EXACT TIME-DEPENDENT PLANE-STRESS SOLUTIONS FOR ELASTIC BEAMS - A NOVEL-APPROACH

We consider an elastically isotropic beam of narrow rectangular cross section governed by the dynamic equations of linearized plane stress t heory and subject to typical boundary and initial conditions associate d with flexure. We use one of the three stress-displacement relations to express the axial stress sigma(x) in terms of the axial displacemen t U and the normal stress sigma. Assuming this latter stress and the s hear stress tau to be given functions of position (x, z) and time t, w e write the remaining two stress-displacement equations as a nonhomoge neous hyperbolic system for U and the normal displacement W. This syst em has a simple, explicit solution in terms of sigma, tau, and V, the value of W on the centerline of the beam introducing certain body forc es f(x) and f(z), we obtain explicit formulas for sigma, tau, U, and W valid in the interior of the beam and satisfying any imposed traction s on the faces of the beam. We satisfy initial conditions by adding ce rtain explicitly computable increments to the initial displacements an d velocities. Satisfaction of end conditions of displacement or tracti on yields a certain consistency condition along the centerline in edge zones (''boundary layers'') of width root nu H, where nu 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 t hese end zones the body forces f(x) and f(z) and the incremental initi al conditions are ''small.'' If V within the edge zones is also identi fied with the solution of elementary beam theory, then a certain incre ment of the order of the dominant longitudinal stress sigma(x) must be added within the edge zones to the prescribed value of tau on the fac e of the beam. (This is consistent with the neglect of two-dimensional end effects in elementary beam theory.) These results should be of us e as analytical benchmarks for checking numerical codes.