Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites

Abstract

A new two-dimensional curved beam element formulation with higher-order transverse shear deformation for linear static analysis of laminated composites is presented. The displacement approximation in the transverse direction can be of arbitrary desired polynomial order p, thereby permitting strains of at least order ( p − 1). This is accomplished by introducing additional nodal degrees of freedom in the element displacement approximation that correspond to Lagrange interpolating polynomials in the transverse direction. The element has an important hierarchical property; the element stiffness matrix, generalized nodal displacement vector and the equivalent nodal load vector corresponding to a polynomial order p are a subset of those corresponding to the approximation order p + 1. The element formulation provides displacement continuity across the interelement boundaries, i.e. C 0 continuity is ensured automatically at the mating element boundaries. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is presented for generally orthotropic material behavior, where the material directions may not be parallel to the global axes, as well as for the laminated composites. The laminated composite material behavior is incorporated in the element formulation through numerical integration for each lamina. For each lamina, the material behavior may be generally orthotropic, and the material direction and lamina thicknesses may vary from point to point within the lamina. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation for laminated composite material behavior. Results obtained from the present formulation are also compared with the available analytical solutions.