The distribution of shearing stresses in a beam is mathematically investigated for two forms of cross-sections, namely, (1) a section bounded by two confocal ellipses. and (2) a sector of a circle. From Saint-Venant's flexure functions for these sections, the solutions of which are given in Appendix I, the shearing stress components are obtained. To illustrate the distribution of shearing stresses, those on the neutral axes and on certain boundaries are worked out, and their distributions are shown in diagrams. It is pointed out that, at a sharp corner projecting outwards in the cross-section of a beam, the shearing stress is zero, and, at the end of a crack and near the corner of a sharp re-entrant in the cross-section, the shearing stresses become very great. But, if the sharp corner is on the vertical symmetrical axis of the section, the shearing stress at the corner of such a re-entrant is zero, and, if the crack is along the vertical symmetrical axis, the distribution of the shearing stresses on the section is not influenced by the presence of such a crack. It is also noticed that, for a thin hollow elliptic cylindrical beam, the resultant shearing stress at any point on the cross-section is in the direction of the tangent at that point to the wall of the beam and can be obtained by the usual practical formula given by the equation, q=Wm/2tI, where W is the vertical load at the free end, t the normal thickness of the wall of the beam, and, m and I have their usual significance. The shearing stress components for a circular section and for a hollow circular section are given in Appendix II, and it is shown that the shearing stress on the neutral axis at the side of a very small circular cavity at the centre of a circular cross-section is twice the shearing stress at the corresponding point in a solid circular section of the same diameter.
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