FROM CONSISTENT VARIATION OF LEVINSON’S HIGHER ORDER BEAM THEORY TO HIGHER ORDER WARPING BEAM THEORY

A new beam theory which includes shear deformation is presented. By assuming the warping displacement of the rectangular cross-section in the form of 3 order Legendre polynomial function, the governing differential equations, which are simple and can be solved easily, was derived using incomplete generalized Hellinger-Reissner variational principle of sub-item in the scope of elasticity theory for plane stress problems. In the new theory, the nonzero shear stress boundary conditions on the lateral surfaces of the beam are strictly satisfied, and it enables analysis of the beams in a unified manner that incorporates tensile force. With definition of a distance between reference points of warping constraint, the warping boundary conditions at the ends of the beam is well-defined, and the theory covering high order shearing beam theory being or not being consistent with variational sense is available. It is also shown that the variational inconsistence of Levinsion’s beam theory is only restricted at the ends of the beam with rotation constraint. Examples compare the accuracy of the new beam theory with that of elasticity theory, Timoshenko beam theory and Levinson’s theory. The comparison shows the merit of the new theory.