EXACT ANALYTIC SOLUTION FOR LATERAL BUCKLING OF THICK CANTILEVER BEAM UNDER A TIP CONCENTRATED LOAD

Abstract

This work presents a mathematical method for solving the lateral buckling (LB) problem of a thick cantilever beam under a tip concentrated load. Lateral buckling problems are governed by Prandtl-Michell equation, a homogeneous ordinary differential equation (ODE) of the second order with variable parameters. The variable parameters are introduced by the bending moment distribution, which in many LB problems would vary along the longitudinal coordinate of the beam. The field equation for LB problem of cantilever beam under tip load is obtained from substitution of the bending moment expression into the Prandtl-Michell LB equation as an ODE with variable parameters. Variable parameters ODEs are difficult to solve mathematically using most common techniques for solving ODEs. However, the field equation obtained for the cantilever LB problem is of Bessel type and methods for solving Bessel equations are used for the problems. The general solution is thus obtained in terms of two Bessel functions of the first kind of order 1/4 and -1/4 and two constants of integration. The boundary conditions are enforced to reduce the problem to an eigenvalue problem for non trivial solutions. The characteristic LB equation obtained is found to be a transcendental equation in terms of the Bessel function of the first kind and of order 1/4. The least eigenvalue is found using methods of computational software and used to find the critical LB load Q_cr, the critical LB bending moment M_cr and the eigenfunction. It is found that the solutions for Q_cr and M_cr are identical with solutions in the literature obtained using analytical and approximate methods. The solutions for the critical LB load Q_cr and the critical LB bending moment M_cr are exact since they were found by requiring that the governing Prandtl-Michell equation be satisfied at all points on the domain as well as on the boundaries.