Exact shape functions of imperfect beam element for stability analysis

In this paper, a versatile and efficient solution for the displacement function of a geometrical imperfect beam element under high axial load is derived. The initial imperfection is included by a sine curve approximation. The aim of this work is to propose a natural extension of the element by stability functions. For this purpose, exact displacement shape functions are solved from the differential equilibrium equation, by using arbitrary boundary conditions. These shape functions offer the possibility to express the equivalent nodal forces for any arbitrary transverse loads, and are further extended to obtain the equivalent nodal forces relating to the effect of the initial curvature without affecting the boundary conditions of the element. The accuracy of this element makes the convergence rate better than the conventional cubic element or other used elements. Euler’s critical load is computed readily with no more than two iterations for each step loading with a single element per member. By using these exact shape functions in computer software, the second-order analysis and the nonlinear integrated design become reliable and easy to use for practical purposes.