CRITICAL BUCKLING LOAD SOLUTION OF THIN BEAM ON WINKLER FOUNDATION VIA POLYNOMIAL SHAPE FUNCTION IN STODOLA-VIANELLO ITERATION METHOD

Abstract

The critical buckling load analysis of thin beam on Winkler foundation (BoWF) is crucial for BoWF subjected to in-plane compressive load. This paper presents the Stodola-Vianello method for the approximate solution of the governing ordinary differential equation (ODE). The Stodola-Vianello method expresses the ODE in iteration form after four successive integrations have been used in reformulating the equation. The problem then reduces to iteration in algebra. By deriving algebraic buckling shape functions that satisfy the boundary conditions, and substitution into the Stodola-Vianello iteration formula derived successive iterations of the buckling shape function would yield better approximations of the buckling shape. The existence condition for convergence which is the identity of the nth and (n +1) iterations are used to find the eigenvalue from which the buckling load is determined. The candidate problem solved is a BoWF with Dirichlet boundary conditions. It is found that one iteration yields sufficiently accurate buckling loads which differs from the exact solution by 0.129% for  0.0807% for  and 0.0567% for  where bl⁴ is a parameter measuring the beam structure – Winkler foundation interaction. The negligible difference in the critical buckling load found is because an approximate algebraic one-parameter shape function was used in the iteration. The paper has demonstrated the effectiveness of the Stodola-Vianello iteration for solving the stability of thin beam on Winkler foundation.