Proc Indian Natn Sci Acad 79 No. 4 December 2013 Spl. Issue, Part A, pp. 563-574 Printed in India. DOI: 10.16943/ptinsa/2013/v79i4/47993 Research Paper Elasto-Plastic Behaviour of Thick Plates with a Higher-Order Shear Deformation Theory T K A N T 1* , R K TRIPATHI2 and V RODE3 1Departmentof Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India 2Departmentof Civil Engineering, National Institute of Technology, Raipur 492 010, India 3D e p a r t m e n t of Civil Engineering, Shri G S Institute of Technology and Science, Indore 452 003, India (Received 18 February 2013; Revised 21 May 2013; Accepted 25 May 2013) A Higher Order Shear Deformation Theory (HOST) is utilized for elasto-plastic analysis of plate bending using incremental finite element formulation. Modified Newton-Raphson method has been used to solve the non-linear equations. Yielding of the material has been modelled using von Mises yield criterion, associated flow rule and isotropic hardening. Results are compared with available benchmark and other solutions. Comparisons clearly demonstrate better performance of the theory for inelastic response. Key Words: Plate-Bending; Higher Order Theory; Elasto-Plastic Response; Incremental Finite Element Method Introduction can be classified on the basis of the assumed field as (i) stress-based theories and (ii) displacement-based Bending behaviour of plates is usually described by theories. Reissner [2, 4] and Mindlin [3] are the two three mathematical models: Poisson-Kirchhoff plate pioneers to provide first order shear deformation theory [1], Reissner-Mindlin first order plate theories theories based on the assumed stress and assumed [2-4] and higher order plate theories [5-9]. Poisson- displacement variations through the thickness of the Kirchhoff theory, which is also known as C l a s s i c a l plate, respectively. These theories provide a first-order Plate Theory (CPT), is based on assumption that basis for the consideration of the transverse shear straight lines perpendicular to mid-surface (i.e., deformation effect and required a C o continuous finite transverse normals) before deformation remain element formulation for the numerical analysis. straight and normal to the mid-surface and undergo However, FOSTs have following limitation: no change in length during deformation. These assumptions imply that the transverse normal strain The transverse shearing strains/stresses turn out z and transverse shear strains and are zero. It can to be constant through the plate thickness and a also be easily shown that CPT is computationally fictitious shear correction coefficient is inefficient (requires C1 continuity of transverse introduced to correct the shear energy. displacement) from the simple finite element Lo et al. [5] have presented a theory for plates formulation’s point of view. based on assumed higher-order displacement field. The first order shear deformation theories Kant [6] has derived an isotropic version of the (FOSTs), which include transverse shear deformation, complete governing equations of such a theory based * Author for Correspondence: E-mails: [email protected], [email protected], [email protected] 564 Tarun Kant et al. on the minimum potential energy principle. A Co finite rectangular plates and therefore the von Mises yield element formulation of this higher order theory is criterion was used for thin plates of perfectly plastic presented, for the first time, by Kant et al. [7]. In this material and presented the bounds for a simply theory, the in-plane displacements are expanded in supported square plate under uniform load as a special the powers of the thickness coordinate (z) by Taylor case. Hodge and Belytschko [15] formulated upper series, which allows: and lower bound problem as a mathematical programming problem, but by using finite element Quadratic variation of the transverse shearing representations of velocity and moment fields insuring strains through the plate thickness avoiding the the satisfaction of yield condition throughout the plate. introduction of a shear correction coefficient. The plastic limit loads of the clamped circular plate Warping of the cross-section is automatically have been presented by Guowel et al. [16] based on incorporated. unified yield criterion. When the plate is loaded beyond elastic limit, The limit analysis is applicable to rigid perfectly the plastic strain occurs, which causes a redistribution plastic behaviour while almost all the materials are of stresses. The computation of this redistribution was elastic before yielding. The current state of stress in not an easy task before digital computers arrived. a yielded material again depends upon the history of Hence modelling of plastic behaviour started as limit loading. Therefore, an analysis starting from the analysis and then switched over to incremental non- loading in the elastic range and gradually increasing linear analysis. Initially for a long span of time, limit to plastic range till failure would be closer to the true analysis was used for the plastic analysis of beams behaviour of the plate. and frames due to their linear idealisation. Simplicity Haythornthwaite [17] was the first to use an of their formulation leads to the collapse loads. Plates, incremental method for the elasto-plastic analysis of due to their two-dimensional idealisation, make the plates. He computed the deflections of plates with problem complicated and in most of the cases lead to circular symmetry for an elastic-plastic material. the upper and lower bounds to the collapse loads Comparison was made with predicted limit loads of instead of collapse loads themselves. The Hopkins and Prager[12]. Armen et al.[18] developed axisymmetrically loaded, circular plates were the a finite element technique for plastic bending by simplest problem to begin with, and therefore attracted additionally considering in-plane stresses and the attention of many researchers. Sokolovsky [10] geometric nonlinearity. They developed the formulation was considered to be the pioneer amongst them. He by interpreting plastic strains as initial strains and using dealt with the non-linear bending of simply supported stress-strain relations from incremental plasticity circular plates with small displacements and thin plate theory. Material behaviour was based on von Mises assumptions. He used the Hencky’s deformation yield criterion with Prager-Zieglar kinematic theory for a strain-hardening material obeying von hardening. Dinis et al. [19] considered the semiloof Mises yield criterion. Pell and Prager [11], Hopkins shell element for elasto-plastic large displacement and Prager[12] and Hopkins and Wang[13] predicted analysis of plates and shells. An initial stiffness the load-carrying capacities of circular plates of a plasticity algorithm and elasto-plastic model was perfectly plastic material and subjected to rotationally considered for the material non-linear analysis. symmetric loads for various support conditions. The initial works were confined to the circular plate case. Owen and Figueiras[20] extended the application Hodge [14] took the research out of this barrier by of semiloof element analysis to include the effect of presenting the procedure of finding the bounds of non- transverse shear by deriving a shear correction circular plates. The previous researches mostly used coefficient for cylindrical bending with the assumption Tresca yield criterion since the directions of principal of constant transverse shear strains. Reddy and moments were already known for circular plates due Chandrashekhara [21] derived a non-linear version to radial symmetry. This was not the case with the of Sander’s first-order shear-deformation shell theory Elasto-Plastic Behaviour of Thick Plates with a Higher-Order Shear Deformation Theory 565 accounting for von Karmann’s strains and Hill’s u(,,)(,)(,) x y z z x y z3 * x y anisotropic yield criterion for the elasto-plastic bending x x v(,,)(,)(,) x y z z x y z3 * x y analysis of plates and shells. y y (1) w(,,)(,) x y z w0 x y Watanabe and Kondo [22] further consolidated their earlier (Kondo and Watanabe [23]) work by The terms have usual meaning except the terms * * considering more numerical examples and better x and y which are the corresponding higher order comparisons. They derived the elastic-plastic terms in the Taylor’s series expansion. The transverse incremental tangent stiffness matrix without numerical displacement component ‘w’ has only one term ‘w0’ integration over the area of the elements using the and thus disregards transverse normal deformation. nodal displacements and nodal forces as fundamental Strain-Displacement Relationships variables instead of generalised strains and stresses respectively. The linear relationships between these displacements and strains can be obtained by using the definitions of Papadopoulos and Taylor [24] developed an strains from the theory of elasticity: inelastic finite element analysis of plates, which includes the effects of transverse shear-deformation. u 3 * They used the Reissner-Mindlin’s plate bending theory z z x x x x and an elasto-plastic constitutive model in the stress- v 3 * z z resultant form based on generalisation of Von Mises y y y y 0 plane stress criterion and associated flow rule. The z u v 3 * plastic constitutive rate equations were integrated by z z xyy x xy xy (2) a return-mapping algorithm and solved incrementally v w 2 * z using a Newton method. Prasad and