Relationships between bending solutions of classical and shear deformation beam theories

The exact relationships between the deflections; slopes/rotations, shear forces and bending moments of a third-order beam theory, and those of the Euler-Bernoulli theory and the Timoshenko beam theory are developed. The relationships enable one to obtain the solutions of the third-order beam theory from any known Euler-Bernoulli or Timoshenko beam theory solutions of beams, for any set of boundary conditions and transverse loads. The relationships can also be used to develop finite element models of the Timoshenko and third-order beam theories, and determine numerical solutions from the finite element model of the Euler-Bernoulli beam theory. The finite element models are free of the shear locking that is found in the conventional shear deformable finite elements