A Layered Boundary Element Nonlinear Analysis of Beams

This work aims to introduce a new layered approach to the nonlinear analysis of initially straight Euler-Bernoulli beams by the Boundary Element Method (BEM). The beam is studied in the context of both geometrical and material nonlinearity. The governing differential equations, derived by applying the principle of minimum total potential energy, are coupled and nonlinear, while the boundary conditions are the most general and may include elastic support or restraint. The boundary value problem, regarding the axial and transverse displacements, is solved using the Analog Equation Method (AEM), a BEM based method, together with an iterative procedure. Although a direct solution to the geometrical nonlinear problem has already been presented, in this work an alternative layered analysis is proposed. The discretization is applied in both the longitudinal direction and the cross-sectional plane, and an iterative process is commenced. First, initial fictitious load distributions are assumed at beam's each cross-section, and the displacements, as well as their derivatives, are computed using the AEM. Second, the two stress resultants, i.e., the axial force and bending moment, are evaluated by appropriate integration over the cross-section. In the end, the derivatives of the stress resultants are evaluated, and the equilibrium of the governing equations is checked. If the equilibrium is satisfied, the process is terminated. Otherwise, the fictitious load distributions are updated, and the procedure starts over again. Several representative examples are studied, and the results are compared with those presented in the literature, validating the reliability and effectiveness of the proposed method

Vibration and buckling of thin-walled composite I-beams with arbitrary lay-ups under axial loads and end moments

A finite element model with seven degrees of freedom per node is developed to study vibration and buckling of thin-walled composite I-beams with arbitrary lay-ups under constant axial loads and equal end moments. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. Numerical results are obtained for thin-walled composite I-beams to investigate the effects of axial force, bending moment and fiber orientation on the buckling moments, natural frequencies, and corresponding vibration mode shapes as well as axial-moment-frequency interaction curves