This item has the following downloads:
1 STRESS RESULTANT BASED ELASTO PLASTIC ANALYSIS OF STRUCTURES By JINSANG CHUNG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTO R OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013
2 2013 Jinsang Chung
3 To my family who ha s always suppo rted me in all that I have done
4 ACKNOWLEDGMENTS I would first and foremost like to thank my advisor Dr. Nam Ho Kim for his valuable insights and patience with me as we faced our challenges. I would like to thank my committee members Dr. Peter G. Ifju, Dr. Bhavani V. Sankar and Dr. Gary R. Consolazio for their guidance. I would also like to thank the members pas t and present of the Multidisciplinary Design and Optimization group at the University of Florida for their help and support. Extra thanks go to Hyungwoo Lee, the CEO of MIDASIT and the development team of MIDASIT who support all my study. F inally, I would like to express my special gratitude to my family members, my wife ( Miyoung Seol) and my sons ( Yongwon and Myoungwon ) for their continuous support and encouragement.
5 TABLE OF CONTENTS page STRESS RESULTANT BASED ELASTO PLASTIC ANALYSIS OF STRUCTURES ..... 1 ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 Motivation and Scope ................................ ................................ ............................. 12 Outline ................................ ................................ ................................ .................... 15 2 FRAME ELEMENT MODEL ................................ ................................ .................... 17 Geometrically N onlinear F rame E lement based on F orce M ethod ......................... 17 Incremental F orm for M aterial N onlinearity based on F orce M ethod ...................... 23 Transformation b etween Reference Systems ................................ ......................... 25 Material Nonlinearity of a Frame Element ................................ ............................... 28 Example P roblems ................................ ................................ ................................ .. 34 An Example for G eometric N onlinearity ................................ ........................... 34 An Example for G eometric and M aterial N onlinearity ................................ ....... 35 Summary ................................ ................................ ................................ ................ 38 3 PLATE ELEMENT MODEL ................................ ................................ ..................... 39 General Concepts of Material Nonlinearity of a Plate ................................ ............. 39 Yield Criterion based on Stress Resultants ................................ ............................. 43 Von Mises Criterion ................................ ................................ .......................... 43 Drucker Prager Criterion ................................ ................................ .................. 45 Parametric S tudy for t he M odified Y ield C riterion of a Concrete Plate ............. 50 Hardening Model ................................ ................................ ................................ .... 53 Flow Rule and Tangent Stiffness ................................ ................................ ............ 56 Plastic C onsistency P arameter, ................................ ................................ .... 58 Newton Raphson Algorithm for the Plastic Consistency Parameter ................. 59 Elasto Plastic Constitutive Equations ................................ ............................... 61 Reinforcement by Rebar ................................ ................................ ......................... 63 Geometry and Con stitutive Equation ................................ ................................ 63 The Smeared Layer Method of Rebar Reinforcement ................................ ...... 66 The Integrated Section Method of Rebar Reinforcement ................................ 68 Tests of a U nit E lement ................................ ................................ ........................... 70
6 A M etal P late B ased on the von Mises Criterion ................................ .............. 70 A C oncrete Plate Based on Drucker Prager Criterion ................................ ...... 76 Practical Examples ................................ ................................ ................................ 81 A Part of Steel Girder Bridge ................................ ................................ ............ 81 A Superstructure of a PSC Girder Bridge ................................ ......................... 84 Neutral Axis Moving of a Concrete Plate under Bending ................................ ........ 88 Summary ................................ ................................ ................................ ................ 91 4 NONLINEAR DYNAMIC ANALYSIS ................................ ................................ ....... 92 Nonlinear D ynamic E quation ................................ ................................ .................. 92 Dynamic Equation u nder Earthquake Loadings ................................ ...................... 95 Practical E xamples of E arthquake L oading s ................................ ........................... 96 A P art of Steel Girder Bridge under Earthquake Loadings ............................... 98 A Superstructure of a PSC Girder Bridge under Earthquake Loadings .......... 1 01 A PSC Girder B ridge under E arthquake L oadings ................................ ......... 104 Summary ................................ ................................ ................................ .............. 110 5 CONCLUSIONS ................................ ................................ ................................ ... 111 APPENDIX A EQUATION FOR FRAME ELEMENT BASED ON FORCE METHOD .................. 113 B THE PLASTIC CONSISTENCY PARAMETER OF VON MISES CRITERION ..... 117 C THE PLASTIC CONSISTENCY PARAMETER OF DRUCKER PRAGER CRITERION ................................ ................................ ................................ .......... 121 LIST OF REFERENCES ................................ ................................ ............................. 124 BIOGRAPHICAL SKETC H ................................ ................................ .......................... 127
7 LIST OF TABLES Table page 3 1 Material nonlinear properties for a steel plate ................................ ..................... 71 3 2 Material coefficient for the Drucker Prager criterion on the integrated section method ................................ ................................ ................................ ............... 77 3 3 Fully plastic yielding forces and moments under uni directional loadings ........... 85 3 4 Comparison of calculation cost between layered and stress resultant model .... 87
8 LIST OF FIGURES Figure page 1 1 Modeling methods for nonlinear simulation of a structural member .................... 13 2 1 Local coordinate of a f rame element. ................................ ................................ 18 2 2 Transformation of deformation with rigid body modes ................................ ........ 25 2 3 Transformation of forces with rigid body mode s ................................ ................. 26 2 4 Flow charts for state determination ................................ ................................ ..... 33 2 5 Snap through behavior by displacement and force based method ..................... 34 2 6 A simple frame with infinitesimal displacement ................................ ................... 35 2 7 A frame element under horizontal specific displacement ................................ .... 37 2 8 Comparison of results between with and without geometric nonlinearity ............ 37 3 1 Nonlinear models for elasto plastic behavior of a pla te ................................ ...... 42 3 2 Initialization of equivalent plastic curvature, ................................ ................. 45 3 3 Yielding stress resultants in c ombined loading cases ................................ ......... 51 3 4 Yield surface moving according to hardening t ypes ................................ ........... 55 3 5 Flow chart for calculating plastic consistency parameter and updating. ............. 62 3 6 Geometric shape and stress strain curve of a r ebar ................................ ........... 64 3 7 Test cases of stress resultants yield criterion ................................ ..................... 70 3 8 Axial stress resultants vs. strains in pure axial loading ................................ ....... 72 3 9 Stress couple resultant vs. rotation in pure bending loading .............................. 73 3 10 Axial stress resultant s vs. strain s in pre moment axial loading ........................... 74 3 11 Stress couple resultant vs. rotation in pre axial bending loading ........................ 75 3 1 2 Geometry of a steel reinforced concrete plate ................................ .................... 77 3 13 Axial stress resultant s vs. strain s in pure axial loading ................................ ....... 78 3 14 Stress couple resultants vs. rotations in pure bending loading w/ rebar ............. 79
9 3 1 5 Axial stress resul tant s vs. strain s in pre moment (M p /2) axial loading ................ 79 3 1 6 Stress couple resultant s vs. rotation s in pre axial (N p (+)/2) bending loa ding ..... 80 3 17 Stress couple resultant s vs. rotation s in pre axial (N p ( )/2) bending loading w/ rebar ................................ ................................ ................................ ................... 80 3 18 A steel box girder bridge ................................ ................................ ..................... 82 3 19 Axial stress resultant s vs. axial strain s with and without pre moment ............... 83 3 20 Bending stress resultant s vs. rotational angles with and without pre axial ........ 83 3 21 A superstructure of a PSC girder bridge ................................ ............................ 84 3 22 Axial stress resultants vs. strains in pure axial loading ................................ ....... 86 3 23 Stress couple resultant s vs. rotation s in pure bending loading ........................... 87 3 24 Vertical displacement under factored self weight with axial pre stress ............... 87 3 25 Mov ing of neutral axis of a concrete plate ................................ .......................... 88 3 26 Results of a concrete plate from cycli c bending ................................ .................. 89 3 27 Extension of membrane under cyclic bending ................................ .................... 90 4 1 A structure under earthquake loading ................................ ................................ 95 4 2 Grou nd acceleration of Tohoku Japanese earthquake 2011 ............................. 97 4 3 Loadings of a n earthquake acceleration. ................................ ............................ 97 4 4 Displacement history results of a part of a steel box girder bridge ..................... 99 4 5 Acceleration and reaction history of a part of a steel box girder bridge ............ 100 4 6 Displacement and acceleration history of a superstructure of PSC girder bridge ................................ ................................ ................................ ............... 102 4 7 Reaction history of a superstructure of a PSC girder bridge ............................. 103 4 8 Displacement history results of a steel box girder bridge ................................ 106 4 9 Displacement and acceleration history results of a PSC girder bridge ............. 107 4 10 Forces hi story results of the left bottom frame element ................................ ... 108 4 11 Reaction history results at the left supports of the PSC girder bridge ............... 109
10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRESS RESULT ANT BASED ELASTO PLASTIC ANALYSIS OF STRUCTURES By J insang C hung August 2013 Chair: Nam Ho Kim Major: Mechanical Engineering The demand of nonlinear analysis for a design of building and civil structures has continuously been increased due to earthquake loadings For a large structure, the nonlinear analysis including both material and geometric nonlinearity is a difficult and time consuming process because of a lot of elements and a complex elasto plastic behavior based on stress strain relation. I n a d esign for civil structures, frame and plate elements are commonly used but they require many sub divided layers in order to describe material nonlinearity over the cross section In this research, stress resultant s based elasto plastic behavior s are studi ed for both elements. Firstly, a force based frame element with stress resultant s based material nonlinearity is studied. Stress resultant s based material nonlinearity was already developed by many researchers to predict behavior under earthquake loading, but it is mostly used in the context of displacement based method In this research, the force based frame element, which is well known for its computational efficiency, is integrated with stress resultant s based elasto plastic model to achieve computatio nal efficiency by reducing the number of elements for modeling a structure. Also, an incremental form based on force method is derived in order to make the integration possible.
11 Secondly a stress resultant based elasto plastic model for a concrete plate is pr oposed as a substitute of a layered model which is commonly used and requires many sub layers in order to describe a nonlinear stress distribution over the thickness Iliushin is extended to a concrete plate based on the Drucker Pr ager yield criterion after a modification from a parametric study. Two new parameters are introduced to the yield function in order to describe the non symmetric, fully plastic moment of a concrete plate and the coupled behavior of membrane and bending act ion s In addition, a n integrated section method using equivalent material coefficients is presented for the stress resultant based concrete plate for steel rebar reinforcement. Thirdly, a stress resultant model for a plate and a frame element based on forc e based method are applied to implicit nonlinear dynamic analysis including geometric and material nonlinearity. S everal practical examples are tested and the various results of prop o sed method s are compared with those of reference methods.
12 CHAPTER 1 INTR ODUCTION Motivation and Scope A lot of numerical calculation is required for a designer to obtain a reasonably accurate resistance against various internal and external loadings. Nowadays, numerous computer systems and numerical methods make it possible fo r structural engineers to accomplish such a large amount of calculation. Generally building and civil structures are designed to be elastic to maintain structural integrity under ordinary and predictable loading scenarios. Sometimes, however, simulations b eyond elasticity are also required to estimate collapse patterns and weak points under unexpected loading, such as earthquake. Finding weak points can be useful for inducing an evenly distributed collapse mode, which creates a safer structure in extreme co nditions. Numerical simulations beyond elasticity require much more calculation. Since failure criteria of materials are defined in local stress strain relations, the solid model in Figure 1 1(a) is the most ideal for simulations beyond the elastic range. The failure status is estimated in each solid, layer, or section, depending on whether the solid model, layered model, or the section model are used. When it comes to a real multi story building or a multi span bridge under earthquake loadings, the solid m odel is nearly impossible due to a huge amount of numerical calculations involved; elasto plastic stress strain relation must be calculated at every integration point of every solid element Therefore, other type nonlinear simulation method is required to reduce the simulation cost. For that purpose, the layered model is currently used for a plate element, and the fiber or stress resultant section models are used for a frame element. However, the layered model and the fiber
13 section model also require lots o f numerical calculation and information storage. To apply the nonlinear analysis to a large structure, more efficient methods are required, and thus, the stress resultant s based methods are studied in this research, especially for frame and plate elements. Figure 1 1 Modeling methods for nonlinear simulation of a structural member. A) Solid model B) Stress based model C) Resultant based model. A frame element has been a major object for the nonlinear behavior by many researchers due to the fact that lots of types of civil structures can be modeled by the frame element. The nonlinear behavior of a frame under earthquake loading has been studied to reduce the damage of civil structure s Sectional material nonlinearity of a frame section can be consider ed by two different models: the resultant section model and the fiber section model. The resultant section model which define s the sectional nonlinear response using the moment curvature relation was researched by Takeda et al. , Hilmy and Abel , Haj jar and Gourley  and El Tawil and Deierlein . The fiber section model estimates the response of section based on the uni axial stress strain relation of each fiber cell consisting a frame section. The uni axial constitutive model of concrete was re searched by Kent and Park , Mander et al. , and El Tawil and Deierlein . Using these sectional material nonlinearities, material nonlinearity of a frame element was
14 developed based on the force based formulation by Spacone et al. . The conce pts of the fiber section model and the resultant section model are shown in Figure 1 1(b) and 1 1(c) respectively. On the other hand, for geometric nonlinearity, displacement based method has been commonly used due to its simple formulation. A frame elemen t based on force based method was enhanced by Neuenhofer and Filippou  using the curvature based displacement, and it has been shown that it required a fewer number of elements to yield results of comparable accuracy, compared to the displacement based method. Shear deformation was incorporated into the force based method by Taylor et al. . In this study a nonlinear frame element incorporating both geometric and material nonlinearity based on previously developed force based methods is implemented. To integrate two nonlinearities, geometric and material nonlinearities, in a frame element, equations of geometric nonlinearity are derived by incremental forms. This element is also applied to nonlinear implicit dynamic analysis In case of plate elemen ts, material nonlinearity can be simulated by either the layere d model or the stress resultant model. In general, the layered model can be used for any type of failure criterion because it is based on the local stress strain relation. On the other hand, th e stress resultant model has only been applied to metal plates based on von Mises criterion. Firstly, the yield function based on stress resultants was suggested by Iliushin  and it was modified to describe the Bauschinger effect by Bieniek and Funaro . The progressive development of plastic zone under bending moment was described by a plastic curvature parameter suggested by Crisfield . The influence of transverse shear forces on the plastic behavior for plates and shells
15 was incorporated into the stress resultant yield function by Shi and Voyiadjis . Most research so far has focused only on a met al plate. In this research, the stress resultant model for a concrete plate which is widely used in civil and building structures is presented An appropriate yield function is proposed using theoretical and parametric studies, and general plasticity rules are applied to the yield function for plastic evolution In addition, the effect of steel rebar s is incorporated by the smeared layer and the inte grated section methods for the layered model and the stress resultant model respectively. T he various test cases of a unit element and several practical examples are investigated for availability and efficiency. Finally, th e proposed model s and methods ar e applied to the nonlinear dynamic analysis for earthquake loadings. T he total Lagrangian formulation is used to include geometric nonlinearity and the Newmark integration scheme is applied for nonlinear dynamic analysis. Several practical examples based o n the proposed model s and methods are tested in terms of accuracy and computational efficiency. Outline Chapter 2 introduces geometric nonlinearity of a frame element based on the force based formulation, which was known for its efficiency. An incremental form of geometric nonlinearity for application of material nonlinearity is derived, which makes it possible that any type of material nonlinearity can be applicable using a strain stress resultants relation. Chapter 3 introduces the difference between the layered model and stress resultant model. For a concrete plate, a yield function using stress resultants based on Drucker Prager failure criterion is proposed. Plastic behaviors such as flow rule and plastic consistency parameter are also derived for the p roposed yield function. In addition, for concrete reinforcement, the smeared layer and the integrated section
16 models are proposed. A phenomenon showing over plastic expansion under cyclic bending is investigated Chapter 4 describes the implicit nonlinear dynamic analysis The proposed models and methods are applied to the nonlinear dynamic analysis under earthquake loadings. Several practical examples are tested for stress resultant yield criteria for metal and concrete plates. Also, a PSC girder bridge is tested for the accuracy and efficiency of the proposed force based frame element and stress resultant based yiel ding model of a concrete plate.
17 CHAPTER 2 FRAME ELEMENT MODEL Geometrically N onlinear F rame E lement based on F orce M ethod Historically, frame finite elements for nonlinear analysis are developed based on the displacement method, in which internal deformations of the element are expressed as functions of nodal displacements. Even though the displacement based method is simple to be implement ed results are not so accurate in nonlinear analysis because the assumption of cubic polynomial is not accurate enough to describe complex behaviors, such as plastic deformation . In the force based element, the beam internal forces are interpolated using shape functions and nodal forces. Different from the displacement based element, the force based element is accurate because the force interpolation is not affected by the plastic response or beam section properties. Therefore, it would be more computation ally efficient to use the force base element method to solve for nonlinear problems with a large plastic deformation. The displacement based formulation yields a stiffness matrix to describe the relation between force and displacement, while the force base d formulation yields a flexibility matrix, which is the i nverse of the stiffness matrix. In this research, the force based element is used for geometrically nonlinear analysis. Since this method yields a flexibility matrix, it is called the force based for mulation, compared to the stiffness formulation based on displacement. The equations for the force based formulation in this research are based on the paper of Neuenhofer and Filippou . The local coordinate of a frame element is shown in Figure 2 1.
18 T he shape function that interpolates internal forces based on the equilibrium equation is expressed as in Eq. 2 1 T he shape function depends on deformation T he axial force is assumed to be constant in t his element. Figure 2 1 Local coordinate of a f rame element (2 1 ) where and are th e sectional force and nodal force vector respectively. The governing equation of force based element is expressed in Eq. 2 2 and the sectional deformation , and the nodal d isplacement, are defined in Eq. 2 3. The relation between both terms is expressed in Eq. 2 4. (2 2 ) (2 3)
19 (2 4) T he flexibility of the frame element can be calculated by Eq. 2 5 where is the section flexibility matrix. Each derivative term in Eq. 2 5 can be directly calculated as in Eq s 2 6, 2 7, 2 8 and 2 9. The terms and are expressed in Eq. 2 10 by a vector form and calculated a derivation against nodal forces. (2 5) (2 6) (2 7)
20 (2 8) (2 9) (2 10) Two curvatures of the frame element are estimated by Eq. 2 1 1 in which they are based on the curvatures at integration point s and Lagrangian polynomials. The Lagrangian polynomials are obtained from the Lagrangian interpolation formula in Eq. 2 1 2 Displacements can be obt ained by integrating the curvatures as shown in Eq. 2 1 3 (2 1 1 ) (2 1 2 )
21 (2 1 3 ) When simply suppo rted boundary conditions ( ) are applied, the unknown coefficients in Eq. 2 1 3 can be obtained as in Eq. 2 1 4 and displacements ( ) are expressed as in Eq. 2 1 5 When n integration points are used, the Lag rangian polynomials can conveniently be determined by using the so called Vandermode matrix, which is from the polynomial least square fitting, Eq. 2 1 6 When the curvatures are expressed using nodal forces, the final equations for each displacement ( ) based on curvature can be expressed as in Eq. 2 17. (2 1 4 ) (2 1 5 ) (2 1 6 )
22 (2 1 7 ) When Eq. 2 18 is applied to Eq. 2 17, the displacement ( ) can be expressed as Eq. 2 19, which are based on nodal element forces, displacements of integration points an d flexibility of the section. (2 1 8 ) (2 19) where and are sectional flexibility of y, z direct ion of a beam, and is the length of the beam.
23 Incremental F orm for M aterial N onlinearity based on F orce M ethod I ncremental form equations are required to solve for nonlinear equations where the constitutive relation is given in the incremental form, such as elasto plasticity T he equilibrium equations based on the total form in the previous section are transformed into incremental form equations for material and geometric nonlinearity. Incremental resultant forces are also affect ed by pre exist ing forces T he incremental sectional force vector is expressed in Eq. 2 20 using incremental and pre existing nodal forces and t he incremental form of interpolation is expressed in Eq. 2 21 The governing equation of Eq. 2 2 is transformed to an incremental form as in Eq. 2 22 using pre exist ing sectional deformation and incremental deformation at the current step The flexibility relation in Eq. 2 5 can be applicable to the incremental form without modification (2 20 ) (2 21) (2 22 ) The incremental displacement s ( ) can be calculated from Eq s 2 2 3 and 2 2 6 The derivatives of displacement agai nst nodal forces ( ) are expressed by Eq s 2 2 4 and 2 2 7 The detailed equations for flexibility are summarized in Appendix A.
24 (2 2 3 ) (2 2 4 ) (2 2 5 ) (2 2 6 ) (2 2 7 ) (2 2 8 )
25 Transformation b etween Reference Systems Since the force based elements in the previous section are formulated in a reference system without transverse displacement modes, a transformation is required between a ref erence system without transverse displacement and that of with transverse displacement. In Fig ure 2 2 the relation of displacement between the two reference systems is shown A B Figure 2 2 Transformation of deformation with rigid body modes A) x z plane transformation B) x y plane transformation The transformation matrix between reference systems can be expressed in Eq. 2 29. The two vectors, and are element displacement vectors without and with rigid boy motion respectively. The variable, is the twisting angle and it is assumed linear between two nodes as in Eq. 2 30.
26 (2 2 9 ) (2 30) In addition, the relation of forces between the two reference systems can be described in Fig ure 2 3 A B Figure 2 3 Transformation of forces with rigid body modes A) x z plane transformation B) x y plane transformation
27 The relation of forces between the two reference systems can be described as in Eq. 2 3 1 and transformation matrix can be written as a matrix form in Eq. 2 3 2 (2 3 1 ) (2 3 2 ) w here and are a nodal force vector with rigid body modes and the element force vector without rigid body modes respectively and is the transformation matrix. The stiffness matrix can be expressed using the transformation matrix the flexibility and the stress stiffness matrix as in Eq. 2 3 3
28 (2 3 3 ) w here and are the flexibility matrix and the stress stiffness matrix respectively and is the transformation matrix. (2 3 4 ) Material Nonlinearity of a Frame Element There are two different groups of stress resultant s based models in a frame The first is the lumped model in which the nonlinear behavior is concentrated at the hinges that are located in the end of the element The hysteretic force deformation relation of the lumped model is based on phenomenological rules or classical plasticity theory for force resultants. Generally it is quite difficult to select the parameters of the hysteretic force deformation relations at the hinges. The second is the distributed model in which the nonlinearity is distributed along the entire element, and the nonlinearity is
29 determined by numerical integration through the element. In this case, the nonlinear constitutive relation is deri ved from phenomenological moment curvature relations, classical plasticity theory for force deformation, or subdivision of the section into fibers and integration of the uniaxial stress strain relation. In this research, for efficient analysis, the distri buted model including phenomenological moment curvature relation is applied for the nonlinear behavior of frame element. The basic reference system is the same with Fig ure 2 1. The frame element formulation needs two requirements; the one is the relation b etween the section forces and deformations, and the other is the relation between nodal displacements and integrated displacements from sectional deformation. The e quation , expresses the linearized relation between the element fo rce increment ( ) and the corresponding deformation increment ( ). The element stiffness is obtained by inverting the flexibility matrix, and the element state determination is based on the equilibrium cond ition rather than the compatibility. The state determination is known as the process of finding the resisting forces based on the given displacements. In state determination, there are two phases; (1) the element state determination, in which the element r esisting forces are estimated based on given displacements, and (2) the structure state determination, in which the element resisting forces are assembled for structure resisting forces. The second structure state determination is the same with the general nonlinear Newton Raphson algorithm. In the displacement based method, the section deformations are estimated from the element displacement with the deformation interpolation functions. The section resisting forces are subsequently determined from force d eformation relation and the
30 element resisting forces are generated through the weighted integration of the section resisting forces. In the force based method, on the other hands, the section forces are estimated from the element forces, which are assumed from the current element deformation and the stiffness of the previous iteration. The first problem is to determine section deformations fr o m section forces and sectional flexibility which can be estimated using the fiber model or the resultant section m odel The second problem is to estimate the change of stiffness caus ed by the change of element forces for given deformations. The first problem can be solved by using a general material nonlinearity rule, but a special process is required for the second p roblem. In this research, the procedure suggested by E. Spacone  is used. The condition of deformation compatibility in the structural level requires that the residual deformation should be corrected. This is accomplished at the element level by applyin g corrective element forces based on the current stiffness. The section forces are determined from the force interpolation function so that the equilibrium is always strictly satisfied within the element. T he section forces cannot change during the section state determination so as not to violate the equilibrium within the element. The linear approximation of the section force deformation relation about the present state results in residual section deformation. The integration of the residual section deform ation yields the update d new r esidual deformation of the element. The process is repeated until the residual deformation become ignorable. At the i th step of a Newton Raphson, the process of state determination is following. The index j is the iteration c ounter for state determination in the element level. T he incremental element deformation is given and the initial stiffness is assumed as a
31 previous converged stiffness. The first iteration for an element is j=1 and the corresponding incremental element fo rces are calculated by Eq. 2 3 7 using previous converged flexibility ( ) and current incremental displacement ( ) The increment al section force s ( ) can be obtained by Eq. 2 3 8 and the incremental section deformation s ( ) are estimated by Eq. 2 3 9 using previous step sectional flexibility ( ) and increment of section forces ( ) The corresponding section resi sting forces ( ) and the new tangent flexibility ( ) are calculated by nonlinear constitutive relations based on the stress resultants or the fiber section models. Generally, in the displacement based metho d, the calculated forces ( ) can be directly converted to element resisting forces H owever, they can cause violation of the equilibrium along the element. In this force based method, the residual section deformation ( ) is calculated using the unbalanced section forces ( ) as in Eq. 2 40 and t he residual element deformation ( ) is obtained by integrati ng the residual section deformation ( ) along to the element in Eq. 2 4 1 The residual element deformation should be applied to the element in order to reduce the compatibility error. The additional forces ( ) are calculated by the residual element deformation ( ) and the updated stiffness ( ) in Eq. 2 4 2 The process from Eq. 2 4 2 to Eq. 2 4 6 is iterated until when a convergence criteri on is satisfied in Eq. 2 4 7 The state determination process is displayed as a f low chart in Figure 2 4. The constitutive law part of the flow chart is for sectional material nonlinear ity, and any nonlinear material models can be used here
32 (2 3 5 ) (2 3 6 ) (2 3 7 ) (2 3 8 ) (2 3 9 ) (2 40 ) (2 4 1 ) (2 4 2 ) (2 4 3 ) (2 4 4 ) (2 4 5 ) (2 4 6 ) ( 2 4 7 )
33 Figure 2 4 Flow charts for state determination
34 Example P roblems An Example for G eometric N onlinearity In order to show the efficiency of the force based method a snap through behavior of a structure is tested as show n in Figure 2 5. The results from the force based method are compared with th ose of the displacement based method. The geometric shape and material parameters are shown in Figure 2 5(a) and the displacement results at the center are compared along with the n umber of elements Figure 2 5(b) shows that the result of force based method with two elements has enough accuracy compared with those of displacement based method with eight elements Therefore, it can be concluded that the force based method can yield almost same accuracy with much fewer e lements. A B Figure 2 5 Snap through behavior by displacement and force based method A) Geometry and material properties B) Force displacement curves
35 An Example for G eometric and M aterial N onlinearity Figure 2 6 shows the comparison between two formulations: the one includes only material nonlinearity and the other includes both material and geometric nonlinearity in a simple beam. The constitutive relati on is given in terms of the moment curvature relation as in Figure 2 6(b), which has infinitesimal curvature and bi linear material nonlinearity. The two cases show almost same results due to the fact that very small displacement cause s only ignorable effe ct of geometric nonlinearity. Although the difference is small, it can be conclude d that the proposed procedure based on force based method for geometric and material nonlinearit y are consistent. A B Figure 2 6 A simple frame with infinitesimal displacement A) Geometry of a simple beam B) Moment c urvature relation C) Result of moment vs. curvature at center
36 C Figure 2 6 Continued The procedure described in the previous section is applied to a column as shown in Figure 2 7 (a) T he cyclic force curve at displacement and the bi linear nonlinearity for bending stiffness shown in Figure s 2 7(b) and (c) are used. T he results of moment vs. tip displacement at top and moment vs. curvature at the bottom are compared betwe en two cases ; with and without geometric nonlinearity in Figures 2 8 ( a ) and ( b ) In the case of considering material nonlinearity only the moment vs. tip displacement curve and the moment vs. curvature at the bottom curve are symmetric and straight in ev ery state changing region. However, when it comes to including geometric non linearit y the curve of moment vs. tip displacement is not straight and the highest positive bottom moment is slightly smaller than that of material nonlinearity only Accordingly the positive curvature of bottom of column is much less than that of other due to the fact that the bending stiffness greatly reduced after yielding. T hese results are caused by the combin ed effect of geome tric and material nonlinearities
37 A B C Figure 2 7 A frame element under horizontal specific displacement A) Geometry shape. (B) Cyclic force curve. C) M oment vs. curvature relation o f the frame element. A B Figure 2 8 Comparison of results between with and without geometric nonlinearity. A ) Moment vs. displacement at top. B ) M oment vs. curvature at bottom.
38 Sum mary In this chapter, a force based method for geometric nonlinearity is introduced to reduce the number of element s for modeling a structure. T he force based method can describe the geometric nonlinearity well with less number of elements than the displac ement based method due to its usage of force interpolation functions which strictly satisfy equilibrium in the deformed configuration. An example with the snap through behavior shows that fewer elements are required to yield comparable accura cy with the i ntroduced method. The i ncremental form of the force based method is derived to be applied to material nonlinearity. S tress resultant s based and stress based material nonlinearity were already developed in the literature for sectional behavior of a frame el ement and thus it is shown that th e se material nonlinearit ies can be incorporated into the force based method.
39 CHAPTER 3 PLATE ELEMENT MODEL General Concepts of Material Nonlinearity of a Plate In this chapter, a n efficient elasto plastic model based on the stress resultants is introduced The model can be integrated with geometrical nonlinearity and applicable to metal and concrete plates. Generally, for the elasto plastic behavior of a plate, the layered model is used to consider stresses distribution at each layer which are calculated based on the strains that are assumed linearly distributed through thickness These stresses are applied to a yield criterion for the elasto plastic behavior and the updated stresses and elasto plastic tangential stiffness of each layered are integrated through the thickness to calculate internal force s and stiffness of the element. The concept of layered model is described in Fig ure 3 1(a) and the basic equations for constitutive relation between stresses and str ains of a single layer are described in Eq. 3 1 The relation of stress resultants and the mid plane strains can be described as in Eq. 3 2 using integration through the thickness Numerical calculation for elasto plastic behavior is performed in the constitutive relation between stress and strain in Eq. 3 1. The accuracy of results depends on t he number of layers, and thus, enough number of layers should be used. In this research, the number of layers is fixed to 15. Most commercial software programs use the layered model for elasto plastic behavior of plates. Reasonable results can be expected in a layered model for all yield criteria. However, a lot of calculation is required for this model because a complicated elasto plastic behavior should be estimated in every layer at each integration point. In case of a large structure using tens of thous ands of plate elements, the cost for analysis c ould be significant.
40 For the efficiency of calculating the elasto plastic behavior of a plate element, the stress  For metal s using the von Mises criterion, a modified yield criterion was suggested by Voyiadjis and Woelke  The concept is described in Fig ure 3 1(b) and the yield criterion is explained in the following sub section. In this model, the yield criterion consists of stress resultants instead of stresses ; therefore, no layer by layer integration is required The stress resultants are estimated by using mid plane strains in Eq s. 3 3 and 3 s for metal (von Mises criterion) are applied for the development of plastic deformations across the thickness. Both kinematic and isotropic hardening rules are derived based on stress resultant s In this research, a modified yield criterion based on Ilius function for concrete plates (Drucker Prager criterion) is proposed for concrete building and civil structures. The basic concepts for hardening and flow rules are based on the paper of Voyiadjis and Woelke  (3 1) (3 2)
41 where and are membrane strains and out of plane bending curvatures. The variables and are membrane stress resultants and bending stress resultants of a plate. The relation between strains and stress resultants are described in Eqs. 3 3 and 3 4. (3 3) (3 4)
42 A B Figure 3 1 Nonlinear models for elasto plastic behavior of a pla te A) Layered mode of a plate B) Stress resultant model of a pl ate
43 Yield Criterion based on Stress Resultants V on Mises Criterion For efficient calculation of the elasto plastic behavior of metal plate s a modified the von M ises criterion and stress expression s based on stress resultant s The stresses at the top and bottom of a plate are expressed as in Eq. 3 5. I f the stresses in Eq. 3 5 are substituted into the von Mises yield criterion, Eq. 3 6 the results can be written Eq. 3 7 . (3 5 ) (3 6 ) (3 7) w here is the thickness of the plate and and are stress, membrane and moment stress resultants resp ectively. The Eq 3 8 is suggested by Voyiadjis and Woelke  which is a concise form of yield surface where interaction terms are deleted and a plastic curvature parameter is added.
44 (3 8 ) w here and are the full y plastic yielding moment and axial force respectively. The stress expression in Eq. 3 5 is valid until the initial yielding in a plate. Yielding by bending is propagated from top and bottom planes to mid plane as the bending moment increases beyond the initial yielding moment. A specific parameter is required to describe th e continuous yielding beyond the initial yielding point. In this research, followed by Crisfield , t he plastic curvature parame ter is designed for progressive development of plastic zone under a bending moment as in Eq. 3 9 (3 9 ) w here is the equivalent plastic curvature, and are the increme ntal plastic curvatures of each direction. The equivalent plastic curvature is accumulated value from initial plastic deformation by bending. T herefore, it should be set by zero when it meet s elastic status which is described in Figure 3 2. The parameter is calculated based on the directional plastic curvature s as in Eq. 3 9 For a metal plate, the coefficient 1/3 in Eq. 3 9 is based on symmetric progressive yielding and it is theoretically calculated. For a concrete plate, the coefficient is not followed the theoretical calculation due to unsymmetric yielding, it will be explained in the Drucker
45 Prager criterion. For a metal plate, y ielding is initiated at top and bottom surface at = 2/3, = 0 and entire section yielding is occurred at = 1 .0 = Figure 3 2 Initializ ation of equivalent plastic c urvature, Drucker Prager Criterion For efficient calculation of the elasto plastic behavior of concrete plate s a modified Drucker Prager yield criterion based on stress resultants is proposed in this research The criterion is derived based on t he stress based Drucker Prager yield criterion. (3 10 ) where and are respectively, the internal friction angle and cohesion . For a concrete material, the constant can be calculated using tensile and compressive
46 yield stresses as The first term on the right hand side of Eq. 3 10 corresponds to the first invarian t of stress tensor, and the second term is the second invariant of stress deviator. When stress es stress resultant s relations in Eq. 3 5 are substituted into the Drucker Prager yield criterion Eq. 3 10 the yield function c ould be ex pressed following Eqs 3 11 and 3 12. (3 11 ) (3 12 ) where and have physical meaning of initial and hardened uni directional yieldi ng stress respectively. The variables , and were explained in the von Mises criterion. Since Eq 3 12 is the criterion for the initial yielding, it should be modified for a continuous elasto plastic behavior of a concrete plate. In the plastic state, the superposition between the membrane and bending actions is not allowed. In addition, the plastic zone starts from the top and bottoms surfaces and gra dually move toward inside until the entire cross section becomes fully plastic. Because of different roles of membrane and bending stress resultants, the moment term of the first invariant of
47 stress resultants, is removed due to the fact that this term has both compressive and tensile stresses at the same time even though it comes from the first invariant of stresses, hydrostatic stress. Indeed, this term show s a discrepancy with the reference model in numerical comparison shown in the next section. In addition, the coupled term between membrane and moment components, is also removed through a parame tric study and numerical tests. This term was also deleted in the previous research of metal plate based on von Mises yield criterion for a kinematic hardening problem by Armstrong and Frederick . In a concrete plate, the neutral plane deviates from the geometric mid plane due to plastic deformation. This happens because the tensile yield stress is much l ess than the compressive one. Three parameters , and are introduced for progressive evolution of plastic zone unsymmetric stress distribution and the coupled behavior of membr ane and bending resultants. The proposed form of stress resultants model based yield function of a concrete plate can thus be written as Eq. 3 13. (3 13 ) where and are nominal yield membrane and moment resultants of the cross section, respectively. The plastic curvature parameter which is explained in Eq. 3 9, is modified for concrete yielding behavior. On the contrary to the metal, a c oncrete plate has
48 unsymmetric stress distribution due to the difference between tensile and compressive yielding strength, and thus the pattern of yielding under pure bending is different with th at of the metal. I n this research, through the numerical test s using the layered model, the plastic curvature parameter , for a concrete plate is proposed as (3 14) w here is the equivalen t plastic curvature which is defined in Eq. 3 9. For a concrete plate under bending, initial yielding always occurs at the tensile part and the value of is around 0.6 and =0. In a concrete plate, the stress distribution through the thickness is not symmetric contrary to the metal under elasto plastic status. The full y plastic moment of a concrete plate cannot be estimated by the nominal fully plastic moment , due to non symmetric distribution of stresse s In this research, the parameter ( ) is added to express the fully plastic moment based on the nominal fully plastic moment in a concrete plate. In order to derive the parameter , it is assumed that the concrete is in yielding under bending without membrane forces. It is also assume that there is no hardening during plastic deformation. The yield function must satisfy when the material is in the fully plastic state, in which the fully plastic bending mome nt is applied and the plastic curvature parameter become s one. Then, the yield function in Eq. 3 13 can be expressed as from which the moment calibration
49 parameter is calculate d as The fully plastic moment can be calculated by numerical analysis; for example, using the layered model. In addition to the two parameters and the yield criterion is further modified in order to match the results with the layered model under combined axial and bending loadings For that purpose, parameter studies, described in the following section, are conducted and t he exponent of the moment term is introduced The uni axial yielding stress resultants can be estimated by Eq. 3 1 5 and the tensile and the compressive yielding stress resultants are expressed in Eq. 3 1 6 ( 3 1 5 ) (3 1 6 ) where and are positive and negative uni axial yielding stress resultant s respectively. In building structures, there are two special purpose elements which are based on the plate element. The one is the w all element which has only the membrane stiffness and the yield function of the w all element can be expressed as in Eq. 3 1 7 The other is the s lab element which has only the bending stiffness and the yield function of the s lab can be expressed as in Eq. 3 1 8 for wall (3 1 7 ) for slab (3 1 8 )
50 Parametric S tudy for t he M odified Y ield C riterio n of a C oncrete P late P arametric stud ies are performed to check the proposed yield function for a concrete plate under combined cases of axial and bending loadings. T he parametric stud ies are executed using pre axial bending loading and pre moment axial lo ading tests. Through comparison between the results of the layered model and those of stress resultant model, appropriate values of par ameters can be found. In this research the results from the layered model are considered as a reference. For the purpose of parameter study, the original failure function in Eq. 3 12 is modified to have three parameters, and as Eq. 3 19. (3 1 9 ) As explained in the previous section, the two un derlined moment terms are expected to be removed The first term (I1_M) is from the first invariant of stress tensor and the second term (J2_NM) is from the second invariant of deviatoric stress tensor T h e se t wo terms are tested and compared with the refe rence results in Figure 3 3(a) with parameter When the yield function includes these term s the yielding moment s under pre axial loading show different patterns compared with the results of the layered model. The I1_M term makes the maximum yielding moment to occur at different pre axial loadings, while the J2_NM term makes the maximum occurs at zero pre axial loading. Although the curve without these two terms is still different from the curve of the layered model, it is found t hat this curve follows the trend correctly. Based on numerical results with and without I1_M and J2_NM terms in Figure 3 3 (a), these
51 two terms are excluded from the failure function Now, In order to calibrate the difference in amplitude between the layere d model and the proposed failure function, an exponent is introduce d for moment term of the second deviatoric stress invariant tensor In the paramet er study of in Figure 3 3(b), it turned out that the value of 0. 39 matches well with that of the layered model, which is used for the following a nalysis. Also, the pre moment axial loading cases are investigated in Figure 3 3 ( c ) and (d). T he results of the modified yield function well coincide with those of the layered is 0 39 In this research, the results of layered mode l are calculated by numerical method but those of other model s are estimated by Eq. 3 19. A Figure 3 3 Yielding stress resultants in combined loading cases A ) Yielding moment on Pre Axial B ) Yielding moment with C) Tensile yiel ding on Pre Moment D) Compressive yielding on pre moment
52 B C Figure 3 3 Continued
5 3 D Figure 3 3 Continued Hardening Model Generally, the material yield strength increases according to plastic deformation, and the yield occurs earlier w hen the loading direction changes. This is caused by dislocation pileups and tangles. To model the hardening behavior of plates subject to cyclic loadings, isotropic and kinematic hardening rules are required. In this research, linear strain hardening rule is applied for the isotropic hardening and strains are estimated on the top and bottom surface s of plates. (3 20 ) where and are the plastic modulus and the e quivalent plastic strain. The plastic modulus is a material constant. The parameter, is used to describe combined hardening; its value can be varied from 0.0 to 1.0. When its value is equal to zero it describes Kinematic harden ing while one for Isotropic hardening.
54 For kinematic hardening, since the yield surface is based on the stress resultant s the back stress resultants should be used instead of back stress which represent the center of the transferred yield surface in th e resultant forces space. The equations of back stress resultants were suggested by Voyiadjis and Woelke  and th e se equations are modified in this research for efficiency. The yield functions of von Mises criterion including both hardening rule s are re presented by Eq. 3 2 1 for von Mises (3 2 1 ) w here are back stress resultants for kinematic hardening The incremental form of back stress resultants fo r kinematic hardening are calculated by the following equation s, and the parameter , is used for different hardening types. The parameters and are material constants for the membrane and bending behavior. For the steel, =1.0 and =0.8 are applied.
55 Figure 3 4 shows the evolution of the yield su rface a ccording to the plastic strain. The concept of isotropic hardening is shown in Figure 3 4(a) and that of kinematic hardening is shown in Figure 3 4(b). A Figure 3 4 Yield surface moving according to hardening t ype s. A) Isotropic hardening Ki nematic Hardening B) Kinematic Hardening
56 B Figure 3 4 Continued Flow Rule and Tangent Stiffness To describe the behavior of elasto plastic plates, the constitutive law and the flow rule are required. The hypo elastic constitutive relation is written in the rate form as in Eq 3 2 2 (3 2 2 ) where the rate of stress resultant and that of strain rate are defined in Eq s 3 2 3 and 3 24 (3 2 3 ) (3 24) In addition, in the Eq 3 2 2 is the rate of plastic strain, and de scribes the relation between the stress resultant s rate and mid plane strains rate in a plate as
57 (3 2 5 ) w here E and h are respectively, the elastic modulus and thickness of a plate. For an elastic response, it is clear that membrane, bending and transverse shear are decoupled The elasto plastic behavior of the stress resultant model follows a similar formulation with the stress based flow rule. In th e associated flow rule t he mid plane plastic strain rate is proportional to the gradient of plastic potential, which is identical to the yield function, as (3 2 6 ) where is the plastic consistency parameter. Plastic deformation is in the direction normal to the yield surface, and the amount of plastic deformation is decided by the plastic consistency parameter.
58 Plast ic C onsistency P arameter, In general, the plastic consistency parameter is non negative : positive d uring plastic deformation and zero for elastic deformation. On the other hand, the yield function is al ways non positive: for the elastic status and for the plastic status. In optimization, this is called the Kuhn Tucker condition and can be expressed as (3 2 7 ) where is the yield function including hardening terms. The non positive property of the yield function is regarded as a constraint, and the plastic consistency parameter plays the role of the Lagrang e multiplier corresponding to the inequality constraint. The Kuhn Tucker condition satisfies all possible stat e of the material. When the state varies, the condition can have three cases. (a) Elastic loading : (b) Neutral loadin g : (c) Plastic loading : When the stress is on the yield surface, is equivalent to In e lastic and neutral loadings, and there is no plastic deformation. During plastic loadings, is zero which means that the yield function remains zero and the following condition can be obtained : (3 2 8 ) where the ra te of equivalent plastic strain can be defined as
59 The plastic consistency parameter is calculated from the condition in Eq. 3 28. To calculate the plastic consistency parameter an iterative procedure is required and thus the local Newton Raphson method is applied. Newton Raphson A lgorithm for the P lastic C onsistency P arameter In numerical analysis, the rate of plastic consistency parameter is converted into an increment by multiplying it with time increment : In a similar way, all rates in the previous section can be considered as increments. In the following derivations, an increment will be used instead of rates. T he plastic consistency parameter can be calculated after substituting Eqs. 3 2 2 and 3 26 into Eq. 3 2 8 as (3 2 9 ) (3 30 ) The equations for and are listed in Appendix B and C. T he equivalent plastic strain is estimated on the top and bottom surface of a plate using linear strain assumption through the thickness and plastic deformation as in Eq. 3 31
60 ( 3 31 ) The derivative of the back stress resultant s against can be estimated by Eq s 3 3 2 and 3 3 3 Membrane & out of plane shear (3 3 2 ) Bending (3 3 3 ) The third term in Eq. 3 2 9 is simply calculated as in Eq. 3 3 4 when the yield surface is assumed linearly hardened against equivalent plastic strain. (3 3 4 )
61 Elasto P lastic C onstitutive E quation s The constitutive relation betw een incremental stress resultant s and mid plane strains are estimated by Eq. 3 3 5 When t he plastic consistency parameter defined by Eq. 3 30 is applied to Eq. 3 35 the elasto plastic tangent stiffness can be obtained as in Eq. 3 37 (3 35) (3 3 6 ) (3 37) where and are elastic and elasto plastic stiffness es respectively. In case of no hardening, the tangent stiffness can be describes as in Eq. 3 38 by simpl y removing the hardening terms. (3 38) The proce dure to decide the plastic consistency parameter is displayed as a f low chart in Figure 3 5.
62 Flow chart for and stresses and strains at each integration point Figure 3 5 Flow chart for calculating plastic consistency parameter and updating
63 Reinforcement by Rebar Geometry and Constitutive Equation Steel rebar reinforcement is a common way of increasing the strength and ductility of a concrete plate. Since co ncrete has much less strength in tension than compression, a steel rebar is used for reinforcement of the tensile part of concrete. In addition, since concrete is a brittle material the steel rebar is also used for increasing ductility. Ductility is an im portant propert y of a structural member to prevent abrupt failure. The rebar are arranged various ways for th at purpose of the plate. For a vertical wall, the reinforcement are arranged symmetry against the mid plane and for a bending slab, the rebar are m ainly located at the tensile part of concrete. Generally the effect of the steel reinforcement of a plate can be considered by two methods. The first is the smeared layer method in which the rebar is assumed as one layer of the plate, and the strains and stresses are calculated with the same way of each concrete ply. The second is an integrated method in which the effect of rebar is considered by modifying material properties of the concrete T he smeared method is good for the layered model in which the s train of a rebar layer can be estimated accura tely The integrated model is applicable to stress resultant model in which global response is focused rather than an individual layer s response. A general configuration of rebar can be shown in Fig ure 3 6 (a) and the stress strain curve of rebar is assumed in Fig ure 3 6 (b). The following common assumption s are often adopted in rebar modeling : (i) a rigid bond between the rebar and concrete; (ii) a linear strain distribution in a section, and (iii) the evenly d istributed steel reinforcement in an element. The equivalent thickness of rebar layer can be calculated as
64 where A xs and A ys are, respectively, section areas of x and y direction rebars; l xs and l ys are space between rebars; a nd t x and t y are equivalent thicknesses of x and y direction rebars A B Figure 3 6 Geometric shape and stress strain curve of a r ebar A) Geometric shape of rebar B) S tress strain curve of steel rebar
65 The rebar has only one directional strength and its direction can be different with the local axis of the plate. When the angle between the local x of a plate and the direction of a rebar array is assumed as the constitutive equation of a reb ar layer is calculated as in Eq. 3 3 9 The 2 d imensional stress and strain transformation matri c es are shown in Eq s 3 3 9 and 3 40 (3 3 9 ) ( 3 40 ) (3 41 ) T he constitutive equation of the rebar layer part can be expressed as in Eq. 3 42 a t an integration point of a plate. This relatio n can be superposed with that of concrete layers.
66 (3 42 ) The S meared L ayer M ethod of R ebar R einforcement Since a rebar has only one directional strength the constitutive equation of the rebar layer p ortio n can be expressed as in Eq. 3 43 In the elastic domain, the modul i of constitutive equation and are either the elastic modulus ( ) and the tangential modulus ( ) is used for those moduli in elasto plastic domain. ( 3 43 ) where and are incremental stress strain and plastic strain vectors respectively. Numerical integration of elasto plastic evolution can be performed assuming that the material state is fully known at step n, and using given incremental strain the state at step n+1 is calculated using t he backward Euler method A nd at the next step, stress, back stress es and plastic deformation are expressed as in Eq. 3 4 5 In the elastic range, the stress and back stress can be easily calculated as in Eq. 3 4 6 In the plastic range, since the yield func tion should remain zero, the incremental plastic deformation and stress es can be calculated as in Eq. 3 4 8 and the stress, back stress
67 and plastic deformation at n+1 step can be estimated as in Eq. 3 4 5 T he elasto plastic tangent modulus is calculated as in Eq. 3 4 8 (3 4 4 ) where , are stress, backstress and plastic strain at step n and are initial yield stress and plastic modulus respectively. (3 4 5 ) Elastic predictor In the elastic range: (3 4 6 ) In the elasto plastic range: ( 3 4 7 ) T he tangential s tiffness is calculated as Eq. 3 4 9 ( 3 4 8 ) ( 3 4 9 )
68 The I ntegrated S ection M ethod of R ebar R einforcement The rebar reinforcement increases the strength of a concrete plate because of its own str ength and b o nding between the rebar and concrete plate. In the elasto plastic behavior of a concrete plate, one directional deformation can cause an expansion in the orthogonal direction s. Due to the assumption of perfect bonding, the steel reinforcement m ay have tension and the concrete plate may ha ve compression at that direction. T he con fining effect of rebar can increase the yield strength of the concrete plate T herefore, the axial yield strength of a reinforced concrete plate is difficult to be estima ted using simple analytical equation due to the binding effect between the rebar and concrete. In th e integrated model, the effect of rebar reinforcement is incorporated with the concrete plate using equivalent material coefficients. From the Eq s 3 50 an d 3 5 1 the relation between two parameters ( ) of Drucker Prager criterion and equivalent material coefficients ( ) can be obtained. (3 50 ) (3 5 1 ) Therefore, it is necessary to calculate the equivalent parameter s, and which can be estimated based on the axial behavior of a reinforce d concrete plate. In this research, those axial behaviors are estimated by a numerical analysis based on the layered model. T he uni axial tensile and compressive stress resultant s of a concrete plate can be expressed as in E q. 3 5 2 using the two parameters ( ) of the Drucker Prager criterion. Th e se parameters are estimated based on the uni axial yield stress resultants
69 as in Eq. 3 5 3 and the equivalent material coefficient s ( ) can be estimated as Eq s 3 50 and 3 5 1 ( 3 5 2 ) ( 3 5 3 ) where and are the tensile and compressive yield strength of a reinforced c oncrete plate, and are the two parameters of the Drucker Prager criterion. W hen a plate has rebar reinforcement, the yielding strength is increased but, the incre ment ratio is not simpl y estimated due t o the steel confinement effect. In case of compressive axial yielding, the amplitude could increase over 30% rather than a simple summation of concrete and steel rebar strength. However, in tensile axial yielding, the increment is almost same with the simp le summation. T herefore, for the estimation of the tensile and compressive yield strength, numerical analysis using the layered model could be a reasonable choice. I n addition, the elastic modulus is also modified to get the effect of reinforcement as in Eq. 3 53 (3 53 ) where and are the elastic modulus of concrete and steel and are effective thickness es of concrete and steel parts respectively.
70 T ests of a U nit E lement A M etal P late B ased on the von Mises Criterion The stress resultant yield model based on the von Mises criterion is used for a metal plate Geometry of metal plate a nd detailed information of loading conditions, p ure axial, pure bending, pre moment axial and pre axial bending are described in Figure 3 7 Results of the stress resultant yield model are compared with those of the layered model as a reference The plast ic modulus ( ) used as a value of E /50 and E /500 which covers wide range s of strain hardening and t hree different type of hardening ( ) are applied as 0.0, 0.5 and 1.0. Those parameters ( ) individually describe kinematic, combined, and isotropic hardening. The detailed information of elasto plastic material properties of the von Mises criterion is listed in Table 3 1. A B C D Figure 3 7 Test cases of stress resultants yield criterion A) Pure axial B) Pure bending C) Pre moment axial D) Pre axial bending
71 Table 3 1 Material nonlinear propert ies for a steel plate Elasto p lastic material properties Elastic modulus (E) Gpa 210 Initial yielding stress ( ) Mpa 210 Hardening Modulus (H) Gpa 4.2(E/50)~42(E/500) Combined hardening parameter ( ) 0.0 ~ 1.0 The results of stress resultant model based on the von Mises criterion are well matched with those of the layered mo del in all test cases. Those comparisons are shown in Fig ure s 3 8 3 9 3 10 and 3 11 The results of pure axial and pure bending tests are almost identical in all types of hardening. The results of pre moment axial case have a slight difference around a transition zone of elastic plastic behavior. It is caused by the feature of the stress resultant s yield function, in which the state of yielding condition is abruptly occurred in axial loading without gradual change. In the yield criterion based on the str ess resultant s axial ly dominant yielding always shows sharply changed yielding curves even though there are pre moment loadings. In spite of the difference, the converged yielding stat e are well matched each other. In case of pre axial bending cases, grad ually developed yielding show that the plastic curvature parameter ( ) have reasonable effect on yielding. In addition, regardless of hardening types such as kinematic, isotropic and combined hardenings, the results of stress resu ltant model are close with those of the layered model. From th ese comparisons it can be conclude that the stress resultant model based on the von Mises criterion can be an alternative of the layered model with a n acceptable error bound.
72 A B C D E F Figure 3 8 A xial stress resultant s vs. strain s in pure axial loading A) = 0.02E and B) = 0.02E and C) = 0.02E and D) = 0.002E and E) = 0.002E and F) = 0.002E and
73 A B C D E F Figure 3 9 S tress couple resultant vs. rotation in pure bending loading A) = 0.02E and B) = 0.02E and C) = 0.02E and D ) = 0.002E and E) = 0.002E and F ) = 0.002E and
74 A B C D E F Figure 3 10 A xial stress resultant s vs. strain s in pre moment axial loading A ) = 0.02E and B ) = 0.02E and C ) = 0.02E and D ) = 0.002E and E) = 0.002E and F) = 0.002e and
75 A B C D E F Figure 3 11 S tress couple resultant vs. rotation in pre axial bendin g loading A) = 0.02E and B) = 0.02E and C) = 0.02E and D ) = 0.002E and E) = 0.002E and F) = 0.002e and
76 A C oncrete P late B ased on Drucker Prager Criterion The r esults of the stress resultant m odel based on the Drucker P rager criterion are compared with th ose of the layered model for a concrete plate Geometry and loading condition s are the same with the metal model in Fig ure 3 7 and steel rebar are added to reinforce the plate as in Fig ure 3 1 2 F or the r ebar reinforcement the smeared layer method is used in the layered plate model and the integrated section method is applied to the stress resultant model. Material nonlinearity of rebar is modeled using von Mises criterion and elasto perfect ly plastic behavior is assumed in this research The elastic modulus and the initial yielding stress of reinforced steel are assumed as E = 2.10 10 5 M P a and = 210 M P a. The hardening effect of reinforced steel rebar is not considered in this research. Generally the concrete does not have strain hardening and thus hardening of concrete is not considered The material parameters of concrete based on the Drucker Prager criterion are initial cohesion ( ) and initial friction angle ( ) ; both coefficients are assumed as = 4. 6 6 M P a and = 59.78 o Th e se coefficients are estimated based on the compressive and tensile yield strengt hs which are calculated by numerical analysis using the layered model T he compressive yield str ess ( ) is assumed 34.5 M P a and the tensile yield str ess ( ) is estimated by 3.64 M P a using a formula of ACI 3 18 08 for a design of concrete floor systems In Table 3 2, t wo equivalent material co efficients for integrat ed section method are listed which are estimated along to the reinforcement ratio s T he data of n o rebar column are u sed for the case of without reinforcement and 1% r ebar column data are
77 applied to the case s of with reinforcement. H ardening is not considered on condition that concrete has no strain hardening. Table 3 2 Material coefficient for the Drucker Prager criter ion on the integrated section method Rebar reinforcement No R ebar 0.5% Rebar 0.75% Rebar 1% Rebar Tensile yield strength ( N/mm ) 728 960 1070 11 75 Compressive yield strength ( N/mm ) 6900 8540 9350 10 1 6 0 Cohesion (Mpa) 4.6 6 5.9 6 6.58 7. 1 9 Internal fricti on angle 59.78 58.82 58.54 58. 38 Modified elastic modulus (Mpa) 27000 280 00 285 40 2906 0 Figure 3 1 2 Geometry of a steel reinforce d concrete plate The results of stress resultant model based on the Drucker Prager yield crite rion have a good agreement with that of the layered model in all test cases. These comparisons are illustrated in Figures 3 13 ~ 3 17 The results of pure axial and pure bending tests under reinforcement show almost identical each other in Figure s 3 13 and 3 14 The tensile yield strength of the reinforced concrete plate is around 1170 N/mm which is similar with a simple summation of that of concrete and steel rebar, 1148 N/mm. The compressive yield strength of the plate is around 10160 N/mm which is much l arger
78 than a simple summation of that of concrete and steel rebar, 7320 N/mm. The increment of 35% in compressive yield strength is due to the confining effect in the orthogonal direction by the rebar. The comparisons of results of pre moment axial and pre axial bending cases are show n in Figures 3 15 and 3 16 Figure s 3 1 5 (a) and 3 16 (a) show the results of no reinforcement and Figure s 3 15 (b) and 3 16 (b) show the results with reinforcement The confining effect in the or thogonal direction can also be foun d in bending tests shown in Figure 3 14 (b) in which the bending yield strength 106000 N mm/mm, is about 10% larger than a simple summation of concrete and steel strength, 96150 N mm/mm. The results from the stress resultant model under several loading tes ts match well with that of the layered model regardless of rebar reinforcement From these results, the stress resultant model can be expected to have acceptable accuracy for modeling the material nonlinear behavior of a concrete plate. A B Figure 3 1 3 A xial stress resultant s vs. strain s in pure axial loading A) W ithout Reinforcement B) With Reinforcement
79 A B Figure 3 1 4 S tress couple resultant s vs. rotation s in pure bending loading w/ rebar A) W ithout Reinforceme nt B) With Reinforcement A B Figure 3 1 5 A xial stress resultant s vs. strain s in pre moment (M p /2) axial loading A) W ithout Reinforcement B) With Reinforcement
80 A B Figure 3 1 6 S tress couple resultant s vs. ro tation s in pre axial (N p (+)/2) bending loading A) Tensile pre axial without rebar. B) Tensile pre axial with rebar A B Figure 3 17 S tress couple resultant s vs. rotation s in pre axial (N p ( )/2) bending loading w/ rebar A) Compressi ve pre axial without rebar. B) Compressive pre axial with rebar.
81 Practical Examples A P art of S teel G irder B ridge The stress resultants yield criterion of von Mises is applied to a part of a steel box girder bridge and the results are compared with the la yered model which was assumed a reference method in this research. A section of a steel box girder bridge which is reinforced by many ribs is described in Figure 3 1 8 (a) and a part of the top surface including U rib is shown in Figure 3 1 8 (b). The e xtended length of th e section is modeled as in Figure 3 1 8 (c) and the results are compared with the layered model. The elasto plastic material properties of steel are assumed as in Table 3 1. In this analysis, i nitial yield stress of the steel is assumed 210 N/mm 2 The h ardening modulus and combined hardening parameter are assumed E/50 and 0.5 respectively for the s train stiffening effect. Numerical results of the stress resultant model are compared with those of the layered model in Figure 3 19 and Figure 3 2 0 In Figure 3 19, results of pure axial and pre moment axial loadings are compared and one half of fully plastic moment (M p /2) is used as a pre moment loading. As expected, the yielding axial force is reduced and the reduced amo unt of yielding axi a l force is around 20%. The results of pure bending and pre axial bending are shown in Figure 3 20. The yielding bending moment is reduced around 35% in pre axial (N p /2) bending loading. In pure bending, positive and negative fully plast ic yielding moments are almost same, but the positive value is less than the negative value under pre axial loading. For each case, the results of the stress resultant model are very close to those of the layered model As a result, the proposed stress
82 re sultant model is reasonable to be applied for von Mises yield criterion under elasto plastic analysis. A B C Figure 3 18 A steel box girder bridge A) Full section of a steel box girder bridge B) A part of top surface C) FEA m odeling of a part of top surface
83 A B Figure 3 19 A xial stress resultant s vs. axial strain s with and without pre moment A) Pure Axial. B) Pre moment Axial. A B Figure 3 20 Bending str ess resultant s vs. r otational angle s with and without pre axial A) Pure bending. B) Pre axial bending.
84 A S uperstructure of a PSC G irder B ridge The proposed stress resultants yield criterion of Drucker Prager is applied to a practical structure, a supers tructure of a PSC g irder b ridge and t he results are compared with those of the layered model which was assume d a reference in this research. The geometry of the half section shape is illustrated in Figure 3 21(a ) with the FEA model in Figure 3 21(b). The section is modeled by two plates with thicknesses of 250 mm and 450 mm, and the span of this bridge is 50m. The elasto plastic material properties are assumed as those of 1% reinforcement of Table 3 2. The fully yielding moment s of each thickness are estim ated by numerical analysis using the layered model whose value s are 170kN m m /m m and 570kN m m/m m, respectively for the two thicknesses A B Figure 3 21 A superstructure of a PSC girder bridge A) Half s ection of a PSC gi rder bridge B) FEA mo deling of a superstructure of a PSC girder bridge
85 Table 3 3 Fully plastic yielding forces and moments under uni directional loading s Type of modeling Layered Stress resultants Reinforcement w/o Rebar w/ Rebar w/o Rebar w/ Rebar Pu re Axial (M N) Positive 31 51 32 52 Negative 300 440 310 450 Pure Bending (M N m) Positive 54 82 57 87 Negative 36 52 37 58 The results of pure axial loading are shown in Figure 3 22. In the case of rebar reinforcement, the fully axial yielding load s are increased more than that of simple summation of concrete and steel. In compressive part, the axial yielding load is about 40% larger than that of simple summation. H owever, only around 3.5% incre ase is occurred in tensile yielding. T he compressive co nfinement in the width direction of rebar causes th ese additional incre ase of yielding axial force in the compressive part. H owever, tensile yielding is increase d only a few percent due to the tensile confinement in the width direction. The results of pure bending are shown in Figure 3 23. T he fully moment yielding of the reinforced case is increased around 45% from the value of the non reinforced case. The results of yielding loading by the proposed stress resultant s yield criterion are similarly estimated with the layered method in both axial and bending loadings. The fully plastic yielding values of tested cases are summarized in T able 3 3. In addition, common loading type s pre axial and selfweight loading, are tested for this structure. P re axial loadi ng simulates the tendon s pre stress loading which is commonly used in a PSC girder bridge, and factored self weight loading describes the i nertia force due to earthquake loadings. The pre axial loading is assumed 20% of
86 ultimate axial loading, which is an acceptable range f or tendon s pre stress and the factor of selfweight loading is increased until a buckling failure occurs T he relation between the factors of selfweight and vertical displacement are shown in Figure 3 24. The buckling due to the fully p lastic failure occurs at 1.8 times of self weight and the result s of the proposed stress resultant model matches well with that of layered model T he efficiency of stress resultant model can be estimated by the ratio of calculation time of element interna l forces and tangential stiffness per iteration between the layered and stress resultant model. In this research, the ratios are tested according to the number of elements, which are shown in Table 3 4 I t can be concluded that the stress resultant model c an reduce the element calculation time around 70% compared to the layered model that is commonly used. The analysis of layered model is carried out by commercial finite element analysis software [ 31 ], MIDAS/NFX. A B Figure 3 22 A xial stress resultants vs. strain s in pure axial loading A) Pure axial loading w/o Rebar B) Pure axial loading w/ Rebar
87 A B Figure 3 23 S tress coupl e resultant s vs. rotation s in pure bending loading A) Pure bending loading w /o Rebar B) Pure bending loading w/ Rebar Figure 3 24 Vertical displacement under factored self weight with axial pre stress Table 3 4 Comparison of calculation cost between layered and stress resultant model Element N umbers 800 4000 8000 16000 32000 Average time for element s calculation for a iteration step (sec) Layered 1.35 4.07 8.01 16.99 32.27 Stress resultant 0.22 1.09 2.24 4.56 8.95 Ratio 0.16 0.27 0.28 0.27 0.2 8
88 Neutral A xis M oving of a C oncrete P late under B ending A phenomenon showing expansion of overly plastic membrane is found under the cyclic bending. Figure 3 25 describes a small cell of a plate under the cyclic bending. Since the concrete material has di fferent compressive and tensile yield strength, when a concrete cell has a bending curvature over partial yielding state, the stres s distribution of thickness is changed as shown in Figure 3 2 6 (c). Therefore, the neutral axis of curvature is moved as Figur e 3 25 (b) or 3 25 (c) ; it moves up and down when loadings cause negative and positive curvature. T he moving of neutral axis is shown in Figure 3 2 6 (a) according to the curvature Sequentially the mid plane is continuously expanded according to the cyclic be nding as shown in Figure 3 2 6 (b). A B C Figure 3 25. Moving of neutral axis of a concre te plate A) Concrete plate (Layered model). B) Negative curvature loading. C) Positive curvature loading.
89 A B C Figure 3 2 6. Results of a concrete plate from cyclic bending A) N eutral axis moving. B) Membrane expansion. C) Stress distribution through thickness. A simpl y supported beam was tested using solid and plate element s based on the Drucker Prager yield criterion. For the solid model 14 solid layers are used and the layere d model is used for the plate model. Figure 3 27 (a) illustrates the loading and
90 boundary conditions, and Figure 3 27 (b) compares the expansion of end points of solid and mid plane of the plate. The difference between solid model and plate is not much and the membrane expansions are around 0.5% of the length per bending cycle. W hen it has 10 cyclic of bending, it could be 5% of the length A B Figure 3 27. Extension of membrane under cyclic bending A) Simple beam under cy clic pressure loading B) Membrane extension under cyclic loading
91 Summary In this chapter, a stress resultant model for material nonlinear analysis for a plate is introduced F or a concrete plate a stress resultant elasto plastic model is proposed, which has a comparable accuracy with the layered model but with much less computational cost. A yield function based on stress resultants is derived for the Drucker Prager criterion which is widely used for a concrete material. Through a parameter study, the in itial yield function is modified by introducing the moment calibration parameter and the exponent of the moment term The generalized plastic flow rule is applied to this failure function. In addition, a steel rebar model is added for the reinforcement of a concrete plate. The smeared layer method is introduced to the layered model and the integrated section method using equivalent material coefficients is proposed for the stress resultant model. Several test s of unit element show that the proposed method has acceptable accuracy under cyclic loading compared with the layered model with and without reinforcement. In the practical example of a superstructure of a PSC girder bridge, the proposed stress resul tant model also shows acceptable accuracy and computational efficiency compared with the layered model.
92 CHAPTER 4 NONLINEAR DYNAMIC ANALYSIS Nonlinear D ynamic E quation The final goal of this research is to develop an efficient analysis method for large structure s under earthquake loading. T herefore, application of the proposed methods to the nonlinear dynamic analysis is required. T he basic nonlinear dynamic equation is described as in Eq. 4 1 (4 1) w here , and are mass matrix, damping matrix, internal force vector and external loading vector respectively. The acceleration, velocity and displacement are expressed as and In this research, the implicit direct integration method based on the Newmark time integration is used f or the nonlinear dynamic analy sis The basic assumption s of the Newmark method are Eq s 4 2 and 4 3 which describe the velocity and displacement at time step using known previous step s acceleration, velocity, and displacement. (4 2 ) (4 3) The values of and are Newmark integration parameters and generally 0.5 and 0.25 are used for the rigid stable condition. The value of is the time interval for integration. Using Eq s 4 2 and 4 3 the velocity and acceleration at time step can
93 be described using previous step s accelera tion, velocity, and current displacement as in Eq. 4 4. (4 4 ) The incremental forms of displacement, velocity, and acceleration from a step to the next step are describe d as Eq. 4 5 (4 5 ) When the iteration of Newton Raphson method is applied the incremental form s of it eration are required and those forms can be described in Eq. 4 6. T he velocity and acceleration of the current iteration step are expressed using previous iteration s results and current displacement as Eq. 4 7.
94 (4 6 ) (4 7 ) When displa cement ( ), velocity ( ), and acceleration ( ) are applied to the governing equation, Eq. 4 1, the nonlinear dynamic equation can be expressed as Eq. 4 8. and in Eq. 4 9 are called the effective stiffness matrix and the effective force vector respectively (4 8) (4 9 ) where the s uperscript k 1 means the previously iteration stage. T herefore, , and are the displacement, velocity and acceleration of the previous stage.
95 Dynamic Equation u nder Earthquake L oadings The external loading of the dynamic equation under an earthquake only depends on the masses and the ground acceleration. A structure under earthquake loading is shown in Figure 4 1 and the dynamic equation can be described as in Eq. 4 10. T he dampi ng and internal force terms only depend on the relative velocities and displacements and thus Eq. 4 10 can be written as in Eq. 4 11. Compared with the general nonlinear dynamic equation, the external force of the dynamic equation under earthquake loading is decided by the multiplication of mass and ground acceleration. (4 10 ) (4 11 ) Figure 4 1. A structure under earthquake loading
96 where , and are the relative displacement, velocity and acceleration against the ground. The and are the displacem ent, velocity and acceleration at the ground. Practical E xamples of E arthquake L oading s Generally, an earthquake loading has dozens of seconds shaking and the record of shaking are normally measured as 0.01 sec time interval. T herefore, t he total time steps for a dynamic analysis of a structure are often over several thousand steps I n addition, the strong shakings of an earthquake which can cause plastic deformation are generally a small part of the whole length of earthquake In this res earch the ground accelerations of the Tohoku Japanese earthquake in 2011 i n Figure 4 2 are used for nonlinear dynamic analysis. Ground accelerations are recorded in three orthogonal directions for earthquake loading Horizontal, t angential and vertical di rection accelerations are shown in Figure 4 2 The distance from epicenter is around 177km and the maximum peak acceleration is 332.8 cm/s 2 and the ground motion has 150 seconds shaking. In this research, three directional ground acceleration s are applied as earthquake loading and the nonlinear dynamic analysis are performed for 10 seconds period, the most severe shaking period from 65 second to 75 second. Time increment for recording of an earthquake is 0.01 sec and the time increment for the nonlinear dy namic analysis is set as 0.005 sec. In the nonlinear dynamic analysis, t he total Lagrangian formulation for frame and plate is used to include geometric nonlinearity In this research, there is no description of geometric nonlinearity because the g eometric nonlinear formulation of element is out of scope However, all nonlinear analyses in this chapter include the geometric and material nonlinearity.
97 A B C Figure 4 2. Ground acceleration of Tohoku Japanese earthquake 2011 A) Horizontal direction acceleration B) Tangential direction acceleration C) Vertical direction acceleration Figure 4 3. Loadings of an earthquake acceleration
98 A P art of Steel Girder Bridge under Earthquake Loadings The nonlinear dynamic analysis of a part of a steel girder bridge section is performed based on the von Mises yield criterion. T he FEA modeling and material properties are the same with the model in Figure 3 18. Three directional earthquake loadings in Figure 4 2 are applied to the str ucture in Figure 4 3 T he model in Figure 3 18 is a small part of a steel box girder bridge and thus the external forces based on the multiplication of mass and ground acceleration are not enough to cause plastic yielding. T herefore, a loading factor of 2 00 is multipli ed to the ground accelerations in order to induce plastic deformation T he tangential and vertical displacements at the center of the length are compared between linear dynamic and nonlinear dynamic using stress resultant and layered models i n Figure 4 4 The discrepancy is developed from around 3.0 seconds. The tangential displacement shows slight shift to the negative direction in Figure 4 4 (a) and the vertical displacement has around 2.0cm shift ed to the positive direction in Figure 4 4 (b). The shifted displacement is accumulated by the plastic strains. After several big peaks causing plastic deformation the pattern of displacement are almost same between linear and nonlinear results. The tangential acceleration at the center of a bridge an d the tangential reaction at the left end of the bridge are shown in Figure s 4 5 (a) and (b) respectively. In the displacement comparison, t he difference between the linear and the nonlinear dynamic is quite distinct, but the difference between the stress resultant and the layered model is ignorable. From these results, it can be concluded that the stress resultant model based on the von Mises criterion is also reasonable for nonlinear dynamic analysis under earthquake loadings
99 A B Figure 4 4. Displ acement history results of a part of a steel box girder bridge A) Tangential displacement history at the center of bridge B) Vertical displacement history at the center of bridge
100 A B Figure 4 5. Acceleration and reaction history of a part of a s teel box girder bridge A) Tangential acceleration history at the center of bridge B) Tangential reaction history at the left suppo rt of bridge
101 A S uperstructure of a PSC G irder B ridge under E arthquake L oadings The nonlinear dynamic analysis of a superst ructure of a PSC girder bridge is performed based on the Drucker Prager yield criterion. T he FEA model and material properties are the same with the model in Figure 3 21. Three directional earthquake loadings in Figure 4 2 are applied to the structure as F igure 4 3 T he model in Figure 3 21 is a full superstructure of a PSC girder bridge and thus the designed section can sustain the external forces based on the moderate ground acceleration in Figure 4 2 within linear elastic range. T herefore, a loading fa ctor of 15 is multiplied to the ground accelerations to cause elasto plastic behavior s T he tangential displacement and acceleration at the center of the length are compared between linear dynamic and nonlinear dynamic analysis using stress resultant and layered models in Figure 4 6 The discrepancy is developed from around 2 .0 seconds. I n the tangential displacement the nonlinear results shows larger fluctuation s in several big peaks than linear results in Figure 4 6 (a) and the acceleration results are s hown in Figure 4 6 (b). In Figure 4 7 the tangential and vertical reactions of the left end of the structure are also shown. T he difference between two elasto plastic models of a plate is not recognizable. The difference between the linear and nonlinear dy namic is quite distinct, but the difference between the stress resultant and the layered model is ignorable From these results, it can be concluded that the stress resultant model based on the Drucker Prager criterion is also reasonable for nonlinear dyna mic analysis under earthquake loadings
102 A B Figure 4 6. Displacement and acceleration history of a superstructure of PSC girder bridge. A) Tangential displacement history at the center of the bridge B) Tangential acceleration history at the center of the bridge
103 A B Figure 4 7. Reaction history of a superstructure of a PSC girder bridge A) Tangential reaction history at the left support of the bridge B) Vertical reaction history at the left support of the bridge
104 A PSC G irder B ridge under E ar thquake L oadings The nonlinear dynamic analysis of a PSC girder bridge is performed based on the frame and the plate element which are proposed in C hapter s 2 and 3. T he geometry shape and FEA model are shown in Figure 4 8 (a). The total length of the bridge is 100 m and span lengths are 34m, 32m and 34m respectively. The superstructure is modeled using plate element s and the substructure is modeled using frame element s T he b ottoms of substructure are fixed and the both ends of the superstructure have a si mply support boundary condition The tangential and vertical translational direction s are connected between t he superstructure and the substructure. T he shape of the superstructure of a PSC g irder bridge is the same with Figure 3 21 and elasto plastic mat erial properties of a plate are assumed as those of 1% reinforcement in Table 3 2 T he elasto plastic behavior of a column is modeled by a moment curvature relation shown in Figure 4 8 (c) which is estimated by numerical calculation of 3 dimensional elasto plastic solid model The Drucker Prager and von Mises criteria are used for material nonlinearity modeling of the column. The moment curvature relation of a column depends on the axial force; therefore, the relation is calculated on the condition that 9,0 00kN compressive axial loading is loaded due to self weight The simplest method considering the material nonlinearity of a frame is used because the relation of moment curvature is not the main interest of this research. However, theoretically any type of material nonlinearity could be used to the frame element in this research. Three directional earthquake loadings in Figure 4 2 are applied to the structure in Figure 4 8 (a) and t he loading factor of 15 is multiplied to the ground accelerations to cause elasto plastic behavior. To describe the pre tension of tendons and self weight, a
105 horizontal a xial force, 60,000kN, and a body force in the negative vertical direction are pre loaded before loading of the ground acceleration s T he tangential displacemen ts and acceleration at the center of the bridge length are compared between the linear and nonlinear dynamic analysis using the stress resultant and layered models in Figure 4 9 The discrepancy is developed from around 1. 5 seconds and the amplitudes are r educed in nonlinear analysis due to the plastic behavior of the structure. The forces results of the left bottom column are compared in Figure 4 1 0 The axial force and the local y direction al moment (horizontal direction in global) show close but the diff erence in the local z direction al moment (tangential direction in global) has 2~3 times. It means that the column has a plastic defor mation in the t angential direction but not in the horizontal and vertical direction s The reactions of the left bottom are compared in F igure 4 1 1 In t he tangential reaction the result of nonlinear analysis shows around a half of that of the linear analysis at the maximum but in the vertical direction, the result shows similar amplitude with that of linear analysis In gen eral bridge behavior under earthquake loadings tangential direction has main damage because the horizontal direction is supported by the superstructure s axial strength and the earthquake loading of vertical direction is relatively weak compared to other two directions. The difference s of various results of the bridge between the linear and the nonlinear dynamic analysis are quite distinct, but the difference s of those results between the stress resultant and the layered model are less than 10 percent F rom these results, it can be concluded that the stress resultant model can be alternative of the layered model for nonlinear dynamic analysis under earthquake loadings
106 A B C Figure 4 8. Displacement history resu lts of a steel box girder bridge A) FEA modeling of a PSC girder bridge B) Section of the column C) Moment Curvature relation
107 A B Figure 4 9. Displacement and acceleration history results of a PSC girder bridge A) Tangential displacement history at the center of bridge B) Tangential acceleration history at the center of bridge
108 A B C Figure 4 1 0. Forces hi story results of the left bottom frame element A) Axial forces history B) Y direction moment history C) Z direction moment histor y
109 A B Figure 4 1 1. Reaction history results at the left supports of the PSC girder bridge A) Tangential reaction history B) Vertical reaction history
110 Summary In this chapter, plate elements based on the stress resultant model ( Chapter 3 ) and fram e element s based on force based method ( Chapter 2 ) are applied to implicit nonlinear dynamic analysis including geometric and material nonlinearity. The Newmark method is implemented for implicit nonlinear dynamic equation. For a frame element, the moment curvature relation is used for elasto plastic behavior. In a plate element the von Mises criterion and the Drucker Prager yield criterion are used for a metal and a concrete plate respectively. S everal practical examples are tested and the various result s from the prop o sed method s are compared with those of the reference method. Also, the steel rebar are incorporated by the smeared layer and the integrated section m ethods for the layered model and the stress resultant model of a concrete plate respectiv el y. From the results of comparison it can be concluded that the stress resultant model is available for nonlinear dynamic analysis under earthquake loadings
111 CHAPTER 5 CONCLUSIONS In C hapter 2 a force based method for geometric nonlinearity is introdu ced to reduce the number of element for modeling a structure. The usage of force interpolation functions, which strictly satisfy the equilibrium in deformed configuration, can reduce the number of elements compared with the displacement based interpolation functions. The i ncremental form of the force based method is derived to be applied for material nonlinearity ; a ny type s of material nonlinearit ies of a section of frame can be used in the proposed force based method. S everal example s show the availability and efficiency in which fewer elements are enough to yield comparable accura cy with the reference method which is the displacement based method. In C hapter 3, a stress resultant s based elasto plastic model for a concrete plate is proposed, which has a co mparable accuracy with the layered model but with much less computational cost. A yield function based on stress resultants is derived for the Drucker Prager criterion which is widely used for modelling a concrete material behavior Through a parameter stu dy, the initial yield function is modified by introducing the moment calibration parameter , and the exponent of the moment term The generalized plastic flow rule is applied to this failure function. I n addition, a steel rebar model is added for the reinforcement of a concrete plate. The smeared layer method is introduced to the layered model and the integrated section method using equivalent material coefficients is proposed for the stress resultant pl ate model. Several tests of unit element show that the proposed method has acceptable accuracy under cyclic loading compared with the layered model with and without reinforcement. In the practical example s of steel and concrete bridge, the proposed stress resultant model
112 also shows acceptable accuracy and computational efficiency compared with the layered model. In C hapter 4 the frame element based on the force method derived in C hapter 2 and the stress resultant model for a plate proposed in C hapter 3 ar e applied to implicit nonlinear dynamic analysis including geometric and material nonlinearity. A steel rebar is also incorporated by the smeared layer and the integrated section methods for a concrete plate. S everal practical examples are tested and the v arious results of prop o sed method s are compared with those of the reference methods. From these studies it can be concluded that the proposed methods can reduce the cost of analysis of elasto plastic behavior for civil and building structures under earthq uake loadings
113 APPENDIX A EQUATION FOR FRAME ELEMENT BASED ON FORCE METHOD The detail equations for the frame element based on force interpolation functions are listed as follow (A 1) (A 2) (A 3)
114 (A 4) (A 5)
115 (A 6) (A 7)
116 (A 8)
117 APPENDIX B THE PLASTIC CONSISTENCY PARAMETER OF VON MISES CRITERION The detail equations for the pla stic consistency parameter based on the von Mises criterion , are listed as follow (B 1) (B 2) (B 3) (B 4)
118 (B 5) (B 6) (B 7)
119 (B 8) (B 9)
120 The equations , are same with the von Mises criterion. (B 10) (B 11)
121 APPENDIX C THE PLASTIC CONSISTENCY PARAMETER OF DRUCKER PRAGER CRITERION The detail equations for the plastic consistency parameter based on the Drucker Prager criterion , are listed as follow (C 1) (C 2) (C 3)
122 The equations , are same with the Drucker Prager criterion. (C 4) (C 5)
123 (C 6)
124 LIST OF REFERENCES  Crisfield, M.A. (1991) Nonlinear Finite Element Analysis of Solid and Structures. Vol. 1. John Wiley & Sons Ltd, New York  Crisfield, M.A. (1997) Nonlinear Finite E lement Analysis of Solid and Structures. Vol. 2 John Wiley & Sons Ltd, New York  Neuenhofer, A., & Filippou, F. C. (1998). Geometrically nonlinear flexibility based frame finite element. Journal of Structural Engineering 124 (6), 704 711.  Spacone, E., & El Tawil, S. (2004). Nonlinear Analysis of Steel Concrete Composite Structures: State of the Art. Journal of Structural Engineering, ASCE, 130(2), 159 168.  Spacone, E., Ciampi, V., & Filippou, F. C. (1996). Mixed formulation of nonlinear beam fi nite element. Computers & structures 58 (1), 71 83.  Bathe, K. J. (1996) Finite element procedures, Prentice Hall, Eglewoodcliffs, N.J.  Spacone, E., Ciampi, V., & Filippou, F. C. A beam element for seismic damage analysis. EERC Report 92/08 Earthq uake Engineering Center, University of California, Berkeley, CA(1992)  Takeda, T. Sozen, M.A., and Nielsen, N. (1970). Reinforced concrete response to simulated earthquakes. J. Struct. Div., ASCE 96(12), 2557 2573  Hilmy, S. I., & Abel, J. F. (1985) A strain hardening concentrated plasticity model for nonlinear dynamic analysis of steel buildings. Proc., NUMETA85, Numerical Methods in Engineering, Theory and Applications 1 303 314.  Hajjar, J. F., and Gourley, B. C. (1997). A cyclic nonlinear model for concrete filled tubes. I: Formulation J. Struct. Eng., 123(6), 736 744.  El Tawil, S., and Deierlein, G. G. (1999). Strength and ductility of concrete encased composite columns. J. Struct. Eng., 125(9), 1009 1019. [1 2 ] El Tawil, S., and Dei erlein, G. G. (2001a). Nonlinear analysis of mixed steel concrete moment frames. Part I: Beam coulomb element formulation. J. Struct. Eng., 127(6), 647 655. [1 3 ] El Tawil, S., and Deierlein, G. G. (2001). Nonlinear analysis of mixed steel concrete moment f rames. Part II: Implementation and verification. J. Struct. Eng., 127(6), 656 665, [1 4 ] Kent, D. C., and Park, R. Flexural members with confined concrete. (1971) J Struct. Div. ASCE, 97(7), 1969 1990.
125  Mander, J. B., Priestly M. N. J., and Park, R. ( 1988). Theoretical stress strain model for confined concrete. J. Struct. Eng., 114(8), 1805 1826.  Taylor, R. L., Filippou, F. C., Saritas, A., & Auricchio, F. (2003). A mixed finite element method for beam and frame problems. Computational Mechanics 31 (1), 192 203.  Iliushin, A.A. (1956). Plastichnost Gostekhizdat, Moscow (in Russian).  Crisfield, M.A. (1981). Finite element analysis for combined material and geometric nonlinearities. In W. Wunderlich et al. (eds). Nonlinear Finite Element A nalysis in Structural Mechanics, Springer Verlag, New York, pp. 325 338.  Shi, G., & Voyiadjis, G. Z. (1992). A simple non layered finite element for the elasto plastic analysis of shear flexible plates. International journal for numerical methods in e ngineering 33 (1), 85 99.  George Z. Voyiadjis & Pawel Woelke. (2008) Elasto Plastic and Damage Analysis of Plates and Shells. Springer  Bieniek and Funaro, 1976, Elasto plastic behavior of plates and shells. Technical report DNA 3584A, Weidlinge r Associates, New York  Frederick, C. O. and Armstrong, P. J. A Mathematical Representation of the Multiaxial Bauschinger Effect. Materials at High Temperatures, 2007; 24(1):1 26.  Sam Lee, 2008, Nonlinear Dynamic Earthquake Analysis of Skyscrape rs. CTBUH8th World Congress  Zhang, Y. X., & Bradford, M. A. (2007). Nonlinear analysis of moderately thick reinforced concrete slabs at elevated temperatures using a rectangular layered plate element with Timoshenko beam functions. Engineering Struc tures 29 (10), 2751 2761  Baskar, K., Shanmugam, N. E., & Thevendran, V. (2002). Finite element analysis of steel concrete composite plate girder. Journal of Structural Engineering 128 (9), 1158 1168.  O. C. Zienkiewicz and R.L. Taylor (1989) Ba sic formulation and linear problems In: The Finite Element Method, Vol 1, 4 th E dn McGraw Hill, London  Bonet, J., & Wood, R. D. (1997). Nonlinear continuum mechanics for finite element analysis. Cambridge university press.  Whitney, J. M. (1987). Structural analysis of laminated anisotropic plates. CRC Press.  Belytschko, T., Moran, B. & Liu, W. K. ( 1999). Nonlinear finite elements for continua and structures. John Wiley & Sons Ltd, New York
126  Lubliner, J. (2008). Plasticity theory Dover Publications.  MIDAS NFX manual. MIDAS Information and Technology Ltd., Seoul, Korea 2011.
127 BIOGRAPHICAL SKETCH Jinsang Chung graduated from the Hanyang University in 1994 with a Bachelor of Science in Civil Engineering and obtained a Master o f Science in Civil Engineering at the KAIST in 1996 He has worked as a FEA software developer in MIDASIT which has several commercial software He joined the University of Florida in 20 1 0 His research interests include: finite element methods, material a nd geometrical nonlinearity. He received his Ph.D. from the University of Florida in the s ummer of 2013.