Issue 50

K. Meftah et alii, Frattura ed Integrità Strutturale, 50 (2019) 276-285; DOI: 10.3221/IGF-ESIS.50.23 277 method (EFGM) was initially applied to fracture problems and subsequently applied to 2D problems and to 3D problems [7, 8]. On the other hand, the incremental cyclic plasticity theory is recently used firstly to determine the actual stress and strain state arising in two-dimensional or axi-symmetric notched components and later is extended by Marangon et al. [9] and Campagnolo et al. [10] to study the three-dimensional effects at the tip of rounded notches in plates of finite thickness. The first order shear deformation theories (FSDTs), which include transverse shear deformation, for bending plates have been initially proposed by Reissner [11] and further developed by Mindlin [12]. These theories are widely employed in the nonlinear elasto-plastic behavior. In this study, a finite element method for analyzing the elasto-plastic plate bending problems is presented. The previous Q4  [13] plate element with transverse energy of shearing is extended to account for nonlinear elasto-plastic. The goal of this work is to present a transverse energy four-node element Q4  with only four corner nodes which is significantly superior to the classical four-node element and is not computationally as expensive as a quadratic quadrangle 9-noded Hétérosis element. Having a finite element method for linear elastic analysis of Reissner-Mindlin bending plates, we further develop the model to investigate the elasto-plastic behavior and plastic zone of the structures under consideration. A modified Newton-Raphson method has been used to solve the non-linear equations. Von-Mises yield criterion has been adopted to deal with yielding of the materials along with the isotropic hardening. A computer program has been developed and a number of plate-bending problems have been solved. As the applications of the present element, the square plates with the various boundary conditions are calculated. The results have been compared with existing benchmark solutions. Results obtained with the Q4  plate element, with the adopted constitutive laws, are compared with those provided by the quadratic quadrangle 9-noded Hétérosis element presented in References [1, 14] with the provision of selective integration and reduced integration. All the computations were carried out in FORTRAN Finite Element code developed by Owen and Hinton [1] and Hinton and Owen [14]. R EISSNER -M INDLIN PLATE THEORY he Reissner-Mindlin plate theory (also designated first order shear deformation theory, FSDT) is more adequate for the analysis of moderately thick plates. The sign convention for stress resultants, the displacement field and the coordinate system are indicated in Fig. 1. General notation is M x , M y bending moments, M xy twisting moments, V x , V y transverse shear forces, w deflection in z -direction and x  , y  rotations of the xz - and yz -planes, respectively. Figure 1 : Schematic of the Reissner-Mindlin plate indicating the sign convention chosen for forces and moments. The displacement field at any point within the element is given by:       , , , x y u z x y v z x y w w x y             (1) The flexural and transverse shear strains in the plate for isotropic homogeneous linear behavior elastic can be written in the concise matrix form as: T