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

Abstract

The problem of a compressed Timoshenko beam of finite length in frictionless and bilateral contact with an elastic half-plane is investigated here. A Chebyshev series solution is found and, for some limiting cases, an analytic form solution is provided. 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 shown that beams with smooth edges cannot exhibit a rigid-body buckling mode. A limit value of the soil compliance is found for beam with sharp edges, below which an analytic buckling load formula is provided without loss of reliability. 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 stiff beams resting on compliant substrates is proposed.

2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force

“…Although the behavior of a finite layer differs from that of an elastic half-plane, in some circumstances a preliminary solution can be obtained based on the 2D Boussinesq solution, owing to the local character of the problem. For instance, the stress analysis and the buckling problem of a Timoshenko beam in contact with an elastic support have been recently performed based on the Green's function for an elastic half-plane (Falope, Lanzoni, & Radi, 2020;. Proper adjustments to extend the 2D Boussinesq solution to layers of finite height can be handled based, for example, on strain energy equivalence arguments (Poulos & Davis, 1974).…”

A loaded beam in full frictionless contact with a couple stress elastic half-plane: Effects of non-standard contact conditions

“…Then, they found the series coefficient after the imposition of the strain compatibility condition along the contact zone by using a numerical collocation procedure. The analysis was also extended to the problem of a shear deformable beam in contact with a classical elastic half-plane by Lanzoni and Radi (2016) and to the beam buckling problem by Falope et al (2020).…”

Third-Order Theory for the Bending Analysis of Laminated Thin and Thick Plates Including the Strain Gradient Effect

“…These aspects have been highlighted by so-called multiscale analysis [ 38 , 39 , 40 , 41 ]. In these circumstances, structural theories based on classical elasticity could turn out to be inadequate to model such innovative mediums [ 42 , 43 , 44 , 45 , 46 , 47 ]. Nonlocal theories have been developed to overcome these issues, as illustrated in the first papers by Eringen [ 48 , 49 ].…”