Effect of element thickness on the eigenvalues of beams

The sensitivities of eigenvalues to the change of element thickness have been calculated for beams with various end conditions. For each beam, the sensitivities along the beam length fluctuate more for higher modes and the number of maxima increases linearly with the mode number. The sensitivities are high and positive at clamped ends, negative at free ends, and very low near simply supported ends. For a cantilever beam there exist positions where the eigenvalue sensitivities to element thickness are zero, which means that the eigenvalues are not affected by the thickness at those points. For a simply supported beam the sensitivities are always positive for all modes.