Bounds for the Shear Center Coordinates of Prismatic Beams

Abstract

The problem of combined torsion and flexure of a cantilever beam is considered. A recent formulation of this problem by Reissner as a three-dimensional mixed boundary value problem where the end of the cantilever is subjected to prescribed translational displacements and a rigid body rotation leads to an unambiguous definition of the center of shear. The coincidence of the centers of shear and of twist follows from the reciprocity relations of linear elasticity. The coordinates of these centers can be expressed in terms of the stiffness coefficients relating the prescribed displacements and rotation to the transverse forces and the torque required. Through use of the principles of minimum potential energy and of minimum complementary energy, upper and lower bounds are obtained for these stiffness coefficients and from this for the coordinates of the shear center. The cross section of the beam is assumed to be arbitrary except having a single axis of symmetry. The effects of material orthotropy and of finite length appear in our potential energy bound. In the case of long beams calculation of the bounds involves the determination of the St. Venant warping function for the section. The results show that when the ratio of the shear modulus to the geometric mean of the Young's moduli tends to zero or when the Poisson's ratio tends to zero, the upper as well as the lower bound approach the shear center location in accordance with the definition given by Trefftz. Numerical results are obtained for the problems of semicircular and equilateral triangular sections.