De Saint-Venant solution for the flexure of cantilevers of cross-section in the form of complete and curtate circular sectors, and on the influence of the manner of fixing the built-in end of the cantilever on its deflection

(1) Hitherto, as far as we are aware, the well-known de Saint-Venant solution of the problem of flexure has been limited to relatively few cases and in all these the cross-section of the beam possessedbiaxial symmetry, there being an axis of symmetry in the plane of loading and also an axis perpendicular to this plane. De Saint-Venant’s view that his solution could be directly applied to asymmetrical sections arises from a misconception. We have recently found it possible to extend the de Saint-Venant solution to certain cases in which there is only one axis of symmetry which may be (1) perpendicular to the plane of loading, or (2) in the plane of loading. The former case has been dealt with by Young, Elderton and Pearson in a memoir published last year. The flexure in this case is accompanied by torsion, and the problem is not without some bearing on the torsion of aircraft propeller blades, for which, indeed, it was worked out. The latter case for the same sections as were dealt with in that paper was then seen to be solvable, the axis of symmetry being now in and not perpendicular to the plane of loading. As far as we are aware no such cases of flexure with uni-axial symmetry of the cross-section have yet been published, although it is clear that a great variety of sections in actual use fall under this category. For the particular section we have dealt with in this paper, that of a “trough” or “gutter-pipe” section, represented for mathematical purposes by a curtate circular annulus loaded in its plane of symmetry, we found it needful to introduce a term of the from C logr in addition to the ordinary Fourier solution. We than obtainedtwo equations for determining C, one from either curved surface of the annulus, and were checked in the course of our investigation. We are very grateful to Mr. W. M. Macaulay of King’s College, Cambridge, for dispelling our difficulty by showing that the two equations could after some reduction be shown to be identical.