GALERKIN METHOD FOR LATERAL BUCKLING SOLUTIONS OF SIMPLY SUPPORTED BEAM UNDER POINT LOAD AT MIDSPAN

Charles Chinwuba Ike

Abstract


This work studies Galerkin method for lateral buckling solutions of a simply supported beam under point load at midspan.The problem is a boundary value problem (BVP) governed by Prandtl-Michell equations. Closed form solutions involve complicated Bessel functions. The Galerkin’s method converts the BVP upon use of truncated series of linearly independent shape functions to a finite dimensional space. Shape functions that apriori satisfy all the boundary conditions are used to express the BVP as an eigenvalue problem. A general multi-parameter shape function formulation is presented. Linear algebra methods are used to find nontrivial solutions to the set of homogeneous algebraic equations. The obtained solutions are the eigenvalues from which the least eigenvalue is used to find the critical buckling load. A one-parameter shape function gave critical buckling load Qcr which is 1.3506% greater than the exact critical buckling load Qcr(exact). A two-parameter shape function gave Qcr which is 0.0553% greater than Qcr(exact). A three-parameter shape function gave Qcr(exact). The results show a rapid convergence of Qcr to the exact solution because exact shape functions that satisfy all the boundary conditions are used. The study has demonstrated the rapid convergence of the Galerkin method solution to the exact solution especially when exact shape functions are used in formulating the stability matrices. The rapid convergence of the Galerkin solution to the exact solution is due to the exact buckling shape function used in formulating the Galerkin equations..

 

KEYWORDS: Characteristic lateral buckling equation, critical lateral buckling load, lateral buckling, Galerkin method, Prandtl-Michell equation.

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