Non-linear dynamic analysis of beams with variable stiffness

In this paper the analog equation method (AEM), a BEM-based method, is employed to the non-linear dynamic analysis of a Bernoulli–Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe non-linear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the dynamic equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled non-linear hyperbolic differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious time-dependent load distributions. A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Beams with constant and varying stiffness are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.