By A. E. H. LOVE
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Additional resources for A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY
When an end is free When an end is fixed the T, N, and G have to vanish there. constraint which fixes taking up is fixed. it may be different directions or such as to allow of the extremity it may be such that the direction In the former case the end will be described as simply supported, in the latter as built-in. BENDING OF RODS IN ONE PLANE. 28 Horizontal 215. Rod very little [215 bent. We shall consider in the first place the very important case of a straight rod resting on rigid supports at the same level, and slightly bent by vertical forces supposed to act in a principal In this case Ijp plane.
BENDING OF RODS IN ONE PLANE. 30 [217 If ^ be the angle which the plane perpendicular to the plane of the curve makes with the principal plane in which the displacement is 2/1, we have tan tan (9 =- B^/B^ = - I,/I^, where /j, I^ are the two principal moments of inertia of the normal section at its centroid, so that this plane and the vertical plane cut the normal section in lines which are conjugate diameters of the ellipse of inertia. This and Bresse given in I. art. 108, is the theorem of Saint-Venant In the remaining problems that will be here considered the plane of flexure will be assumed to be a principal plane of the rod.
G. elastic central-line is a principal plane. Stress System. 213. Suppose that the rod is held bent in a principal plane by forces and couples suitably applied, and consider the stress across any normal section. Let P be any point on the strained elastic The central-line, and suppose the normal section at P drawn. actions of the particles on one side of this section upon the particles on the other side can be reduced to a force and couple at P. The T along the tangent force can be resolved into two components : to the strained elastic central-line at P, and N along the normal same line drawn inwards towards the centre of curvature.