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In the engineering beam theory shear deformation was neglected. No unified beam theory compensating the inconsistency caused by this has been yet proposed. On the basis of St. Venant's theory, the authors have studied so far on the coupled shearing and torsional deformation in the beams with arbitrary cross-section, and showed that these problems become the boundary value problems governed by Poisson's equation. In this study, these problems of coupled shearing and torsional deformation were analyzed by using the boundary element method in which the matrix of shear rigidity and torsional rigidity was determined. This matrix indicates the coupled shearing and torsional deflection. The shear center can be obtained determining the coordinate axes so as to eliminate the nondiagonal terms.

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One of the problems in beam theory is the incompletion of the theory for shear deformation. The authors previously proposed the method analyzing the problems of coupling shear and torsion in beams having arbitrary cross-sectional forms using St. Venant's theory by the finite element method, and the stiffness matrix for space framework elements, taking shear deformation into account, was determined. However, since the restraint and warping connection at beam ends cannot be taken into account in order to approximate these generalization and extension of the warping rigidity of Wagner's equation, in which the mode of warping is left as it is, and the magnitude is axially changed was attempted.

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The elementary beam theory is based on the Bernoulli-Euler hypothesis, in which no shear deformation is allowed to occur excluding the case of torsional deformation. This theory involves some theory to be improved. We consider here a refined beam theory taking into account the effect of shear deformation. This theory may be applied to any cross section of beam using the finite element method. Next, we asked for the stiffness matrix of an element of space framework taking into account the effect of the shear deformation. Some numerical examples of the present theory are shown and compared with values obtained by other methods.

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