ELASTIC of the disk free from pressure, and the terms in l and z enable us to avoid shearing stress on any cylindrical surface. The system of stresses and strains thus expressed satisfies all the conditions except that there is a small radial tension on the boundS surface of amount per unit area 32>(! + «,-)/(! _ ,) Thf resultant of these stresses on any part of the edge of the disk ai d th stress m A stresses ! .involved question verydisk smallis &thin in comparison vith the other whenis the • we may conclude that for a thin disk the expressions given represent the actua! condition at all points which are not vmy close to the edge t of the i § faces t V effecslightly longitudinal contraction is that the 1. plane become concave (Fig. 21). In . Problems of the kindPjust where the stressQ,consists simply of a radial tension andconsidered, a circumferential tension which are functions of r and z, the stress _ components, besides satisfying the equations of equilibrium, are also subject to two conditions of compatibility which can be expressed in the forms P-Q_3Q 0P dr ^dr’ ’•@(Q-rf)=4(p+® 29. The corresponding solution for a disk with a circular axle-hole (radius b) will be obtained from that given in the last section by superposing the Fi g- 21. lollowing system of additional stresses : (1) radial tension of amount ^ w2p£2(l + (2) tension along the circular filaments of amount ’ 2 2 H wy> (l + a jr~)(3 + o-); and the corresponding additional strains are (1) radial contraction of amount 3 + (r f a? "l "sir {(1+-11 ~ ^ j (2) extension along the circular filaments of amount 3 + <r f + a2 ,1 1 ~8E" { ^ + C ~ ^ | w2pS2, (3) contraction of the filaments parallel to the axis of amount g'fS + g-) 2 ,2
SYSTEMS
741 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 (§ 26) 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 to resultant forces. Whatever forces are applied to bend the plate, these couples are always expressible 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 j and the requisite couple per unit of length of the circular edges is of amount Ccr/R, the latter being required to resist the tendency to anticlastic curvature (cf. § 1). 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(l/R1 + a/R2) and +0 "/®,i)) where Rj and R2 are the principal radii of curvature. The value of C for a plate of small thickness 2A is gEA3/(l — cr2). Exactly as in the problem of the beam (§§ 2, 10), 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. 32. lo express this theory analytically, let the middle plane of the plate in the unstrained position be taken as the plane of (a;, ?/) 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 oi this point in the direction perpendicular to the plane, marked p in Fig. 22. If the axes of x and y were parallel to the lines of
Agam, the greatest extension is the circumferential extension at the inner surface, and, when the hole is very small, its amount is nearly double what it would be for a complete disk. .In the problem of the rotating shaft we have the following& stress-system: (1) radial tension of amount ia2p(a2-r2)(3 - 2<r)/(l - od (2) circumferential tension of amount wp M3 - 2<r)/(l - <7) - r2(l +2 2<r)/(l - <r)}, (3) longitudinal tension of amount £w p(a2- 2ry/(l - a). he resultant longitudinal tension at any normal section vanishes, and the radial tension vanishes at the bounding surface ; and thus the expressions here given may be taken to represent the actual condition at all points which are not very close to the ends ot the shaft. The contraction of the longitudinal filaments is uniform and equal to .jw'-paV/E. The greatest extension in the rotating shaft is the2 circumferential extension close to the axis and its amount is ^w /ia2(3 - 5o-)/E(l - a). e Talue ,°^ anformulae y theoryisofdiminished the strength rotating shafts loundnd on these by of thelong circumstance that at sufficiently high speeds the shaft may rtend to take up a curved form, the straight form being unstable. The shaft is then said to whirl. This occurs when the period of rotation of the shaft is very nearly coincident with one of its periods of lateral vibration. (See Greenhill, Proc. Inst. Mech. Engineers, April 1883.) The lowest speed at which whirling can take place in a shaft of length l freely supported at its ends, is given by the formula w2p = £Ea2(7r/Z)4. As in § 14, this formula should not be applied unless the length of tne shaft is a considerable multiple of its diameter. It implies that whirling is to be expected whenever o> approaches this critical value. 31. Thin Plate under Pressure.—The theory of the deFig. 22. formation of plates, whether plane or curved, is very intricate, partly because of the complexity of the kinematical curvature at the point, the flexural couple acting across the section normal to a; (or y) would have the axis of y (or x) for its axis ; but relations involved. We shall here indicate the nature of the when the lines of curvature are inclined to the axes of co-ordinates,
