Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order

TLDR: For arbitrary polynomial loading and a sufficient finite number of nodal points N, the solution for the 3D Timoshenko beam differential equations was given in this paper, where Ii and Ji were the Lagrangian polynomials of order N−1 and N, respectively.

Abstract: For arbitrary polynomial loading and a sufficient finite number of nodal points N, the solution for the 3D Timoshenko beam differential equations is polynomial and given as \({{\varvec \theta} = \sum_{i=1}^N I_i {\varvec \theta}_i}\) for the rotation field and \({{\bf u} = \sum_{i=1}^{N+1} J_i {\bf u}_i}\) for the displacement field, where Ii and Ji are the Lagrangian polynomials of order N−1 and N, respectively. It has been demonstrated in this work that the exact solution for the displacement field may be also written in a number of alternative ways involving contributions of the nodal rotations including \({{\bf u} = \sum_{i=1}^N I_i \left[ {\bf u}_i + \frac 1 N ( {\varvec \theta} - {\varvec \theta}_i ) \times {\bf R}_i \right]}\), where Ri are the beam nodal positions.