| θ = | Wbτ (1 + σ) | . | ||||
| 2Et3 ( 1 + | (1 + σ) | b4τ2 | ) | |||
| 30 | t² | |||||
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 ½Wbτ (1+σ) / Et3, 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 2⁄3Eh3 / (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 corresponding couples on an element of a section at right angles to the axis of y, estimated per unit of length of the axis of x, are of amounts −C [∂²w/∂y² + σ (∂²w/∂x²)], = G2 say, and −H. The resultant S1 of the shearing stresses on the element ABCD, estimated as before, is given by the equation S1 = ∂G1/∂x − ∂H/∂y (cf. § 57), and the corresponding resultant S2 for an element perpendicular to the axis of y is given by the equation S2= −∂H/∂x − ∂G2/∂y. If the plate is bent by a pressure p per unit of area, the equation of equilibrium is ∂S1/∂x + ∂S2/∂y = p, or, in terms of w,
| ∂4w | + | ∂4w | + 2 | ∂4w | = | p | . |
| ∂x4 | ∂y4 | ∂x2∂y2 | C |
This equation, together with the special conditions at the rim, suffices for the determination of w, and then all the quantities here introduced are determined. Further, the most important of the stress-components are those which act across elements of normal sections: the tension in direction x, at a distance z from the middle plane measured in the direction of p, is of amount −3Cz/2h3 [∂²w/∂x² + σ (∂²w/∂y²)], and there is a corresponding tension in direction y; the shearing stress consisting of traction parallel to y on planes x = const., and traction parallel to x on planes y = const., is of amount [3C(1 − σ)z/2h3] · (∂²w/∂x∂y); these tensions and shearing stresses are equivalent to two principal tensions, in the directions of the lines of curvature of the surface into which the middle plane is bent, and they give rise to the flexural couples.
69. In the special example of a circular plate, of radius a, supported at the rim, and held bent by a uniform pressure p, the value of w at a point distant r from the axis is
| 1 | p | (a² − r²) ( | 5 + σ | a² − r² ), | |
| 64 | C | 1 + σ |
and the most important of the stress components is the radial tension, of which the amount at any point is 3⁄32(3 + σ) pz (a² − r)/h³; the maximum radial tension is about 1⁄3(a/h)²p, and, when the thickness is small compared with the diameter, this is a large multiple of p.