A new beam element for analysis of planar large deflection

Abstract

In the present study, a simple and efficient finite element approach is presented for large deflection analysis of both the straight and curved Euler–Bernoulli beams in the planar static problems. The linear stress–strain relationship is assumed for the Euler–Bernoulli beam. In some of finite element methods, displacements together with rotations, and in some others, positions are considered as the main fields of interpolation. However, in the present study, the main idea for the interpolation is using the dimensions of the deformed element instead of the displacements. Therefore, the slope angle (like the previous works) and the length of beam centroidal axis (unlike the previous works) are used as the main field parameters. This treatment creates simplicity in the constitutive equations. Next, using the weighted residual method, the constitutive equations are applied to the element. Using the equilibrium equations and kinematic equations, the position coordinates of the nodes are related to the internal forces and the main field parameters. In the present study, three-node element and, consequently, Simpson’s 1/3 rule is used for integration. For solving nonlinear equations of the beam, the Newton–Raphson method is used. Finally, several numerical examples are presented and compared with the previous works to illustrate the validity and efficiency of the new element.