A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of spatial beam structures

This paper concerns a novel isogeometric Timoshenko beam formulation for a geometrically nonlinear analysis of spatial beams using the total Lagrangian description. Constitutive laws for hyperelastic materials, whose behavior varies with their deformations, are widely defined by using strain energy density functions that are written in terms of the Green–Lagrange strain tensor. Many finite element beam formulations for geometrically nonlinear analyses of spatial beams are developed using the Green–Lagrange strain tensor and its energy conjugate, the second Piola–Kirchhoff stress tensor. Unfortunately, there virtually exist no isogeometric Timoshenko beam formulations for this type of analysis that are derived by using this energy conjugate pair. To allow the possibility of considering hyperelastic materials, the present isogeometric beam formulation is developed in the total Lagrangian description using the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The proposed formulation is capable of simulating beam structures that are subjected to large displacements and rotations, without any restriction in magnitude. Three-dimensional beam configurations are reduced into one-dimensional structures using the beam axis and director vectors of the cross sections. The cross-sectional rotation along the beam axis is represented by an orthogonal tensor, which is parameterized by a vector-like parameter. Updating the cross-sectional rotations is performed purely by natural exponentiation and superposition of relevant rotational quantities. To show the accuracy and efficiency of the proposed beam formulation, some benchmark and well-established numerical examples with various types of beam, i.e., straight, curved, pre-twisted beams, are analyzed. The obtained results are compared with those results in the literature, obtained from both analytical and numerical methods.