would also be untwisted, τ is (sin α cos α) / r, and then, restoring the values of A and C, we have
| R = | Eπc4 | σ | sin α cos2 α, | |
| 4r 2 | 1 + σ |
| K = | Eπc4 | 1 + σ cos2 α | cos α. | |
| 4r | 1 + σ |
65. The theory of spiral springs affords an application of these results. The stress-couples called into play when a naturally helical spring (α, r) is held in the form of a helix (α′, 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
| C ( | sin α′ cos α′ | − | sin α cos α | ). |
| r ′ | r |
and the flexural couple is
| A ( | cos2 α′ | − | cos2 α | ). |
| r ′ | r |
The wrench (R, K) along the axis by which the spring can be held in the form (α′, r ′) is given by the equations
| R = A | sin α′ | ( | cos2 α′ | − | cos2 α | ) − C | cos α′ | ( | sin α′ cos α′ | − | sin α cos α | ), |
| r ′ | r ′ | r | r ′ | r ′ | r |
| K = A cos α′ ( | cos2 α′ | − | cos2 α | ) + sin α′ ( | sin α′ cos α′ | − | sin α cos α | ). |
| r ′ | r | r ′ | r |
When the spring is slightly extended by an axial force F, = −R, and there is no couple, so that K vanishes, and α′, r ′ differ very little from α, r, it follows from these equations that the axial elongation, δx, is connected with the axial length x and the force F by the equation
| F = | Eπc4 | sin α | δx | , | ||
| 4r 2 | 1 + σ cos2 α | x |
and that the loaded end is rotated about the axis of the helix through a small angle
| 4σFxr cos α | , |
| Eπc4 |
the sense of the rotation being such that the spring becomes more tightly coiled.
66. 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 W. E. Ayrton and J. 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. G. H. 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.
If b stands for the breadth and t for the thickness of the strip, and τ for the permanent twist, the approximate formula for the angle θ through which the strip is untwisted on the application of a load W was found to be
| θ = | Wbτ (1 + σ) | . | ||||
| 2Et 3 ( 1 + | (1 + σ) | b4τ2 | ) | |||
| 30 | t 2 | |||||
The quantity bτ 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 same order as t/b, or a higher order, the formula becomes 12Wbτ (1+σ) / Et 3, with sufficient approximation, and this result appears to be in agreement with observations of the behaviour of such strips.
67. Thin Plate under Pressure.—The theory of the deformation of plates, whether plane or curved, is very intricate, partly because of the complexity of the kinematical relations involved. We shall here indicate the nature of the effects produced in a thin plane plate, of isotropic material, which is slightly bent by pressure. This theory should have an application to the stress produced in a ship’s plates. In the problem of the cylinder under internal pressure (§ 77 below) the most important stress is the circumferential tension, counteracting the tendency of the circular filaments to expand under the pressure; but in the problem of a plane plate some of the filaments parallel to the plane of the plate are extended and others are contracted, so that the tensions and pressures along them give rise to resultant couples but not always to resultant forces. Whatever forces are applied to bend the plate, these couples are always expressible, at least approximately in terms of the principal curvatures produced in the surface which, before strain, was the middle plane of the plate. The simplest case is that of a rectangular plate, bent by a distribution of couples applied to its edges, so that the middle surface becomes a cylinder of large radius R; the requisite couple per unit of length of the straight edges is of amount C/R, where C is a certain constant; and the requisite couple per unit of length of the circular edges is of amount Cσ/R, the latter being required to resist the tendency to anticlastic curvature (cf. § 47). If normal sections of the plate are supposed drawn through the generators and circular sections of the cylinder, the action of the neighbouring portions on any portion so bounded involves flexural couples of the above amounts. When the plate is bent in any manner, the curvature produced at each section of the middle surface may be regarded as arising from the superposition of two cylindrical curvatures; and the flexural couples across normal sections through the lines of curvature, estimated per unit of length of those lines, are C (1/R1 + σ/R2) and C (1/R2 + σ/R1), where R1 and R2 are the principal radii of curvature. The value of C for a plate of small thickness 2h is 23Eh3 / (1 − σ2). Exactly as in the problem of the beam (§§ 48, 56), the action between neighbouring portions of the plate generally involves shearing stresses across normal sections as well as flexural couples; and the resultants of these stresses are determined by the conditions that, with the flexural couples, they balance the forces applied to bend the plate.
 |
| Fig. 28. |
68. To express this theory analytically, let the middle plane of the plate in the unstrained position be taken as the plane of (x, y), and let normal sections at right angles to the axes of x and y be drawn through any point. After strain let w be the displacement of this point in the direction perpendicular to the plane, marked p in fig. 28. If the axes of x and y were parallel to the lines of curvature at the point, the flexural couple acting across the section normal to x (or y) would have the axis of y (or x) for its axis; but when the lines of curvature are inclined to the axes of co-ordinates, the flexural couple across a section normal to either axis has a component about that axis as well as a component about the perpendicular axis. Consider an element ABCD of the section at right angles to the axis of x, contained between two lines near together and perpendicular to the middle plane. The action of the portion of the plate to the right upon the portion to the left, across the element, gives rise to a couple about the middle line (y) of amount, estimated per unit of length of that line, equal to C (∂2w∂x2 + σ ∂2w∂y2), = G1, say, and to a couple, similarly estimated, about the normal (x) of amount −C (1 − σ) ∂2w∂x∂y, H, say. The
