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

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 I _i and J _i 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 R _i are the beam nodal positions.