Buckling of elastic beams embedded in granular media

In this paper, an experimental and theoretical study of the buckling response of slender elastic beams within granular media is performed. Buckling loads of beams with different flexural rigidity, length, and boundary conditions within granular media of different depths are determined. The Ritz approximate method is implemented to model the buckling response of the beams based on the concept of an overhanging beam on an elastic foundation, using a series of springs whose spring constants change linearly with respect to the depth of the grains. There is good agreement between the experimental results and the theoretical model. There is a characteristic penetration ratio where the beams are not able to sense the boundary condition at the embedded end, resulting in a convergence of the buckling loads. This condition happens when the rigidity of the beam is lower than the effective stiffness of granular support, leading to the confinement of the lower portion of the beam inside the grains, and acting as a secondary boundary condition that is independent of the condition at the end of the beam. We derive a scaling law to characterize this characteristic penetration ratio in terms of a dimensionless stiffness parameter, allowing for the characterization of three distinct interactions between the beam and medium based on the ratio of granular support effective stiffness to the beam's effective stiffness.