We select this case for consideration. The problem of determining the form of the curve (cf. fig. 23) is mathematically identical with the problem of determining the motion of a simple circular pendulum oscillating through a finite angle, as is seen by comparing the differential equation of the curveEId²φ+ W sin φ = 0ds²with the equation of motion of the pendulumld²φ+ g sin φ = 0.dt²The length L of the curve between two inflections corresponds to the time of oscillation of the pendulum from rest to rest, and we thus haveL √(W / EI) = 2K,Fig. 24.where K is the real quarter period of elliptic functions of modulus sin ½α, and α is the angle at which the curve cuts the line of action of the applied forces. Unless the length of the rod exceeds π√(EI / W) it will not bend under the force, but when the length is great enough there may be more than two points of inflection and more than one bay of the curve; for n bays (n + 1 inflections) the length must exceed nπ √(EI / W). Some of the forms of the curve are shown in fig. 24.For the form d, in which two bays make a figure of eight, we haveL√(W / EI) = 4.6, α = 130°approximately. It is noteworthy that whenever the length and force admit of a sinuous form, such as α or b, with more than two inflections, there is also possible a crossed form, like e, with two inflections only; the latter form is stable and the former unstable.
We select this case for consideration. The problem of determining the form of the curve (cf. fig. 23) is mathematically identical with the problem of determining the motion of a simple circular pendulum oscillating through a finite angle, as is seen by comparing the differential equation of the curve
with the equation of motion of the pendulum
The length L of the curve between two inflections corresponds to the time of oscillation of the pendulum from rest to rest, and we thus have
L √(W / EI) = 2K,
where K is the real quarter period of elliptic functions of modulus sin ½α, and α is the angle at which the curve cuts the line of action of the applied forces. Unless the length of the rod exceeds π√(EI / W) it will not bend under the force, but when the length is great enough there may be more than two points of inflection and more than one bay of the curve; for n bays (n + 1 inflections) the length must exceed nπ √(EI / W). Some of the forms of the curve are shown in fig. 24.
For the form d, in which two bays make a figure of eight, we have
L√(W / EI) = 4.6, α = 130°
approximately. It is noteworthy that whenever the length and force admit of a sinuous form, such as α or b, with more than two inflections, there is also possible a crossed form, like e, with two inflections only; the latter form is stable and the former unstable.
61. The particular case of the above for which α is very small is a curve of sines of small amplitude, and the result in this case has been applied to the problem of the buckling of struts under thrust. When the strut, of length L′, ismaintained upright at its lower end, and loaded at its upper end, it is simply contracted, unless L′²W > ¼π²EI, for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflections is also vertical, and thus the length must be half of one bay (fig. 25, a). For greater lengths or loads the strut tends to bend or buckle under the load. For a very slight excess of L′²W above ¼π²EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central-line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place when the length or the force exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. § 56).
The condition of buckling, corresponding to the above, for a long strut, of length L′, when both ends are free to turn is L′²W > π²EI; for the central-line forms a complete bay (fig. 25, b); if both ends are maintained in the same vertical line, the condition is L′²W > 4π²EI, the central-line forming a complete bay and two half bays (fig. 25,c).
62. In our consideration of flexure it has so far been supposed that the bending takes place in a principal plane. We may remove this restriction by resolving the forces that tend to produce bending into systems of forces acting in the two principal planes. To each plane there corresponds a particular flexural rigidity, and the systems of forces in the two planes give rise to independent systems of stress, strain and displacement, which must be superposed in order to obtain the actual state. Applying this process to the problem of §§ 48-54, and supposing that one principal axis of a cross-section at its centroid makes an angle θ with the vertical, then for any shape of section the neutral surface or locus of unextended fibres cuts the section in a line DD′, which is conjugate to the vertical diameter CP with respect to any ellipse of inertia of the section. The central-line is bent into a plane curve which is not in a vertical plane, but is in a plane through the line CY which is perpendicular to DD′ (fig. 26).
63.Bending and Twisting of Thin Rods.—When a very thin rod or wire is bent and twisted by applied forces, the forces on any part of it limited by a normal section are balanced by the tractions across the section, and these tractions are statically equivalent to certain forces and couples at the centroid of the section; we shall call them thestress-resultantsand thestress-couples. The stress-couples consist of two flexural couples in the two principal planes, and the torsional couple about the tangent to the central-line. The torsional couple is the product of the torsional rigidity and the twist produced; the torsional rigidity is exactly the same as for a straight rod of the same material and section twisted without bending, as in Saint-Venant’s torsion problem (§ 42). The twist τ is connected with the deformation of the wire in this way: if we suppose a very small ring which fits the cross-section of the wire to be provided with a pointer in the direction of one principal axis of the section at its centroid, and to move along the wire with velocity v, the pointer will rotate about the central-line with angular velocity τv. The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section; the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only.
The stress-resultants across any section are tangential forces in the two principal planes, and a tension or thrust along the central-line; when the stress-couples and the applied forces are known these stress-resultants are determinate. The existence, in particular, of the resultant tension or thrust parallel to the central-line does not imply sensible extension or contraction of the central filament, and the tension per unit area of the cross-section to which 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.
64. 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 ¼Eπc4/ (1 + σ); and the flexural rigidity, which is the same for all planes through the central-line, is ¼Eπc4; 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 τ, and then the torsional couple is Cτ, and the flexural couple in the osculating plane is A/ρ, where ρ 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 α be the angle and r the radius of the helix, so that ρ is r sec²α; and let R and K be the force and couple of the wrench (fig. 27).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 KR = Asin α cos³ α−cos α,r²rK = Acos³ α+ Cτ sin α.rThe thrust across any section is R sin α parallel to the tangent to the helix, and the shearing stress-resultant is R cos α at right angles to the osculating plane.When the twist is such that, if the rod were simply unbent, itwould also be untwisted, τ is (sin α cos α) / r, and then, restoring the values of A and C, we haveR =Eπc4σsin α cos² α,4r²1 + σK =Eπc41 + σ cos² αcos α.4r1 + σ65. 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 isC(sin α′ cos α′−sin α cos α).r′rand the flexural couple isA(cos² α′−cos² α).r′rThe wrench (R, K) along the axis by which the spring can be held in the form (α′, r′) is given by the equationsR = Asin α′(cos² α′−cos² α)− Ccos α′(sin α′ cos α′−sin α cos α),r′r′rr′r′rK = A cos α′(cos² α′−cos² α)+ sin α′(sin α′ cos α′−sin α cos α).r′rr′rWhen 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 equationF =Eπc4sin αδx,4r²1 + σ cos² αxand that the loaded end is rotated about the axis of the helix through a small angle4σFxr cos α,Eπc4the sense of the rotation being such that the spring becomes more tightly coiled.
Let α be the angle and r the radius of the helix, so that ρ is r sec²α; and let R and K be the force and couple of the wrench (fig. 27).
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
The thrust across any section is R sin α parallel to the tangent to the helix, and the shearing stress-resultant is R cos α at right angles to the osculating plane.
When the twist is such that, if the rod were simply unbent, itwould also be untwisted, τ is (sin α cos α) / r, and then, restoring the values of A and C, we have
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
and the flexural couple is
The wrench (R, K) along the axis by which the spring can be held in the form (α′, r′) is given by the equations
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
and that the loaded end is rotated about the axis of the helix through a small angle
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 ofplatingin 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 beθ =Wbτ (1 + σ).2Et3(1 +(1 + σ)b4τ2)30t²The 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.
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
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 R1and R2are the principal radii of curvature. The value of C for a plate of small thickness 2h is2⁄3Eh3/ (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.
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. Thecorresponding 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)], = G2say, and −H. The resultant S1of 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 S2for 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∂y2CThis 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 is1p(a² − r²)(5 + σa² − r²),64C1 + σand the most important of the stress components is the radial tension, of which the amount at any point is3⁄32(3 + σ) pz (a² − r)/h³; the maximum radial tension is about1⁄3(a/h)²p, and, when the thickness is small compared with the diameter, this is a large multiple of p.
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. Thecorresponding 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)], = G2say, and −H. The resultant S1of 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 S2for 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,
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
and the most important of the stress components is the radial tension, of which the amount at any point is3⁄32(3 + σ) pz (a² − r)/h³; the maximum radial tension is about1⁄3(a/h)²p, and, when the thickness is small compared with the diameter, this is a large multiple of p.
70.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 (§ 27) stored up in the strained body leads, by a general property of quadratic functions, 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 by an isotropic body of any form4pressed between two parallel planes distant l apart (fig. 29), and denoting the resultant pressure by p, we find that the diminution of volume -δv is given by the equation−δv = lp / 3k,where k is the modulus of compression, equal to1⁄3E / (1 − 2σ). Again, take the problem of the changes produced in a heavy body by different ways of supporting it; when the body is suspended from one or more points in a horizontal plane its volume is increased byδv = Wh / 3k,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 vertical pressures at one or more points in a horizontal plane the volume is diminished by−δv = Wh′ / 3k,where h′ is the height of the centre of gravity above the plane; if the body is a cylinder, of length l and section A, standing with its base on a smooth horizontal plane, its length is shortened by an amount−δl = Wl / 2EA;if the same cylinder lies on the plane with its generators horizontal, its length is increased by an amountδl = σWh′ / EA.
Taking for example the problem presented by an isotropic body of any form4pressed between two parallel planes distant l apart (fig. 29), and denoting the resultant pressure by p, we find that the diminution of volume -δv is given by the equation
−δv = lp / 3k,
where k is the modulus of compression, equal to1⁄3E / (1 − 2σ). Again, take the problem of the changes produced in a heavy body by different ways of supporting it; when the body is suspended from one or more points in a horizontal plane its volume is increased by
δv = Wh / 3k,
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 vertical pressures at one or more points in a horizontal plane the volume is diminished by
−δv = Wh′ / 3k,
where h′ is the height of the centre of gravity above the plane; if the body is a cylinder, of length l and section A, standing with its base on a smooth horizontal plane, its length is shortened by an amount
−δl = Wl / 2EA;
if the same cylinder lies on the plane with its generators horizontal, its length is increased by an amount
δl = σWh′ / EA.
71. In recent years important results have been found by considering the effects produced in an elastic solid by forces applied at isolated points.
Taking the case of a single force F applied at a point in the interior, we may show that the stress at a distance r from the point consists of(1) a radial pressure of amount2 − σFcos θ,1 − σ4πr²(2) tension in all directions at right angles to the radius of amount1 − 2σFcos θ,2(1 − σ)4πr²(3) shearing stress consisting of traction acting along the radiusdron the surface of the cone θ = const. and traction acting along the meridian dθ on the surface of the sphere r = const. of amount1 − 2σFsin θ,2(1 − σ)4πr²where θ is the angle between the radius vector r and the line of action of F. The line marked T in fig. 30 shows the direction of the tangential traction on the spherical surface.Fig. 30.Fig. 31.Thus the lines of stress are in and perpendicular to the meridian plane, and the direction of one of those in the meridian plane is inclined to the radius vector r at an angle½ tan−1(2 − 4σtan θ).5 − 4σThe corresponding displacement at any point is compounded of a radial displacement of amount1 + σFcos θ2(1 − σ)4πErand a displacement parallel to the line of action of F of amount(3 − 4σ) (1 + σ)F1.2(1 − σ)4πErThe effects of forces applied at different points and in different directions can be obtained by summation, and the effect of continuously distributed forces can be obtained by integration.
Taking the case of a single force F applied at a point in the interior, we may show that the stress at a distance r from the point consists of
(1) a radial pressure of amount
(2) tension in all directions at right angles to the radius of amount
(3) shearing stress consisting of traction acting along the radiusdron the surface of the cone θ = const. and traction acting along the meridian dθ on the surface of the sphere r = const. of amount
where θ is the angle between the radius vector r and the line of action of F. The line marked T in fig. 30 shows the direction of the tangential traction on the spherical surface.
Thus the lines of stress are in and perpendicular to the meridian plane, and the direction of one of those in the meridian plane is inclined to the radius vector r at an angle
The corresponding displacement at any point is compounded of a radial displacement of amount
and a displacement parallel to the line of action of F of amount
The effects of forces applied at different points and in different directions can be obtained by summation, and the effect of continuously distributed forces can be obtained by integration.
72. The stress system considered in § 71 is equivalent, on the plane through the origin at right angles to the line of action of F, to a resultant pressure of magnitude ½F at the origin and a [1 − 2σ/2(1 − σ)] · F/4πr², and, by the application of this system of tractions to a solid bounded by a plane, the displacement just described would be produced. There is also another stress system for a solid so bounded which is equivalent, on the same plane, to a resultant pressure at the origin, and a radial traction proportional to 1/r², but these are in the ratio 2π : r−2, instead of being in the ratio 4π(1 − σ) : (1 − 2σ)r−2.
The second stress system (see fig. 31) consists of:(1) radial pressure F′r−2,(2) tension in the meridian plane across the radius vector of amountF′r−2cos θ / (1 + cos θ),(3) tension across the meridian plane of amountF′r−2/ (l + cos θ),(4) shearing stress as in § 71 of amountF′r−2sin θ / (1 + cos θ),and the stress across the plane boundary consists of a resultant pressure of magnitude 2πF′ and a radial traction of amount F′r−2. Ifthen we superpose the component stresses of the last section multiplied by 4(1 − σ)W/F, and the component stresses here written down multiplied by −(1 − 2σ)W/2πF′, the stress on the plane boundary will reduce to a single pressure W at the origin. We shall thus obtain the stress system at any point due to such a force applied at one point of the boundary.In the stress system thus arrived at the traction across any plane parallel to the boundary is directed away from the place where W is supported, and its amount is 3W cos²θ / 2πr². The corresponding displacement consists of(1) a horizontal displacement radially outwards from the vertical through the origin of amountW (1 + σ) sin θ(cos θ −1 − 2σ),2πEr1 + cos θ(2) a vertical displacement downwards of amountW (1 + σ){2 (1 − σ) + cos²θ }.2πErThe effects produced by a system of loads on a solid bounded by a plane can be deduced.
The second stress system (see fig. 31) consists of:
(1) radial pressure F′r−2,
(2) tension in the meridian plane across the radius vector of amount
F′r−2cos θ / (1 + cos θ),
(3) tension across the meridian plane of amount
F′r−2/ (l + cos θ),
(4) shearing stress as in § 71 of amount
F′r−2sin θ / (1 + cos θ),
and the stress across the plane boundary consists of a resultant pressure of magnitude 2πF′ and a radial traction of amount F′r−2. Ifthen we superpose the component stresses of the last section multiplied by 4(1 − σ)W/F, and the component stresses here written down multiplied by −(1 − 2σ)W/2πF′, the stress on the plane boundary will reduce to a single pressure W at the origin. We shall thus obtain the stress system at any point due to such a force applied at one point of the boundary.
In the stress system thus arrived at the traction across any plane parallel to the boundary is directed away from the place where W is supported, and its amount is 3W cos²θ / 2πr². The corresponding displacement consists of
(1) a horizontal displacement radially outwards from the vertical through the origin of amount
(2) a vertical displacement downwards of amount
The effects produced by a system of loads on a solid bounded by a plane can be deduced.
The results for a solid body bounded by an infinite plane may be interpreted as giving the local effects of forces applied to a small part of the surface of a body. The results show that pressure is transmitted into a body from the boundary in such a way that the traction at a point on a section parallel to the boundary is the same at all points of any sphere which touches the boundary at the point of pressure, and that its amount at any point is inversely proportional to the square of the radius of this sphere, while its direction is that of a line drawn from the point of pressure to the point at which the traction is estimated. The transmission of force through a solid body indicated by this result was strikingly demonstrated in an attempt that was made to measure the lunar deflexion of gravity; it was found that the weight of the observer on the floor of the laboratory produced a disturbance of the instrument sufficient to disguise completely the effect which the instrument had been designed to measure (see G.H. Darwin,The Tides and Kindred Phenomena in the Solar System, London, 1898).
73. There is a corresponding theory of two-dimensional systems, that is to say, systems in which either the displacement is parallel to a fixed plane, or there is no traction across any plane of a system of parallel planes. This theory shows that, when pressure is applied at a point of the edge of a plate in any direction in the plane of the plate, the stress developed in the plate consists exclusively of radial pressure across any circle having the point of pressure as centre, and the magnitude of this pressure is the same at all points of any circle which touches the edge at the point of pressure, and its amount at any point is inversely proportional to the radius of this circle. This result leads to a number of interesting solutions of problems relating to plane systems; among these may be mentioned the problem of a circular plate strained by any forces applied at its edge.
74. The results stated in § 72 have been applied to give an account of the nature of the actions concerned in the impact of two solid bodies. The dissipation of energy involved in the impact is neglected, and the resultant pressure between the bodies at any instant during the impact is equal to the rate of destruction of momentum of either along the normal to the plane of contact drawn towards the interior of the other. It has been shown that in general the bodies come into contact over a small area bounded by an ellipse, and remain in contact for a time which varies inversely as the fifth root of the initial relative velocity.
For equal spheres of the same material, with σ = ¼, impinging directly with relative velocity v, the patches that come into contact are circles of radius(45π)1⁄5(v)2⁄5r,256Vwhere r is the radius of either, and V the velocity of longitudinal waves in a thin bar of the material. The duration of the impact is approximately(2.9432)(2025π²)1⁄5r.512v1/5V4/5For two steel spheres of the size of the earth impinging with a velocity of 1 cm. per second the duration of the impact would be about twenty-seven hours. The fact that the duration of impact is, for moderate velocities, a considerable multiple of the time taken by a wave of compression to travel through either of two impinging bodies has been ascertained experimentally, and constitutes the reason for the adequacy of the statical theory here described.
For equal spheres of the same material, with σ = ¼, impinging directly with relative velocity v, the patches that come into contact are circles of radius
where r is the radius of either, and V the velocity of longitudinal waves in a thin bar of the material. The duration of the impact is approximately
For two steel spheres of the size of the earth impinging with a velocity of 1 cm. per second the duration of the impact would be about twenty-seven hours. The fact that the duration of impact is, for moderate velocities, a considerable multiple of the time taken by a wave of compression to travel through either of two impinging bodies has been ascertained experimentally, and constitutes the reason for the adequacy of the statical theory here described.
75.Spheres and Cylinders.—Simple results can be found for spherical and cylindrical bodies strained by radial forces.
For a sphere of radius a, and of homogeneous isotropic material of density ρ, strained by the mutual gravitation of its parts, the stress at a distance r from the centre consists of(1) uniform hydrostatic pressure of amount1⁄10gρa (3 − σ) / (1 − σ),(2) radial tension of amount1⁄10gρ (r²/a) (3 − σ) / (1 − σ),(3) uniform tension at right angles to the radius vector of amount1⁄10gρ (r²/a) (1 + 3σ) / (1 − σ),where g is the value of gravity at the surface. The corresponding strains consist of(1) uniform contraction of all lines of the body of amount1⁄30k−1gρa (3 − σ) / (1 − σ),(2) radial extension of amount1⁄10k−1gρ (r²/a) (1 + σ) / (1 − σ),(3) extension in any direction at right angles to the radius vector of amount1⁄30k−1gρ (r²/a) (1 + σ) / (1 − σ),where k is the modulus of compression. The volume is diminished by the fraction gρa/5k of itself. The parts of the radiivectorswithin the sphere r = a {(3 − σ) / (3 + 3σ)}1/2are contracted, and the parts without this sphere are extended. The application of the above results to the state of the interior of the earth involves a neglect of the caution emphasized in § 40, viz. that the strain determined by the solution must be small if the solution is to be accepted. In a body of the size and mass of the earth, and having a resistance to compression and a rigidity equal to those of steel, the radial contraction at the centre, as given by the above solution, would be nearly1⁄3, and the radial extension at the surface nearly1⁄6, and these fractions can by no means be regarded as “small.”76. In a spherical shell of homogeneous isotropic material, of internal radius r1and external radius r0, subjected to pressure p0on the outer surface, and p1on the inner surface, the stress at any point distant r from the centre consists of(1) uniform tension in all directions of amountp1r1³ − p0r0³,r0³ − r1³(2) radial pressure of amountp1− p0r0³r1³,r0³ − r1³r³(3) tension in all directions at right angles to the radius vector of amount½p1− p0r0³r1³.r0³ − r1³r³The corresponding strains consist of(1) uniform extension of all lines of the body of amount1p1r1³ − p0r0³,3kr0³ − r1³(2) radial contraction of amount1p1− p0r0³r1³,2μr0³ − r1³r³(3) extension in all directions at right angles to the radius vector of amount1p1− p0r0³r1³,4μr0³ − r1³r³where μ is the modulus of rigidity of the material, = ½E / (1 + σ). The volume included between the two surfaces of the body is increased by the fraction (p1r1³ − p0r0³) / k(r0³ − r1³) of itself, and the volume within the inner surface is increased by the fraction3 (p1− p0)r0³+p1r1³ − p0r0³4μr0³ − r1³k (r0³ − r1³)of itself. For a shell subject only to internal pressure p the greatest extension is the extension at right angles to the radius at the inner surface, and its amount ispr1³(1+1r0³);r0³ − r1³3k4μr1³the greatest tension is the transverse tension at the inner surface, and its amount is p (½ r0³ + r1³) / (r0³ − r1³).77. In the problem of a cylindrical shell under pressure a complication may arise from the effects of the ends; but when the ends are free from stress the solution is very simple. With notation similar to that in § 76 it can be shown that the stress at a distance r from the axis consists of(1) uniform tension in all directions at right angles to the axis of amountp1r1² − p0r0²,r0² − r1²(2) radial pressure of amountp1− p0r0²r1²,r0² − r1²r²(3) hoop tension numerically equal to this radial pressure.The corresponding strains consist of(1) uniform extension of all lines of the material at right angles to the axis of amount1 − σp1r1² − p0r0²,Er0² − r1²(2) radial contraction of amount1 + σp1− p0r0²r1²,Er0² − r1²r²(3) extension along the circular filaments numerically equal to this radial contraction,(4) uniform contraction of the longitudinal filaments of amount2σp1r1² − p0r0².Er0² − r1²For a shell subject only to internal pressure p the greatest extension is the circumferential extension at the inner surface, and its amount isp(r0² + r1²+ σ);Er0² − r1²the greatest tension is the hoop tension at the inner surface, and its amount is p (r0² + r1²) / (r0² − r1²).78. When the ends of the tube, instead of being free, are closed by disks, so that the tube becomes a closed cylindrical vessel, the longitudinal extension is determined by the condition that the resultant longitudinal tension in the walls balances the resultant normal pressure on either end. This condition gives the value of the extension of the longitudinal filaments as(p1r1² − p0r0²) / 3k (r0² − r1²),where k is the modulus of compression of the material. The result may be applied to the experimental determination of k, by measuring the increase of length of a tube subjected to internal pressure (A. Mallock,Proc. R. Soc. London, lxxiv., 1904, and C. Chree,ibid.).
For a sphere of radius a, and of homogeneous isotropic material of density ρ, strained by the mutual gravitation of its parts, the stress at a distance r from the centre consists of
(1) uniform hydrostatic pressure of amount1⁄10gρa (3 − σ) / (1 − σ),
(2) radial tension of amount1⁄10gρ (r²/a) (3 − σ) / (1 − σ),
(3) uniform tension at right angles to the radius vector of amount
1⁄10gρ (r²/a) (1 + 3σ) / (1 − σ),
where g is the value of gravity at the surface. The corresponding strains consist of
(1) uniform contraction of all lines of the body of amount
1⁄30k−1gρa (3 − σ) / (1 − σ),
(2) radial extension of amount1⁄10k−1gρ (r²/a) (1 + σ) / (1 − σ),
(3) extension in any direction at right angles to the radius vector of amount
1⁄30k−1gρ (r²/a) (1 + σ) / (1 − σ),
where k is the modulus of compression. The volume is diminished by the fraction gρa/5k of itself. The parts of the radiivectorswithin the sphere r = a {(3 − σ) / (3 + 3σ)}1/2are contracted, and the parts without this sphere are extended. The application of the above results to the state of the interior of the earth involves a neglect of the caution emphasized in § 40, viz. that the strain determined by the solution must be small if the solution is to be accepted. In a body of the size and mass of the earth, and having a resistance to compression and a rigidity equal to those of steel, the radial contraction at the centre, as given by the above solution, would be nearly1⁄3, and the radial extension at the surface nearly1⁄6, and these fractions can by no means be regarded as “small.”
76. In a spherical shell of homogeneous isotropic material, of internal radius r1and external radius r0, subjected to pressure p0on the outer surface, and p1on the inner surface, the stress at any point distant r from the centre consists of
(1) uniform tension in all directions of amount
(2) radial pressure of amount
(3) tension in all directions at right angles to the radius vector of amount
The corresponding strains consist of
(1) uniform extension of all lines of the body of amount
(2) radial contraction of amount
(3) extension in all directions at right angles to the radius vector of amount
where μ is the modulus of rigidity of the material, = ½E / (1 + σ). The volume included between the two surfaces of the body is increased by the fraction (p1r1³ − p0r0³) / k(r0³ − r1³) of itself, and the volume within the inner surface is increased by the fraction
of itself. For a shell subject only to internal pressure p the greatest extension is the extension at right angles to the radius at the inner surface, and its amount is
the greatest tension is the transverse tension at the inner surface, and its amount is p (½ r0³ + r1³) / (r0³ − r1³).
77. In the problem of a cylindrical shell under pressure a complication may arise from the effects of the ends; but when the ends are free from stress the solution is very simple. With notation similar to that in § 76 it can be shown that the stress at a distance r from the axis consists of
(1) uniform tension in all directions at right angles to the axis of amount
(2) radial pressure of amount
(3) hoop tension numerically equal to this radial pressure.
The corresponding strains consist of
(1) uniform extension of all lines of the material at right angles to the axis of amount
(2) radial contraction of amount
(3) extension along the circular filaments numerically equal to this radial contraction,
(4) uniform contraction of the longitudinal filaments of amount
For a shell subject only to internal pressure p the greatest extension is the circumferential extension at the inner surface, and its amount is
the greatest tension is the hoop tension at the inner surface, and its amount is p (r0² + r1²) / (r0² − r1²).
78. When the ends of the tube, instead of being free, are closed by disks, so that the tube becomes a closed cylindrical vessel, the longitudinal extension is determined by the condition that the resultant longitudinal tension in the walls balances the resultant normal pressure on either end. This condition gives the value of the extension of the longitudinal filaments as
(p1r1² − p0r0²) / 3k (r0² − r1²),
where k is the modulus of compression of the material. The result may be applied to the experimental determination of k, by measuring the increase of length of a tube subjected to internal pressure (A. Mallock,Proc. R. Soc. London, lxxiv., 1904, and C. Chree,ibid.).
79. The results obtained in § 77 have been applied to gun construction; we may consider that one cylinder is heated so as to slip over another upon which it shrinks by cooling, so that the two form a single body in a condition of initial stress.