ELASTIC sinja, and a is the angle at which the curve cuts the line of action of the applied forces. Unless the length of the rod exceeds tt V(EI/W) it will not bend under the force, but when the length is great enough there may be more than two points of inflexion and more than one bay of the curve ; for n bays (n + 1 inflexions) the length must exceed nir ^/(EI/W). Some of the forms of the curve are shown in Fig. 14. ior the iorm d, in which two bays make a figure of eight we have ° ’ L v/(W/EI) = 4'6, a = 130° approximately (see Hess, Math. Ann. xxiii., 1884). It is noteworthy that whenever the length and force admit of a sinuous form, such as a or b, with more than two inflexions, there is also possible a crossed form, like e, with two inflexions only; it is probable that the latter form is stable and the former unstable. 14. The particular case of the above for which a is very small is a curve of sines of small amplitude, and the result in this case has been applied to the problem of the buckling of struts under thrust.1 When the strut, of length I/, is maintained upright at its lower end, and loaded at its upper end, it is simply compressed, unless L'2W > ^7r2EI; for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflexions is also Fig. 14 vertical, and thus the length must be half of one bay (Fig. 15, a). For greater lengths or loads the strut tends to bend or buckle under the load ; for a very slight excess of L'2W above ^7t2EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place when the length or the force exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. § 10). The condition of buckling, corresponding to the above, for a long strut, of length L', when both ends are free to turn is L'2W>7t2EI ; for the central line forms a complete bay (Fig. 15, b); if both ends are maintained in the same Fig. 15. vertical line, the condition is L'2W > 47t2EI, the central line forming a complete bay and two half bays (Fig. 15, c). 15. In our consideration of flexure it has so far been supposed that the bending takes place in a principal plane. We may remove this restriction by resolving the forces that tend to produce bending into systems of forces acting in the two principal planes. To each plane there corresponds a particular flexural rigidity, and the systems of forces in the two planes give rise to independent systems of stress, strain, and displacement, which must be superposed in order to obtain the actual state. Applying this process to the problem of §§ 2-8, and supposing that
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one principal axis of a cross-section at its centroid makes an angle 6 with the vertical, then for any shape of section the neutral surface or locus of unextended fibres cuts the section in a line DD', which is conjugate to the vertical diameter CP with respect to any ellipse of inertia of the section. The central line is bent into a plane curve which is not in a vertical plane, but is in a plane through the line CY which is perpendicular to DD' (Fig. 16). 16. Bending and Twisting of Thin Bods.—When a very thin rod or wire is bent and twisted by applied forces, the forces on any part of it limited by a normal section are balanced by the stresses across the section, and these stresses are statically equivalent to certain forces and couples at the centroid of the section; we shall call them the stress-resultants and the stress-couples. The stresscouples consist of two flexural couples in the two principal
planes, and the torsional couple about the tangent to the central line. The torsional couple is the product of the torsional rigidity and the twist produced; the torsional rigidity is exactly the same as for a straight rod of the same material and section twisted without bending, as in Saint-Yenant’s torsion problem (Elasticity, Ency. Brit. vol. vii. p. 812). The twist r is connected with the deformation of the wire in this way: if we suppose a very small ring which fits the cross-section of the wire to be provided with a pointer in the direction of one principal axis, and to move along the wire with velocity v, the pointer will rotate about the central line with angular velocity tv. The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section • the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only. The stress-resultants across any section are shearing stresses in the two principal planes, and a tension or thrust along the central line; when the stress-couples and the applied forces are known these stress-resultants are determinate. The existence in particular of the resultant tension or thrust parallel to the central line does not imply sensible extension or contraction of the central filament, and the tension per unit area of the cross-section to which S. III. - 93
