When beam theories fail

The free vibration of completely unrestrained prismatic rectangular parallelepipeds and the static response of planar elastic rectangles under transverse loads are examined to determine the relative accuracy of several widely used beam theories.The Euler-Bernoulli, Rayleigh, and Timoshenko theories are applied to both isotropic and orthotropic beams and compared with elasticity-based solutions.The resulting frequencies, energies, displacements, and stresses are compared with solutions to the linear theory of elasticity for solids with the same geometries and material constitutions.Sources of the resulting differences in response are identified, and guidelines are suggested.