INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 11, 875-881 (1977) zyx ELASTO-PLASTIC FINITE ELEMENT ANALYSIS I. KALEV zyxwvu AND J. GLUCK zyxwv Department of Structures, Faculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa, Israel SUMMARY An elasto-plastic finite element small displacement procedure to predict the behaviour of structures under cyclic loads is presented. The solution combines incremental procedure with correction for equilibrium and iterative scheme. Applications are made for a rectangular strip and a notched bar under cyclic axial loads. Flow theory zyxwvutsrq pf plasticity with either isotropic hardening or kinematic linear hardening is used. INTRODUCTION The equation of equilibrium for the material non-linear analysis used in the finite element small displacements formulation may be written as follows: [K"]. {AU}={AP}+[KP] zyxw . {AU}+{R} (1) where [K"] is the structural elastic stiffness matrix, {AU} is the unknown incremental nodal displacement vector, {AP} is the incremental applied load vector, [KP] is the plastic stiffness matrix which is a function of the instantaneous stress state, and {R} is the vector of equilibrium corrections due to the nature of the solution procedure. Numerous procedures have been used for solution of the non-linear equation (1). An extensive list of references on the subject can be found in References 1-3. In all these procedures the loads are applied in small steps which correlates with the incremental flow theory of plasticity. The first class of solution procedure is that in which the equilibrium correction vector {R} is kept smaller than some prescribed value. It is obtained by repeated iterations, for each load step, using either successive approximations, the Newton-Raphson or the modified Newton- Raphson techniques. It was shown in Reference 3 that the Newton-Raphson procedure fails to converge in many cases when unloading exists. The second class of solution procedure is the incremental procedure in which the equilibrium corrections vector zyxwvut {I?} is excluded from equation (1) or is treated as residual force vector for the next load step. A frequently used incremental procedure is the Euler forward difference method, commonly referred to as the tangent modulus method. The term [KP]. {AU} in equation (1) is transformed to the left-hand side of the equation and {A v) is directly solved. It requires evaluation and 'inversion' of a new stiffness matrix zyxwv ([K"] - [KP]) at each load step, which results in an increase in computer cost. However, relatively large load steps can be used. It was pointed out in Reference 4 that this procedure is not applicable in strain softening or non-associated plasticity situations when the stiffness matrix ceases to be positive definite. In the present paper a solution procedure that combines 'linear' iterations with incremental procedure and equilibrium corrections are used. It requires only a single 'inversion' of the elastic stiffness matrix and the iterations are performed on the equilibrium equation without altering Received 9 April 1975 RevisedZFebruary 1976and 15June 1976 @ 1977byJohnWiley&Sons,Ltd. zyxwvutsrqp 815