Buckling of non-uniform beams by a direct functional perturbation method

This study is divided into two parts. In the first, the buckling load ( P ) of heterogeneous columns is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering 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 (Fréchet) 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. Both deterministic and stochastic examples of simply supported columns are solved and discussed. Results are compared with solutions found in the literature for validation. second part, the Optimized DFPM (ODFPM) is presented. It is based on finding a new material property (which is a function of K or S ) around which the DFPM solution is more accurate. The new material property is found by requiring that the second order perturbation term in the Fréchet series is minimized. As a result, a non-linear differential equation is obtained which relates the new material property with K through morphology. An exact solution for this equation is found, in a power form K f , where f depends on morphology. Calculating P with respect to this new property yields more accurate results for the statistical characteristics of P .