Analysis of laminated general shells undergoing finite rotations and large motion

Abstract

The primary objective of this dissertation is the development of a modeling tool, using the finite element method, for static and dynamic analysis of general refined laminated shell structures undergoing finite rotations and large motion with strains assumed to remain small.

A kinematic model based on the material representation is presented leading to a third order shear deformation theory with large rotation capabilities and quadratic transverse shear stress distribution across the thickness. A singularity-free parametrization of the rotation field is adopted with it the exponential mapping for configuration update. A Total Lagrangian formulation is used with the second PiolaKirchhoff stresses, Green-Lagrange strains and constitutive equations defined with respect to laminate with general curvilinear coordinates. The developed shell element is composed of an arbitrary number of layers where the fiber directions are allowed to vary in any way from layer to layer.

The finite element discretization is carried out using a four-node isoparametric laminated shell element with seven degrees of freedom per node. The transverse shear locking problem is avoided by applying the Assumed Natural Strain concept to the constant part of the transverse shear strain. A consistent linearization of the weak form of equilibrium equations (static case) or equations of motion (dynamic case) is undertaken to achieve a quadratic rate of convergence.

The dynamic part consists of designing and implementing an energy-momentum conserving time stepping algorithm. This algorithm is based on a general methodology for the design of exact energymomentum schemes, which was recently proposed in the literature and applied successfully to nonlinear shells based on the first order shear deformation theory. Here it is extended, for the first time, to the third order shear deformation theory.

The developed finite rotation shell element is then implemented in two independent computer programs, one for static and the other for dynamic analysis. Then it is tested on some challenging linear and nonlinear problems, recently reported in the literature, and the results show its excellent performance and robustness. A couple of examples show the discrepancy in prediction between between third and first order shear deformation theories and this raises the need for such refined theories.