An analysis of an Euler-Bernoulli beam-column with arbitratry initial crookedness by transfer matrix methods

Abstract

The effects of initial crookedness, often neglected in the analysis of structural members under axial as well as transverse loads, are considered. The transfer matrix method is chosen as an approach to the problem because it provides a way to analyze both simple and complex systems with relative ease. It is assumed that the initial crookedness can be discribed by a mathematical function. This function is incorporated into the basic beam equations, which are then used in the derivation of a transfer matrix for a beam-column. Only the extension column of the obtained transfer matrix contains information relating to the initial crookedness. General formulas for the elements of the extension column are given. These formulas are used to obtain extension columns for the case where the initial crookedness is described by an infinite sine series. A very specific half-sine initial crookedness with eccentric-end loads is shown to collapse to the well known secand solution. Implementation of these transfer matrices must be completed with extreme care since the general transfer matrix method allows arbitrary placement of transverse loads and/or supports. This paper outlines the matrix manipulation procedures that are to be used to implement these new matrices in the presence of in-span loads or manipulation procedures that are to be used to implement these new matrices in the presence of in-span loads or supports.