Buckling of a Timoshenko beam bonded to an elastic half-plane: Effects of sharp and smooth beam edges

The problem of a compressed Timoshenko beam of finite length in frictionless and bilateral contact with an elastic half-plane is investigated here. The problem formulation leads to an integro-differential equation which can be transformed into an algebraic system by expanding the rotation of the beam cross sections in series of Chebyshev polynomials. An eigenvalue problem is then obtained, whose solution provides the buckling loads of the beam and, in turn, the corresponding buckling mode shapes. Beams with sharp or smooth edges are considered in detail, founding relevant differences. In particular, it is proofed that beams with smooth edges cannot exhibit a rigid-body buckling mode. A characteristic value of the stiffness ratio dimensionless parameter has been found for sharp edges, under which without loss of reliability, an analytic buckling load formula is provided. Finally, in agreement with the Galin solution for the rigid flat punch on a half-plane, a simple relation between the half-plane elastic modulus and the Winkler soil constant is found. Thus, a straightforward formula predicting the buckling loads of high stiff beams resting on elastic compliant substrates is proposed.