Analysis of Beam Vibrations by Hu‐Washizu Finite Elements

The utility of the Hu-Washizu variational principle as basis for finite elements is illustrated by application to vibrations of beams. Within the framework of Timoshenko beam theory a finite element is formed in which transverse displacement and cross-section rotation vary linearly with the spatial coordinate. The differential-difference equations for a uniform element-length model of a uniform beam are developed. Fundamental natural frequencies of simply-supported beams, as predicted by the spatially discretized model, are compared with those deduced from the partial differential equations of the Timoshenko theory. Agreement is excellent even when the beam is modeled by fewer than ten elements and when the element-length is significantly greater than the depth of the beam. Comparison is also made with the results of finite-element modeling where the strain-displacement equations are satisfied a priori. Not surprisingly this displacement-compatible model produces poor results, owing to the excess stiffness generated by the strain energy of shear in conjunction with the linearly varying displacements.