Nonlinear Dynamic Analysis of Structures Having Different Moduli in Tension and Compression

Abstract

ynamic analysis of structures composed of materials having different moduli in tension and compression is presented in this work. Linearised dynamic analysis of bimodular transversely loaded members with zero axial force is presented, first. The formulae governing the neutral surface position and normal stress are derived. The position of the neutral surface was rendered independent of the spatial and temporal coordinates by introducing a special assumption which reduced the coupled nonlinear problem of the flexure of such a member into a linear one. The actual position then became a function of section geometry and the two elastic moduli and was determined by the equivalent section method. The elemental dynamic stiffness matrix is derived based on the exact displacement function governed by the governing partial differential equation and the structural stiffness matrix is formulated according to fundamental structural mechanics. Symbolic and numerical examples were solved to show the error incurred when the same modulus theory is still used. Also, Nonlinear dynamic analysis of a an Euler-Bernoulli beam-column member is presented. The cubic equation governing the neutral surface position of such a member is derived based on equilibrium conditions of the dynamic moments and applied axial forces. The perturbation solution of the cubic equation is used for determining the neutral surface position of a beam-column member. Especially, for a beam member, the cubic equation governing the neutral surface position reduces to a quadratic equation, the solution of which is exactly consistent with the linearised dynamic analysis presented earlier in this thesis. For a beam-column member, the combined action of the dynamic moments and axial forces makes the neutral surface position dependent on the external transverse dynamic loading as well as inertia forces a matter that yields a nonlinear integro-partial differential equation governing the flexural vibration of a bimodular beam-column member. This equation is solved by using the finite difference method with the aid of Maple 12 symbolic mathematics package for accomplishing the solution of this unique equation. A numerical study (through examples) was conducted on such members revealing, among other findings, a curious phenomenon, termed intrinsic damping herein. The origin of this phenomenon and other results are illustrated and commented upon.