Bending relationships between the modified couple stress-based functionally graded Timoshenko beams and homogeneous Bernoulli–Euler beams

摘要

The Bernoulli–Euler and Timoshenko beam theories are reformulated using a modified couple stress theory and through-thickness power-law variation of a twoconstituent material [functionally graded material (FGM)]. The model contains a material length scale parameter that can capture the size effect in a FGM. The equations are then used to develop algebraic relationships for the deflections, slopes, stress resultants of the Timoshenko beam theory (TBT) for microstructure-dependent FGM beams in terms of the same quantities of the conventional Bernoulli–Euler beam theory (BET). The relationships allow determination of the solutions of the TBT for microstructure-dependent FGM beams whenever solutions based on the BET are available. Examples of the use of the relationships are presented using straight beams with simply supported and clamped boundary conditions.