Dynamic analysis of the axially-loaded Timoshenko beam using quadratic matrix equations

Abstract

The purpose of this thesis is to obtain a power series expansion of the general stiffness matrix for a beam-column element. Solutions are obtained using the general Timoshenko beam theory including the combined effects of bending and shear stress, together with axial force and transverse and rotary inertia. The general stiffness matrix of this continuous, elastic, vibrating, structural element possesses components which contain complex hyperbolic and trigonometric functions. The idealization of this continuous system into a discrete element system is performed utilizing a power series expansion technique which produces a family of matrices containing algebraic components. The latter matrix form is extremely efficient for matrix computer operations. The matrix series expansion produces a two term expansion of elastic stiffness matrix, a two term expansion of the consistent mass matrix, and a two term expansion of the geometric stiffness matrix. In addition, all coupling matrices of the latter three parameters are obtained.