Non-standard boundary conditions in classical plane elasticity

The old problem of the proper modelling of displacements boundary conditions at built-in ends of beams is readdressed within the framework of linear plane elasticity. Focusing on the cantilever loaded by an end force, as an example, we first review some common methods of describing the geometrical constraint at the fixed end. A new, non-standard, boundary condition is introduced next. The idea is to make the displacements at the 'rigid' boundary as small as possible by imposing simple least squares integral conditions on the displacements. Comparison of the distortion profiles at the built-in edge, obtained from the various sets of boundary data, points in favour of the least squares conditions. Further support is gained by making contact with Timoshenko's classical formula for the effect of shearing stresses on the deflection of the beam. This turns out to be almost identical with the prediction obtained with the new boundary conditions.