Consistent Nonlocal Integral and Gradient Formulations for Force-Based Timoshenko Elements with Material and Geometric Nonlinearities

Both integral and implicit gradient consistent nonlocal formulations are developed for a force-based beam element with material and geometric nonlinearities. The element is based on the Timoshenko beam theory, which accounts for shear deformations. Material nonlinearity is considered by using inelastic constitutive relationships, and geometric nonlinearity is considered by using the corotational formulation in the global system and a curvature-shear-based displacement interpolation (CSBDI) in the local system. Integration point dependency for strain-softening responses is addressed by using the section deformation as the nonlocal variable. The weak form of the implicit gradient-type governing equation is derived, and an efficient strategy is proposed to solve it. Consistent element flexibilities for both the integral and implicit gradient formulations are derived. To implement the proposed elements, a new and simplified state determination algorithm is developed. Finally, four illustrative numerical examples are presented to demonstrate the utility of proposed element and validate it. The results indicate that the proposed element can accurately capture both material and geometric nonlinearities, and offers consistent response predictions for any number of integration points due to its nonlocal regularization.