The linearized three-dimensional beam theory of naturally curved and twisted beams: The strain vectors formulation

This paper presents the equations of the linearized geometrically exact three-dimensional beam theory of naturally curved and twisted beams. A new finite-element formulation for the linearized theory is proposed in which the strain vectors are the only unknown functions. The linear form of the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal, is enforced to be satisfied at chosen points. An arbitrary curved and twisted axis of the beam is taken into account which demands proper consideration of the non-linearity of spatial rotations. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by comparing present numerical results with various analytical and numerical results.