Analysis of 3D Timoshenko frames having geometrical and material nonlinearities

• A novel force-based finite element for analysis of 3D frames is presented.
• Both material and geometric nonlinear behavior are considered in this study.
• The formulations are based on the two-field Hellinger–Reissner functional.
• Shear-locking problem is removed in this formulation.
• This element applicable for structures with arbitrarily large rigid body displacements and rotations.

A three-dimensional Timoshenko beam is formulated based on the force method and the Hellinger–Reissner variational principle. Material nonlinearities are taken into account by the fiber discretization scheme. The element kinematic is adopted such that makes it appropriate for large displacement analysis, but deformations can only be moderately large. Like the other force-based elements, the proposed beam is first formulated in a basic coordinate system without rigid body motions. Then, the resulted flexibility matrix and resisting force vector are transformed to the global coordinates using a corotational scheme. This finite element analysis is locking-free, because of the exact force interpolation functions. In addition, a 3D extension of the “Curvature and Shear Based Displacement Interpolation” or CSBDI, proposed by Jafari et al., is presented in the paper [10]. To facilitate the numerical implementation of the element, a clear algorithm for state determination of the element is presented. Various numerical examples evaluate the accuracy and efficiency of the suggested element.