A boundary condition correction for the clamped constraint of elastic plate/beam theory

Abstract Elastic plate/beam theory is used by mechanical and civil engineers to model a variety of structures which are flat in two/one dimension(s) and thin in the other direction(s). If the plate is clamped at an end by being sandwiched between two rigid surfaces, it is customary to model this constraint (i.e. boundary condition) as a rigid constraint which does not permit either displacement or rotation at that edge. Here it is shown that an elastic plate/beam, with an end perfectly bonded at its top and bottom surfaces to a rigid constraint, deflects an additional amount due to elastic rotation effects within the support. It is shown that this correction can be accommodated with a simple modified boundary condition to standard plate theory without the need for a more complicated theory which includes a warping function. The analysis presented here is valid for plate bending theory but is equally valid for beam theory with minor modification. The finite rotational stiffness within the support is determined by applying a distributed load (statically equivalent to a couple moment) to such a constrained infinite elastic strip. The displacement profile is determined analytically and approximated by a straight line. The ratio of the applied moment to the slope of this line provides the rotational stiffness. Examples are given for a cantilever and a clamped-clamped plate/beam as well as for a clamped circular plate. For a beam we use Timoshenko beam theory, while for a plate Reissner-Mindlin is used. For each of these cases the correction due to the effect of rotational compliance is often greater than the correction due to shear deformation. Finite element analyzes, using two-dimensional elasticity, of these configurations show excellent agreement with the analytical results of this modified plate/beam theory.