Nonlinear analysis of functionally graded microstructure-dependent beams

Nonlinear finite element models of functionally graded beams considering the von Kármán geometric nonlinearity, power-law variation of material through the beam height, and microstructure length scale parameter are developed for the Euler–Bernoulli beam theory and the Timoshenko beam theory. To capture the size effect, a modified couple stress theory with one length scale parameter is used. Such theories play crucial role in predicting accurate deflections of micro- and nano-beam structures. Numerical results are presented to show the effect of nonlinearity, shear deformation, power-law index, microstructural length scale, and boundary conditions on the bending response of beams under mechanical loads. In general, the effect of microstructural parameter is to stiffen the beam, while shear deformation has the effect of modeling realistically as flexible beams.