Single-variable formulations and isogeometric discretizations for shear deformable beams

We present numerical formulations of Timoshenko beams with only one unknown, the bending displacement, and it is shown that all variables of the beam problem can be expressed in terms of it and its derivatives. We develop strong and weak forms of the problem. The strong form of the problem involves the fourth derivative of the bending displacement, whereas the symmetric weak form involves, somewhat surprisingly, third and second derivatives. Based on these, we develop isogeometric collocation and Galerkin formulations, that are completely locking-free and involve only half the degrees of freedom compared to standard Timoshenko beam formulations. Several numerical tests are presented to demonstrate the performance of the proposed formulations.