Geometrically nonlinear Euler–Bernoulli and Timoshenko micropolar beam theories

Two ways of incorporating moderate rotations of planes normal to the axis of a straight beam into the Euler–Bernoulli and the Timoshenko micropolar beam theories are presented. In the first case, the von Kármán nonlinear strains are used to incorporate the moderate rotations of normal planes into the beam theories. In the second case, appropriate approximations are made on the nonlinear Cosserat deformation gradient to reflect the condition of moderate rotations of the normal planes. The governing nonlinear differential equations and corresponding natural boundary conditions in both cases are derived using the principle of virtual displacements. A weak-form Galerkin displacement finite element formulation is presented for the developed nonlinear beam theories. The phenomenon of locking usually encountered in beam displacement finite elements is eliminated using higher-order finite elements with nodes located at spectral points. Finally, numerical examples are presented to illustrate the effect of coupling number and bending characteristic length scale on deflections and microrotations when a micropolar beam is modeled with the developed nonlinear beam theories.