p-Approximation beam elements for elastostatics and elastodynamics with isotropic, generally orthotropic and laminated composite materials

Abstract

This dissertation presents finite element formulations for two and three dimensional beam elements for elastostatic and elastodynamic analyses. The formulations possess hierarchical property, i.e., the element approximation functions and the nodal variables corresponding to the lower order polynomials are a subset of those corresponding to the higher order polynomials. The hierarchical interpolation functions are derived directly from the Lagrange interpolating polynomials. The element formulation ensures continuity of displacements across the inter-element boundaries, thus C$\sp\circ$ continuity is automatically guaranteed.

The element properties, i.e., element stiffness matrix, mass matrix and the equivalent nodal load vectors, are established by using the principle of virtual work and the hierarchical element approximation. Complete stress-strain states are retained in the element derivation, thus the beam element formulation can be used to simulate complex stresses and deformations at or near singularities. The element formulation is further extended to account for orthotropic and laminated composite materials. There are no restrictions on the number of laminas, their thicknesses, or the layup pattern. The lamina properties are incorporated in the element formulation through numerically integrating the element matrices and the equivalent load vectors for each lamina.

The geometries of the beam elements are defined by the coordinates of the nodes located on the axes of the elements and the node point vectors representing the beam nodal cross-sections. Since the node point vectors are not necessarily normal to the beam axes, the formulations permit non-differentiable geometries, i.e., sharp junctions and re-entrant corners where the two elements meet.

Numerical examples are provided to demonstrate the accuracy, simplicity of modelling, faster convergence rate, and the overall superiority of the present formulation for linear static and linear dynamic analyses with isotropic and laminated composite material properties. The linear dynamic characteristics of the element are demonstrated through eigen-solutions. The results obtained from the present formulation are compared with, where available, analytical solutions, and h-models using isoparametric elements.