ON SHEAR DEFORMABLE BEAM THEORIES: THE FREQUENCY AND NORMAL MODE EQUATIONS OF THE HOMOGENEOUS ORTHOTROPIC BICKFORD BEAM

Abstract

This paper provides a step forwards the construction and documentation of the frequency equations and the characteristic functions of a general three-degrees-of-freedom theory that describes the plane motion of shear deformable elastic beams. The governing equations of this shear deformable beam theory (G3DOFBT) involve a general shape function of the transverse beam co-ordinate parameter, the a posteriori choice of which specifies the distribution of the transverse shear strain or stress along the beam thickness. Different choices of this shape function produce, as particular cases, the corresponding governing equations of different beam theories. These include the differential equations of the Euler–Bernoulli beam theory as well as the corresponding equations of the shear deformable theories due to Timoshenko and Bickford. Other examples can also be found by considering the shear deformable beam theories produced as one-dimensional versions of relevant refined plate theories. Since corresponding developments of the Timoshenko beam theory are already available in the literature, the Bickford theory is considered as the pilot beam theory in this study. The frequency equations, the characteristic functions and the orthogonality conditions of this theory, which assumes a through-thickness parabolic distribution of the transverse shear strain or stress, are constructed analytically for all the classical sets of boundary conditions applied at the beam ends. Some preliminary numerical results are also presented and discussed for beams having both their ends simply supported or clamped.