and if the bending a con depends on two one ading to the bending at e lengths of the sport two funiculars thata 8 Fig . 12 sil dation of the si Fig , in rii ve sides of the tot Complicated as ing to F, by 23 Enown, the interélia sinja, and a is the angle at which the curve cuts the line of one principal axis of a cross-section at its centroid makes action of the applied forces. Unless the length of the rod exceeds an angle A with the vertical, then for any shape of section 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) section in a line DD', which is conjugate to the vertical g. 11 ilustrates the the length must exceed nt (EI/W). Some of the forms of the diameter CP with respect to any ellipse of inertia of the e supports A, B, C, curve are shown in Fig. 14. There will be an main For the form d, in which two bays make a figure of eight, we section. The central line is bent into a plane curve which sistem of fictiñor have is not in a vertical plane, but is in a plane through the AB, HBC* EI L/(W/EI)=4.6, a=130° line CY which is perpendicular to DD' (Fig. 16). g and - L,BCT: approximately (see Hess, Math. Ann. xxiii., 1884). It is noteworthy 16. Bending and i'wisting of Thin Rods. When a very on Dearer to B of the that whenever the length and force admit of a sinuous form, such thin rod or wire is bent and twisted by applied forces, = two latter is a ficha as a or b, with more than two inflexions, there is also possible a f the figure is the the forces on any part of it limited by a normal section crossed form, like e, with two inflexions only; it is probable that CE to represent to the latter form is stable and the former unstable. are balanced by the stresses across the section, and these E are fred puz stresses are statically equivalent to certain forces and 14. The particular case of the above for which a is very couples at the centroid of the section; we shall call them small is a curve of sines of small amplitude, and the result the stress-resultants and the stress-couples. The stress in this case has been ap- couples consist of two flexural couplės in the two principal end, it is simply come sides are 2, 3,47 pressed, unless L'2W>7EI; d in the figuren for the lower end corre- line of inflexions is also 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 The equatia load ; for a very slight excess of L'2W above 47EI, the theory on which the above discussion is founded, is not oral rinity quite adequate, as it assumes the central line of the strut Fig. 16. . it is derived is founded torsional rigidity and the twist produced; the torsional on the assumption that rigidity is exactly the same as for a straight rod of the the length is great com same material and section twisted without bending, as in pared with the diameter Saint-Venant's torsion problem (ELASTICITY, Ency. Brit. vol. (cf. $ 10). vii. p. 812). The twist is connected with the deformation The condition of of the wire in this way: if we suppose a very small ring which buckling, corresponding fits the cross-section of the wire to be provided with a to the above, for a long pointer in the direction of one principal axis, and to move strut, of length L', when along the wire with velocity v, the pointer will rotate about , both ends are free to the central line with angular velocity Tv. The amount of turn is L'W>PEI ; for the flexural couple for either principal plane at any section the central line forms a is the product of the flexural rigidity for that plane, and the complete bay (Fig. 15, resolved part in that plane of the curvature of the central b); if both ends are 6 line at the centroid of the section; the resolved part of maintained in the same the curvature along the normal to any plane is obtained vertical line, the condi Fig. 15. by treating the curvature as a vector directed along the tion is L'OW> 472EI, the central line forming a complete normal to the osculating plane and projecting this vector. bay and two half bays (Fig. 15, c). The flexural couples reduce to a single couple in the oscu15. In our consideration of flexure it has so far been lating plane proportional to the curvature when the two supposed that the bending takes place in a principal flexural rigidities are equal, and in this case only. plane. We may remove this restriction by resolving the The stress-resultants across any section are shearing forces that tend to produce bending into systems of forces stresses in the two principal planes, and a tension or acting in the two principal planes. To each plane there thrust along the central line; when the stress-couples and corresponds a particular flexural rigidity, and the systems the applied forces are known these stress-resultants are of forces in the two planes give rise to independent determinate. The existence in particular of the resultant systems of stress, strain, and displacement, which must be tension or thrust parallel to the central line does not imply superposed in order to obtain the actual state. Applying sensible extension or contraction of the central filament, this process to the problem of $S 2–8, and supposing that I and the tension per unit area of the cross-section to which W of the flerei wire , for it e forces to One das elastica bila family is appliers attachela ne forma points action | cos3 a 30 к it would be equivalent is small compared with the tensions 40 Fær cos a and pressures in longitudinal filaments not passing through Епс the centroid of the section; the moments of the latter the sense of the rotation being such that the spring becomes moro tensions and pressures constitute the flexural couples. tightly coiled. 17. We consider, in particular, the case of a naturally 19. A horizontal pointer attached to a vertical spiral straight spring or rod of circular section, radius c, and of spring would be made to rotate by loading the spring, homogeneous isotropic material. The torsional rigidity is and the angle through which it turns might be used to Enc4/(1+0); and the flexural rigidity, which is the same measure the load, at any rate, when the load is not too for all planes through the central line, is įETC4; we shall great; but a much more sensitive contrivance is the denote these by C and A respectively. The rod may be twisted strip devised by Ayrton and Perry. A very thin, held bent by suitable forces into a curve of double curva- narrow rectangular strip of metal is given a permanent ture with an amount of twist t, and then the torsional twist about its longitudinal middle line, and a pointer is couple is Ct, and the flexural couple in the osculating attached to it at right angles to this line. When the plane is Alp, where p is the radius of circular curvature. strip is subjected to longitudinal tension the pointer Among the curves in which the rod can be held by forces rotates through a considerable angle. Bryan (Phil. Mag., and couples applied at its ends only, one is a circular December 1890) has succeeded in constructing a theory helix; and then the applied forces and couples are equi- of the action of the strip, according to which it is regarded valent to a wrench about the axis of the helix. as a strip of plating in the form of a right helicoid, which, Let a be the angle and r the radius of the helix, so that p is after extension of the middle line, becomes a portion of a r seca ; and let R and K be the force and couple of the wrench slightly different helicoid ; on account of the thinness of (Fig. 17). Then the couple formed by R and an equal and opposite force at the strip, the change of curvature of the surface is conany section and the couple K are equivalent to the torsional and siderable, even when the extension is small, and the flexural couples at the section, and this gives the equations for pointer turns with the generators of the helicoid. R and K Taking b for the breadth and t for the thickness of the strip, and – for the permanent twist, the approximate formula for the p2 angle 8 through which the strip is untwisted on the application of a load W was found to be K=A + Ct sin a. Wbr(1+0) 0= The thrust across any section is R sin a parallel to the tangent 2E131+ (1+0) 642 ta The quantity by which occurs in the formula is the total twist in a When the twist is such that, if the rod length of the strip equal to its breadth, and this will generally be were simply unbent, it would also be very small; if it is small of the same order as t/b, or a higher order, untwisted, 7 is sin a cos a/r, and then, re the formula becomes ]Wbr(1+0)/Et, with sufficient approximation, storing the values of A and C, we have and this result appears to be in agreement with observations of the behaviour of such strips. 20. General Theorems.—Passing now from these ques472 1to tions of flexure and torsion, we consider some results that ETA 1+o cos” a can be deduced from the general equations of equilibrium 4r 1+o of an elastic solid body. 18. The theory of spiral springs The form of the general expression for the potential affords an application of these results. energy (ELASTICITY, Ency. Brit. vol. vii. p. 823) stored up The stress-couples called into play in the strained body leads, by a general property of quadwhen a naturally helical spring (a, r) ratic functions, to a reciprocal theorem relating to the effects is held in the form of a helix (a', r"), produced in the body by two different systems of forces, are equal to the differences between viz.: The whole work done by the forces of the first system R those called into play when a straight acting over the displacements produced by the forces of the rod of the same material and sec- second system is equal to the whole work done by the forces Fig. 17. tion is held in the first form, and of the second system acting over the displacements produced those called into play when it is held in the second form. by the forces of the first system. By a suitable choice of Thus the torsional couple is the second system of forces the average values of the comsin a' cos a' sin a cosa ponent stresses and strains produced by given forces, conand the flexural couple is sidered as constituting the first cosa a' cosa system, can be obtained, even A when the distribution of the stress and strain cannot be The wrench (R, K) along the axis by which the spring can be held in the form (a', r') is given by the equations determined. sin a' cos' a' cosa a cos a' sin a' cos a' sin a cos a Taking for example the problem R=A C presented by an isotropic body of r' any form pressed between two cosa a' cosa a sin a' cos a' sin a cos a K=A cos a + C sin a' parallel planes distant 1 apart r' (Fig. 18), and denoting the result. When the spring is slightly extended by an axial force F, = -R, ant pressure by p, the diminution and there is no couple, so that K vanishes, and a', r', differ very of volume - dv is given by the little from a, r, it follows from these equations that the axial equation elongation, ox, is connected with the axial length x and the force - Sv=lp/3K, Fig. 18. F by the equation where k is the modulus of comEmc4 pression, equal to ļE/(1-20). Again, taking the problem of the dx F= K COS a. a (sin in a' (sin :). r changes produced in a heavy body by different ways of supporting it; 42 1to cosa and that the loaded end is rotated about the axis of the helix 1 The line joining the points of contact must be normal to the through a small angle planes. sin a . when the body is suspended from one or more points in a horizontal so bounded which is equivalent, on the same plane, to plane its volume is increased by a pressure at the origin, and a radial shearing stress prodv=Wh/3k, portional to 1/r2, but these are in the ratio 27:7-2, instead where W is the weight of the body, and h the depth of its centre of gravity below the plane; when the body is supported by upward of being in the ratio 47(1-0):(1 – 20)r-2 vertical pressures at one or more points in a horizontal plane the The second stress system consists of volume is diminished by (1) radial pressure F'n-, - 8v=Wh/3k, (2) tension in the meridian plane across the radius vector of amount Fry cos 0/(1+cose), F7-2/(1+cos ), (4) shearing stress as in the last section of amount F7-? sin 0/(1+cos ), and the stress across the plane boundary consists of a pressure of 21. In recent years important results have been found magnitude 24F' and a radial shearing stress of amount Fr-4. If then we superpose the component stresses of the last section multiby considering the effects produced in an elastic solid by plied by, 4(1 - 0)W/F, and the component stresses here written down forces applied at isolated points. multiplied by - (1 - 20)W/27F', the stress on the plane boundary will reduce to a single pressure W at the origin. We shall thus Taking the case of a single force F applied at a point in the obtain the stress system at any point due to such a force applied interior, it can be shown that the stress at a distance r from the at one point of the boundary. point consists of In the stress system thus arrived at the stress across any plane (1) a radial pressure of amount parallel to the boundary is directed away from the place where W 2-0 F cos e is supported, and its amount is 3Wcos2 //2r. The corresponding 1 displacement consists of (2) tension in all directions at right angles to the radius of amount (1) a horizontal displacement radially outwards from the vertical through the origin of amount W(1+0) sin 0 1-20 cos - 1+cos W(1+0) {2(1 - 0)+cos2 0}. 27 Er The effects produced by a system of loads on a solid bounded by a surface. F Thus the principal of energy involved in the impact is neglected, and the stresses are in and perpen- pressure between the bodies at any instant during the dicular to the meridian impact is equal to the rate of destruction of momentum plane, and the direction of of either along the normal to the plane of contact drawn one of those in the meri towards the interior of the other. It has been shown that dian plane is inclined to the radius vector r at an in general the bodies come into contact over a small area angle bounded by an ellipse, and remain in contact for a time 12-40 which varies inversely as the fifth root of the initial 5-40 The corresponding dis- For equal spheres of the same material, with or =, impinging placement at any point is directly with relative velocity v, the patches that come into compounded of a radial contact are circles of radius 757 256 where r is the radius of cither, and V is the velocity of longi. tudinal waves in a thin bar of the material. The duration of the (3 – 40)(1+0) F 1 impact is approximately 2(1-0) 4пE r 37572 | 1/5 The effects of forces applied at different points and in different 128 vl6y46 directions can be obtained by summation, and the effect of continuously distributed forces can be obtained by integration. For two steel spheres of the size of the earth impinging with a velocity of 10 mm. per second the duration of the impact would be 22. The stress system considered in the last section is about twenty-seven hours. The fact that the duration of impact equivalent, on the plane is, for moderate velocities, a considerable multiple of the time through the origin at taken by wave of compression to travel through either of two impinging bodies has been ascertained experimentally, and conright angles to the line stitutes the reason for the adequacy of the statical theory here of action of F, to a described. pressure of magnitude 24. Spheres and Cylinders. — Simple results can be IF at the origin and a found for spherical and cylindrical bodies strained by radial shearing stress of radial forces. For a sphere of radius a, and of homogeneous isotropic material 2(1-0) 47m2) of density e, strained by the mutual gravitation of its parts, the by the application of stress at a distance r from the centre consists of this system of tractions Fig. 20. (1) uniform hydrostatic pressure of amount Dogpa(3-0)/(1-0), to a solid bounded by a (2) radial tension of amount Bogp(rea)(3-0)/(1-0), plane, the displacement just described would be produced. 1 Cf. Auerbach in Winkelmanu's Handbuch der Physik, i. 303. There is also another (Fig. 20) stress system for a solid | Breslau, 1891. tan-1 - ( ko tano). relative velocity. . (2-0432) 1 amount 2 2 to 2 1/2 2 2 3 2 2 z ro 2 3 2 and hoop tension - Peau Tho hoop tension at the inner (3) uniform tension at right angles to the radius vector of (4) uniform contraction of the longitudinal filaments of amount amount togp(72]a)(1+30)/(1-0), 20 Pr.2 – Pori ? where g is the value of gravity at the surface. The corresponding g E ro -ri strains consist of For a shell subject only to internal pressure p the greatest (1) uniform contraction of all lines of the body of amount extension is the circumferential extension at the inner surface, Hok-gpa(3-6)/(1-0), and its amount is (2) radial extension of amount 15k-p(72/a)(1+0)/(1-0), pr+r (3) extension in any direction at right angles to the radius E 72 vector of amount the greatest tension is the hoop tension at the inner surface, and ork-gp(r</a)(1+0)/(1-0), its amount is pro2+r?)/(r.2 – 7,2). where kc is the modulus of compression. The volume is diminished 27. The results just obtained have been applied to gan by the fraction gpa/5lc of itself. The parts of the radii vectores within the sphere r=a{(3-0)/(3+30)}?! are contracted, and the parts construction ; we may consider that one cylinder is heated without this sphere are extended. The application of the above so as to slip over another upon which it shrinks by cooling, results to the state of the interior of the earth is restricted by the so that the two form a single body in a condition of circumstance that, unless the modulus of compression is much initial stress. greater than that of any known material, the stresses and strains expressed above would, in a sphere of the size of the earth, greatly We take P as the measure of the pressure between the two, and exceed the elastic limits. p for the pressure within the inner cylinder by which the system 25. In a spherical shell of homogeneous isotropic material, of surface. To obtain the stress at any point we superpose the is afterwards strained, and denote by pl the radins of the common internal radius r, and external radius 10, subjected to pressure Po ri? 702 — 92 on the outer surface, and p, on the inner surface, the stress at any system consisting of radial pressure poi m-4 and hoop tension point distant r from the centre consists of r.2 + ga? (1) uniform tension in all directions of amount Pir, 8 – Potos upon a system which, for the outer cylinder, consists r.2-722 r3r3 - 2 (2) radial prossure of amount P1 – Po roør, 3 of radial pressure P10 – go? and hoop tension P' 23 - 123 z (3) tension in all directions at right angles to the radius vector and, for the inner cylinder consists of radial 22. pressure P of amount gel2 r,2 P1 – Po To Sr 3 + ri? 2 g 3 = r, 3 gu3 guit '2 - ri The corresponding strains consist of surface is less than it would be for a tube of equal thickness without initial stress in the ratio (1) uniform extension of all lines of the body of amount 1 P.78-Por P 202 72-91 1 : 3k r,3 – r Pro2+7;272-7:1. 2 2222 This shows how the strength of the tube is increased by the initial (2) radial contraction of amount stress. 2urr3 23 (3) extension in all directions at right angles to the radius 28. The problem of determining the distribution of vector of amount stress and strain in a circular cylinder, rotating about its 1 Pi-Po r@ry 3 axis, has not yet been completely solved, but solutions 4μ ro3 -η, zu have been obtained which are sufficiently exact for the two where u is the modulus of rigidity of the material, = E/(1+0). special cases of a thin disk and a long shaft. The volume included between the two surfaces of the body is Piri-Porn increased by the fraction Suppose that a circular disk of radins a and thickness 21, and of k(r.3-5,3) of itself, and the volume density p, rotates about its axis with angular velocity w, and consider within the inner surface is increased by the fraction the following systems of superposed stresses at any point distant, from the axis and 2 from the middle plane : 3(P1-P) 7,8 Pir-Pori (1) uniform tension in all directions at right angles to the axis 4μ ro-r1 1c(r.3 -713) of amount fw pa?(3+0), of itself. For a shell subject only to internal pressure p the (2) radial pressure of amount fwpr(3+0), greatest extension is the extension at right angles to the radius at (3) pressure along the circular filaments of amount fw pr*(1 +30), the inner surface, and its amount is (4) uniform tension in all directions at right angles to the axis of amount fwap(72 – 324)o(1+0)/(1-0). pris 1 + 3k4u r3 7.3 -ra The corresponding strains may be expressed as (1) uniform extension of all filaments at right angles to the axis the greatest tension is the transverse tension at the inner surface, of amount and its amount is p(kr.3 +7,3%)/(r.2-r). 126. In the problem of a cylindrical shell under pressuro a com fw pa°(3+0), E plication may arise from the effects of the ends ; but when the (2) radial contraction of amount ends are free from stress the solution is very simple. With nota 1-02 tion similar to that in § 25 it can be shown that the stress at a E gwapra, distance r from the axis consists of (1) uniform tension in all directions at right angles to the axis (3) contraction along the circular filaments of amount 1-03 (4) extension of all filaments at right angles to the axis of (2) radial pressure of amount Pi-Po roar;? amount 802 - r,2 g 1 (3) hoop tension numerically equal to this radial pressure. Etwap(72 - 3)(1+0), The corresponding strains consist of (1) uniform extension of all lines of the material at right angles (5) contraction of the filaments normal to the plane of the disk to the axis of amount of amount (1+0) E r2 = r22 E 1-0 (2) radial contraction of amount The greatest extension is the circumferential extension near the 1+0 Po-Po rozry? centre, and its amount is E rr, 2 p3 (3+0)(1-0) 0(1+0) ωθρα2 + wapi?. 8E (3) extension along the circular filaments numerically equal to 6E this radial contraction, The longitudinal contraction is required to make the plane faces a stress (1+0)P-P_Q_ @P. Q_aQ of the disk free from pressure, and the terms in l and enable us effects produced in a thin plane plate, of isotropic material, to avoid shearing stress on any cylindrical surface. The system of which is slightly bent by pressure. This theory should stresses and strains thus expressed satisfies all the conditions, except that there is a small radial tension on the bounding have an application to the stress produced in a ship's surface of amount per unit area w?p(72 – 322)o(1+)/(1-0). The plates. In the problem of the cylinder ($ 26) the most resultant of these stresses on any part of the edge of the disk important stress is the circumferential tension, countervanishes , and the stress in question is very small in comparison acting the tendency of the circular filaments to expand with the other stresses involved when the disk is thin ; we may conclude that for a thin disk the expressions given represent the under the pressure, but in the problem of a plane plate actual condition at all points which are not very close to the edge some of the filaments parallel to the plane of the plate are (cf. $ 9). The effect of the longitudinal contraction is that the extended and others are contracted, so that the tensions plane faces become slightly concave (Fig. 21). and pressures along them give rise to resultant couples but In problems of the kind just considered, where the stress consists not to resultant forces. Whatever forces are applied to simply of a radial tension P and a circumferential tension Q, which are functions of r and 2, the bend the plate, these couples are always expressible in terms components, besides of the principal curvatures produced in the surface which, satisfying the equations of equi. before strain, was the middle plane of the plate. The librium, are also subject to simplest case is that of a rectangular plate, bent by a distwo conditions of compatibility which can be expressed in the tribution of couples applied to its edges, so that the forms middle surface becomes a cylinder of large radius R; the requisite couple per unit of length of the straight ar ôr' edges is of amount C/R, where C is a certain constant; 22 and the requisite couple per unit of length of the circular maz(Q-oP)=o (P+Q) edges is of amount Co/R, the latter being required to 29. The corresponding solu. resist the tendency to anticlastic curvature (cf. § 1). If tion for a disk with a circular normal sections of the plate are supposed drawn through axle-hole (radius b) will be obtained from that given in the the generators and circular sections of the cylinder, the last section by superposing the action of the neighbouring portions on any portion so Fig. 21. following system of additional bounded involves flexural couples of the above amounts. stresses : When the plate is bent in any manner, the curvature (1) radial tension of amount } wapb2(1 – u?/m2)(3+0), (2) tension along the circular filaments of amount produced at each section of the middle surface may be * wapb*(1 +ao/r2)(3+0); regarded as arising from the superposition of two cylin drical curvatures; and the flexural couples across normal and the corresponding additional strains are sections through the lines of curvature, estimated per (1) radial contraction of amount unit of length of those lines, are C(1/R, +o/R,) and 3+0 a2 C(1/R, +o/R), where R, and R, are the principal radii 8E of curvature. The value of C for a plate of small thick(2) extension along the circular filaments of amount 3+o ness 2h is Eks/(1 – 02). Exactly as in the problem of the beam (S$ 2, 10), the action between neighbouring por(3) contraction of the filaments parallel to the axis of amount tions of the plate generally involves shearing stresses 0(3+0) across normal sections as well as flexural couples; and -w2pb?. 4E the resultants of these stresses are determined by the Again, the greatest extension is the circumferential extension at conditions that, with the flexural couples, they balance the inner surface, and, when the hole is very small, its amount is the forces applied to bend the plate. nearly double what it would be for a complete disk. 30. In the problem of the rotating shaft we have the following the plate in the unstrained position be taken as the plane of (x, y), 32. To express this theory analytically, let the middle plane of stress-system: (1) radial tension of amount swpla? – 7-2)(3 – 20)/(1-0), and let normal sections at right angles to the axes of u and y be (2) circumferential tension of amount drawn through any point. After strain let w be the displacement of this point in the direction perpendicular to the plane, marked fwap{a2(3 – 20)/(1 - 0) – 72(1 +20)/(1-0)}, p in Fig. 22. If the axes of ac and y were parallel to the lines of (3) longitudinal tension of amount fw-plaa – 2r2)/(1 - 0). The 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 of the shaft. The contraction of the longitudinal filaments is uniform and equal to 1w2paʻg/E. The greatest extension in the rotating shaft is the circumferential extension close to the axis, and its amount is fwépa’(3 – 50)/E(1-0). The value of any theory of the strength of long rotating shafts w?p=+Eao(+1) By Fig. 22. formation of plates, whether plane or curved, is very intri curvature at the point, the flexural couple acting across the section cate, partly because of the complexity of the kinematical normal to x (or y) would have the axis of y (or 2) 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, {(1+0)-(1-0)} wapb?. 8E BA |