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A finite element model has been introduced for static bending analysis of thick plates based on a mixed plate variational formulation. A refined Reissner's mixed variational theoy is employed to derive the governing equations, in terms of the introduced transverse normal stress and displacement variables. The in-plane displacement components of the plate are described by a combination of polynomial and exponential terms. Concerning the transverse displacement component, a first-order polynomial is adopted. A second-order expansion is considered for the variations of the transverse normal component of the stress tensor along the thickness direction of the plate. The boundary conditions of shear and normal tractions on the top and bottom surfaces of the plate are exactly satisfied. Based on the proposed mixed plate theory, a four nodded compatible Hermitian rectangular element which ensures C‌1-type continuity of all unknown parameters of the plate along in-plane directions is employed. An arbitrary free parameter, called the splitting factor, appears in the functional of the proposed variational formulation. In the numerical part of the present paper, a simple formulation has been proposed for selecting the splitting factor which leads to the results of higher precision. Comparison of present bending results for thin and thick plates with results of the three-dimensional theory of elasticity and other plate theories available in literature reveals efficiency of the proposed parametrized mixed plate theory. Moreover, the proposed model has a high convergence rate and is computationally low cost.

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In the new sheet metal forming process as incremental sheet forming and spinning forming, this is not perfectly true in Marciniak-Kuczyinski model to assume that sheet deformation occurs in the plane-stress state indispose there are normal compressive stress and through-thickness stress. In this type of forming processes, the obtained limit strains refer to improving the sheet forming. However, in researches the effects of through-thickness shear stresses, also known as out-of-plane shear, has been studied less. The generalized forming limit diagram is a great curve that includes all six components of the stress tensor. In this paper, the effect of normal comprehensive and through-thickness shear stresses on the limit strain AA6011 aluminum sheet using a modified M-K and the anisotropic Yield function, Hill 48 and by using numerical solutions of nonlinear equations, Newton-Raphson method. The first the forming limit diagram was drawn with the assumption that the through-thickness shear stresses and then the effects of normal comprehensive stress and through-thickness shear stress on the limit strains were proved and the generalized forming limit curves were obtained. The results show that forming limits can be increased significantly by both normal compressive stress and through-thickness shear stresses. Also, the effects of normal stress on increasing the formability of sheet compared with the effects of through-thickness shear stress is greater.