid cross section (Solid elements) Transition cross section (Solid elements) 46 60 2810 394 2810 60 46 Thrust block Cell Hollow cross section (Beam elements) Outer segment Outer segment Inner segment Segment joint 1180 Chamfers not modeled Typ. #9 rebar Typ. posttensioned multistrand tendon Typ. #5 rebar Rebar within chamfers is modeled Front face (Impact face) 66 13.7 26 56.5 13 10 PAGE 59 59 A B Figure 39. Post tensioned tendon locations. A) Solid cross section at ends of wall B) Hollow cro ss section used throughout the interior portion of the wall Figure 310. Post tensioned tendon material model 14 19 19 111 14.2 20 13 33 67 23 13 Typ. posttensioned multistrand tendon Front face (impact face) A B C D F G J I H E 07 16 16 39 09.2 26 13 33 31 26 06 Typ. posttensioned multistrand tendon Front face (impact face) A B C D F G J E H I Engineering strain (in/in)Engineering stress (ksi) 0 0.01 0.02 0.03 0.04 0.05 0.06 0 50 100 150 200 250 300 PAGE 60 60 Figure 311. Schematic of solid cross section A B Figure 312. Schematic of transition cross section. A) Adjacent to solid cross section B) Adjacent to hollow cros s section 81.2 100 Solid concrete cross section Front face (impact face) 23 36 24.2 46 36 20 Transition cross section adjacent to solid cross seciton Front face (impact face) 26 36.5 10 50 06 10 13.7 06 46.5 06 13 Transition cross section adjacent to hollow cross seciton Front face (impact face) PAGE 61 61 A B Figure 313. Rendering of solid element modeling. A) Without reinforcement beam elements visible B) With reinforcement beam elements visible Transition cross section Solid cross section Stop block Front face (impact face) Transition cross section Solid cross section Stop block Typ. #9 mild steel rebar Typ. #5 mild steel rebar Typ. PT multistrand tendon Front face (impact face) PAGE 62 62 Figure 314. Division of hollow concrete cross section. Figure 315. Cross section integration scheme (Depiction of integration poi nts simplified for clarity) Front face Top face Bottom face Rear face Cross section discretized into integration points iintegration pointth dAiiNeutral axis ii X Y Z Z Y zi yi Strain sampled at each integration point to develop nonlinear stress profile Cross section integrated to obtain section forces Y A Z A AM z dA M y dA P dA PAGE 63 63 Figure 316. Nonlinear concrete material model fit Strain (in/in)Stress (ksi) 0.003 0.002 0.001 0 0.001 0.002 0.003 0.004 0.005 1 0 1 2 3 4 5 Compression (Collins and Mitchell) Tension (Hsu) LSDYNA material model (fit) PAGE 64 64 A B Figure 317. Contact ring schematic. A) Section view B) Elevation view (some contact rings removed for clarity) Front face beam element Top face beam element Bottom face beam element Rear face beam element Contact ring 100 81.2 Contact ring Top face beam element Front face/rear face beam elements Bottom face beam elements 6 ring spacing 6.1 ring length PAGE 65 65 Figure 318. Interaction between barge and contact rings (plan view only a single beam element shown for clarity) Typ. contact ring Typ. 6 discretization of beam elements 6.1 contact ring length Typ. impacting barge bow corner 6 contact ring spacing Typ. beam element PAGE 66 66 A B Figure 319. Segment contact face on outer segment of flexible wall. A) Outer segment B) Outer segment viewing beam element region Figure 320. Grout cracking upon joint opening Posttensioning continuous through segment joints Contact rings Transition cross section (solid elements) Solid cross section (solid elements) Segment contact face Segment contact face Contact rings Solid cross section (solid elements) Transition cross section (solid elements) Concrete and mild steel beam elements Posttensioning tendon Impact load Compression zone remains in contact Tension side loses contact and gap forms Typ. posttensioned tendon Strain compatibility is lost as grout cracks when gap forms Flexible wall inner segment Flexible wall outer segment PAGE 67 67 Figure 321. Sensitivity of debonded length of post tensioning at segment joints Figure 322. Midspan deflection of flexible wall for 3x3 flotilla impact at 17 with initial velocity of 2.9 fps Deflection (in)Load (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 300 600 900 1200 1500 1800 2100 No debonded length 54" debonded length (1/2 AASHTO) 108" debonded length (AASHTO) Time (sec)Deflection (in) 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 PAGE 68 68 Figure 323. Frequency analysis of displacement time history of flexible wall. Figure 324. Deflection time history for damped pluc k test of flexible wall at mid span Frequency (Hz)M a gni t ude of F ouri e r c oe ffi c i e nt s 0 5 10 15 20 25 30 35 40 45 50 0 5E5 0.0001 0.00015 0.0002 Mode 1 Mode 2 Mode 3 Low frequencies (~1 Hz) related to flotilla oscillation Time (sec)Deflection (in) 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 PAGE 69 69 Figure 325. Overview of numerical process used to model post tensioning Flexible wall Typ. posttension tendon (cable elements) Cable elements contract through time to create PT force (t = 0 sec. to t = 0.5 sec.) P(t) P(t) L LL(t) Force held constant until equilibrium is achieved (t = 0.5 sec. to t = 1.0 sec.) Flexible wall initially in equilibrium (at t = 0 sec.) 1 2 3 P P LL PAGE 70 70 CHAPTER 4 VERIFICATION AND VALIDATION OF FLEXIB LE WALL MODEL Model V erification U sing I ndependent C ross S ectional A nalysis Flexural behavior of the flexible wall FE model is verified by comparing the momentcurvature (M curvature data cal culated by an independent cross sectional analysis tool. The cross sectional analysis tool utilizes the numerical procedure developed by Consolazio et al. (2004), and has the capability of analyzing simultaneous biaxial bending and axial loading applied to an arbitrary concrete cross section. This numerical procedure employs user defined nonlinear material models for mild steel, high strength steel, and concrete. Additionally, the procedure has been amended with the approach of Consolazio et al. (2007) to i ncorporate tensionstiffening into the calculation of concrete section forces. Two verification simulations, each considering different boundary conditions, are conducted. One simulation is conducted wherein the flexible wall is considered with idealized s imply supported boundary conditions (no foundation supports are modeled). A subsequent simulation considers the moment curvature relationship of the wall section with inclusion of the effects of the rigid cell and thrust block foundation supports. Simplified S ectional A nalysis A single span FE model (Figure 41 ) of the flexible wall is quasistatically loaded to f ailure with idealized pin roller boundary conditions. Cross sectional forces and section curvatures, at the mid span of the flexible wall, are obtained from the FE simulation results data from which the moment curvature relationship is plotted. The flexibl e wall FE model used in this verification case includes all nonlinear beam elements for concrete and steel reinforcement as well as solid element regions at each end of the flexible wall. However, foundation supports (cells, thrust PAGE 71 71 blocks, etc.) are not modeled and the supports are instead modeled using simple pinroller boundary conditions. The moment curvature relationship for the wall is also calculated independently using a validated cross sectional analysis software package (Consolazio et al. 2004, 2007). The concrete cross section, mild steel rebar, and post tensioned tendons are all considered in the analysis with the appropriate nonlinear material models. A comparison of the FE and sectional analysis moment curvature relationships is provided in Figure 42. Strong agreement is observed between the cross sectional analysis and the FE model. The ultimate moment obtained from FE analysis (631000 kipin) differs only by 2% from the value obtained from sectional analysis (645000 kipin). Sectional A nalysis with F oundation S upports An enhanced moment curvature analysis is also conducted that considers the contribution of the foundation supports to the moment curvature relationship. As the flexible wall responds to loading by flexure, the integrated stop blocks react upon the thrust blocks, consequently introducing axial force in the wall (Figure 43). As a result of this phenomenon, approximately 2000 kips of additional axial force (i.e., beyond the axial force produced by post tensioning) is observed in the flexible wall at the failure moment. This add ition of axial load increases the flexural failure capacity of the flexible wall, which is observed through the moment curvature relationship. To develop a corresponding moment curvature relationship for these boundary conditions using the cross sectional analysis tool, increases in axial load produced by the stop blocks must be taken into account. Since the axial load in the wall changes as a function of the applied lateral load at mid span, a series of individual cross sectional analyses are performed to obtain moment curvature data at a series of discretely sampled axial load levels. Following the procedure PAGE 72 72 described in Figure 44, a moment curvature relationship is obtained from FE analysis of the flexible wall and foundation supports under the applied lateral loading at midspan. Using this moment curvature relationship, a series of sample curvatures is created by discretizing the curve 0) and the curvature at ui) caused by the stop i) at midspan is also extracted from the same FE simulation, and is used to quantify the axial load that occurs at each of the discretized curvature levels. i i, a cross sectional analysis is conducted to compute the full moment curvature i. From each such curve, the moment Mi i of interest is quantified. Plotting the resulting set of points produced by this procedure generates a moment curvature relationship f or the flexible wall with inclusion of the effects of changing axial load. In Figure 45, a comparison of the moment curvature data computed from FE analysis of the wall and foundation model is compared to corresponding data computed from the independent sectional analyses. Good agreement is observed between the point sampled cross sectional analysis data and the FE data. The ultimate moment predict ed by the FE model (722000 kipin) differs only by 3% from the ultimate moment calculated by the cross sectional analysis (699000 kipin). Model V alidation U sing Winfield Lock and Dam E xperimental T est D ata In August 2008, the USACE conducted a series of twenty three full scale 3x3 barge flotilla impact experiments at the Winfield Lock and Dam in Red House, WV. As defined by the parameters illustrated in Figure 46, impact conditions tested at Winfield included angles of 0) ranging from 0.7 ft/sec (fps) to 3.6 fps. Although the final report documenting the se experiments (Barker et al., i n press) is still in production at the time of this writing, two measured impact force time histories from the test PAGE 73 73 program have been made available by the USACE for purposes of validating the flexible wall model developed in the present study. Impa ct conditions for these cases consisted of impacts by a 3x3 flotilla at: 17 with a velocity of 2.9 fps (denoted Experiment 10); and 13.6 with a velocity of 2.7 fps (denoted Experiment 20). To assess the accuracy of the flexible wall FE model, impact simu lations are conducted at impact angles and velocities that correspond to Winfield impact Experiments 10 and 20. Finite E lement M odeling of L oad M easurement S ystem Each impact experiment conducted at the Winfield Lock and Dam utilized a custom designed and fabricated load measurement beam (Figure 47 ) attached to the corner of the impacting barge in the flotilla to quantify impact forces. The complete load measurement system consists of a solid 9 in. x 5 in. curved steel beam, two bi axial 400 kip clevis pin (or shear pin) load cells, and two clevis mounts. Each shear pin is inserted through one end of the loadbeam and through a clevis mount that is welded to the ba rge bow. Upon impact, force is transmitted from the load beam to the shear pins, then to the clevis mounts, and finally into the corner of the barge bow. Since the structural configuration, stiffness, and distribution of impact forces associated with the l oad measurement beam (and the associated pins and clevis mounts) differ substantially from those of a typical, unmodified (uninstrumented) barge corner, it is important to determine whether impact forces generated through the use of such a load beam will differ significantly from impact forces that would be generated by collision from an uninstrumented barge. If the instrumented (loadbeam) and uninstrumented impact forces are significantly different, this has important implications for the design of naviga tion guidance structures. Furthermore, if significant differences exist, then validation of the finite element model can only achieved if a representation of the load beam itself is included in the barge model (since this is the manner in which the experim ental data were collected). Consequently, a finite element model PAGE 74 74 of the loadmeasurement system used in the USACE impact experiments is developed for use in conducting validation impact simulations. The FE model of the loadmeasurement system is shown in F igure 48 and is composed primarily of three dimensional solid elements representing the two clevis mounts and the load measurement beam. All such solid elements utilize the same nonlinear steel material model that is used in the barge finite element model (recall Figure 27). Other aspects of this material model include the use of a unit weight of 0.490 kip/ft3 and the inclusion of strain rate effects. Rather than modeling the shear pins with solid elements and using computationally expensive contact definitions between the load beam, pins, and cl evis mounts, instead a series of nodal rigid bodies are used to emulate the function of the pins. A nodal rigid body in an LS DYNA finite element model consists of a collection of nodes that are constrained to move as a single rigid body (i.e., the distanc es between the nodes within the body remain constant, even as the overall body translates and rotates through space). As illustrated in Figure 4 9, a t each end of the loadmeasurement beam, a nodal rigid body is defined to tie (constrain) nodes within the footprint of the clevis pin to a single line of nodes lying inside the clevis mount. Note that the nodal line inside the clevis mount corresponds the hypothetical longitudinal axis of the shear pin. This modeling technique results in a computationally efficient means of allowing the end of the load beam to rotate within the clevis mesh in a way that emulates the rotation that would be permitted by a pin. For purposes of attaching the clevis mounts to the corner of the barge bow model, translational constraints are introduced between nodes on the rear faces of the clevis mounts and the nearest corresponding nodes on the bow corner plate (Figure 410). PAGE 75 75 Simulations of I n strumented Barge I mpacts on the Winfield F lexible Wall To validate the flexible wall model, as well as the 3x3 barge flotilla model described earlier, simulations of impact Experiments 10 and 20 from the Winfield Lock and Dam test program are simulated numerically. Each experiment is simulated twice: once with a model of the load measurement beam attached to the corner of the im pacting barge, and once without the load measurement beam. Impact force and deflection timehistories obtained from these simulated conditions are shown in Figures 411 and 412 for Winfield Experiment 10, and Figures 413 and 414 for Experiment 20. All forces and deflections plotted i n these figures, experimental and numerical, are normal (perpendicular) to the longitudinal axis of the wall. In Figure 4 11 (Experiment 10), the peak impact force measured experimentally, using the load beam pictured in Figure 47, was 517 kips. Numerical impact simulation of the same condition, with inclusi on of the load beam finite element model depicted in Figure 410, predicts an impact force of 466 kips (10% less than the experimental value). In Figure 413 (Experiment 20), the peak impact force measured experimentally was 411 kips, whereas the numerical simulation predicts a for ce of 377 kips (8% less than the experimental value). Hence, when the numerical simulation models account for the manner in which the physical tests were conducted (i.e., use of the load measurement beam), good agreement is observed between test data and s imulation data. This observation also extends beyond just matching peak force magnitudes. In both Figures 411 and 413, the overall durations of loading agree well between experiment and simulation. Further, the dynamic frequency of oscillation in the experimental and simulated impact force time histories are also in reasonable agreement. In addition to the agreement of impact forces, peak deflections at mid span shown in Figures 412 and 414 are also in good agreement. The peak deflection for Experiment 10 is 0.52 in. whereas the finite element simulation predicts a peak of 0.51 in. Similarly, peak deflection for Expe riment 20 is 0.43 in. PAGE 76 76 whereas the finite element simulation predicts a peak of 0.41 in. Based on these observations, the flexible wall model and barge flotilla model are both considered to be validated to a level of accuracy ( 10% error in predicted peak force and 5% error in predicted peak deflection) that is acceptable for use in this study. In Figures 4 11 and 413, it is also important to note that peak impact forces (386 kips for the Experiment 10, and 313 kips for Experiment 20) predicted from simulations in which the load beam is omitted from the flotilla model (i.e., impact from an uninstrumented barge flotilla) are considerably smaller than the forces predicted when the load measurement beam is included. For both the Experim ent 10 and Experiment 20 conditions simulated here, inclusion of a loadmeasurement beam produces an approximately 20% increase in the impact forces that are predicted. This observed force increase is attributed to the fact that the stiffness of the loadm easurement beam is considerably larger than the stiffness of a typical uninstrumented barge bow corner. This finding indicates that unnecessary conservatism could be unintentionally introduced into design impact loads specified in design standards if such standards are developed strictly based on force measurements made using devices such as the load measure ment beam. Since the goal in the remainder of this study is to quantify peak impact forces for collisions from typical uninstrumented commercial barge flotillas, the impact simulations described in the following chapter are conducted using flotilla models (3x3 and 3x5) in which the loadmeasurement beam is excluded. PAGE 77 77 Figure 41. Schematic of moment curvature FE simulation. Figure 42. Moment curvature relationships obtained from cross sectional analysis and finite element analysis Midspan point load Pinroller boundary conditions Momentcurvature computed at midspan Nonlinear beam elements Solid elements Curvature ( *106)Moment (kipin) 10 10 30 50 70 90 110 130 150 170 180 0 200000 400000 600000 800000 Consolazio et al. 2004, ACI Structural Journal Flexible wall FE model PAGE 78 78 A B Figure 43. Mechanism of axial load increase w ithin flexible wall. A) Undeformed B ) Deformed under flexure Rigid cell Thrust block Flexible wall Stop block Rigid cell Thrust block Stop block reacting against thrust block N Midspan point load Flexible wall N PAGE 79 79 Figure 44. Point sampled moment curvature procedure Moment (M) Curvature ( Mresponse of FE model Curvatures discretized between initial curvature () and curvature at ultimate moment () uCurvature ( Axial load increase (N) u N pointsampled at each Moment (M) Curvature ( Mresponse of FE model i ui ui Mcomputed at each (N,)for specific (M, ii ii) PAGE 80 80 Figure 45. Comparison of moment curvature relationships from cross sectional analysis and finite element analysis of the wall and foundation model Figure 46. Definition of impact conditions at Winfield Lock and Dam Curvature ( *106)Moment (kipin) 10 10 30 50 70 90 110 130 150 0 100000 200000 300000 400000 500000 600000 700000 800000 Flexible wall FE model Consolazio et al. 2004 (Pointsampled N) D i r e c t i o n o f i n i t i a l v e l o c i t y B a r g e f l o t i l l a V0 Flexible wall Thrust block Rigid cell PAGE 81 81 Figure 47. Loadmeasurement system (beam, shear pins, and clevis mounts) attached to corner of barge (Photo credit: U.S. Army C orps of Engineers). Figure 48. Overview of finite element model of loadmeasurement system Clevis mount Loadmeasurement beam PAGE 82 82 A B Figure 49. Use of noda l rigid bodies in finite element model of load measurement system. A) Elevation view B) Plan view Figure 410. Rendering of loadmeasuring beam constrained to barge. Loadmeasurement beam Clevis mount Nodal rigid body Diameter of clevis pin load cells Nodally merged to solids Nodally merged to loadmeasurement beam Loadmeasurement beam Area of clevis pin load cells Clevis mount Center of rotation for clevis pin Nodal rigid body Loadmeasurement beam Clevis mount Bow corner plate Impacting barge bow corner (mesh not shown for clarity) Rear face of clevis mount constrained to exterior face of bow corner plate PAGE 83 83 Figure 411. Force time history for Experiment 10 and finite element simulations with and without the loadmeasurement beam Figure 412. Deflection time history for Experiment 10 and finite element simulations with and without the loadmeasurement beam Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 100 200 300 400 500 600 Winfield Exp. 10: 3x3 17 2.9 fps FEA (3x3 17 2.9 fps) FEA (3x3 17 2.9 fps with loadmeasurement beam) Time (sec)Deflection (in) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Winfield Exp. 10: 3x3 17 2.9 fps FEA (3x3 17 2.9 fps) FEA (3x3 17 2.9 fps with loadmeasurement beam) PAGE 84 84 Figure 413. Force time history for Experiment 20 and finite element simulations with and without the loadmeasurement beam Figure 414. Deflection time history for Experiment 20 and finite element simulations with and without the loadmeasurement beam Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 100 200 300 400 500 600 Winfield Exp. 20: 3x3 13.6 2.7 fps FEA (3x3 13.6 2.7 fps) FEA (3x3 13.6 2.7 fps with loadmeasurement beam) Time (sec)Deflection (in) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Winfield Exp. 20: 3x3 13.6 2.7 fps FEA (3x3 13.6 2.7 fps) FEA (3x3 13.6 2.7 fps with loadmeasurement beam) PAGE 85 85 CHAPTER 5 IMPACT SIMULATIONS A ND RESULTS Overview Using the barge flotilla and flexible wall finite element models verified and validated in the previous chapter, a core set of fifteen dynamic impact simulations are conduct ed at angles of 0) ranging from 2 fps to 5 fps with impact occurring at mid span (Figure 51, Table 51). Each impact simulation utilizes either a 3x3 barge flot illa model (three strings of three barges each; recall Figure 24) or a 3x5 barge flotilla model (three strings of five barg es each) to impact the high resolution FE model of the Winfield flexible wall. Force T ime H istories of I mpact S imulations In Figures 5 2 through 57, impact force time histories are plotted for each case noted in Table 51. Each force plot presents results of a specific barge flotilla size and impact angle wherein the individual data traces represent different impact velocities. Forces shown are normal (perpendicular) to the impact face of the flexible wall and have not been filtered or smoothed in any way (i.e., they are raw simulation results). Each impact simulation is carried out (continued) through time until both a zero impact force is achieved and the velocity vector of the flotilla indicates movemen t away from the flexible wall, thereby indicating impact has ceased and subsequent re impact is extremely unlikely. Peak impact forces and peak mid span wall deflections (resolved normal to the impact face of the wall) are summarized in Tables 52 and 53 for each flotilla size (3x3 and 3x5). When the data presented in Tables 52 and 53 are considered independently of each other, the trends indicated are essentially as expected in that increases of impact velocity and/or impact angle are accompanied by increases of impact force. However, when comparisons are made PAGE 86 86 between impact forces generated by different bar ge flotilla sizes (masses) impacting under the same conditions (approach angle and impact velocity), the results are not as expected. As the data summarized in Table 54 demonstrate, peak impact forces produced by 3x3 and 3x5 flotilla impacts are nearly identical (less than 3% difference) as long as the impact conditions (angle, velocity) are identical. This phenomenon of increased barge flotilla size (mass ) not being accompanied by increased impact force, while unexpected, can be understood by considering the longitudinal flexibility of multibarge flotillas. A flotilla is comprised of individual barges (each with a relatively high stiffness) that are inter connected by a series of lashings (with a relatively low stiffness). In most flotilla collision scenarios, peak impact force occurs during the initial load pulse that is generated during initial contact with the wall. During this phase, only the fore most portion of the flotilla is redirected by the flexible wall. In Figure 58, a simplified illustration of flotilla deformation dur ing this phase is presented. The portion of flotilla mass effectively engaged by the wall when peak impact force is generated is essentially the same regardless of flotilla length (3x3 or 3x5). This effect is clearly evident upon examination of the finite element simulations results. Figure 59 shows the deformed configurations (scaled by a factor of 50 so that deflections are easi ly visible) of the 3x3 and 3x5 barge flotillas at time t=0.4 sec (when peak impact force occurs for the 20, 3 fps impact condition). Clearly apparent in the figure is the fact that, at the point in time when peak impact force is generated, only the first (fore most) row of barges has deflected in both the 3x3 and 3x5 impact cases. This explanation is further corroborated by the force time histories shown in Figures 510 to 514, which compare the 3x3 and 3x5 barge flotilla results for matching impact conditions. PAGE 87 87 Evident in each plot is the fact that the force time histories for the 3x3 and 3x5 flotillas are nearly identical through the first impact pulse, particularly for shallow angle impacts (10 and 20). In the steeper angle (30) impact cases, impact force time histories agree not only through t he first pulse, but also through the second pulse as well. In the 30 cases, multiple pulses of significant impact load occur, but forces generated during the first pulse are still reasonable approximations of overall maximum force arising during the entir e collision event. The relationship revealed by this study between flotilla size, flotilla deformation, and impact force has important implications for the future development of an equivalent static impact force prediction method (i.e., a new design stand ard). Previous design methodologies developed by the USACE, such as ETL (Engineer Technical Letter) 11102 563 (USACE 2004), have correlated impact force to total momentum (where both the force and momentum are measured in the direction normal to the impac ted surface). However, the findings of this study suggest that flotilla momentum alone may not be an adequate predictor of impact force. Figure 515 p resents results for all 3x3 and 3x5 barge flotilla impacts considered in this study by plotting peak impact force as a function of momentum (where both force and momentum are in the direction normal to the wall). In the figure, impact cases that generate t he same impact force (due to having the same impact angle and speed, but differing momentums) are bounded by a dashed box. As evidenced by the figure, peak impact forces cannot be uniquely correlated to momentum. In particular, at high levels of impact momentum (i.e., severe, high energy impacts), a wide range of differing momentums may result in the same peak impact force. Sensitivity of B arge F lotilla I mpact F orces to F lotilla S ize Considering the findings discussed above, it is desirable to determine the minimum barge flotilla length required to generate a peak impact force that is equivalent to that produced by the 3x3 and 3x5 barge flotillas. Hence, four additional impact simulations are conducted, each using PAGE 88 88 a un ique barge flotilla size (1x3, 2x3, 3x2, and 3x1) to further explore the relationship between flotilla size and peak impact force. Each of these additional flotilla impact simulations is conducted at an angle of 20 and at an impact speed of 3 fps. An ill ustration of each barge flotilla utilized (1x3, 2x3, 3x2, and 3x1) is provided in Figure 516. Note that each of these flotilla finite element models is created by deleting appropriate portions of the 3x3 barge flotilla model presented earlier. Time history force results from the four additional simulations are presented in Figures 517 and 518, which illustrate the effects of varying flotilla length and flotilla width, respectively. Impact forces presented are in the direction normal to the impact face of the flexible wall. Peak force results are also summarized in Tables 55 and 56. From Figure 517 and Table 55, it is evident that flotilla length has little influence on peak impact force generated. Even the single row (three wide) 3x1 flotilla is capable of producing a peak impact force that is within 10% of that produced by a 3x3 flotilla. This result further corroborates the notion that the magnitude of the initial pulse of impact force generated during a collision is primarily attributable to redirection of the first row of barges, regardless of the overall length of the flotilla. Results presented in Figure 518 and Table 56, reveal that, unlike flotilla length, flotilla width has a significant influence on peak force generated during impact. Since changing the flotilla width involves changing the mass contained within the first row, it is logical that peak impact force will decrease as the width (and therefore mass) of the lead row decreases. An interesting consequence of this finding may be that since two flotillas of equal mass (and therefore eq ual cargo carrying capacity) but differing widths (e.g. a 3x2 and a 2x3) produce different impact forces for the same impact angle and speed, in selected circumstances, it might PAGE 89 89 be possible to limit the impact forces generated on navigational structures by limiting the width of barge flotillas that operate on a particular section of waterway. PAGE 90 90 Table 51. Barge flotilla impact conditions Flotilla type Angle Velocity (V 0 ) (ft/sec) 3x3 10 2 3x3 10 3 3x3 10 4 3x3 20 3 3x3 20 4 3x3 20 5 3x3 30 3 3x3 30 5 3x5 10 2 3x5 10 3 3x5 20 2 3x5 20 3 3x5 20 4 3x5 30 4 3x5 30 5 Table 52. Summary of results from 3x3 barge flotilla impact simulations Impact condition Peak normal impact force (kip) Peak normal mid span deflection (in) 10 2 fps 229 0.259 10 3 fps 276 0.313 10 4 fps 313 0.388 20 3 fps 483 0.554 20 4 fps 595 0.636 20 5 fps 658 0.735 30 3 fps 778 0.761 30 5 fps 906 0.992 Table 53. Summary of results from 3x5 barge flotilla impact simulations Impact condition Peak normal impact force (kip) Peak normal mid span deflection (in) 10 2 fps 229 0.252 10 3 fps 278 0.319 20 2 fps 378 0.439 20 3 fps 492 0.572 20 4 fps 598 0.630 30 4 fps 831 0.884 30 5 fps 881 0.965 PAGE 91 91 Table 54. Comparison of results between 3x3 and 3x5 flotilla impact simulations Impact condition Normal impact force (kip) 3x3 flotilla 3x5 flotilla 10 2 fps 229 229 (+0%) 10 3 fps 276 278 (+1%) 20 3 fps 483 492 (+2%) 20 4 fps 595 598 (+1%) 30 5 fps 906 881 ( 3%) Table 55. Effect of flotilla length on peak impact forces Flotilla configuration Flotilla weight (kip) Peak normal impact force (kip) Total impulse of impact (kip sec) 3x3 36000 483 540 3x2 24000 ( 33%) 474 ( 2%) 372 ( 31%) 3x1 12000 ( 67%) 440 ( 9%) 195 ( 64%) Table 56. Effect of flotilla width on peak impact forces Flotilla configuration Flotilla weight (kip) Peak normal impact force (kip) Total impulse of impact (kip sec) 3x3 36000 483 540 2x3 24000 ( 33%) 414 ( 14%) 359 ( 34%) 1x3 12000 ( 67%) 319 ( 34%) 177 ( 67%) PAGE 92 92 Figure 51. Schematic of impact condition for flotilla impact simulations Figure 52. Force time histories for 3x3 barge flotillas impacting at 10. D i r e c t i o n o f i n i t i a l v e l o c i t y B a r g e f l o t i l l a V0 Rigid cell Impact occurs at midspan Thrust block Flexible wall Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 10 2 fps 3x3 10 3 fps 3x3 10 4 fps PAGE 93 93 Figur e 53. Force time histories for 3x3 barge flotillas impacting at 20. Figure 54. Force time histories for 3x3 barge flotill as impacting at 30 Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 20 3 fps 3x3 20 4 fps 3x3 20 5 fps Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 30 3 fps 3x3 30 5 fps PAGE 94 94 Figure 55. Force time histories for 3x5 barge flotillas impacting at 10. Figure 56. Force time histories for 3x5 barge flotillas impacting at 20. Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x5 10 2 fps 3x5 10 3 fps Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x5 20 2 fps 3x5 20 3 fps 3x5 20 4 fps PAGE 95 95 Figure 57. Force time histories for 3x5 barge flotillas impacting at 30. A B Figure 58. Simplified illustration of flotilla deformation during impact. A) 3x3 barge flotilla B) 3x5 barge flotilla Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x5 30 4 fps 3x5 30 5 fps Initial impact redirects mass of foremost row Remaining rows of flotilla unaffected during initial impact pulse Remaining rows of flotilla unaffected during initial impact pulse Initial impact redirects mass of foremost row PAGE 96 96 A B Figure 59. Barge flotilla deformation for impact condition 20 and 3 fps (displacements scaled by factor of 50 and flexible wall removed for clarity). A) 3x3 barge flotilla B) 3x5 barge flotilla Figure 510. Forc e time histories for 3x3 and 3x5 barge flotillas impacting at 10 with 2 fps initial velocity Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 10 2 fps 3x5 10 2 fps PAGE 97 97 Figure 511. Force time histories for 3x3 and 3x5 barge flotillas impacting at 10 with 3 fps initial velocity Figure 512. Force time histories for 3x3 and 3x5 barge flotillas impacting at 20 with 3 fps initial velocity Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 10 3 fps 3x5 10 3 fps Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 20 3 fps 3x5 20 3 fps PAGE 98 98 Figure 513. Force time histories for 3x3 and 3x5 barge flotillas impacting at 20 with 4 fps initial velocity Figure 514. Force time histories for 3x3 and 3x5 barge fl otillas impacting at 30 with 5 fps initial velocity Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 20 4 fps 3x5 20 4 fps Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 100 200 300 400 500 600 700 800 900 1000 3x3 30 5 fps 3x5 30 5 fps PAGE 99 99 Figure 515. Peak impact forces plotted as a function of momentum (both quantities correspond to the direction normal to the flexible wall) A B C D Figure 516. Barge flotilla configurations considered in sensitivity study. A) 3x2 flotilla B) 3x1 flotilla C) 2x3 flotilla D) 1x3 flotilla Momentum (kipsec)F o r c e ( k i p ) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 400 500 600 700 800 900 1000 3x3 barge flotilla 3x5 barge flotilla Cases with same impact angle and speed Outline of 3x3 flotilla 3x2 flotilla Flexible wall Outline of 3x3 flotilla 3x1 flotilla Flexible wall Outline of 3x3 flotilla 2x3 flotilla Flexible wall Outline of 3x3 flotilla 1x3 flotilla Flexible wall PAGE 100 100 Figure 517. Effect of flotilla length on impact force time histories Figure 518. Effect of flotilla width on impact force time histories Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 100 200 300 400 500 600 3x3 20 3 fps 3x2 20 3 fps 3x1 20 3 fps Time (sec)Force (kip) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 100 200 300 400 500 600 3x3 20 3 fps 2x3 20 3 fps 1x3 20 3 fps PAGE 101 101 CHAPTER 6 CONCLUSIONS In this study, a finite element model of a flexible guide wall currently in service at the Winfield Lock and Dam, WV, was developed, verified, validated, and subjected to simulat ed barge flotilla impacts for the purpose of quantifying impact forces. The finite element wall model accounts for the segmental post tensioned nature of the physical wall at Winfield Lock and Dam, the variable cross sectional shape of the wall, nonlinear concrete and mild reinforcing steel behavior, the effects of post tensioning (including elastic shortening), and approximation of bearing pad stiffness, among other issues. Nonlinear static flexural response of the finite element wall model was verified ag ainst results obtained using an independent (and previously validated) sectional analysis calculation procedure. The flexible wall finite element model was then combined with previously developed models of 3x3 and 3x5 barge flotillas to facilitate dynamic barge wall impact simulations. Validation of the combined wall and barge flotilla models was carried out by conducting simulations of two full scale impact experiments conducted by the US Army Corps of Engineers (USACE) at the Winfield Lock and Dam. Becau se the physical experiments conducted at Winfield utilized a relatively stiff steel load measurement beam that was affixed to the impacting barge, a finite element model of this same load measurement device was developed in this study and included in the validation impact simulations. Force time histories from impact simulations were compared to data gathered during the Winfield impact experiments, and, for the two impact conditions examined, peak impact forces from the simulations were within 10% of the pe ak forces measured during the experiments. Additionally, the stiffness of the loadmeasurement beam was found to alter the magnitude of forces generated during impact. A comparison of results from finite element impact simulations of barge flotillas with a nd without a load PAGE 102 102 measurement beam revealed that the presence of the load measurement beam artificially increased the impact forces by approximately 20%. Following validation, a set of fifteen (15) dynamic, nonlinear finite element impact simulations were conducted on the flexible wall model using both 3x3 and 3x5 barge flotilla models. Impacts were conducted at angles ranging from 10 to 30 and for impact velocities ranging from 2 ft/ sec to 5 ft/sec Unexpectedly, simulation results revealed that 3x3 and 3x5 barge flotillas produce nearly equal peak impact forces when impacting the flexible wall at the same angle of obliquity and impact velocity. To aid in understanding this phenomenon, four (4) additional impact simulations were conducted for reduced barg e flotilla sizes of 3x2, 3x1, 2x3, and 1x3. By comparing data from impact simulations involving 3x5, 3x3, 3x2, 3x1, 2x3, and 1x3 barge flotillas, it was determined that peak impact forces are primarily a function of the mass contained within the first (for e most) row of barges in the flotilla Due to the relative flexibility of the lashings, during a flotilla wall collision event, only barges in the fore most row of the flotilla are redirected immediately by the wall. Because maximum impact forces generally occur during the first load pulse, when only the fore most barge row is being redirected, peak impact forces correlate to the mass of the lead row, not the mass of the entire flotilla. This finding has important implications for the future development of an updated equivalent static load prediction method for use in the design of navigational guidance structures. The practice of relying on linear momentum as a predictor of impact force, as has been used in the development of past design methodologies, is not likely to lead to the most efficient and economical design standards if applied to multi barge flotilla impact scenarios. A more refined approach in which flotilla momentum is supplemented with additional items of PAGE 103 103 data, such as geometric configuration i nformation, will be required if a simple and conservative, yet accurate (and therefore economical) impact load prediction procedure is to be developed. PAGE 104 104 REFERENCES ACI Committee 318 (2008). Building Code Requirements for Structural Concrete (ACI 318 08), American Concrete Institute, Farmington Hills, MI AASHTO. (2007). AASHTO LRFD Bridge Design Specifications 4th Ed ., AASHTO, Washington D.C. Barker, B.C., Ebeling, R.M., Gray, C.E., Haskins, R.W., and Warren, T.W. (In press). Barge impact experi ments at Winfield Lock and Dam, Winfield, West Virginia USACE: ERDC, Vicksburg, MS. Collins, M. P., and Mitchell, D. (1991). Prestressed Concrete Structures, Prentice Hall, Englewood Cliffs, NJ C onsolazio, G.R., Fung, J., Ansley M. (2004). M P Diagrams for Concrete Sections Under Biaxial Flexure and Axial Compression, American Concrete Institute (ACI) Structural Journal 101(1), 114123. Consolazio, G.R., Woolf, E. (2007). Advanced Algorithm s for Computing Moment Curvature Relationships for Concrete Sections. Structures Research Rep No. 2007/65163, University of Florida, Gainesville, FL Consolazio, G.R., Walters, R.A., Harper, Z.S. (2010). Development of Finite Element Models for Studying Multibarge Flotilla Impacts. Structures Research Report No. 2010/87754, University of Florida, Gainesville, FL Gent, A. N. (2001) Engineering with Rubber: How to Design Rubber Components Hanser Gardner Publications, Cincinnati, OH. Hsu, Thomas T.C. ( 1993). Unified Theory of Reinforced Concrete, CRC Press, Boca Raton, FL Jones, N. (1997). Structural Impact Cambridge University Press, Cambridge, U.K. LSTC (2009). LS DYNA Keyword Users Manual: Version 971, Livermore Software Technology Corporation, L ivermore, CA Nilson, A. H. (1987). Design of Prestressed Concrete John Wiley & Sons, Hoboken, NJ Patev, R.C., B arker, B.C., Koestler III, L.V. (2003). Full Scale Barge Impact Experiments, Robert C. Byrd Lock and Dam, Gallipolis Ferry, West Virginia, USACE: ERDC, Vicksburg, MS. Tedesco, J.W., McDougal, W.G., Ross, C.A. (1999) Structural dynamics: theory and applications Addison Wesley Longman, Inc., Menlo Park, CA U SACE. (2004). Barge Impact Analysis for Rigid Walls, Engineer Technical Letter 11102563, USACE, Washington D.C PAGE 105 105 USACE. (2007). 2007 Flood Control and Navigation Maps: Mississippi River, U SACE Washington D.C. PAGE 106 106 BIOGRAPHICAL SKETCH The author was born in 1989 in Fort Pierce, FL. He pursued an Associate of Arts degree in e ng ineering while completing high school and then attended Florida State University in August 2007 to study c ivil e ngineering. After receiving a Bachelor of Science degree in c ivil e ngineering, the author began graduate school at the University of Florida in January 2010. He plans to receive a Master of Engineering degree in ci vil e ngineering in May 2012, with an emphasis in civil structures. After completion of the degree, the author will pursue a career in the design of civil structures. 