Buckling of stochastically heterogeneous beams, using a functional perturbation method

The buckling load and its probabilistic nature (average and variance) of Bernoulli beams with stochastic material (bending stiffness) properties is derived analytically by a new functional perturbation method (FPM). A buckling shape function is assumed, based on the homogeneous solution and additional terms to account for the morphology effects. The buckling load in the transcendental equation is treated as a functional of the bending modulus (stiffness or compliance) field. Applying a functional perturbation to the above equation, the buckling load is found analytically to any desired degree of accuracy, as a function of material morphology. The FPM is executed using both stiffness and compliance statistical data. The impact of each of the two data sources on the solution accuracy is examined, showing that compliance based solutions are accurate for small correlation lengths. Statically indeterminate problems can be treated with no additional effort. An example of a simply supported beam is solved in detail. Comparison with previous studies, where stochastic finite element and Monte Carlo simulation were used, showed the relative accuracy and insight capabilities of the method. The clamped-free case is also studied to demonstrate that symmetry conditions, used for homogeneous beams to find the buckling load on the basis of a simply supported case, are not valid for heterogeneous beams.