A novel application of the FPM to the buckling differential equation of non-uniform beams

Abstract

In this study the buckling load ( P ) of non-uniform, deterministic and stochastically heterogeneous beams, is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering the unknown P and the transverse deflection ( W ) as functionals of heterogeneity, i.e., the elastic bending stiffness “ K ” (or the compliance S = 1 / K ). The BDE is expanded functionally, yielding a set of successive differential equations for each order of the (Frèchet) functional derivatives of P and W . The obtained differential equations differ only in their RHS, and therefore a single modified Green function is needed for solving all orders. Consequently, an approximated value for the buckling load is obtained for any given morphology. Four examples of simply supported columns are solved and discussed. In the first three, deterministic realizations of K are considered, whereas in the fourth, K is assumed to be the stochastic field. The results are compared with solutions found in the literature for validation.