738
ELASTIC
it would be equivalent is small compared with the tensions and pressures in longitudinal filaments not passing through the centroid of the section; the moments of the latter tensions and pressures constitute the flexural couples. 17. We consider, in particular, the case of a naturally straight spring or rod of circular section, radius c, and of homogeneous isotropic material. The torsional rigidity is ^E7tc4/(1 + cr); and the flexural rigidity, which is the same for all planes through the central line, is ^Ettc4 ; we shall denote these by C and A respectively. The rod may be held bent by suitable forces into a curve of double curvature with an amount of twist r, and then the torsional couple is Cr, and the flexural couple in the osculating plane is A/p, where p is the radius of circular curvature. Among the curves in which the rod can be held by forces and couples applied at its ends only, one is a circular helix; and then the applied forces and couples are equivalent to a wrench about the axis of the helix. Let a be the angle and r the radius of the helix, so that p is r sec2a ; and let E, and K be the force and couple of the wrench (Fig. 17). Then the couple formed by R and an equal and opposite force at •any section and the couple K are equivalent to the torsional and flexural couples at the section, and this gives the equations for R and K . sin a cos2 a cos a It — A Ct r > r*.» K = AC^ + Cr sina. r The thrust across any section is R sin a parallel to the tangent to the helix, and the shearing stressresultant is R cos a at right angles to the osculating plane. When the twist is such that, if the rod were simply unbent, it would also be untwisted, r is sin a cos a/r, and then, restoring the values of A and C, we have 4 cr R = Ettc 2 4r 1 + cr sm a cos^ a, 4 cr cos2 acos a. Tr R = Ettc — 1 +— 4r 1 + cr 18. The theory of spiral springs affords an application of these results. The stress-couples called into play when a naturally helical spring (a, r) is held in the form of a helix (a, r'), are equal to the differences between those called into play when a straight rod of the same material and section is held in the first form, and those called into play when it is held in the second form. Thus the torsional couple is ^ /sin a' cos a' sin a cos a ^ C V r' r )’ and the flexural couple is 2 ./cos a' cos2a A V r' r )' The wrench (R, K) along the axis by which the spring can be held in the form (a', r') is given by the equations „K A sin a' (cos2 a! cos2 ct ^ cos a Y sin a' cos a! sin aT cos a ^ = vq-7—rd-°—i ? ~ ~ ,/cos2 a' cos2a „ . ,/sm ^ cos‘F sin a cos a K = A cos a ^ — J + Csma J-
SYSTEMS 4crFaT cos a Etc3 ’ the sense of the rotation being such that the spring becomes more tightly coiled. 19. A horizontal pointer attached to a vertical spiral spring would be made to rotate by loading the spring, and the angle through which it turns might be used to measure the load, at any rate, when the load is not too great; but a much more sensitive contrivance is the twisted strip devised by Ayrton and Perry. A very thin, narrow rectangular strip of metal is given a permanent twist about its longitudinal middle line, and a pointer is attached to it at right angles to this line. When the strip is subjected to longitudinal tension the pointer rotates through a considerable angle. Bryan (Phil. Mag., December 1890) has succeeded in constructing a theory of the action of the strip, according to which it is regarded as a strip of plating in the form of a right helicoid, which, after extension of the middle line, becomes a portion of a slightly different helicoid; on account of the thinness of the strip, the change of curvature of the surface is considerable, even when the extension is small, and the pointer turns with the generators of the helicoid. Taking b for the breadth and t for the thickness of the strip, and r for the permanent twist, the approximate formula for the angle 6 through which the strip is untwisted on the application of a load W was found to be W&t(1 + <t)
The quantity br which occurs in the formula is the total twist in a length of the strip equal to its breadth, and this will generally be very small; if it is small of the same3 order as £/&, or a higher order, the formula becomes iW&r(l + <r)/E< , with sufficient approximation, and this result appears to be in agreement with observations of the behaviour of such strips. 20. General Theorems.—Passing now from these questions of flexure and torsion, we consider some results that can be deduced from the general equations of equilibrium of an elastic solid body. The form of the general expression for the potential energy (Elasticity, Ency. Brit. vol. vii. p. 823) stored up in the strained body leads, by a general property of quadratic fungtions, to a reciprocal theorem relating to the effects produced in the body by two different systems of forces, viz.: The whole work done by the forces of the first system acting over the displacements produced by the forces of the second system is equal to the whole work done by the forces of the second system acting over the displacements produced by the forces of the first system. By a suitable choice of the second system of forces the average values of the component stresses and strains produced by given forces, considered as constituting the first system, can be obtained, even when the distribution of the stress and strain cannot be determined. Taking for example the problem presented 1by an isotropic body of any form pressed between two parallel planes distant l apart (Fig. 18), and denoting the resultant pressure by p, the diminution of volume — bv is given by the equation -dv = lplSJc> Fig. 18. where Jc is the modulus of compression, equal to 4E/(1 - 2<r). Again, taking the problem of the changes produced in a heavy body by different ways of supporting it,
When the spring is slightly extended by an axial force F, = - R, and there is no couple, so that K vanishes, and a', r', differ very little from a, r, it follows from these equations that the axial elongation, 5*, is connected with the axial length x and the force F by the equation j, Ettc2 4 sin a 2 Sx ~ if 1+cr cos a x ’ 1 The line joining the points of contact must be normal to the and that the loaded end is rotated about the axis of the helix i planes. through a small angle
