An asymptotic analysis of composite beams with kinematically corrected end effects

Abstract

A finite element-based beam analysis for anisotropic beams with arbitrary-shaped cross-sections is developed with the aid of a formal asymptotic expansion method. From the equilibrium equations of the linear three-dimensional (3D) elasticity, a set of the microscopic 2D and macroscopic 1D equations are systematically derived by introducing the virtual work concept. Displacements at each order are split into two parts, such as fundamental and warping solutions. First we seek the warping solutions via the microscopic 2D cross-sectional analyses that will be smeared into the macroscopic 1D beam equations. The variations of fundamental solutions enable us to formulate the macroscopic 1D beam problems. By introducing the orthogonality of asymptotic displacements to six beam fundamental solutions, the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method. The boundary conditions obtained herein are applied to composite beams with solid and thin-walled cross-sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method-based beam analysis (FAMBA) presented in this paper. The numerical results are compared to those reported in literature as well as 3D FEM solutions.

An asymptotic analysis of anisotropic heterogeneous plates with consideration of end effects

“…This actually yields the same form as the orthogonality condition of asymptotic displacements to the fundamental displacement [Kim et al 2008]. The displacement condition given in (55) was proven to be asymptotically correct up to ᏻ( 2 ) for a transversely isotropic semiinfinite beam [Horgan and Simmonds 1991].…”

“…where the 0th order displacement needs special attention, because it is related to the asymptotic convergence [Buannic and Cartraud 2001;Kim et al 2008]. Each order displacement is given by…”

“…Its solution can be easily found by σ (0) = ε (0) = 0 since it is well posed [Buannic and Cartraud 2001;Kim et al 2008]. From ε (0) = 0 the particular solution is obtained by…”

“…One can solve (22) by applying the orthogonality condition to a rigid body displacement [Cesnik et al 1996;Kim et al 2008]. Consequently its solution is represented bȳ…”

“…where the matrix K I is related to the inverse matrix of K and the orthogonality condition to a rigid body displacement [Kim et al 2008]. Note that each column of (1) represents the warping distribution through the thickness of a plate, which corresponds to six warping functions due to two in-plane extensions, one in-plane shear, two bending curvatures, and one twisting curvature.…”