In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem in beams of arbitrary simply or multiply connected constant cross section. The shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. A boundary value problem is formulated with respect to a warping function and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of this function, while the coordinates of the shear center are obtained from this function using only boundary integration. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one, using again only boundary integration. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows.
The proposed displacement solution constitutes the first step to the solution of the non uniform shear problem avoiding the use of stress functions.
All basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal axes.
The shear deformation coefficients are evaluated using an energy approach instead of Timoshenko’s and Cowper’s definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values.
The present formulation is also applicable to multiple connected domains without fulfillment of further constraints.
The developed procedure retains the advantages of a BEM solution over a pure domain discretization method since it requires only boundary discretization. Numerical examples illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of both the thin tube theory and the engineering beam theory is examined through examples with great practical interest.
Prismatic beam of an arbitrary cross-section occupying the 2-D region Ω.