Approximate methods for contact problems involving beams

A refined beam model is developed by solving the exact elasticity problem of a free layer with a sinusoidal load on the top surface and then expanding the amplitude of the resulting displacement in powers of the dimensionless wavenumber. The truncated series then implies that the beam deflection approximately satisfies an ordinary differential equation of the type developed by Kerr in the context of more general elastic foundation models, but the expansion process provides a convenient criterion for determining the range of length scales in which the ODE can be expected to be accurate. The method is illustrated for the problem of a beam loaded only by contact with an indenting body and/or with end supports – in particular, for the case of two contacting cantilever beams, for which the analytical results show good agreement with a finite element solution, except for the asymptotic region near edge of contact.