Nonlinear stress analysis of shell structures in buckling and snapping problems by exact geometry solid-shell elements through sampling surfaces formulation

Abstract

In this paper, the nonlinear three-dimensional (3D) stress analysis of shell structures in buckling and snapping problems is presented. The exact geometry or geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method. The SaS formulation is based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. The SaS are located at the Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree N), that is, the outer surfaces are not included into a set of SaS. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-the-thickness distributions of displacements, strains and stresses leads to an efficient higher-order shell formulation. The incremental equilibrium equations are solved by the Newton–Raphson method combined with the Crisfield arc-length algorithm. The tangent stiffness matrix is evaluated by effective 3D analytical integration. As a result, the proposed GeX hybrid-mixed solid-shell element exhibits superior performance for coarse meshes and allows much larger load increments than is possible with existing displacement-based solid-shell elements. This can be useful for the 3D stress analysis of thin and thick shells in different states such as pre-buckling, bifurcation and post-buckling.