In his most important point Professor Stephan is completely correct. That is, his shear coefficient and mine are identical. I was aware of his work when it first came out and discounted it because it was derived by solving a static problem for a specific type of loading. It implied that the shear coefficient for the static problem was a function of the type of loading and, further, that by choosing the right type of static loading one could find the best value of the shear coefficient for dynamic loads. As it turns out both of these implications are correct.
Professor Stephan states, “The motivation of Hutchinson appears to be the construction of a theory in which the shear coefficient takes on the “best” value. I did not mean to convey that impression. My motivation was simply to construct a simple, consistent, dynamic theory which did not require guessing a shear coefficient. This simple consistent theory allowed me to find an expression for the shear coefficient in the Timoshenko beam theory. That this shear coefficient agreed with the “best” values simply validated my approach.
Professor Stephen states, “the values of the coefficient for the circular cross section, both solid, hollow and thin-walled, and for the elliptic cross section calculated in , are identical to those given in [3,4].” What he doesn’t note is that the expressions for the rectangular cross section in both his and my paper also produce identical results. He further comments on the fact that the shear coefficient goes to zero for the rectangular cross section. Actually, as shown in my Fig. 4, it is the reciprocal of the shear coefficient that goes to zero which means the shear coefficient would have a pole at that point. Professor Stephens conclusion that the beam is stiffened with an increase in width is correct. I do not understand, however, his remark that, “one would not normally employ Timoshenko theory for a beam having large width to depth ratio.” For a Poisson’s ratio of 0.3 the reciprocal of the shear coefficient goes to zero at a width to depth ratio of about 3. This is definitely within the range I would expect Timoshenko theory to be applied. Also the lowest natural frequency would probably occur (depending on boundary conditions) about a neutral axis in the width direction. I should probably also note that the shear coefficient is very sensitive to the assumed shear stress distribution. If one were to assume a simple parabolic shear stress distribution through the thickness of the rectangular beam one would get a shear stress of independent of the aspect ratio.
I recently presented a paper entitled “Shear Coefficients for Thin-Walled Timoshenko Beams” at the Third International Symposium on the Vibrations of Continuous Systems, July 23–27, 2001 at Jackson Lake Lodge, WY. In that paper I considered all the thin-walled cases treated by Cowper plus two additional cases. As in all other cases the “best” shear coefficient agreed with Cowper’s values only for Poisson’s ratio equal zero.
As to Professor Stephen’s final remark, “it is noted that while the above values for the coefficient may be widely accepted as best, paradoxically Cowper’s values appear to be more widely used; it is hoped the investigators will in the future make greater use of these “best” values,” I fully agree.