Exact Time-Dependent Plane Stress Solutions for Elastic Beams: A Novel Approach

Abstract

We consider an elastically isotropic beam of narrow rectangular cross section governed by the dynamic equations of linearized plane stress theory and subject to typical boundary and initial conditions associated with flexure. We use one of the three stress-displacement relations to express the axial stress σx in terms of the axial displacement U and the normal stress σ. Assuming this latter stress and the shear stress τ to be given functions of position (x, z) and time t, we write the remaining two stress-displacement equations as a nonhomogeneous hyperbolic system for U and the normal displacement W. This system has a simple, explicit solution in terms of σ, τ, and V, the value of W on the centerline of the beam. Introducing certain body forces fx and fz, we obtain explicit formulas for σ, τ, U, and W valid in the interior of the beam and satisfying any imposed tractions on the faces of the beam. We satisfy initial conditions by adding certain explicitly computable increments to the initial displacements and velocities. Satisfaction of end conditions of displacement or traction yields a certain consistency condition along the centerline in edge zones (“boundary layers”) of width vH, where v is Poisson’s ratio and 2H is the depth of the beam. In particular, if V is taken as a solution of the equations of elementary beam theory, then outside these end zones the body forces fx and fz and the incremental initial conditions are “small.” If V within the edge zones is also identified with the solution of elementary beam theory, then a certain increment of the order of the dominant longitudinal stress σx must be added within the edge zones to the prescribed value of T on the face of the beam. (This is consistent with the neglect of two-dimensional end effects in elementary beam theory.) These results should be of use as analytical benchmarks for checking numerical codes.