Geometrically nonlinear formulation of 3D finite strain beam element with large rotations

Abstract The geometrically nonlinear formulation of three-dimensional (3D) curved beam elements with large rotations has recently aroused much interest. The element geometry is usually constructed using coordinates of the nodes of the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions. The element displacement field is described using three translations at the element nodes and three rotations about the local axes. These types of 3D beam element formulations are, however, restricted to small nodal rotations between two successive load increments. The beam element formulation presented in this paper removes such restrictions; this is accomplished by defining beam geometry using nodal displacements and tangent vectors instead of rotation angles at the two ends of the curved beam element. The fact that, unlike rotation angles, vectors can be added without difficulty, allows large rotations be made within a load increment. Removing all the transcendental trigonometrical cosine and sine functions, the formulation has the advantage of being simple and purely algebraic.