Abstract
In this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time.
Highlights
Shell elements which account for the layer sequence of a laminated structure are able to predict the deformation behaviour of the reference surface in an accurate way
In several publications the equilibrium equations are exploited within post-processing procedures to obtain the interlaminar stresses, e.g. [1,2] for the transverse shear stresses and e.g. [3] for the thickness normal stresses
The purpose of this paper is to present a shell model which is able to compute the load-deflection behaviour and the complicated three-dimensional stress-state of geometrical nonlinear, elasto-plastic, layered shells
Summary
Shell elements which account for the layer sequence of a laminated structure are able to predict the deformation behaviour of the reference surface in an accurate way. The purpose of this paper is to present a shell model which is able to compute the load-deflection behaviour and the complicated three-dimensional stress-state of geometrical nonlinear, elasto-plastic, layered shells. (ii) The proposed nonlinear variational formulation leads to Euler-Lagrange equations which include besides the usual shell equations in terms of stress resultants, the local equilibrium in terms of stresses, a constraint which enforces the correct shape of the displacement fluctuations through the thickness, the geometric field equations and the constitutive equations. Quadratic functions allow the computation of second derivatives These are necessary when rewriting the local equilibrium equation for the thickness direction with the derivatives of the transverse shear strains. We compare with the displacements and stresses computed with Reissner–Mindlin shell elements
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