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

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 θ = ∑_i=1^N I_i θ_i for the rotation field and u = ∑_i=1^N+1 J_i 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 u = ∑_i=1^N I_i [ u_i + 1/N ( θ - θ_i ) × R_i ] , where R i are the beam nodal positions.