TWO FORMULI OF THE SHEAR CENTER

Abstract

In literature there are essentially two different formuli for the coordinates of the shear center. One of them - which can be derived anaIitically from the solution of the Saint-Venant problems - contains two terms: one of these contains the torsion stress function, the other contains the warping function at twisting. The other formula was derived by Trefftz on the base of energetic considerations, and differs from the previous one in that the term containing the stress function is missing. Why do these formuli differ when both of them were derived from the analitical solution of the Saint-Venant problem? In the Saint-Venant problems the Saint-Venant principle has been applied, i.e. the distributed forces on the end-section z = l, have been replaced by a concentrate force and couple statically equivalent with it. In analytically deriving the formula of the shear center, only this statical equivalence is necessary. But Trefftz's conception implies the consideration, that the equivalence also holds for the energy i.e. the work done by the two, statically equivalent force systems is unchanged in the course or deformation. The paper deals with this problem, studies the rightfulness and conditions of using the energy and work theorems in the Saint-Venant problems. It verifies, that considering these conditions, Betti's theorem (the way suggested by Trefftz) gives the same formula for the shear center as the analytical solution. In addition it shows how to determine the coordinates of the shear center, when the origin of the system of coordinates in solving the boundary problem for the warping function is an arbitrary point of the cross section.