INFLUENCE OF PIER NONLINEARITY, IMPACT ANGLE, AND COLUMN SHAPE ON PIER RESPONSE TO BARGE IMPACT LOADING

INFLUENCE OF PIER NONLINEARITY, IMPACT ANGLE, AND COLUMN SHAPE ON PIER RESPONSE TO BARGE IMPACT LOADING By BIBO ZHANG 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

ACKNOWLEDGEMENTS I would like to thank my research advisor, Dr. Gary Consolazio for providing continuous guidance, excellent research ideas, detailed teaching and all this with a lot of patience. I am thankful for being able to learn so much during the past year and a half. I would also like to extend my gratitude to Florida Department of Transportation for providing funding for this project. I would like to express my heartfelt thanks to all the graduate students who worked on this project, especially Ben Lehr, David Cowan, Alex Biggs and Jessica Hendrix. Their research helped me enormously in completing my thesis. My family and friends have been very supportive throughout this effort. I wish to thank them for their understanding and support. ii

TABLE OF CONTENTS page ACKNOWLEDGEMENTS... ii LIST OF TABLES...v LIST OF FIGURES... vi ABSTRACT... ix CHAPTER INTRODUCTION.... Overview.... Background of AASHTO Guide Specification....3 Objective... AASHTO BARGE AND BRIDGE COLLISION SPECIFICATION...5 3 FINITE ELEMENT BARGE IMPACT SIMULATION...9 3. Introduction...9 3. Background Study... 3.3 Pier Model Description... 3. Barge Finite Element Model...9 3.5 Contact Surface Modeling...6 NON-LINEAR PIER BEHAVIOR DURING BARGE IMPACT...3. Case Study...3. Analysis Results...3 5 SIMULATION OF OBLIQUE IMPACT CONDITIONS...37 5. Effect of Strike Angle on Barge Static Load-Deformation Relationship...38 5. Effect of Strike Angle on Dynamic Loads and Pier Response... 5.3 Dynamic Simulation Results... iii

6 EFFECT OF CONTACT SURFACE GEOMETRY ON PIER BEHAVIOR DURING IMPACT...5 6. Case Study...5 6. Results...5 7 COMPARISON OF AASHTO PROVISIONS AND SIMULATION RESULTS...63 8 CONCLUSIONS...67 LIST OF REFERENCES...69 BIOGRAPHICAL SKETCH...7 iv

LIST OF TABLES Table page 3- Comparison of original and adjusted section properties...6 3- Input data in LS-DYNA simulations...8 3-3 Comparison of plastic moment and displacement using properties of pier cap...9 3- Comparison of plastic moment and displacement using properties of pier column...9 3-5 General modeling features of the testing barge...5 - Dynamic simulation cases...3 5- Dynamic simulation cases... 7- Peak forces computed using finite element impact simulation...66 v

LIST OF FIGURES Figure page - Relation between impact force and barge damage depth according to Meir- Dornberg s Research (after AASHTO [])...3 - Collision energy to be absorbed in relation with collision angle and the coefficient of friction (after AASHTO [])...8 3- Global modeling of San-Diego Coronado Bay Bridge (after Dameron [])... 3- Pier model used for local modeling (after Dameron [])... 3-3 Global pier modeling for seismic retrofit analysis (after Dameron [])... 3- Mechanical model for discrete element (after Hoit [])...3 3-5 Bilinear expression of moment-curvature and stress-strain curve...7 3-6 Moment-curvature derivation...8 3-7 Main deck plan of the construction barge... 3-8 Outboard profile of the construction barge... 3-9 Typical longitudinal truss of the construction barge... 3- Typical transverse frame (cross bracing section) of the construction barge... 3- Dimension and detail of barge bow of the construction barge... 3- Layout of barge divisions... 3-3 Meshing of internal structure of zone-...3 3- Buoyancy spring distribution along the barge...6 3-5 Pier and contact surface layout...7 3-6 Rigid links between pier column and contact surface...7 3-7 Exaggerated deformation of pier column and contact surface during impact...8 vi

3-8 Comparison of impact force versus crush depth for rigid and concrete contact models...9 3-9 Overview of barge and pier model for dynamic simulation...3 - Comparison of impact force history for severe impact case...3 - Comparison of impact force history for non-severe case...3-3 Impact force and crush depth relationship comparison for severe impact case...35 - Comparison of impact force crush depth relationship for non-severe case...35-5 Comparison of pier displacement for severe impact case...36-6 Comparison of pier displacement for non-severe case...36 5- Static crush between pier and open hopper barge...38 5- Results for static crush analysis conducting with a ft. wide pier...39 5-3 Results for static crush analysis conducting with a 6 ft. wide pier...39 5- Results for static crush analysis conducting with a 8 ft. wide pier... 5-5 Layout of barge head-on impact and oblique impact with pier... 5-6 Impact force in X direction for high speed impact on rectangular pier... 5-7 Impact force in X direction for high speed impact on circular pier.... 5-8 Impact force in X direction for low speed impact on rectangular pier...5 5-9 Impact force in X direction for low speed impact on circular pier...5 5- Impact force in Y direction for high-speed oblique impact...6 5- Impact force in Y direction for low speed oblique impact...6 5- Force-deformation results for high speed impact on rectangular pier...7 5-3 Force deformation results for high speed impact on circular pier...7 5- Force-deformation results for low speed impact on rectangular pier...8 5-5 Force-deformation results for low speed impact on circular pier...8 5-6 Pier displacement in X direction for high speed impact on rectangular pier...9 5-7 Pier displacement in X direction for low speed impact on rectangular pier...9 vii

5-8 Pier displacement in X direction for high speed impact on circular pier...5 5-9 Pier displacement in X direction for low speed impact on circular pier...5 5- Pier displacement in Y direction for high-speed oblique impact...5 5- Pier displacement in Y direction for low speed oblique impact....5 6- Impact force in X direction for high speed head-on impact...5 6- Impact force in X direction for high speed oblique impact...55 6-3 Impact force in X direction for low speed head-on impact...55 6- Impact force in X direction for low speed oblique impact...56 6-5 Impact force in Y direction for high speed oblique impact...56 6-6 Impact force in Y direction for low speed oblique impact...57 6-7 Pier displacement in X direction for high speed head-on impact...57 6-8 Pier displacement in X direction for high speed oblique impact...58 6-9 Pier displacement in X direction for low speed head-on impact...58 6- Pier displacement in X direction for low speed oblique impact...59 6- Pier displacement in Y direction for high speed oblique impact...59 6- Pier displacement in Y direction for low speed oblique impact....6 6-3 Vector-resultant force-deformation results for high speed head-on impact...6 6- Vector-resultant force-deformation results for high speed oblique impact...6 6-5 Vector-resultant force-deformation results for low speed head-on impact...6 6-6 Vector-resultant force-deformation results for low speed oblique impact...6 7- AASHTO and finite element loads in X direction...6 7- AASHTO and finite element loads in Y direction....65 viii

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 INFLUENCE OF PIER NONLINEARITY, IMPACT ANGLE, AND COLUMN SHAPE ON PIER RESPONSE TO BARGE IMPACT LOADING By Bibo Zhang December Chair: Gary R. Consolazio Major Department: Civil and Coastal Engineering Current bridge design specifications for barge impact loading utilize information such as barge weight, size, and speed, channel geometry, and bridge pier layout to prescribe equivalent static loads for use in designing substructure components such as piers. However, parameters such as pier stiffness and pier column geometry are not taken into consideration. Additionally, due to the limited experimental vessel impact data that are available and due to the dynamic nature of incidents such as vessel collisions, the range of applicability of current design specifications is unclear. In this thesis, high resolution nonlinear dynamic finite element impact simulations are used to quantify impact loads and pier displacements generated during barge collisions. By conducting parametric studies involving pier nonlinearity, impact angle, and impact zone geometry (pier-column cross-sectional geometry), and then subsequently comparing the results to those computed using current design provisions, the accuracy and range of applicability of the design provisions are evaluated. The comparison of AASHTO provisions and ix

simulation results shows that for high energy impacts, peak predicted barge impact forces are approximately 6% of the equivalent static AASHTO loads. For low energy impacts, peak dynamic impact forces predicted by simulation can be more than twice the magnitude of the equivalent static AASHTO loads. However, because the simulationpredicted loads are transient in nature whereas the AASHTO loads are static, additional research is needed in order to more accurately compare results from the two methods. x

CHAPTER INTRODUCTION. Overview Barge transportation in inland waterway channels and sea coasts has the potential to cause damage to bridges due to accidental impact between barges and bridge substructures [-]. Recently, two impact events caused damage serious enough to collapse bridges and unfortunately result in the loss of lives as well. To address the potential for such situations, loads due to vessel impacts must be taken into consideration in substructure (pier) design using the American Association of State Highway and Transportation Officials (AASHTO) Highway Bridge Design Specifications [5] or the AASHTO Guide Specification for Vessel Collision Design for Highway Bridges []. In design practice, the magnitude and point of application of the impact load are specified in the AASHTO provisions []. The focus of this thesis is on the evaluation of whether the loads specified in the AASHTO provisions [] are appropriate given the variety of barge types, pier geometries and impact angles that are possible. This goal may be approached in several ways: analytical methods, experimental methods, or both. This thesis focuses on the analytical approach: nonlinear finite element modeling to dynamically simulate barge collisions with bridge piers. Of interest is to estimate the range of the impact load due to different impact conditions and other considerations that might affect the peak value of impact load and the impact duration time. The dynamic analysis code LS-DYNA [6] was employed for all impact simulations presented in this thesis.

. Background of AASHTO Guide Specification The AASHTO Guide Specification For Vessel Collision Design [] covers the following topics: Part : General provision (ship and barge impact force and crush depth) Part : Design vessel selection Part 3: Bridge protection system design Part : Bridge protection planning Part is directly related to the goal of this thesis: checking the sufficiency of the design barge impact forces specified by AASHTO. Therefore, only Part is discussed in this section. The method to determine impact force due to barge collision of bridges in AASHTO is based on research conducted by Meir-Dornberg in West Germany in 983 []. Very little research has been presented in the literature with respect to barge impact forces. The experimental and theoretical studies performed by Meir-Dornberg were used to study the collision force and the deformation when barges collide with lock entrance structures and with bridge piers. Meir-Dornberg s investigation also studied the direction and height of climb of the barge upon bank slopes and walls due to skewed impacts and groundings along the sides of the waterway. Meir-Dornberg s study included dynamic loading with a pendulum hammer on three barge partial section models in scale :.5; static loading on one barge partial section model in scale :6; and numerical computations. The results show that no significant difference was found between the static and dynamic forces measured and that impact force and barge bow damage depth can be expressed in a bilinear curve as shown

3 in Figure -. The study further proposed that barge bow damage depth can be expressed as a function of barge mass and initial speed. 3 5 P B (kips) 5 5 6 8 a B (feet) Figure -. Relation between impact force and barge damage depth according to Meir- Dornberg s Research (after AASHTO []) AASHTO adopted the results of Meir-Dornberg s study with a modification factor to account for effect of varying barge widths. In Meir-Dornberg s research, only European barges with a bow width of 37. ft were considered, which compares relatively closely with the jumbo hopper barge bow width of 35. ft. The jumbo hopper barge is the most frequent barge size utilizing the U.S. inland waterway system. The width modification factor adopted by AASHTO is intended to permit application of the design provisions to barges with different bow widths. Impact load is then defined as an equivalent static force that is computed based on impact energy and barge characteristics. A detailed description of the calculation of the equivalent static force according to AASHTO is included in Chapter of this thesis.

.3 Objective The finite element based analysis method described in this thesis is part of a project funded by FDOT [] to study the uncertainties in the basis of the barge impact provisions of the AASHTO. The project consists of a combination of analytical modeling and fullscale impact testing of the St. George Island Causeway Bridge. The results from this thesis provide analytically based estimations of impact forces and barge damage levels, and may be used for comparison to results from the full-scale impact tests. The structure of the remainder of this thesis is as follows: Chapter explains the AASHTO design method for computing impact force and bow damage depth. Chapter 3 describes nonlinear finite element modeling of the impact test barge and piers of the St. George Island Causeway Bridge. Chapter investigates the effect of non-linearity of pier material on impact force and barge damage depth by comparing pier behavior predicted by linear and nonlinear material models. Chapter 5 examines the effect of impact surface geometry on impact force and dynamic pier behavior. Two types of geometry are considered: rectangular and circular pier cross sections. Chapter 6 examines the effect of impact angle on impact force and pier behavior. Head-on impacts and 5 degree oblique impacts are investigated for both rectangular and circular piers. Comparisons between finite element impact simulations results and the AASHTO provisions are presented in Chapter 7. Finally, Chapter 8 summarizes results from the preceding chapters and offers conclusions.

CHAPTER AASHTO BARGE AND BRIDGE COLLISION SPECIFICATION As stated in the previous chapter, the AASHTO provisions concerning barge and bridge collision are based on the Meir-Dornberg study []. The barge collision impact force associated with a head-on collision is determined by the following procedure given by AASHTO: For a <. 3 ft., B For a. 3 ft., B For above equations, a B and B P = a R (kips) (.) B B B P = ( 39 + a ) R (kips) (.) B R B are expressed as B a B / KE. (.3) + 567 R = B R = / 35 (.) B B B C KE = H ( ) W V 9. (.5) in which P B is impact force (kip); a B is barge bow damage depth (ft); R is barge width B modification factor; B is barge width (ft); KE is kinetic energy of a moving barge (kipft.);w is barge dead weight tonnage (tonnes);v is barge impact speed (ft/sec); C represents the hydrodynamic mass coefficient. B H 5

6 The hydrodynamic mass coefficient C H accounts for the mass of water surrounding and moving with the barge so that the inertia force from this mass of water needs to be added to the total mass of barge. C H varies depending on many factors such as water depth, under-keel clearance, distance to obstacles, shape of the barge, barge speed, currents, position of the barge, direction of barge travel, stiffness of bridge and fender system, and the cleanliness of the barge s hull underwater. For a barge moving in a straight-line motion, the following values of otherwise by accepted analysis procedures: C H may be used, unless determined C =.5 for large under-keel clearances (.5draft ) H C =.5 for small under-keel clearances (.5draft ) H The expression of vessel kinetic energy comes from general expression of kinetic energy of a moving object: mv KE = WV = g (.6) where m is the mass of the barge; g is the acceleration of gravity;w is the barge dead weight tonnage;v is the barge impact speed. Expressing KE in kip-ft., W in tonnes ( tonne =. ton =.5 kips), V in ft/sec, g = 3. ft/sec, and including the hydrodynamic mass coefficient, C H, Equation.6 results in the AASHTO equation:.5c HWV KE = 3. C HWV = 9. (.7) The impact force calculation described above is for head-on impact conditions. The AASHTO provisions specify that for substructure design, the impact force shall be applied as a static force on the substructure in a direction parallel to the alignment of the

7 centerline of the navigable channel. In addition, a separate load condition must also be considered in which fifty percent of the load computed as described above shall be applied to the substructure in a direction perpendicular to the navigation channel. These transverse and longitudinal impact forces shall not be taken to act simultaneously. Commentary given in the AASHTO provisions also suggests the following equation to calculate impact energy due to an oblique impact. Though this equation is not a requirement, it provides a useful means of computing the collision energy to be absorbed either by the barge or the bridge. E = η * KE (.8) Values of η are shown in Figure - as a function of the impact angle (α ) and coefficient of friction ( µ ) based on research by Woisin, Saul and Svensson [7]. This method is from a theoretical derivation of energy dissipation of ship kinetic motion, and assumes that the ship bow width is smaller than the impact contact surface. Thus sliding between the ship bow and the pier contact surface is possible, the friction force can be derived based on coefficient of friction, and the change of impact energy can be derived. Though this method provides a very useful way to find the energy to be dissipated during an oblique impact of a barge with a pier, it is not applicable to the oblique impact simulations included in the thesis because the barge bow is much larger than pier width, and impact takes place at center zone of barge bow, so pier cuts into the bow during impact, thus sliding between the barge and the pier is not likely to happen. However, for cases when impact doesn t occur at center zone of barge bow, and barge bow corners

8 slide along the pier surface, this method may provide an alternative means to calculate kinetic energy to be dissipated during the impact. Figure -. Collision energy to be absorbed in relation with collision angle and the coefficient of friction (after AASHTO [])

CHAPTER 3 FINITE ELEMENT BARGE IMPACT SIMULATION 3. Introduction Nonlinearity in structural behavior can take two forms: material nonlinearity and geometric nonlinearity. When the stiffness of a structure changes with respect to load induced strain, material nonlinearity takes place. When displacements in a structure become so large that equilibrium must be satisfied in the deformed configuration, then geometric nonlinearity has occurred [8]. For modeling of structural nonlinearity, both material nonlinearity and geometric nonlinearity may be taken into account. For the finite element code LS-DYNA [6], material nonlinearity can be accounted for by defining a piecewise linear stress-strain relationship or by defining the parameters of an elastic, perfectly plastic material model. Geometric nonlinearity is always included in LS-DYNA when using beam elements, shell elements and brick elements for structural modeling. Geometric nonlinearity is included in the element formulation for beam element. For shell element and brick element, when mesh is refined enough, geometric nonlinearity is also included in element internal forces. Dynamic simulation of barge impacts with bridge piers involves generating two separate models: barge and pier/soil. The barge is made of steel plates, channel beams and angle beams. Non-linearity in these elements can be approached by modeling the steel plate and channel beams using shell elements and a corresponding nonlinear stressstrain model. However in nonlinear pier modeling, the concrete pier cap and pier columns 9

are heavily reinforced with steel bars. During impact, it is possible for the steel bars to yield at certain locations and form plastic hinges in the reinforced concrete elements. Nonlinear material modeling may be used to study this type of inelastic response and investigate the locations at which plastic hinges form during impact. 3. Background Study Many researchers have published papers on nonlinear analysis of bridges, bridge substructures [9,,], and other types of reinforced concrete structures. Researchers focusing on the behavior of high-strength reinforced concrete columns subjected to blast loading have used solid elements to model concrete and beam elements to model the reinforcement [9]. The Winfrith concrete material model available in LS-DYNA was adopted by Ngo et al. in modeling the concrete. This approach enables the generation of information such as crack locations, directions, and width. The solid elements used were mm in each dimension for both concrete and reinforcement. For unconfined concrete, the Hognestad [] stress-strain curve was used; for confined concrete, modified Scott s model [9] was employed in the modeling to include confined concrete and to incorporate the effect of relatively high strain rate [9]. The concrete column was subjected to a blast load that had a time duration of approximately.3 milliseconds. Researchers studying bridge behavior under seismic loading developed a global nonlinear model of the San Diego-Coronado Bay Bridge. Figure 3- shows the global nonlinear model, developed by the California Department of Transportation (Caltrans). The model was analyzed using the commercially available finite element code ADINA [3]. San Diego-Coronado Bay Bridge is.6 miles long and extends across San Diego Bay. The model included the entire.6-mile long bridge (see Figure 3-). Modeling included two steps: local modeling and global modeling. An example of local modeling is

that the detailed finite-element analyses of three typical bridge piers were performed using experimentally-verified structural models and concrete material models to predict stiffness, damage patterns and ultimate capacity of the pier. The finite element model of an individual bridge pier is shown in Figure 3-. Data were then used to idealize the pier column stiffness and plastic-hinge behavior in the global-model piers. Pier modeling in the global bridge model is shown in Figure 3-3. Nonlinearities ultimately included in the global model were global large displacements (primarily to capture P- effects in the towers), contact between spans at the expansion joints and at the abutment wall, nonlinear-plastic behavior of isolation bearings, post-yield behavior of pier column plastic hinges, and nonlinear overturning rotation of the pile cap []. Figure 3-. Global modeling of San-Diego Coronado Bay Bridge (after Dameron [])

Pier Cap Pier Column Pile C ap Figure 3-. Pier model used for local modeling (after Dameron []) Figure 3-3. Global pier modeling for seismic retrofit analysis (after Dameron [])

3 Developers of the commercially available pier analysis software FB-Pier [], use three-dimensional nonlinear discrete elements to model pier columns, pier cap, and piles. The discrete elements (see Figure 3-) use rigid link sections connected by nonlinear springs []. The behavior of the springs is derived from the exact stress-strain behavior of the steel and concrete in the member cross-section. Geometric nonlinearity is accounted for by using P- moments (moments of the axial force times the displacement of one end of an element to the other ). Since the piles are subdivided into multiple elements, the P-δ moments (moments of axial force times internal displacements within members due to bending) are also taken into account. Figure 3-. Mechanical model for discrete element (after Hoit []) Figure 3- shows the mechanical model of the discrete element. The model consists of four main parts. There are two segments in the center that can both twist torsionally and extend axially with respect to each other. Each of these center segments is connected by a universal joint to a rigid end segment. The universal joints permit bending at the quarter points about two flexural axes by stretching and compressing of the appropriate springs. The center blocks are aligned and constrained such that springs aligned with the

axis of the element provide torsional and axial stiffness. Discrete angle changes at the joints correspond to bending moments and a discrete axial shortening corresponds to the axial thrust []. 3.3 Pier Model Description Consolazio et al. [] discussed dynamic impact simulations of jumbo open hoppers barge with piers of the St. George Island Causeway Bridge. In their report, the pier is modeled with a combination of solid elements to model pier column, pier cap and pile cap, beam elements to model steel piles and discrete non-linear spring elements to model nonlinear soil behavior. The solid elements are used to accurately describe the distribution of mass in the pier. In the present study, similar approaches to modeling have been used for several components of the simulation models developed. A linear elastic material with density, stiffness and Poison s ratio corresponding to concrete is assigned to the solid elements. Material properties for the beam elements are described in the following paragraph. Nonlinear spring properties (for both lateral springs and axial springs) derived using the FB-Pier software [] are assigned to the soil springs. In this thesis, beam elements are employed to model pier columns and pier caps, while solid elements are used to model pile caps. Both pier columns and pier caps are heavily reinforced concrete elements consisting of numerous steel bars compositely embedded within a concrete matrix. When a pier column or pier cap yields during dynamic impact, plastic hinges may form in the pier column or pier cap that may affect impact force history and structural pier response. Using beam elements to model pier columns and the pier caps permits the use of a nonlinear material model capable to representing plastic hinge formation.

5 LS-DYNA includes a nonlinear material called *MAT_RESULTANT_PLASTICITY, which is an elastic, perfectly plastic model. Assigning this material model to beam elements requires specification of mass density, Young s modulus, Poison s ratio, yield stress, cross sectional properties (including area, moment of inertia with respect to strong axis, moment of inertia with respect to weak axis, torsional moment of inertia and shear deformation area). Based on these properties, LS-DYNA assumes a rectangular cross section [6], and internally calculates the normal stress distribution on the cross section. Normal stress from axial deformation, bending of strong axis and bending of weak axis are combined and checked for the possibility of plastic flow. By checking for plastic flow at each time step, element stiffnesses may be updated accordingly. Work hardening is not available in this material model. For nonlinear modeling of pier, the steel piles are also modeled by this material type. For HP x73 steel piles, a test model was set up. Comparison of independently calculated theoretical results and LS-DYNA results show that error percentages for strong axis plastic moment capacities are less than 8% and error percentages for weak axis bending are less than 8%. Analysis cases considered in the thesis include both headon impacts and oblique impacts. For head-on impact, weak axis bending dominates; for oblique impact, plastic bending moment about both axes will occur. Therefore, the pile cross section properties are adjusted to produce the same error percentage in both strong axis and weak axis bending. Adjusted pile properties are applied to both head-on impact and oblique impact to keep comparison conditions the same when results from the two conditions are compared. To keep the pile bending stiffness unaltered, only the cross-

6 sectional area is changed. Table 3- shows the original and adjusted cross-sectional properties. Table 3-. Comparison of original and adjusted section properties Case Original Adjusted Trial Value of Area (m ) Plastic Moment (Strong Axis Bending) (N*m) Plastic Moment (Weak Axis Bending) (N*m) Error Percentage (Area) Error Percentage (Plastic Moment) (Strong Axis) Error Percentage (Plastic Moment) (Weak Axis).38 x -.5 x - 5.86 x 5.83 x 5 3. x 5.5 x 5 9.5 % 8. %.9 % 7.9 %.7 % An alternative to modeling the effect of reinforcement on bending moment capacity involves the use of moment curvature relationships. However LS-DYNA does not support direct specification of moment-curvature for beam elements. Results from tests making use of material models *MAT_CONCRETE_BEAM, *MAT_PIECEWISE_LINEAR_- PLASTICITY, and *MAT_FORCE_LIMITED showed that these models do not represent reinforced beam bending moment capacity to a satisfying extent. Moment-curvature relationships may be sufficiently approximated using the *MAT_RESULTANT_PLASTICITY model. Usually, a moment-curvature relationship is a curve described by a series of points. The shape of the curve is similar to a bilinear curve. A stress strain curve for an elastic, perfectly plastic material is also a bilinear curve. Figure 3-5 shows similarities

7 between a simplified moment-curvature curve and a stress-strain curve for an elastic, perfectly plastic material. M σ M y σ y EI E Φ y Φ ε y ε a) moment-curvature b) stress-strain Figure 3-5. Bilinear expression of moment-curvature and stress-strain curve For an arbitrary cross section, Mc σ = (3-) I g M E = (3-) φ Material parameters for elastic, perfectly plastic material are: young s modulus and yield stress. Young s modulus can be derived from the bilinear moment-curvature curve based on Equation 3-, however yield stress is unknown due to the fact that LS-DYNA assumes rectangular cross section and internally calculate the dimension (width and height) of the rectangular cross section based on input cross section properties. Thus a yield stress is assumed first and input into LS-DYNA. Based on output yield moment from LS-DYNA and Equation 3-, c value (dimension of rectangular cross section) is calculated. This correct c value (dimension of rectangular cross section) is plugged into I g c

8 Equation 3- using the known yield moment to get the corresponding yield stress. This yielding stress is used for data input for elastic, perfectly plastic material type. To simplify the moment-curvature relationships used, the following rule is used for both pier columns and pier caps. The yield moment (M y ) for the bilinear curve is equal to half the summation of yielding moment M o y and ultimate moment M o u from the original moment-curvature relationship. Initial stiffness for the simplified bilinear momentcurvature relationship stays the same as that of the original moment-curvature relationship (see Figure 3-6). Data used in the LS-DYNA simulations for the pier columns and pier cap are given in Table 3-. Μ u o M Original Moment-Curvature M y Μ y o Bilinear Moment-Curvature Μ cr o Φ y Φ Figure 3-6. Moment-curvature derivation Table 3-. Input data in LS-DYNA simulations Pier E (N/ m ) σ y (N/ m ) Pier Column.86 x.9 x 6 Pier Cap.86 x 6. x 6 Moment-curvature relationships for the pier column and the pier cap are developed based on steel reinforcement layout and material properties. Tables 3-3 and 3- show the

9 error percentage of a test model for both strong axis bending and weak axis bending, for the pier cap and the pier column respectively. The test model is a 8-meter simply supported beam with a concentrated load at mid-span. Plastic moment and displacement at mid span calculated by LS-DYNA are compared with those from theoretical calculations. Table 3-3. Comparison of plastic moment and displacement using properties of pier cap Pier Cap LS-DYNA Theoretical Error Results Value Percentage Strong Axis Plastic Moment (N*m). x 6. x 6 7% Displacement at Mid-span at Yielding (m) 6. 6. 3% Weak Axis Plastic Moment (N*m) 6.3 x 6 5.3 x 6 8% Displacement at Mid-span at Yielding (m) 9. 8. % Table 3-. Comparison of plastic moment and displacement using properties of pier column LS-DYNA Theoretical Error Pier Column Results Value Percentage Plastic Moment 9.9 x (N*m) 6.6 x 6 6% Strong Axis Displacement at Mid-span 5. 5. % at Yielding (m) Plastic Moment 8.8 x (N*m) 6 9. x 6 % Weak Axis Displacement at Mid-span 5.5 5.9 6% at Yielding (m) 3. Barge Finite Element Model The impact vessel of interest in this thesis is a construction barge, 5.5 ft. in length and 5 ft. in width. Figure 3-7 through 3- describe the dimensions and the internal structure of the construction barge.

'-" 5'-" Transverse Frame Barge Bow Longitudinal Truss *3 Panel Longitudinal Truss Longitudinal Truss 8'-6" 5'-6" 7'-" Figure 3-7. Main deck plan of the construction barge Transverse Frame Serrated Channel 8'-6" 7'-" Figure 3-8. Outboard profile of the construction barge Transverse Frame C Channel L Beam 35'-" 35'-" Figure 3-9. Typical longitudinal truss of the construction barge L x 3 x / C 8 x 3.75 Top & Bottom L 3.5 x 3.5 x 5/6 typ. Figure 3-. Typical transverse frame (cross bracing section) of the construction barge

'-6" '-" 35'-" Figure 3-. Dimension and detail of barge bow of the construction barge The construction barge is made up of steel plates, standard steel angles (Lsections), channels (C-sections) and serrated channel beams. The bow portion of the barge is raked. Twenty-two internal longitudinal trusses span the length of the barge and nineteen trusses span transversely across the width of the barge. The twenty-two longitudinal trusses are made up of steel angles, while the nineteen transverse trusses are made up of steel channels. Serrated channel beams are used at the side walls to provide stiffness to the wall plates. Reference [] gives a very detailed description of modeling of an open hopper barge, in which the barge is divided into three zones and consequently treated in three different ways with respect to mesh resolution. The three zones are called zone-, zone- and zone-3 respectively. For modeling of the construction barge that is of interest here, the same concept was applied. The construction barge was divided into three longitudinal zones, as is illustrated in Figure 3-.

6'-" 9'-" 5'-6" Zone- Zone- Zone-3 Figure 3-. Layout of barge divisions For centerline, head-on impacts, the central portion of barge zone- (see Figure 3-3) is where most plastic deformation occurs and impact energy is dissipated. This area is thus the critical part in modeling dynamic collisions of barges with piers. Since all simulations described in this thesis are for centerline impacts, internal structures in the central area of zone- are modeled with a refined mesh of shell elements to capture large deformations, material failure, and thus to dissipate energy. Internal trusses in the port and starboard off-center portion of the bow are modeled using lower-resolution beam elements since only small deformations are expected and material failure is not likely to occur during centerline impacts of the barge. Unlike zone-, structures in zone- and -3 construction barge will sustain relatively minor deformations that will cause primarily elastic stress distributions in the outer plates, inner trusses and frame structures. Material failure is not expected in these zones. Zone- is modeled using shell elements for outer plate and beam elements for internal trusses and frames. Compared to the size of the shell elements of zone-, those in zone- are considerably larger in size. Use of relatively simple beam elements reduces the computing time required to perform impact analysis.

3 Zone- 9'-.5" Port Zone (Lower Resolution) 5'-" Width of Barge Headlog of Barge 3'-3" Central Zone (High Resolution) 9'-.5" Starboard Zone (Lower Resolution) Figure 3-3. Meshing of internal structure of zone- In zone-3, the aft portion of the construction barge functions to carry the cargo weight of the barge and is not expected to undergo significant deformation during dynamic impact. Thus the barge components in this zone are modeled with solid elements. Density of the solid elements was selected to achieve target payload conditions. All shell elements in the model are assigned a piecewise linear plastic material model for A36 steel. A detailed description of this material type is provided in the research report by Consolazio et al.[]. Solid elements are assigned an elastic material property since no plastic deformation in zone-3 is expected. Mass density of the solid element represents the fully loaded payload condition based on a total barge plus payload weight of 9 tons as is described in the AASHTO provisions.

Beam elements in the barge model are assigned elastic, perfectly plastic material type. LS-DYNA material model number 8, *MAT_RESULTANT_PLASTICITY is employed to do so. For this material type, the required input of cross sectional properties are: area, moment of inertia with respect to the strong axis, moment of inertia with respect to the weak axis, torsional moment of inertia, shear deformation area. Though LS-DYNA assumes a rectangular cross section and internally calculates cross sectional dimensions based on area, flexural moment of inertia, and torsional moment of inertia, a test model of a L x3x/ angle prepared by the author showed that the plastic moment predicted by LS-DYNA can be as accurate as 99% for strong axis bending and 95% for weak axis bending. A test model was developed and the plastic moment capacity for both strong axis bending and weak axis bending for a non-symmetric angle section were computed. For other types of beams such as channels and wide flange members, plastic moment capacity can be derived from cross section properties available in the AISC Manual of Steel Construction []. Channels and wide flange beams showed error percentages varying up to 8% when the plastic moment was computed using the *MAT_RESULTANT_- PLASTICITY material in LS-DYNA. Contact definition *CONTACT_AUTOMATIC_SINGLE_SURFACE (self contact) is assigned to the barge bow to capture the fact that under impact loading, the internal members within the barge bow may not only contact each other, but also fold over on themselves due to buckling. During an impact simulation, LS-DYNA checks for the possibility for elements contacting each other within a defined contact area, thus a large self contact area will increase computing time drastically. To minimuze computational time, the area in the barge bow where contact is likely to occur is carefully chosen.

5 Table 3-5. General modeling features of the testing barge Model Features 8-node brick elements 8 -node shell elements 8, -node beam elements 8,3 -node Discrete Spring elements 9 -node point mass elements 9 Model Dimensions Length 5.5 Ft Width 5. Ft Depth.5 Ft Contact Definitions CONTACT_AUTOMATIC_SINGLE_SURFACE CONTACT_AUTOMATIC_NODES_TO_SURFACE Table 3-6 General modeling features of the jumbo hopper barge Model Features 8-node brick elements 3 -node shell elements,87 -node beam elements,6 -node Discrete Spring elements 8 -node point mass elements 8 Model Dimensions Length 95 Ft Width 35 Ft Depth Ft Contact Definitions CONTACT_AUTOMATIC_SINGLE_SURFACE CONTACT_AUTOMATIC_NODES_TO_SURFACE CONTACT_TIED_NODES_TO_SURFACE Welds are used in the barge to connect the head log plate, top plate and the bottom plate. These welds are modeled by the *CONSTRAINED_SPOTWELD constraint type. Computationally, the spotwelds consist of rigid links between nodes of the head log, top plate and bottom plate. Detailed descriptions of self contact definition and weld modeling are given in the research report developed by Consolazio et al. []. Connection between zone-, zone-, and zone-3 are made with nodal rigid body constraints. For the connection of zone- to zone-, the transition between internal trusses modeled by shell elements and internal trusses modeled by beam elements is approached by using rigid links to connect nodes from shell element and beam element to transfer

6 internal section forces in a distributed manner. For the connection of zone- to zone-3, nodal rigid bodies are defined to connect small elements in zone- with those in zone-3. Buoyancy Spring with Zero Gap Buoyancy Spring with Non-zero Gap Figure 3-. Buoyancy spring distribution along the barge A pre-compressed buoyancy spring model is applied to the barge to simulate buoyancy effects. The buoyancy spring stiffness was formulated based on tributary area and draft depth of each spring and a gap was added to the spring formulation. Since different positions on the barge hull have different draft depths, the buoyancy spring formulation varies with longitudinal location. Gaps between the water level and barge hull are determined from the geometry of the bottom surface of the barge (see Figure 3- ). The pre-compression of buoyancy spring is calculated using Mathcad worksheet. The comparison of general modeling features of construction barge and open hopper barge is provided in Table 3-5 and 3-6. 3.5 Contact Surface Modeling When pier columns and pier caps are modeled using beam elements, contact surfaces need to be modeled and added to the pier column to enable contact detection during impact (see Figure 3-5). Also in Figure 3-5, since shear wall is modeled by beam elements, rigid body is defined at connection of shear wall, pier column and pile cap. In this region, only very small deformation could likely occur due to thickness of shear wall. So it is treated as rigid body. Modeling of contact surface needs to be done

pier column 7 carefully since the contact surface may add extra stiffness to the pier column, thus changing the original stiffness of the pier and affect the simulation results. pier cap barge motion contact surface water line rigid body shear wall pile cap Figure 3-5. Pier and contact surface layout pier column rigid link rigid contact surface Figure 3-6. Rigid links between pier column and contact surface

8 pier column contact surface impact force Figure 3-7. Exaggerated deformation of pier column and contact surface during impact To make sure that contact surface will not add extra stiffness to the pier, it is divided into separate elements. Each separate element is assigned rigid material properties and is connected to the pier column through rigid links (see Figure 3-6). Under bending of the pier column, these elements will act independently, and transfer the impact force to the pier column beam elements. Figure 3-7 shows an exaggerated depiction of deformation of the contact surface during impact. Though friction on the contact surface may add extra bending moment to the pier column, studies shows that when the element size of pier column is set to approximately 6 inches, the extra bending moment transmitted to the pier column is less than 5% compared to the primary bending moment sustained during impact for the most severe cases considered here (6 knots, full load). Though the contact surface in a real pier is made of concrete, use of a rigid material model is verified by comparing the impact force versus crush depth relationships from static barge crush analysis. Figure 3-8 shows a comparison of impact force versus crush depth relationships computed using rigid contact surfaces and concrete contact surfaces.

9 Though the impact forces differ slightly after the crush depth exceeds inches, overall, the curves are in good agreement. Crush depth (m).5.5 Impact force (MN) 6 5 3 rigid material elastic material 8 6 Impact force (kip) 3 5 6 Crush depth (in) Figure 3-8. Comparison of impact force versus crush depth for rigid and concrete contact models The concrete cap seal is not modeled explicitly but its mass is added to that of the pile cap to account for increased inertial resistance. Soil springs are assigned spring stiffnesses derived from the FB-Pier program, and nodal constraints are added to the soil springs. Detailed descriptions of soil springs and constraints of nodes are available in the research report by Consolazio et al. []. A typical impact simulation model in which a pier model has been combined with a barge model is shown in Figure 3-9. As the figure illustrates, resultant beam elements are used to model the pier columns and cap and the contact surface representation described above is used to detect contact between the barge and the pier.

3 Figure 3-9. Overview of barge and pier model for dynamic simulation

CHAPTER NON-LINEAR PIER BEHAVIOR DURING BARGE IMPACT Non-linear pier behavior, barge deformation and energy dissipation are several of the issues that are relevant when considering barge-pier collisions. The answer to questions of how much the non-linearity in modeling affects these considerations, if nonlinearity causes fundamental changes to pier behavior helps understand barge and pier behavior during impact, thus when impact cases are considered, whether non-linearity should be included in modeling or not will be justified and thus facilitate the dynamic simulation modeling procedure.. Case Study In the barge and the pier impacts modeled here, the barge is selected to have fully loaded weight of 9 tons (per the AASHTO provisions). This loaded weight is chosen to be the same as that of fully loaded open hopper barge to enable comparison with results of dynamic simulations previously conducted using a hopper barge finite element model. The rectangular columns of the pier are used to define the contact surface. Two barge impact velocities are considered: 6 knots and knot. Barge with a 6 knot speed and fully loaded condition represents the most critical impact scenario and thus the most severe nonlinear pier behavior. Barge impact with a knot speed and fully loaded condition represents the scenario that only a very small region of pier shows nonlinearity. These two cases cover a large range of impact scenarios, thus results from these two cases can reasonably cover the effect of non-linearity. All cases included in this chapter are listed in Table -. 3

3 Table -. Dynamic simulation cases Case Contact Surface Speed Impact Angle Material Property Loading Condition A Rectangular 6 knot Head-on Linear Full B Rectangular 6 knot Head-on Nonlinear Full C Rectangular knot Head-on Linear Full D Rectangular knot Head-on Nonlinear Full. Analysis Results For both severe impact case and non-severe impact case, Figures - through -6 show that using nonlinear pier material and using linear pier material generate the same impact force peak value and almost the same impact duration time since after the internal structure in the barge bow yields, it cannot exert a larger impact force. Also, for both non-severe impact condition and severe impact condition, approximately the same amount of energy is dissipated (area under barge impact force vs. crush depth curve) using nonlinear pier material and linear pier material respectively. It is shown that for both severe impact case and non-severe impact case, barge crush depth after impact for linear pier is always larger than barge crush depth after impact for nonlinear pier (Figure -3, Figure -). During impact, for the severe impact case, all steel piles yield; even for the non-severe impact case, part of the steel piles yield during impact. Yielding of steel piles prevents the pier structure from generating increased resistance to the barge, thus the pier structure cannot create larger crush depth in barge bow. Also yielding of piles generates residual deformation of pier structure after impact as shown in Figure -5. The residual deformation can be as large as - at the point for measurement (the impact point). The pier column and pier cap do not yield

33 during impact even for the most severe impact case. For the barge with knot impact speed and fully loaded condition, the pier residual deformation is almost negligible. Plots of pier column bending moment shows that the peak value of the pier column bending in the impact zone of the pier exceeds the cracking moment of pier column cross section. Since the moment-curvature is simplified as a bilinear curve with initial stiffness the same as that of the un-cracked cross section, the cracking moment is not reflected in the bilinear moment-curvature curve. There is very little difference between pier behavior using linear pier and using nonlinear pier material for the barge with a knot speed, fully loaded condition. Partially yielded piles during impact caused very little effect on pier behavior. For this case, the effect of non-linearity of pier material can be ignored almost completely. For the barge with 6 knot speed, fully loaded condition, though non-linearity of pier material does have an effect on impact force history, impact force vs. crush depth relationship, and pier displacement, the influence is limited. The results drawn here are based specifically on impact simulations of a barge impacting a channel pier of the St. George Island Causeway bridge. The piles of this pier are HPx73 steel piles. As a result, the characteristics of these piers are quite different from the concrete piles as are also often employed in bridges. Different pile properties may have a substantial effect on impact force and pier behavior during impact. Thus additional work needs to be done for impacts of different pier types to comprehensively study the effect of pier material nonlinearity on barge impact force and pier behavior.