Buckling mode localisation in randomly disordered continuous beams using a simplified model

Buckling mode localisation in large randomly disordered one-dimensional structures is studied. Furstenbergʹs theorem on the limiting behaviour of the product of random matrices is employed to determine the Lyapunov exponent and the localisation factor. Greenʹs function formulation is applied to show that although the buckling loads are different for different sample structures, the buckling loads satisfy a probability distribution which depends only on the disorder parameters and is independent of the specific sample realisations for large structures. Due to the positivity of the Lyapunov exponent it is found that localised modes (bulges) may be visible for an arbitrary value of load close to the buckling loads if there exist perturbations or imperfections.