Analytical Finite Element for Timoshenko Beams

To improve the calculation accuracy and efficiency of structural force and deformation of deep beams, the deflection control equation of a deep beam was built by Timoshenko beam theory, and analytical displacement shape functions for deflection, section flexural angle and shear angle of deep beam were constructed. Then, potential energy functions for the beam model were established using the potential energy principle; analytical element formulations for beams and the total element stiffness matrix were obtained via the variational principle of potential energy stationary value. Finally, the proposed analytical finite element method was applied to calculate the end deflections of a cantilever deep beam and a simply supported deep beam; and the calculation results were compared with those by theoretical solution and interpolation polynomial method. The results show that the solutions of end deflection and rotation obtained from the proposed analytical element by one element number is in accordance with the theoretical solutions; the maximum relative error between the results calculated from interpolation shape function method and the theoretical solution is 19.785%. To verify the influence of shear deformation on the deflection, the proposed element was also compared with the Euler beam element. The comparison results show that, for cantilever beams subjected to distributed load, the relative error between the results calculated from the Euler beam theory and the proposed element derived by the Timoshenko beam theory is 50%. For simply supported beams subjected to a concentrated bending moment at the end, the relative error is 10.769%. It is proved that the proposed analytical beam element can satisfy the high accuracy and efficiency requirement and avoid shear locking problems.