STRENGTH OF MATERIALS
1. The name “strength of materials” is given to that part of the theory of engineering which deals with the nature and effects of stresses in the parts of engineering structures. Its principal object is to determine the proper size and form of pieces which have to bear given loads, or, conversely, to determine the loads which can be safely applied to pieces whose dimensions and arrangement are already given. It also treats of the relation between the applied loads and the changes of form which they cause.
The subject comprises experimental investigation of the properties of materials as to strength and elasticity, and mathematical discussion of the stresses in ties, struts, beams, shafts, and other elements of structures and machines.
2. Stress is the mutual action at the surface of contact between two bodies, or two imaginary parts of a body, whereby each of the two exerts a force upon the other. Thus, when a stone lies on the ground there is at the surface of contact a stress, one aspect of which is the force directed downwards with which the stone pushes the ground, and the other aspect is the equal force directed upwards with which the ground pushes the stone. A body is said to be in a state of stress when there is a stress between the two parts which lie on opposite sides of an imaginary surface of section. A pillar or block supporting a weight is in a state of stress because at any cross section the part above the section pushes down against the part below, and the part below pushes up against the part above. A stretched rope is in a state of stress, because at any cross section the part on each side is pulling the part on the other side with a force in the direction of the rope’s length. A plate of metal that is being cut in a shearing machine is in a state of stress, because at the plane which is about to become the plane of actual section the portion of metal on each side is tending to drag the portion on the other side with a force in that plane.
3. In a solid body which is in a state of stress the direction of stress at an imaginary surface of division may be normal, oblique, or tangential to the surface. When oblique it is often conveniently treated as consisting of a normal and a tangential component. Normal stress may be either push (compressive stress) or pull (tensile stress). Stress which is tangential to the surface is called shearing stress. Oblique stress may be regarded as so much push or pull along with so much shearing stress. The amount of stress per unit of surface is called the intensity of stress. Stress is said to be uniformly distributed over a surface when each fraction of the area of surface bears a corresponding fraction of the whole stress. If a stress P is uniformly distributed over a plane surface of area S, the intensity is P/S. If the stress is not uniformly distributed, the intensity at any point is δP∕δS, where δP is the amount of stress on an indefinitely small area δS at the point considered. For practical purposes intensity of stress is usually expressed in tons weight per square inch, pounds weight per square inch, or kilogrammes weight per square millimetre or per square centimetre. Οne ton per sq. in.=2240 lb per sq. in.= 1·511 kilos. per sq. mm.
4. The simplest possible state of stress is that of a short pillar or block compressed by opposite forces applied at its ends, or that of a stretched rope or other tie. In these cases the stress is wholly in one direction, that of the length. These states may be distinguished as simple longitudinal push and simple longitudinal pull. In them there is no stress on planes parallel to the direction of the applied forces.
A more complex state of stress occurs if the block is compressed or extended by forces applied to a pair of opposite sides, as well as by forces applied to its ends,—that is to say, if two simple longitudinal stresses in different directions act together. A still more complex state occurs if a third stress be applied to the remaining pair of sides. It may be shown that any state of stress which can possibly exist at any point of a body may be produced by the joint action of three simple pull or push stresses in three suitably chosen directions at right angles to each other. These three are called principal stresses, and their directions are called the axes of principal stress. These axes have the important property that the intensity of stress along one of them is greater, and along another it is less, than in any other direction.
These are called respectively the axes of greatest and least principal stress.
5. Returning now to the case of a single simple longitudinal stress, let AB (fig. 1) be a portion of a tie or a strut which is being pulled or pushed in the direction of the axis AB with a total stress P. On any plane CD taken at right angles to the axis we have a normal pull or push of intensity p =P∕S, S being the area of the normal cross-section. On a plane EF whose normal is inclined to the axis at an angle θ we have a stress still in the direction of the axis, and therefore oblique to the plane EF, of intensity P∕S', where S' is the area of the surface EF, or S/cos θ. The whole stress P on EF may be resolved into two components, one normal to EF, and the other a shearing stress tangential to EF. The normal component (P n , fig. 2) is PcosØ; the tangential component (P t ) is Psin θ . Hence the intensity of normal pull or push on EF, or p n , is p cos 2 θ , and the intensity of shearing stress EF, or p t , is p sin θ cos θ .
This expression makes p t a maximum when 0 = 45°: planes inclined at 45° to tho axis are called planes of maximum shearing stress; the intensity of shearing stress on them is ½ p .
6. Shearing stress in one direction is necessarily accompanied by an equal intensity of shearing stress in another direction at right angles to the first. To prove this it is sufficient to consider the equilibrium of an indefinitely small cube (fig. 3), with one pair of sides parallel to the direction of the shearing stress P t . This stress, acting on two opposite sides, produces a couple which tends to rotate the cube. No arrangement of normal stresses on any of the three pairs of sides of the cube can balance this couple; that can be done only by a shearing stress Q t whose direction is at right angles to the first stress P t and to the surface on which P t acts, and whose intensity is the same as that of P t . The shearing stresses P t and Q t may exist alone, or as components of oblique stress.
7. If they exist alone, the material is said to be in a state of simple shearing stress. This state of stress may be otherwise described by reference to the stresses on diagonal planes of the cube ABCD. Thus P t aud Q t produce a normal stress R on a diagonal plane, and the equilibrium of the triangular prism
(fig. 4) requires that
R=P t √2. But R acts on a surface which is greater than each of the sides in the ratio of √2: 1. The inten sity of normal stress on the diagonal plane
AC is therefore the same as the intensity of shearing stress on AB or BC. The same considerations apply to the other diagonal plane BD at right angles to AC, with this difference, that the stress on it is normal pull instead of push. Hence we may regard a state of simple shearing stress as compounded of two simple longitudinal stresses, one of push and one of pull, at right angles to each other, of equal intensity, and inclined at 45° to the direction of the shearing stress.
8. Strain is the change of shape produced by stress. If the stress is a simple longitudinal pull, the strain consists of lengthening in the direction of the pull, accompanied by contraction in both directions at right angles to the pull. If the stress is a simple push, the strain consists of shortening in the direction of the push and expansion in both directions at right angles to that; the stress and the strain are then exactly the reverse of what they are in the case of simple pull. If the stress is one of simple shearing, the strain consists of a distortion such as would be produced by the sliding of layers in the direction of the shearing stresses.
A material is elastic with regard to any applied stress if the strain disappears when the stress is removed. Strain which persists after the stress that produced it is removed is called permanent set. For brevity, it is convenient to speak of strain which disappears when the stress is removed as elastic strain.
9. Actual materials are generally very perfectly elastic with regard to small stresses, and very imperfectly elastic with regard to great stresses. If the applied stress is less than a certain limit, the strain is in general small in amount, and disappears wholly or almost wholly when the stress is removed. If the applied stress exceeds this limit, the strain is, in general, much greater than before, and the principal part of it is found, when the stress is removed, to consist of permanent set. The limits of stress within which strain is wholly or almost wholly elastic are called limits of elasticity.
For any particular mode of stress the limit of elasticity is much more sharply defined in some materials than in others. When well defined it may readily be recognized in the testing of a sample from the fact that after the stress exceeds the limit of elasticity the strain begins to increase in a much more rapid ratio to the stress than before. This characteristic goes along with the one already mentioned, that up to the limit the strain is wholly or almost wholly elastic.
10. Within the limits of elasticity the strain produced by a stress of any one kind is proportional to the stress producing it. This is Hooke’s law, enunciated by him in 1676.
In applying Hooke’s law to the case of simple longitudinal stress, —such as the case of a bar stretched by simple longitudinal pull,— we may measure the state of strain by the change of length per unit of original length which the bar undergoes when stressed. Let the original length be l, and let the whole change of length be δ l when a stress is applied whose intensity p is within the elastic limit. Then the strain is measured by Sl∕l, and this by Hooke’s law is proportional to ρ. This may be written δl: l :: p : E, where E is a constant for the particular material considered. The same value of E applies to push and to pull, these modes of stress being essentially continuous, and differing only in sign.
11. This constant E is called the modulus of longitudinal extensibility, or Young’s modulus. Its value, which is expressed in the same units as are used to express intensity of stress, may be measured directly by exposing a long sample of the material to longitudinal pull and noting the extension, or indirectly by measuring the flexure of a loaded beam of the material, or by experiments on the frequency of vibrations. It is frequently spoken of by engineers simply as the modulus of elasticity, but this name is too general, as there are other moduluses applicable to other modes of stress. Since E =pl/δl, the modulus may be defined as the ratio of the intensity of stress p to the longitudinal strain δl∕l .
12. In the case of simple shearing stress, the strain may be measured by the angle by which each of the four originally right angles in the square prism of fig. 3 is altered by the distortion of the prism. Let this angle be φ in radians; then by Hooke’s law p∣φ=C, where p is the intensity of shearing stress and C is a constant which measures the rigidity of the material. C is called the modulus of rigidity, and is usually determined by experiments on torsion.
13. When three simple stresses of equal intensity p and of the same sign (all pulls or all pushes) are applied in three directions, the material (provided it be isotropic, that is to say, provided its properties are the same in all directions) suffers change of volume only, without distortion of form. If the volume is V and the change of volume δV, the ratio of the stress p to the strain δV∕Y is called the modulus of cubic compressibility, and will be denoted by K. The state of stress here considered is the only one possible in a fluid at rest. The intensity of stress is equal in all directions.
14. Of these three moduluses the one of most importance in engineering applications is Young’s modulus E. When a simple longitudinal pull or push of intensity p is applied to a piece, the longitudinal strain of extension or compression is p ∕E. This is accompanied by a lateral contraction or expansion, in each transverse direction, whose amount may be written p ∕σE, where σ is the ratio of longitudinal to lateral strain. It is shown in the article Elasticity, § 47, that E= 9CK / 3K+C and σ= 2(3K+C) / 3K-2C
15. Beyond the limits of elasticity the relation of strain to stress becomes very indefinite. Materials then exhibit, to a greater or less degree, the property of plasticity. The strain is much affected by the length of time during which the stress has been in operation, and reaches its maximum, for any assigned stress, only after a long (probably an indefinitely long) time. Finally, when the stress is sufficiently increased, the ratio of the increment of strain to the increment of stress becomes indefinitely great if time is given for the stress to take effect. In other words, the substance then assumes what may be called a completely plastic state; it flows under the applied stress like a viscous liquid.
16. The ultimate strength of a material with regard to any stated mode of stress is the stress required to produce rupture. In reckoning ultimate strength, however, engineers take, not the actual intensity of stress at which rupture occurs, but the value which this intensity would have reached had rupture ensued without previous alteration of shape. Thus, if a bar whose original cross-section is 2 square inches breaks under a uniformly distributed pull of 60 tons, the ultimate tensile strength of the material is reckoned to be 30 tons per square inch, although the actual intensity of stress which produced rupture may have been much greater than this, owing to the contraction of the section previous to fracture. The convenience of this usage will be obvious from an example. Suppose that a piece of material of the same quality he used in a structure under conditions which cause it to bear a simple pull of 6 tons per square inch; we conclude at once that the actual load is one-fifth of that which would cause rupture, irrespective of the extent to which the material might contract in section if overstrained. The stresses which occur in engineering practice are, or ought to be, in all cases within the limits of elasticity, and within these limits the change of cross-section caused by longitudinal pull or push is so small that it may be neglected in reckoning the intensity of stress.
Ultimate tensile strength and ultimate shearing strength are well defined, since these modes of stress (simple pull and simple shearing stress) lead to distinct fracture if the stress is sufficiently increased. Under compression some materials yield so continuously that their ultimate strength to resist compression can scarcely be specified; others show so distinct a fracture by crushing (§ 43 below) that their compressive strength may be determined with some precision. In what follows, the three kinds of ultimate strength will be designated by the symbols f t , f s , and f c , for tension, shearing, and crushing respectively.
Some of the materials used in engineering, notably timber and wrought-iron, are so far from being isotropic that their strength is widely different for stresses in different directions. In the case of wrought-iron the process of rolling develops a fibrous structure on account of the presence of streaks of slag which become interspersed with the metal in puddling; and the tensile strength of a rolled plate is found to be considerably greater in the direction of rolling than across the plate. Steel plates, being rolled from a nearly homogeneous ingot, have nearly the same strength in both directions.
17. In applying a knowledge of the ultimate strength of materials to determine the proper sizes of parts in an engineering structure, these parts are proportioned so that the greatest intensity of stress (which for brevity is called the working stress) will be only a certain fraction of the ultimate strength. The ratio ultimate strength / working stress is called the factor of safety. The choice of a factor of safety depends on many considerations, such as the probable accuracy of the theory on which the calculation of working stress has been based; the uniformity of the material dealt with, and the extent to which its strength may be expected to conform to the assumed value or to the values determined by experiments on samples; the deviations from the specified dimensions which may be caused by bad workmanship; the probable accuracy in the estimation of loads; the extent to which the materials will deteriorate in time. The factor is rarely less than 3, is very commonly 4 or 5, and is sometimes as much as 12, or even more.
The ultimate strength for any one mode of stress, such as simple pull, has been found to depend on the time rate at which stress is applied; this will be noticed more fully later (§§ 28-34). It has also been found to depend very greatly on the extent and frequency of variation in the applied stress. A stress considerably less than the normal ultimate strength will suffice to break a piece when it is frequently applied and removed; a much smaller stress will cause rupture if its sign is frequently reversed; and hence in a structure which has to bear what is called live load the permissible intensity of stress is less than in a structure which has to bear only load and also on its frequency of variation (§§ 45, 46 below).
18. From an engineering point of view, the structural merit of a material, especially when live loads and possible shocks have to be sustained, depends not only on the ultimate strength but also on the extent to which the material will bear deformation without rupture. This characteristic is shown in tests made to determine tensile strength by the amount of ultimate elongation, and also by the contraction of the cross-section which occurs through the flow of the metal before rupture. It is often tested in other ways, such as by bending and unbending bars in a circle of specified radius, or by examining the effect of repeated blows. Tests by impact are generally made by causing a weight to fall through a regulated distance on a piece of the material supported as a beam.
19. Ordinary tests of strength are made by submitting the piece to direct pull, direct compression, bending, or torsion. Testing machines are frequently arranged so that they may apply any of these four modes of stress; tests by direct tension are the most common, and next to them come tests by bending. When the samples to be tested for tensile strength are mere wires, the stress may be applied directly by weights; for pieces of larger section some mechanical multiplication of force becomes necessary. Owing to the plasticity of the materials to be tested, the applied loads must be able to follow considerable change of form in the test-piece: thus in testing the tensile strength of wrought-iron or steel provision must be made for taking up the large extension of length which occurs before fracture. In most modern forms of large testing machines the loads are applied by means of hydraulic pressure acting on a piston or plunger to which one end of the specimen is secured, and the stress is measured by connecting the other end to a lever or system of levers provided with adjustable weights. In small machines, and also in some large ones, the stress is applied by screw gearing instead of by hydraulic pressure. Springs are sometimes used instead of weights to measure the stress, and another plan is to make one end of the specimen act on a diaphragm forming part of a hydrostatic pressure-gauge (§ 23 below).
20. Figs. 5 and 6 show an excellent form of single-lever testing machine designed by Mr J. H. Wicksteed, in which the stress is applied by an hydraulic plunger and is measured by a lever or steelyard and a movable weight. The illustration shows a 30-ton machine, but machines of similar design have been built to exert a force of 100 tons or more. AA is the lever, on which there is a graduated scale. The stress on the test-piece T is measured by a weight W of 1 ton (with an attached vernier scale), which is moved along the lever by a screw-shaft S; this screw-shaft is driven by a belt from a parallel shaft R, which takes its motion, through bevel-wheels and a Hooke’s joint in the axis of the fulcrum, from the hand-wheel H. (The Hooke’s joint in the shaft R is shown in a separate sketch above the lever in fig. 6.) The holder for the upper end of the sample hangs from a knife-edge three inches from the fulcrum of the lever. The lower holder is jointed to a crosshead C, which is connected by two vertical screws to a lower crosshead B, upon which the hydraulic plunger P, shown in section in fig. 5, exerts its thrust. G is a counterpoise which pushes up the plunger, when the water is allowed to escape. Hydraulic pressure may be applied to P by pumps or by an accumulator. In the present instance it is applied by means of an auxiliary plunger Q, which is pressed by screw gearing into an auxiliary cylinder. Q is driven by a belt on the pulley D. This puts stress on the specimen, and the weight W is then run out along the lever so that the lever is just kept floating between the stops E, E. Before the test-piece is put in, the distance between the holders is regulated by means of the screws connecting the upper and lower crossheads C and B, these screws being turned by a handle applied at F. Fig. 7 is a section of one of the holders, showing how the test-piece T is gripped by serrated wedges. The knife-edges are made long enough to prevent the load on them ever exceeding 5 tons to the linear inch.
21. Another example of the single-lever type is the Werder testing machine, much used on the continent of Europe. In it the specimen is horizontal; one end is fixed, the other is attached to the short vertical arm of a bell-crank lever, whose fulcrum is pushed out horizontally by an hydraulic ram. In many other testing machines a system of two, three, or more levers is employed to reduce the force between the specimen and the measuring weight. Probably the earliest machine of this class was that of Major Wade, in which one end of the specimen was held in a fixed support, and the stretch was taken up by screwing up the fulcrum plate of one of the levers. In most multiple-lever machines, however, the fulcrums are fixed, and the stress is applied to one end of the specimen by hydraulic power or by screw gearing, which of course takes up the stretch, as in the single-lever machines already described. Mr Kirkaldy, who was one of the earliest as well as one of the most assiduous workers in this field, applies in his 1,000,000 fb machine a horizontal hydraulic press directly to one end of the horizontal test-piece. The other end of the piece is connected to the short vertical arm of a bell-crank lever; the long arm of this lever is horizontal, and is connected to a second lever to which weights are applied. In some of Messrs Fairbanks’s machines the multiple-lever system is carried so far that the point of application of the weight moves 24,000 times as far as the point of attachment to the test-piece. The same makers have employed a plan of adjusting automatically the position of the measuring weight, by making the scale lever complete an electric circuit when it rises or falls so that it starts an electric engine which runs the weight out or in. Generally the measuring weight is adjusted by hand. In some, chiefly small, machines, the weight adjusts itself by means of another device. It is fixed at one point of a lever which is arranged as a pendulum, so that, when the test-piece is pulled by force applied at the other end, the pendulum lever is deflected from its originally vertical position and the weight acts with increasing leverage.
Multiple-lever machines have the advantage that the measuring weight is reduced to a conveniently small value, and that it can be easily varied to suit test-pieces of different strengths. On tho other hand, their multiplicity of joints makes the leverage somewhat uncertain and increases friction. Another drawback is the inertia of the working parts. It is impossible to avoid oscillations of the levers: and, to prevent them from producing important errors in the recorded stress, the inertia of the oscillating system should be minimized. In a testing machine in which the specimen is directly loaded the inertia is simply that of the suspended weight M. In a lever machine, which multiplies the weight n times, the weight applied to the lever is reduced to M ∕n , but its inertia, when referred to the test-piece, is (M /n) × n 2 or Mn. The inertia which is effective for producing oscillation is thus increased n times, so far as the weight alone is concerned, and this detrimental effect of leverage is increased by the inertia of the levers themselves. Tho effect will be more serious the greater is the leverage n.
22. Whitworth and others employ machines in which one end of the specimen is held in a fixed support; an hydraulic press acts on the other end, and the stress is calculated from the pressure of fluid in the press, this being observed by a pressure-gauge. Machines of this class are open to the obvious objection that the friction of the hydraulic plunger causes a large and very uncertain difference between the force exerted by the fluid on the plunger and the force exerted by the plunger on the specimen. It appears, however, that in the ordinary conditions of packing the friction is very nearly proportional to the fluid pressure, and its effect may therefore be allowed for with some exactness. The method is not to be recommended for work requiring precision, unless the plunger be kept in constant rotation on its own axis during the test, in which case the effects of friction are almost entirely eliminated.
23. In another important class of testing machines the stress (applied as before to one end of the piece, by gearing or by hydraulic pressure)
is measured by con necting the other end to a flexible diaphragm, on which a liquid acts whose pressure is determined by a gauge.
Fig. 8 shows a simple machine of this class (used in 1873 for testing wire by Sir W. Thomson and the late Prof. F.
Jenkin). The wire is stretched by means of a screw at the top, and pulls up the lower side of a hydrostatic bellows; water from the bellows rises in the gauge-tube G, and its height measures the stress. Fig. 9 is Thomasset's testing machine, in which one end of the specimen is pulled by an hydraulic press A. The other end acts through a bell-crank lever B on a horizontal diaphragm C, consisting of a metallic plate and a flexible ring of india-rubber. The pressure on the diaphragm causes a column of mercury to rise in the gauge-tube D. The same principle is made use of in the testing machines of Chauvin and Marin-
Darbel, Maillard, and Bailey. It has found its most important application in the remarkable testing machine of Watertown arsenal, built in 1879 by the U.S. Government to the designs of Mr A. H. Emery. This is a horizontal machine, taking specimens of any length up to 30 feet, and exerting a pull of 360 tons or a push of 480 tons by an hydraulic press at one end. The stress is taken at the other end by a group of four large vertical diaphragm presses, which communicate by small tubes with four similar small diaphragm presses in the scale case. The pressure of these acts on a system of levers which terminates in the scale beam. The joints and bearings of all the levers are made frictionless by using flexible steel connecting plates instead of knife-edges. The total multiplication at the end of the scale beam is 420,000.
24. The results of tests are very commonly exhibited by means of stress-strain diagrams, or diagrams showing the relation of strain to stress. A few typical diagrams for wrought-iron and steel in tension are given in fig. 10, the data for which are taken from tests of long rods by Mr Kirkaldy. Up to the elastic limit these diagrams show sensibly the same rate of extension tor all the materials to which they refer. Soon after the limit of elasticity is passed, a point, which has been called by Prof. Kennedy the yield-point, is reached, which is marked by a very sudden extension of the specimen. After this the extension becomes less rapid; then it continues at a fairly regular and gradually increasing rate; near the point of rupture the metal again begins to draw out rapidly.
When this stage is reached rupture will occur through the flow of the metal, even if the load be somewhat decreased. The diagram may in this way be made to come back towards the line of no load, by withdrawing a part of the load as the end of the test is approached (§ 29 below).
25. Fig. 11 is a stress-strain diagram for cast-iron in extension and compression, taken from Hodgkinson’s experiments. The extension was measured on a rod 50 feet long; the compression was also measured on a long rod, which was prevented from buckling by being supported in a trough with partitions. The full line gives the strain produced by loading; it is continuous through the origin, showing that Young’s modulus is the same for pull and push. (Similar experiments on wrought-iron and steel in extension and compression have given the same result.)
The broken line shows the set produced by each load. Hodgkinson found that some set could be detected after even the smallest loads had been applied. This is probably due to the existence of initial internal stress in the metal, produced by unequally rapid cooling in different portions of the cast bar. A second loading of the same piece showed a much closer approach to perfect elasticity. The elastic limit is, at the best, ill defined; but by the time the ultimate load is reached the set has become a more considerable part of the whole strain. The pull curves in the diagram extend to the point of rupture; the compression curves are drawn only up to a stage at which the bar buckled (between the partitions) so much as to affect the results.
26. Testing machines are now frequently fitted with autographic appliances for drawing strain diagrams. When the load is measured by a weight travelling on a steelyard, the diagram may be drawn by connecting the weight with a drum by means of a wire or cord, so that the drum is made to revolve through angles proportional to the travel of the weight. At the same time another wire, fastened to a clip near one end of the specimen, and passing over a pulley near the other end, draws a pencil through distances proportional to the strain, and so traces a diagram of stress and strain on a sheet of paper stretched round the drum. Apparatus of this kind has been used by Messrs Fairbanks, Unwin, Aspinall, and others. In Mr Wicksteed’s autographic recorder the stress is determined by reference, not to the load on the lever, but to the pressure in the hydraulic cylinder by which stress is applied. The main cylinder is in communication with a small auxiliary hydraulic cylinder, the plunger of which is kept rotating to avoid friction at its packing. This plunger abuts against a spring, so that the distance through which it is pushed out varies with the pressure in the main cylinder. A drum covered with paper moves with the plunger under a fixed pencil, and is also caused to rotate by a wire from the specimen through distances proportional to the strain. The scale of loads is calibrated by occasional reference to the weighted lever. In Prof. Kennedy’s machine autographic diagrams are drawn by applying the stress to the test-piece through an elastic master-bar of larger section. The master-bar is never strained beyond its elastic limit, and within that limit its extension furnishes an accurate measure of the stress; this gives motion to a pencil, which writes on a paper moved by the extension of the test-piece. In Prof. Thurston’s pendulum machine for torsion tests, a cam attached to the pendulum moves a pencil through distances proportional to the stress, while a paper drum attached to the other end of the test-piece turns under the pencil through distances proportional to the angle of twist.
27. The elastic extension or compression of a test-piece of ordinary dimensions is so small as to require for its measurement refined methods which are seldom employed in everyday practical testing. Measurements of this class must be made simultaneously on opposite sides of the test-piece, to guard against error through the bending of the piece. Microscopes and also various forms of micrometer calipers are used for the purpose. A method capable of great delicacy, which has been used by Bauschinger and others, is to measure the strain by light reflected from a pair of small mirrors attached to rollers which turn as the specimen extends or contracts. With apparatus of this kind it may be shown that iron, steel, or other materials with a well-defined yield-point begin to show a marked defect of elasticity at a somewhat lower stress. The true elastic limit comes considerably earlier in the test than the point which usually passes by that name.
28. In testing a plastic material such as wrought-iron or mild steel it is found that the behaviour of the metal depends very materially on the time rate at which stress is applied. When once the elastic limit is passed the full strain corresponding to a given load is reached only after a perceptible time, sometimes even a long time. If the load be increased to a value exceeding the elastic limit, and then kept constant, the metal will be seen to draw out (if the stress be one of pull), at first rapidly and then more slowly. When the applied load is considerably less than the ultimate strength of the piece (as tested in the ordinary way by steady increment of load), it appears that this process of slow extension comes at last to an end. On the other hand, when the applied load is nearly equal to the ultimate strength, the flow of the metal continues until rupture occurs. Then, as in the former case, extension goes on at first quickly, then slowly, but, finally, instead of approaching an asymptotic limit, it quickens again as the piece approaches rupture. The same phenomena are observed in the bending of timber and other materials when in the form of beams. If, instead of being subjected to a constant load, a test-piece is set in a constant condition of strain, it is found that the stress required to maintain this constant strain gradually decreases.
29. The gradual flow which goes on under, constant stress— approaching a limit if the stress is moderate in amount, and continuing without limit if the stress is sufficiently great—will still go on at a diminished rate if the amount of stress be reduced. Thus, in the testing of soft iron or mild steel by a machine in which the stress is applied by hydraulic power, a stage is reached soon after the limit of elasticity is passed at which the metal begins to flow with great rapidity. The pumps often do not keep pace with this, and the result is that, if the lever is to be kept floating, the weight on it must be run back. Under this reduced stress the flow continues, more slowly than before, until presently the pumps recover their lost ground and the increase of stress is resumed. Again, near the point of rupture, the flow again becomes specially rapid; the weight on the lever has again to be run back, and the specimen finally breaks under a diminished load.
These features are well shown by fig. 12, which is copied from the autographic diagram of a test of mild steel.
30. But it is not only; through what we may call the viscosity of materials that the time rate of loading affects their behaviour under test. In iron and steel, and probably in some other metals, time has another effect of a very remarkable kind. Let the test be carried to any point a (fig. 13) past the original limit of elasticity. Let the load then be removed; during the first stages of this removal the material continues to stretch slightly, as has been explained above. Let the load then be at once replaced and loading continued. It will then be found that there is a new yield-point b at or near the value of the load formerly reached; up to this point there is little other than elastic strain.
The full line be in fig. 13 shows the subsequent behaviour of the piece. But now let the experiment be repeated on another sample, with this difference, that an interval of time, of a few hours or more, is allowed to elapse after the load is removed and before it is replaced. It will then be found that a process of hardening has been going on during this interval of rest; for, when the loading is continued, the new yield-point appears, not at b as formerly, but at a higher load d.
Other evidence that a change has taken place is afforded by the fact that the ultimate extension is reduced and the ultimate strength is increased (c, fig. 13).
31. A similar aud even more marked hardening occurs when a load (exceeding the original elastic limit), instead of being removed and replaced, is kept on for a sufficient length of time without change. When loading is resumed a new yield-point is found only after a considerable addition has been made to the load. The result is, as in the former case, to give greater ultimate strength and less ultimate elongation, Fig. 14 exhibits two experiments of this kind, made with annealed iron wire. A load of 23½ tons per square inch was reached in both cases; ab shows the result of continuing to load after an interval of five minutes, and acd after an interval of 45½ hours, the stress of 23½ tons being maintained during the interval in both cases.
32. It must not be supposed that in a material hardened by strain the elasticity is perfect up to the yield-points which are shown in fig. 13 at b and d or in fig. 14 at c. In experiments made for this article, it has been found that, after a piece of very soft iron wire has been hardened (as in fig. 14) by the continued application of a load which had caused stretching, if a small addition be made to the load (bringing it to a value between a and the new yield-point), although there is at first no apparent drawing out, nevertheless if time be given the wire begins again to draw, and a large amount of stretching at an increased pace may ensue. In this way wires have been broken with loads considerably short of those which would have been required had the process of loading, from the point a onward, been continued at a moderately rapid rate. A slow process of viscous deformation may in fact be occurring at the same time that the metal shows a quasi-elasticity with respect to rapid alteration of stress. Bauschinger’s micrometric experiments have shown that after a piece has been hardened by rest the true limit of elasticity, or the point at which Hooke’s law begins to fail, comes far short of the yield-point. He has also shown that a long interval of rest after the set has taken place produces a slow rise of the true limit of elasticity, apparently a slower rise than the lapse of time causes in the yield-point itself.
33. In the testing of iron and steel the time during which any state of (pull) stress (exceeding the original elastic limit) exists affects the result in two somewhat antagonistic ways. It augments extension, by giving the metal leisure to flow. This may be called the viscous effect. But, on the other hand, it reduces the amount of extension which subsequent greater loads will cause, and it increases the amount of load required for rupture in the way which has just been described. This may be called the hardening effect. When a piece is broken by continuous gradual increment of load, these two effects are occurring at all stages of the test. If the viscous effect existed alone, or if the hardening effect were small, the material would show to greater advantage as regards elongation, and to less advantage as regards ultimate strength, the more slowly the load were applied. Tin and lead may be cited as materials for which this is the case. But when the hardening effect is relatively great, as in iron and steel, the material shows less elongation and a higher breaking strength the more slowly it is tested. An excellent illustration of this is given by the following experiment of Mr Bottomley. Pieces of iron wire, annealed and of exceptionally soft quality, when loaded at the rate of 1 lb in 5 minutes, broke with 44¼lb and stretched 27 per cent. of their original length. Other pieces of the same wire, loaded at the rate of 1 lb in 24 hours, broke with 47 lb and stretched less than 7 per cent. Again, it has been found that an excessively rapid application of stress (by the explosion of gun-cotton) makes soft steel stretch twice as much as in ordinary testing. The case is very different, however, if the material has been previously hardened by strain. It does not appear that such variations in the rate of loading as are liable to occur in practical tests of iron or steel have much influence on the extension or the strength, great as the effects of time are when the metal is loaded either much more slowly or much more quickly. In fig. 15 the results are shown of tests of two similar pieces of soft iron wire, one loaded to rupture in 4 minutes and the other at a rate about 5000 times slower.
34. The hardening effect which intervals of rest ... , . from load or of constant load produce, once the primitive elastic limit is passed, has been examined by Beardsley, Thurston, Bauschinger, Ewing, and others. The effect of even a few minutes pause is perceptible, an hour or two of constant stress has a very marked influence, and after 24 hours or so there appears to be little further hardening. The American Board found that iron bars, previously stressed to about 50,000 lb per square inch, gained in strength, by intervals of rest from stress, to the extent of about 9 per cent. in one day, 16 per cent. in three days, and 18 per cent. in six months.
35. It may be concluded that, when a piece of metal has in any way received a permanent set by stress exceeding its limits of elasticity, it is hardened, and (in some cases at least) its physical properties go on slowly changing for days or even months. Instances of the hardening effect of permanent set occur when plates or bars are rolled cold, hammered cold, or bent cold, or when wire is drawn. When a hole is punched in a plate the material contiguous to the hole is severely distorted by shear, and is so much hardened in consequence that when a strip containing the punched hole is broken by tensile stress the hardened portion, being unable to extend so much as the rest, receives an undue proportion of the stress, and the strip breaks with a smaller load than it would have borne had the stress been uniformly distributed. This bad effect of punching is especially noticeable in thick plates of mild steel. It disappears when a narrow ring of material surrounding the hole is removed by means of a rimer, so that the material that is left is homogeneous. Another remarkable instance of the same kind of action is seen when a mild-steel plate which is to be tested by bending has a piece cut from its edge by a shearing machine. The result of the shear is that the metal close to the edge is hardened, and, when the plate is bent, this part, being unable to stretch like the rest, starts a crack or tear which quickly spreads across the plate on account of the fact that in the metal at the end of the crack there is an enormously high local intensity of stress (see Elasticity, § 72). By the simple expedient of planing off the hardened edge before bending the plate homogeneity is restored, and the plate will then bend without damage.
36. The hardening effect of strain is removed by the process of annealing, that is, by heating to redness and cooling slowly. In iron, very mild steel, and most other metals the rate of cooling is a matter of indifference; but in steel that contains more than about 0·2 per cent. of carbon another kind of hardening is produced if the metal, after being heated to redness, is cooled suddenly. When the proportion of carbon is considerably greater than this, steel may be rendered excessively hard and brittle (“glass-hard ”) by sudden cooling from a red heat. Further, by being subsequently heated to a moderate temperature, it may be deprived of some of this hardness and rendered elastic through a wide range of strain. This process is called the tempering of steel; its effects depend on the temperature to which the steel is heated after being hardened, and the grade of temper which is acquired is usually specified by the colour (blue, straw, &c.) which appears on a clean surface of the metal during this heating, through the formation of a film of oxide. In the ordinary process of rolling plates or bars of iron or mild steel the metal leaves the rolls at so high a temperature that it is virtually annealed, or pretty nearly so. The case is different with plates and bars that are rolled cold: they, like wire supplied in the hard-drawn state (that is, without being annealed after it leaves the draw-plate), exhibit the higher strength and greatly reduced plasticity which result from permanent set.
37. The extension which occurs when a bar of uniform section is pulled is at first general, and is distributed with some approach to uniformity over the length of the bar. Before the bar breaks, however, a large additional amount of local extension occurs at and near the place of rupture. The material flows in that neighbourhood much more than in other parts of the bar, and the section is much more contracted there than elsewhere. The contraction of area at fracture is frequently stated as one of the results of a test, and is a useful index to the quality of materials. If a flaw is present sufficient to determine the section at which rupture shall occur the contraction of area will in general be distinctly diminished as compared with the contraction in a specimen free from flaws, although little reduction may be noted in the total extension of the piece. Local extension and contraction of area are almost absent in cast-iron and hard steel; on the other hand they are specially prominent in wrought-iron, mild steel, and other metals that combine plasticity with high tensile strength. An example is shown in fig. 16, which is copied from a photograph of a broken test-piece of Whitworth soft fluid-compressed steel.
38. Experiments with long rods show that the general extension which occurs in parts of the bar not near the break is somewhat irregular; it exhibits here and there incipient local stretching, which has stopped without leading to rupture. This is of course due in the first instance to want of homogeneity. It may be supposed that when local stretching begins at any point in the earlier stages of the test it is checked by the hardening effect of the strain, until, finally, under greater load, a stage is reached in which the extension at one place goes on so fast that the hardening effect cannot keep pace with the increase in intensity of stress which results from diminution of area; the local extension is then unstable, and rupture ensues. Even at this stage a pause in the loading, and an interval of relief from stress, may harden the locally stretched part enough to make rupture occur somewhere else when the loading is continued.
39. Local stretching causes the percentage of elongation which a test-piece exhibits before rupture (an important quantity in engineers’ specifications) to vary greatly with the length and section of the piece tested. It is very usual to specify the length which is to exhibit an assigned percentage of elongation. This, however, is not enough; the percentage obviously depends on the relation of the transverse dimensions to the length. A fine wire of iron or steel, say 8 inches long, will stretch little more in proportion to its length than a very long wire of the same quality. An 8-inch bar, say 1 inch in diameter, will show something like twice as much the percentage of elongation as a very long rod. The experiments of M. Barba show that, in material of uniform quality, the percentage of extension is constant for test-pieces of similar form, that is to say, for pieces of various size in which the transverse dimensions are varied in the same proportion as the length. It is to be regretted that in ordinary testing it is not practicable to reduce the pieces to a standard form, with one proportion of transverse dimensions to length, since an arbitrary choice of length and cross-section gives results which are incapable of direct comparison with one another.
40. The form chosen for test-pieces in tension tests affects not only the extension but also the ultimate strength. In the first place, if there is a sudden or rapid change in the area of cross section at any part of the length under tension (as at AB, fig. 17), the stress will not be uniformly distributed there.
The intensity will be greatest at the edges A and B, and the piece will, in consequence, pass its elastic limit at a less value of the total load than would be the case if the change from the larger to the smaller section were gradual. In a non-ductile material, rup ture will for the same reason take place at AB, with a less total load than would otherwise be borne. On the other hand, with a sufficiently ductile material, although the section AB is the first to be permanently deformed, rupture will preferably take place at some section not near AB, because at and near AB the con traction of sectional area which precedes rupture is partly prevented by the presence of the projecting portions C and D. Hence, too, with a ductile material samples such as those of fig. 18, in which the part of smallest section between the shoulders or enlarged ends of the piece is short, will break with a greater load than could be borne by long uni form rods of the same section. In good wrought-iron and mild steel the flow of metal pre ceding rupture and causing local con traction of section extends over a length six or eight times the width of the piece; and, if the length throughout which the section is uniform be materially less than this, the process of flow will be rendered more difficult and the breaking load of the sample will be raised.
These considerations have of course a wider application than to the mere interpretation of special tests. An important practical case is that of riveted joints, in which the metal left between the rivet-holes is subjected to tensile stress. It is found to bear, per square inch, a greater pull than would be borne by a strip of the same plate, if the strip were tested in the usual way with uniform section throughout a length great enough to allow complete freedom of local flow.
41. The tensile strength of long rods is affected by the length in quite a different way. With a perfectly homogeneous material, no difference should be found in the strength of rods of equal sectional area and of different lengths, provided the length of both were great enough to prevent tho action described in § 40 from affecting the result. But, since no material is perfectly homogeneous, the longer rod will in general be the weaker, offering as it does more chances of a weak place; and the probable defect of strength in the long rod will depend on the degree of variability of the material. When this has been established by numerous tests of short samples, the strength which a rod of any assigned length may be expected to possess can be calculated by an application of the theory of probabilities. A theory of the strength of long bars has been worked out on this basis by Prof. Chaplin, and has been experimentally confirmed by tests of long and short samples of wire. The theory does not apply when the length is so small that the action of § 40 enters into the case, and the experimental data on which it is based must be taken from tests of samples long enough to exclude that action.
42. In tension tests, rupture may occur, as in fig. 19, by direct separation over a surface which is nearly plane and normal to the line of stress. This is usual in hard steel aud other comparatively non-ductile materials. Or it may occur by shearing along an oblique plane, as in fig. 20, which shows the fracture of a piece of steel softer than the specimen of fig. 19. In very ductile samples these two modes of rupture are frequently found in combination, as in fig. 21, where a central core is broken by direct tension while round it is a ring over which separation has taken place by shearing. In this instance the ring is in two parts, one above and one below the surface of rupture of the central flat core. In other instances, such as that of the sample shown in fig. 16, the shorn ring forms a continuous cone or crater round a flat core.
43. In compression tests of a plastic material, such as mild steel, a process of flow may go on without limit; the piece (which must of course be short, to avoid buckling) shortens and bulges out in the form of a cask. This is illustrated by fig. 22 (from one of Fairbairn’s experiments), which shows the compression of a round block of steel (the original height and diameter of which are shown by the dotted lines) by a load equal to 100 tons per square inch of original sectional area. The surface over which the stress is distributed becomes enlarged, and the total load must be increased in a corresponding degree to maintain the process of flow. The bulging often produces longitudinal cracks, as in the figure, especially when the material is fibrous as well as plastic (as in the case of wrought-iron). A brittle material, such as cast-iron brick, or stone, yields by shearing on inclined planes as in figs. 23 and 24, which are taken from Hodgkinson’s experiments on cast-iron. The simplest fracture of this kind is exemplified by fig. 23, where a single surface (approximately a plane) of shear divides the compressed block into two wedges. With cast-iron the slope of the plane is such that this simple mode of fracture can take place only if the height of the block is not less than about 3 / 2 the width of the base. When the height is less the action is more complex. Shearing must then take place over more than one plane, as in fig. 24, so that cones or wedges are formed by which the surrounding portions of the block are split off. The stress required to crush the block is consequently greater than if the height were sufficient for shearing in a single plane.
44. The inclination of the surfaces of shear, when fracture takes place by shearing under a simple stress of pull or push, is a matter of much interest, throwing some light on the question of how the resistance whieh a material exerts to stress of one kind is affected by the presence of stress of another kind,—a question scarcely touched by direct experiment. At the shorn surface there is, in the case of tension tests, a normal pull as well as a shearing stress, and in the case of compression tests a normal push as well as shearing stress. If this normal component were absent the material (assuming it to be isotropic) would shear in the surface of greatest shearing stress, which, as we have seen in § 5, is a surface inclined at 45° to the axis. In fact, however, it does not shear on this surface. Hodgkinson’s experiments on the compression of castiron give surfaces of shear whose normal is inclined at about 55° to the axis of stress, and Kirkaldy’s, on the tension of steel, show that when rupture takes place by shear the normal to the surface is inclined at about 25° to the axis. These results show that normal pull diminishes resistance to shearing and normal push increases resistance to shearing. In the case of cast-iron under compression, the material prefers to shear on a section where the intensity of shearing stress is only 0·94 of its value on the surface of maximum shearing stress (inclined at 45°), but where the normal push is reduced to 0·66 of its value on the surface of maximum shearing stress.
45. Fatigue of Metals.— A matter of great practical as well as scientific interest is the weakening which materials undergo by repeated changes in their state of stress. It appears that in some if not in all materials a limited amount of stress-variation may be repeated time after time without appreciable deterioration in the strength of the piece; in the balance-spring of a watch, for instance, tension and compression succeed each other some 150 millions of times in a year, and the spring works for years without apparent injury. In such cases the stresses lie well within the elastic limits. On the other hand, the toughest bar breaks after a small number of bendings to and fro, when these pass the elastic limits, although the stress may have a value greatly short of the normal ultimate strength. A laborious research by Wohler, extending over twelve years, has given much important information regarding the effects on iron and steel of very numerous repeated alternations of stress from positive to negative, or between a higher and a lower value without change of sign. By means of ingeniously contrived machines he submitted test-pieces to direct pull, alternated with complete or partial relaxation from pull, to repeated bending in one direction and also in opposite directions, and to repeated twisting towards one side and towards opposite sides. The results show that a stress greatly less than the ultimate strength (as tested in the usual way by a single application of load continued to rupture) is sufficient to break a piece if it be often enough removed and restored, or even alternated with a less stress of the same kind. In that case, however, the variation of stress being less, the number of repetitions required to produce rupture is greater. In general, the number of repetitions required to produce rupture is increased by reducing the range through which the stress is varied, or by lowering the upper limit of that range. If the greatest stress be chosen small enough, it may be reduced, removed, or even reversed many million times without destroying the piece. Wohler’s results are best shown by quoting a few figures selected from his experiments. The stresses are stated in centners per square zoll; in the case of bars subjected to bending they refer to the top and bottom sides, which are the most stressed parts of the bar. [table]
IV. Iron bar bent by supporting at one end, the other end being loaded; alternations of stress from pull to push caused by rotating the bar :— [table]
Stress.
From + to —
Stress.
From + to —
46. From these and other experiments Wohler concluded that the wrought-iron to which the tests refer could probably bear an indefinite number of stress changes between the limits stated (in round numbers) in the following table (the ultimate tensile strength was about 19½ tons per square inch) :— [table]
Hence it appears that the actual strength of this material varies in a ratio which may be roughly given as 3: 2: 1 in the three cases of (a ) steady pull, (b ) pull alternating with no stress, very many times repeated, and (c ) pull alternating with push, very many times repeated. Factors of safety applicable to the three cases might therefore rationally stand to one another in the ratio of 1: 2: 3. For steel Wohler obtained results of a generally similar kind. His experiments were repeated by Spangenberg, who extended the inquiry to brass, gun-metal, and phosphor-bronze. On the basis of Wohler’s results formulas have been devised by Launhardt, Weyrauch, and others to express the probable actual strength of metals under assigned variations of stress; these are, of course, of a merely empirical character, and the data are not yet extensive enough to give them much value.
47. Wohler’s experiments, dealing, as all experiments must, with a finite number of stress-changes, leave it an open question whether there are any limits within which a state of stress might be indefinitely often varied without finally destroying the material. It is natural to suppose that a material possessing perfect elasticity would suffer no deterioration from stress-changes lying within limits up to which the elasticity is perfect. But these limits, if they exist at all, are probably very narrow. Indeed, in the case of iron, there is indirect evidence that all alteration of stress whatsoever affects the molecular structure in a way not consistent with the notion of perfect elasticity. When the state of stress in iron is varied, however slowly aud however little, the magnetic and thermo-electric qualities of the metal are found to change in an essentially irreversible manner. Every variation leaves its mark on the quality of the piece; the actual quality at any time is a function of all the states of stress in which the piece has previously been placed. It can scarcely be doubted that sufficiently refined methods of experiment would detect a similar want of reversibility in the mechanical effects of stress, even when alterations of stress occur slowly enough to escape the effects of viscosity which have been examined by Sir William Thomson and discussed under Elasticity (vol. vii. pp. 802 sq. ). In any case, the viscosity investigated by Thomson causes such stress-changes as occur rapidly to do work on the material, and the destructive effect of repeated changes may be due in great part to this cause. His experiments show that rapid stress-changes often repeated do produce a cumulative effect in reducing the modulus of elasticity; and it is very probable that this fatigue of elasticity is associated with fatigue of strength.
There are as yet no experiments showing how far fatigue of strength is affected by the frequency, as distinguished from the mere number, of the stress-changes, nor whether a period of rest, after fatigue has been induced, restores strength. That it does so may be conjectured from Thomson’s discovery that rest restores elasticity after elastic fatigue. The conjecture is strengthened by Bauschinger’s discovery that, after a permanent set has been produced and a period of rest follows, the apparent limit of elasticity (in the strict sense of that term) rises slowly with the lapse of time. Both questions are of obvious practical interest.
48. When a strain is produced within the limits to which Hooke’s law applies, the work done in producing it is half the product of the stress into the strain. A load applied to a piece suddenly, but without impact, does an amount of work in straining the piece which is measured by the weight of the load into the distance it sinks in consequence of the strain. Hence, provided this strain falls within the elastic limit, the strain and the stress are twice as great as the same load would produce when in equilibrium. Instances of load applied with complete suddenness, and yet without shock, are rare; but it is a common case for loads to be applied so rapidly that the stress reaches a value intermediate between that due to a static load and the double stress due to the same load applied at once. Thus the Railway Commissioners found that certain bridges were deflected by a train passing at a speed of 50 miles per hour 1 /7 more than by the same load at rest. The fact that a “live” load produces greater stress than a dead load is of course to be distinguished from the question Wohler’s experiments deal with—the greater destructiveness of the intermitted or varied stress which a live load causes. In many cases engineers allow in one operation for these quite independent influences of a live load by choosing a higher factor of safety for the live than for the dead part of the whole load on a structure, or (what is the same thing) by multiplying the live load by a coefficient (often 1½), adding the product to the dead load, and treating the sum as if all were dead load.
49. A useful application of diagrams showing the relation of strain to stress is to determine the amount of work done in straining a piece in any assigned way. The term “resilience” is conveniently used to specify the amount of work done when the strain just reaches the corresponding elastic limit. Thus a rod in simple tension or simple compression has a resilience per unit of volume = f 2 ∕2E, where f is the greatest elastic pull or push. A blow whose energy exceeds the resilience (reckoned for the kind of stress to which the blow gives rise) must in the most favourable case produce a permanent set; in less favourable cases local permanent set will be produced although the energy of the blow is less than the resilience, in consequence of the strain being unequally distributed. In a plastic material a strain exceeding the limit of elasticity absorbs a relatively large amount of energy, and generally increases the resilience for subsequent strains. Fracture under successive blows, as in the testing of rails by placing them as beams on two supports, and allowing a weight to fall in the middle from a given height, results from the accumulated set which is brought about by the energy of each blow exceeding the resilience
50. In an important paper which is reprinted in the article Elasticity, and should be carefully studied in this connexion, Prof. James Thomson has pointed out that the effect of any externally applied load depends, to a very material extent, on whether there is or is not initial internal stress, or, in other words, whether the loaded piece is initially in what Prof. Hearson has called a state of ease. Internal stress, existing without the application of force from without the piece, must satisfy the condition that its resultant vanishes over any complete cross-section. It may exist in consequence of set caused by previously applied forces (a case of which instances are given below), or in consequence of previous temperature changes, as in cast-iron, which is thrown into a state of internal stress by unequally rapid cooling of tho mass. Thus in (say) a spherical casting an outside shell solidifies first, and has become partially contracted by cooling by the time the inside has become solid. The inside then contracts, and its contraction is resisted by the shell, which is thereby compressed in a tangential direction, while the metal in the interior is pulled in the direction of the radius. Allusion has already been made to the fact, pointed out by J. Thomson, that the defect of elasticity under small loads which Hodgkinson discovered in cast-iron is probably due to initial stress. In plastic metal a nearly complete state of ease is brought about by annealing; even annealed pieces, however, sometimes show, in the first loading, srnall defects of elasticity, which are probably due to initial stress, as they disappear when the load is reapplied.
51. Little is exactly known with regard to the effect of temperature on the strength of materials. Some metals, notably iron or steel containing much phosphorus, show a marked increase in brittleness at low temperatures, or “cold shortness.” Experiments on the tensile strength of wrought-iron and steel show in general little variation within the usual atmospheric range of heat and cold. The tensile strength appears to be slightly reduced at very low temperatures, and to reach a maximum when the metal is warmed to a temperature between 100° C. and 200° C. When the temperature exceeds 300° C. the tensile strength begins to fall off rapidly, and at 1000° C. it is less than one-tenth of the normal value. Reference may be made, in this connexion, to the effect which a “blue heat,” or temperature short of red heat, is believed to have on the plasticity and strength of iron, and more especially of mild steel. It appears that steel plates and bars bent or otherwise worked at a blue heat not only run a much more serious risk of fracture in the process than when worked either cold or red-hot, but become deteriorated so that brittleness may afterwards show itself when the metal is cold.
52. The following table gives a few representative data regarding the strength of the more important materials used in engineering (the figures are gathered from the writings of Barlow, Hodgkinson, Kirkaldy, Thurston, Rankine, Unwin, Clark, and others) :— [table]
Ultimate Strength.
Tons per Square Inch.
Young's
Modulus
E.
22 along ,,
19 across
Steel—
Mild steel plates (Siemens or
Steel wire, ordinary, about
Tempered steel rope wire
13,000
13,000
7000
1700
2400
53. Space admits of no more than a short and elementary account of some of the more simple straining actions that occur in machines and engineering structures.
The stress which acts on any plane surface AB (fig. 25), such as an imaginary cross-section of a strained piece, may be represented by a figure formed by setting up ordinates A a , B5,
&c., from points on the surface, the length of these being made proportional to the intensity of stress at each point.
This gives an ideal solid, which may be called the stress figure, whose height shows the distribution of stress over the surface which forms its base. A line drawn from g, the centre of gravity of the stress figure, parallel to the ordinates A a , &c., determines the point C, which is called the centre of stress, and is the point through which the resultant of the distributed stress acts. In the case of a uniformly distributed stress, ab is a plane surface parallel to AB, and C is the centre of gravity of the surface AB. When a bar is subjected to simple pull applied axially—that is to say, so that the resultant stress passes through the centre of gravity of every cross-section,—the stress may be taken as (sensibly) uniformly distributed over any section not near a place where the form of the cross-section changes, provided the bar is initially in a state of ease and the stress is within the limits of elasticity.
54. Uniformly varying stress is illustrated by fig. 26. It occurs (in each case for stresses within the elastic limit) in a bent beam, in a tie subjected to non-axial pull, and in a long strut or column where buckling makes the stress become non-axial. In uniformly varying stress the intensity p at any point P is proportional to the distance of P from a line MN, called < the neutral axis, whieh lies in the plane of the stressed surface and at right angles to the direction AB, which is assumed to be that in which the intensity of stress varies most rapidly. There is no varia tion of stress along lines parallel to MN. If MN passes through
C, the centre of gravity of the sur face, as in fig. 27, it may easily be shown that the total pull stress on one side of the neutral axis is equal to the total push stress on the other side, whatever be the form of the surface AB. The resultant of the whole stress on AB is in that case a couple, whose moment may be found as follows. Let d S be an indefinitely small part of the surface at a distance x from the neutral axis through C, and let p be the intensity of stress on d S. The moment of the stress on d S is xpd S But p=p 1 x∕x 1 =p 2 x/x 2 (see fig. 27). The whole moment of the stress on AB is ∫xpd S = (p 1 ∕ x ι ∫ x 2 d S= p 1 I∕ x 1 or p 2 I∕ x 2 , where I is the moment of inertia of the surface AB about the neutral axis through C.
55. A stress such as that shown in fig. 26 or fig. 28 may be regarded as a uniformly distributed stress of intensity p 0 (which is the intensity at the centre of gravity of the surface C) and a stress of the kind shown in fig. 27. The resultant is p 0 S, where S is the whole area of the surface, and it acts at a distance CD from C such that the moment p 0 S . CD =
(p 2 - p 0 )I/ x 1 = (p 1 + p 0 )I∕ x 2 . Hence p 2 = p 0 (1+ x 2 S . CD∕I), and p 1 = p 0 (1- x 1 S. CD∕I).
56. Simple bending occurs when a beam is in equilibrium under equal and opposite couples in the plane of the beam. Thus if a beam (fig. 29), supported at its ends, be loaded at two points so that W 1 l 1 = W 2 l 2 , the por tion of the beam lying between W 1 and W 2 is subjected to a simple bending stress. On any section AB the only stress consists of pul 1 and push, and has for its resultant a couple whose moment M=W 1 Z 1 =W 2 Z 2 . This is called the bending moment at the section. If the stress be within the elastic limits it will be distributed as in fig.
30, with the neutral axis at the centre of gravity of the section. The greatest intensities of push and of pull, at the top and bottom edge respectively, are (by § 54) p 1 = M y 1 ∕I and p 2 = M y 2 /I , and the intensity at any point at a distance y above or below C is p=M y ∕I.
57. Let the bending moment now be increased; non-elastic strain will begin as soon as either p 1 or p 2 exceeds the corresponding limit of elasticity, and the distribution of stress will be changed in consequence of the fact that the outer layers of the beam are taking set while the inner layers are still following Hooke’s law. As a simple instance we may consider the case of a material strictly elastic up to a certain stress, and then so plastic that a relatively very large amount of strain is produced without further change of stress, a case not very far from being realized by soft wrought-iron and mild steel. The diagram of stress will now take the form sketched in fig. 31. If the elastic limit is (say) less for compression than for tension, the diagram will be as in fig. 32, with the neutral axis shifted towards the tension side. When the beam is relieved from external load it will be left in a state of internal stress, represented, for the case of fig. 31, by the dotted lines in that figure.
58. In consequence of the action which has been illustrated (in a somewhat crude fashion) by figs. 31 and 32, the moment required to break the beam (M 1 ) cannot be calculated from the ultimate tensile or compressive strength of the material by using the formula M 1 = f c I∕ y 1 , or f c I∕ y 2 . When experiments are made on the ultimate strength of bars to resist bending, it is not unusual to apply a formula of this form to calculate an imaginary stress f which receives the name of the modulus of trans verse rupture. Let the section be such that y 1 =
y 2 . Then the modulus of transverse rupture is defined as f = M 1 y 1 ∕I.
This mode of stating the results of experiment on transverse strength is unsatisfactory, inasmuch as the modulus of rup ture thus determined will vary with different forms of section. Thus a plastic material for which f c and f t are equal, if tested in the form of an I beam in which the flanges form practically the whole area of section, will have a modulus of rapture sensibly equal to f t or f c , On the other hand, if the material be tested in the form of a rectangular bar, the modulus of rupture may approach a value one and a half times as great. For in the latter case the distribution of stress may approach an ultimate condition in which half the section is in uniform tension f t , and the other half in uniform compression of the same intensity. The moment of stress is then ¼ f t bh 2 , b being the breadth and h the depth of the section; but by definition of the modulus of rupture f, M= 1 / 6 fbh 2 . In tables of the modulus of transverse rupture the values are generally to be understood as referring to bars of rectangular section. Values of this modulus for some of the principal materials of engineering are given in the article Bridges, vol. iv. p. 292.
59. The strain produced by bending stress in a bar or beam is, as regards any imaginary filament taken along the length of the piece, sensibly the same as if that filament were directly pulled or compressed by itself. The resulting deformation of the piece consists, in the first place and chiefly, of curvature in the direction of the length, due to the longitudinal extension and compression of the filaments, and, in the second place, of transverse flexure, due to the lateral compression and extension which go along with their longitudinal extension and compression (see Elasticity, § 57). Let l 1 , fig. 33, be a short portion of the length of a beam strained by a bending moment M (within the limits of elasticity). The beam, which we assume to be originally straight, bends in the direction of its length to a curve of radius R, such that R∕Z = y 1 ∕ δl , δl being the change of l by extension or compression, at a distance y 1 from the neutral axis. But δl = lp 1 ∕E by § 10, and p 1 =M y 1 ∕I. Hence R = EI∕M. The transverse flexure is not, in general, of practical importance. The centre of curvature for it is on the opposite side from the centre for longitudinal flexure, and the radius is Rσ, where σ is the ratio of longitudinal extension to lateral contraction under simple pull.
60. Bending combined with shearing is the mode of stress to which beams are ordinarily subject, the loads, or externally applied forces, being applied at right angles to the direction of the length. Let AB, fig. 34, be any cross-section of a beam in equilibrium. The portion V of the beam, whieh lies on one side of AB, is in equi librium under the joint action of the external forces F 1 , F 2 , F 3 , &c., and the forces which the other por tion U exerts on V in consequence of the state of stress at AB. The forces F 1 , F 2 , F 3 , &c. may be referred to AB by introducing couples whose moments are F 1 x 1 , F 2 x 2 , F 3 x 3 , &c. Hence the stress at AB must equilibrate, first, a couple whose moment is ΣF x , and, second, a force whose value is 2F, which tends to shear V from U. In these summations regard must of course be had to the sign of each force; in the diagram the sign of F 3 is opposite to the sign of F 1 and F 2 , Thus the stress at AB may be regarded as that due to a bending moment M equal to the sum of the moments about the section of the externally applied forces on one side of the section (ΣF x ), and a shearing force equal to the sum of the forces about one side of the section (2F). It is a matter of convenience only whether the forces on V or on U be taken in reckoning the bending moment and the shearing force. The bending moment causes a uniformly varying normal stress on AB of the kind already discussed in § 56; the shearing force causes a shearing stress in the plane of the section, the distribution of which will be investigated later. This shearing stress in the plane of the section is (by § 6) accompanied by an equal intensity of shearing stress in horizontal planes parallel to the length of the beam.
61. The stress due to the bending moment, consisting of longitudinal push in filaments above the neutral axis and longitudinal pull in filaments below the neutral axis, is the thing chiefly to be considered in practical problems relating to the strength of beams. The general formula p 1 = M y 1 ∕I becomes, for a beam of rectangular section of breadth b and depth h , p 1 = 6M∕ bh 2 = 6M∕SA, S being the area of section. For a beam of circular section it becomes p 1 = 32M∕π h 3 = 8M ∕Sh. The material of a beam is disposed to the greatest advantage as regards resistance to bending when the form is that of a pair of flanges or booms at top and bottom, held apart by a thin but stiff web or by cross-bracing, as in I beams and braced trusses. In such cases sensibly the whole bending moment is taken by the flanges; the intensity of stress over the section of each flange is very nearly uniform, and the areas of section of the tension and compression flanges (S 1 and S 2 respectively) should be proportioned to the value of the ultimate strengths f t and f c , so that S 1 f t =S 2 f c . Thus for cast-iron beams Hodgkinson has recommended that the tension flange should have six times the sectional area of the compression flange. The intensity of longitudinal stress on the two flanges of an I beam is approximately M∕S 1 h and M∕S 2 h , h being the depth from centre to centre of the flanges.
62. In the examination of loaded beams it is convenient to represent graphically the bending moment and the shearing force at various sections by setting up ordinates to represent the values of these quantities. Curves of bending moment and shearing force for a number of important practical cases of beams supported at the ends will be found in the article Bridges, with expressions for the maximum bending moment and maximum shearing force under various distributions of load. The subject may be briefly illustrated here by taking the case of a cantilever or projecting bracket—(1) loaded at the end only (fig. 35); (2) loaded at the end and at another point (fig. 36); (3) loaded over the whole length with a uniform load per foot run. Curves of bending moment are given in full lines and curves of shearing force in dotted lines in the diagrams.
The area enclosed by the curve of shearing force, up to any ordinate, such as ab (fig. 37), is equal to the bending moment at the same section, represented by the ordinate ac. For let x be increased to x + δx, the bending moment changes to 2F(x + δ x ), or δM = δ xΣ F. Hence the shearing force at any section is equal to the rate of change of the bending moment there per unit of the length, and the bending moment is the integral of the shearing force with respect to the length. In the case of a continuous distribution of load, it should be observed that, when x is increased to x +δ x , the moment changes by an additional amount which depends on (δx) 2 and may therefore be neglected.
63. To examine the distribution of shearing stress over any vertical section of a beam, we may consider two closely adjacent sections AB and DE (fig. 38), on which the bending moments are M and M + δM respectively. The resultant horizontal force due to the bending stresses on a piece ADHG enclosed between the adjacent sections, and bounded by the horizontal plane GH at a distance y 0 from the neutral axis, is shown by the shaded figure. This must be equilibrated by the horizontal shearing stress on GH, which is the only other horizontal force acting on the piece. At any height y the intensity of resultant horizontal stress due to the difference of the bending moments is y δM∕I, and the whole horizontal force on GH is ⅛lfP vι yzdy, z being the breadth. If q be the intensity of horizontal shearing stress on the section GH, whose breadth is z 0 , we have qzjδx=^-∕ yzdy . 1 √i'o But δM∕δ x is the whole shearing force Q on the section of the beam. Hence Q Γ Vl j 9 , =∑⅞ ∕ y-dy; and this is also the intensity of vertical shearing stress at the distance y 0 from the neutral axis. This expression may conveniently be written q = QA y∕z 0 I , where A is the area of the surface AG and y the distance of its centre of gravity from the neutral axis. The intensity q is a maximum at the neutral axis and diminishes to zero at the top and bottom of the beam. In a beam of rectangular section the value of the shearing stress at the neutral axis is q max. = 3 / 2 Q/ bh. In other words, the maximum intensity of shearing stress on any section is ⅜ of the mean intensity. Similarly, in a beam of circular section the maximum is 4/ 3 of the mean. This result is of some importance in application to the pins of pin-joints, which may be treated as very short beams liable to give way by shearing.
In the case of an I beam with wide flanges and a thin web, the above expression shows that in any vertical section q is nearly constant in the web, and insignificantly small in the flanges. Practically all the shearing stress is borne by the web, and its intensity is very nearly equal to Q divided by the area of section of the web.
64. The foregoing analysis of the stresses in a beam, which resolves them into longitudinal pull and push, due to bending moment, along with shear in longitudinal and transverse planes, is generally sufficient in the treatment of practical cases. If, however, it is desired to find the direction and greatest intensity of stress at any point in a beam, the planes of principal stress passing through the point must be found by an application of the general method given in the article Elasticity, chapter iii. In the present case the problem is excep tionally simple, from the fact that the stresses on two planes at right angles are known, and the stress on one of these planes is wholly tan gential. Let AC (fig. 39) be an indefinitely small portion of the horizontal section of a beam, on which there is only shearing stress, and let AB be an indefinitely small portion of the vertical section at the same place, on which there is shearing and normal stress. Let q be the intensity of the shearing stress, which is the same on AB and AC, and let p be the in tensity of normal stress on AB: it is required to find a third plane BC, such that the stress on it is wholly normal, and to find r, the in tensity of that stress. Let θ be the angle (to be determined) which BC makes with AB. Then the equilibrium of the triangular wedge ABC requires that r BC cos θ=p. AB+ q. AC, and rBC sin θ=q . AB; or (r-p) cos θ = q sin θ, and r sin θ = q cos θ.
Hence, q 2 =r(r-p), tan 2θ = 2q∕p, r = ½ p ±√ q 2 +¼ p 2 .
The positive value of r is the greater principal stress, and is of the same sign as p . The negative value is the lesser principal stress, which occurs on a plane at right angles to the former. The equation for θ gives two values corresponding to the two planes of principal stress. The greatest intensity of shearing stress occurs on the pair of planes inclined at 45° to the planes of principal stress, and its value is √ q 2 + ¼ p 2 (by § 5).
65. The above determination of r, the greatest intensity of stress due to the combined effect of simple bending and shearing, is of some practical importance in the case of the web of an I beam. We have seen that the web takes practically the whole shearing force, distributed over it with a nearly uniform intensity q. If there were no normal stress on a vertical section of the web, the shearing stress q would give rise to two equal principal stresses, of pull and push, each equal to q, in directions inclined at 45° to the section. But the web has further to suffer normal stress due to bending, the intensity of which at points near the flanges approximates to the intensity on the flanges themselves. Hence in these regions the greater principal stress is increased, often by a considerable amount, which may easily be calculated from the foregoing formula. What makes this specially important is the fact that one of the principal stresses is a stress of compression, which tends to make the web yield by buckling, and must be guarded against by a suitable stiffening of the web.
The equation for θ allows the lines of principal stress in a beam to be drawn when the form of the beam and the distribution of loads are given. An example has been shown in the article Bridges (§ 13, fig. 12), vol. iv. p. 290.
66. The deflexion of beams is due partly to the distortion caused by shearing, but chiefly to the simple bending which occurs at each vertical section. As regards the second, which in most cases is the only important cause of deflexion, we have seen (§ 59) that the radius of curvature R at any section, due to a bending moment M, is EI∕M, which may also be written E y 1 ∕ p 1 . Thus beams of uniform strength and depth (and, as a particular case, beams of uniform section subjected to a uniform bending moment) bend into a circular arc. In other cases the form of the bent beam, and the resulting slope and deflexion, may be determined by integrating the curvature throughout the span, or by a graphic process (see Bridges, § 25), which consists in drawing a curve to represent the beam with its curvature greatly exaggerated, after the radius of curvature has been determined for a sufficient number of sections. In all practical cases the curvature is so small that the arc and chord are of sensibly the same length. Calling i the angle of slope, and u the dip or deflexion from the chord, the equation to the curve into which an originally straight beam bends may be written
<ύί_ .. d 2 w di El dx ’ dx 2 dx~ M
Integrating this for a beam of uniform section, of span L, supported at its ends and loaded with a weight W at the centre, we have, for the greatest slope and greatest deflexion, respectively, i 1 = WL 2 ∕16EI, u 1 =WL 3 ∕48EI. If the load W is uniformly distributed over L, i 1 = WL 2 ∕24EI and u 1 = 5WL 3 ∕384EI. For other cases, see Bridges, §24.
The additional slope which shearing stress produces in any originally horizontal layer is q ∕C, where q is, as before, the intensity of shearing stress and C is the modulus of rigidity. In a round or rectangular bar the additional deflexion due to shearing is scarcely appreciable. In an I beam, with a web only thick enough to resist shear, it may be a somewhat considerable proportion of the whole.
67. Torsion occurs in a bar to which equal and opposite couples are applied, the axis of the bar being the axis of the couples, and gives rise to shearing stress in planes perpendicular to the axis. Let AB (fig. 40) be a uniform circular shaft held fast at the end A, and twisted by a couple applied in the plane BB. As suming the strain to be within the limits of elasticity, a radius
CD turns round to
CD', and a line AD
drawn at any dis tance r from the axis, and originally straight, changes into the helix AD'. Let θ be the angle which this helix makes with lines parallel to the axis, or in other words the angle of shear at the distance r from the axis, and let i be the angle of twist DCD'. Taking two sections at a distance dx from one another, we have the arc θdx = rdi. Hence q, the intensity of shearing stress in a plane of cross-section, varies as r, . di since q =C θ = Cr di / dx . The resultant moment of the whole shearing stress on each plane of cross-section is equal to the twisting moment M. Thus ∫2πr 2 qdr = M .
Calling r 1 the outside radius (where the shearing stress is greatest) and q 1 its intensity there, we have q = rq 1 /r 1 , and hence, for a solid shaft, q 1 = 2M∕π r 1 3 . For a hollow shaft with a central hole of radius r 2 the same reasoning applies: the limits of integration are now r 1 and r 2 , and ιr (r ι 4 - r 2 4 )
The lines of principal stress are obviously helices inclined at 45° to the axis.
If the shaft has any other form of section than a solid or symmetrical hollow circle, an originally straight radial line becomes warped when the shaft is twisted, and the shearing stress is no longer proportional to the distance from the axis. The twisting of shafts of square, triangular, and other sections has been investigated by M. de St Venant (see Elasticity, § 66-71, where a comparison of torsional rigidities is given). In a square shaft (side = h ) the stress is greatest at the middle of each side, and its intensity there is q 1 = M∕0∙281 h 3 .
For round sections the angle of twist per unit of length is i = .-γL = in solid and r ,. —- in hollow shafts.
68. In what has been said above it is assumed that the stress is within the limit of elasticity. When the twisting couple is increased so that this limit is passed, plastic yielding begins in the outermost layer, and a larger proportion of the whole stress falls to be borne by layers nearer the centre. The case is similar to that of a beam bent beyond the elastic limit, described in § 57. If we suppose the process of twisting to be continued, and that after passing the limit of elasticity the material is capable of much distortion without further increase of shearing stress, the distribution of stress on any cross section will finally have an approximately f* y *l uniform value q ’ , and the moment of torsion will be ^. 2πr~qdr = f^'(r ι 3-r 2 3 )∙ In the case of a solid shaft this gives for M a value greater than it has when the stress in the outermost layer only reaches the intensity q' , in the ratio of 4 to 3. It is obvious from this consideration that the ultimate strength of a shaft to resist torsion is no more deducible from a knowledge of the ultimate shearing strength of the material than the ultimate strength of a beam to resist bending is deducible from a knowledge of f t and f c . It should be noticed also that as regards ultimate strength a solid shaft has an important advantage over a hollow shaft of the same elastic strength, or a hollow shaft so proportioned that the greatest working intensity of stress is the same as in the solid shaft.
69. Twisting combined with Longitudinal Stress.— When a rod is twisted and pulled axially, or when a short block is twisted and compressed axially, the greatest intensity of stress (the greater principal stress) is to be found by compounding the longitudinal and shearing stresses as in § 64. In a circular rod of radius r 1 , a total longitudinal force P in the direction of the axis gives a longitudinal normal stress whose intensity p 1 =P∕π r 1 2 . A twisting couple M applied to the same rod gives a shearing stress whose greatest intensity q 1 =2M∕π r 1 3 . The two together give rise to a pair of principal stresses of intensities r = P∕2π r 1 2 ±∖∕(2M∕π r 1 3 ) 2 +(P∕2π r 1 3 ) 2 , their inclinations to the axis being defined by the equation tan 20 = 2M∕r 1 P, and the term under the square root is the greatest intensity of shearing stress.
70. Twisting combined with Bending.— This important practical case is realized in a crank-shaft (fig. 41). Let a force P be applied at the crank-pin A at right angles to the plane of the crank.
At any section of the shaft C
(between the crank and the bearing) there is a twisting mo ment M 1 = P . AB, and a bend ing moment M 2 =P. BC. There is also a direct shearing force P, but this does not require to be taken into account in calculating the stress at points at the top or bottom of the circumference (where the intensity is greatest), since (by § 63) the direct shearing stress is dis tributed so that its intensity is zero at these points. The stress there is consequently made up of longitudinal normal stress (due to bending), p 1 = 4M 2 ∕πr 1 3 , and shearing stress (due to torsion), q 1 = 2M 1 ∕π r 1 3 . Combining these, as in § 64, we find for the principal stresses r=2(M 2 ±∖∕M 1 2 + M 2 2 )∕π r 1 3 , or r = 2P(BC±AC)∕π r 1 3 . The greatest shearing stress is 2P . AC∕πr 1 3 ∙, and the axes of principal stress are inclined so that tan 20 = M 1 ∕M 2 = AB∕BC. The axis of greater principal stress bisects the angle ACB.
71. Long Columns and Struts—Compression and Bending.— A long strut or pillar, compressed by forces P applied at the ends in the direction of the axis, becomes unstable as regards flexure when P exceeds a certain value. Under no circumstances can this value of P be exceeded in loading a strut. But it may happen that the intensity of stress produced by smaller loads exceeds the safe compressive strength of the material, in which case a lower limit of load must be chosen. If the applied load is not strictly axiai, if the strut is not initially straight, if it is subject to any deflexion by transverse forces, or if the modulus of elasticity is not uniform over each cross-section,—then loads smaller than the limit which causes instability will produce a certain deflexion which increases with increase of load, and will give rise to a uniformly varying stress of the kind illustrated in figs. 26 and 28. We shall first consider the ideal case in which the forces at the ends are strictly axial, the strut perfectly straight and free from transverse loads and perfectly symmetrical as to elasticity. Two conditions have to be distinguished—that in which the ends are left free to bend, and that in which the ends are held fixed. In what follows, the ends are supposed free to bend. The value of the load which causes instability will be found by considering what force P applied to each end would suffice to hold an originally straight strut in a bent state, supposing it to have received a small amount of elastic curvature in any way. Using u as before to denote the deflexion at any part of the length, the bending moment is Pu, and (taking the origin at the middle of tho chord) the equation to the elastic curve is d 2 u _ - Pu
<‰Γ 2- ^~ËÎ’
from which, for a strut of uniform section, u = u 1 cos x √P∕EI, u 1 being the deflexion at the centre, Now u = 0 when x =½L (the half length), and therefore ½L∖∕P∕EI = 1 /2πr or an integral multiple of ½π. The smallest value (½π) corresponds to the least force P. Thus the force required to maintain the strut in its curved state is P = π 2 EI∕L 2 , and is independent of u 1 . This means that with this particular value of P (which for brevity we shall write P 1 ) the strut will be in neutral equilibrium when bent; with a value of P less than P 1 it will be stable: with a greater value it will be unstable. Hence a load exceeding P 1 will certainly cause rupture. The value
π 2 EI∕L 2 applies to struts with round ends, or ends free to turn. If the ends are fixed the effective length for bending is reduced by one half, so that P 1 then is 4π 2 EI∕L 2 . When one end is fixed and the other is free P 1 has an intermediate value, probably about 9π 2 EI∕4L 2 .
72. The above theory, which is Euler’s, assigns P 1 as a limit to the strength of a strut on account of flexural instability; but a stress less than P 1 may cause direct crushing. Let S be the area of section, and f c the strength of the material to resist crushing. Thus a strut which conforms to the ideal conditions specified above will fail by simple crushing if f c S is less than P 1 , but by bending if f c S is greater than P 1 . Hence with a given material and form of section the ideal strut will fail by direct crushing if the length is less than a certain multiple of the least breadth (easily calculated from the expression for P 1 ), and in that case its strength will be independent of the length; when the length is greater than this the strut will yield by bending, and its strength diminishes rapidly as the length is increased.
But the conditions which the above theory assumes are never realized in practice. The load is never strictly axial, nor the strut absolutely straight to begin with, nor the elasticity uniform. The result is that the strength is in all cases less than either f c S or P 1 . The effect of deviations from axiality, from straightness, and from uniformity of elasticity may be treated by introducing a term expressing an imaginary initial deflexion, and in this way Euler’s theory may be so modified as to agree well with experimental results on the fracture of struts, and may be reconciled with the observed fact that the deflexion of a strut begins gradually and passes through stable values before the stage of instability is reached. In consequence of this stable deflexion the stress of compression on the inside edge becomes greater than P∕S, the stress on the outside edge becomes less than P∕S, and may even change into tension, and the strut may yield by one or the other of these stresses becoming greater than f c or f t respectively. As regards the influence of length and moment of inertia of section on the deflexion of struts, analogy to the case of beams suggests that the greatest deflexion consistent with stability will vary as L 2 ∕ b , b being the least breadth, and the greatest and least stress, at opposite edges of the middle section, will consequently be
J√ 1 ,∈U∖ n s (1 ± &a j, where a is a coefficient depending on the material and the form of the section. This gives, for the breaking load, P = S f c ∕(1 + a L 2 ∕ b 2 ) or -S f t ∕(1 -a L 2 /b 2 ), the smaller of the two being taken.
This formula, which is generally known as Gordon’s, can be made to agree fairly with the results of experiments on struts of ordinary proportions, when the values of f as well as a are treated as empirical constants to be determined by trial with struts of the same class as those to which the formula is to be applied. Gordon’s formula may also be arrived at in another way. For very short struts we have seen that the breaking load is f c S, and for very long struts it is π 2 EI∕L 2 . If we write P=∕S∕(1+ f c SL 2 ∕π 2 EI), we have a formula which gives correct values in these two extreme cases, and intermediate values for struts of medium length. By writing this P=√S∕(1 + c SL 2 /I), and treating f and c as empirical constants, we have Gordon’s formula in a slightly modified shape. Gordon’s formula is largely used; it is, however, essentially empirical, and it is only by adjustment of both constants that it can be brought into agreement with experimental results. For values of the constants, see Bridges. In the case of fixed ends, c is to be divided by 4.
73. Bursting Strength of Circular Cylinders and Spheres.— Space remains for the consideration of only one other mode of stress, of great importance from its occurrence in boilers, pipes, hydraulic and steam cylinders, and guns.
The material of a hollow cylinder, subjected to pressure from within, is thrown into a stress of circumferential pull. When the thickness t is small compared with the radius R, we may treat this stress as uniformly distributed over the thickness. Let p be the intensity of fluid pres sure within a hollow circular cylinder, and let f be the intensity of circumferential stress.
Consider the forces on a small rectangular plate
(fig. 42), with its sides parallel and perpen dicular to the direction of the axis, of length l and width Rδ θ , δ θ being the small angle it subtends at the axis. Whatever forces act on this plate in the direction of the axis are equal and opposite. The remaining forces, which are in equilibrium, are P, the total pressure from within, and a force Tat each side due to the circumferential stress. P= pl Rδ θ and T= flt . But by the triangle of forces (fig. 43) P = Tδ θ . Hence f=p R ∣t.
The ends of the cylinder may or may not be held together by longitudinal stress in the cylinder sides; if they are, then, whatever be the form of the ends, a transverse section, the area of which is 2πR t , has to bear a total force p πR 2 . Hence, if f' be the intensity of longitudinal stress, f' =p R ∣ 2 t = ½ f.
74. A thin hollow sphere under internal pressure has equal circumferential pull in all directions. To find its value consider the plate of fig. 42. There are now four equal forces T, on each of the four sides, to equilibrate the radial force P. Hence P = 2Tδ θ and f=p R/2 t
75. When the thickness is not small compared with the radius, the radial pressure is transmitted from layer to layer with reduced intensity, and the circumferential pull diminishes towards the outside. In the case of a thick cylinder with free ends we have to deal at any point with two principal stresses, radial and circumferential, which may be denoted by p and p ’ respectively. Supposing (as we may properly do in dealing with a cylinder which is not very short) that a transverse section originally plane remains plane, the longitudinal strain is uniform. Since there is no longitudinal stress this strain is due entirely to the lateral action of the stresses p and p', and its amount is (p +p')∕σE . Hence at all points p + p'≈ constant. Further, by considering the equilibrium of any thin layer, as we have already considered that of a thin cylinder, we have d / dx (pr )= p'.
These two equations give by integration, p = C + C'∕ r 2 , and p ' = C - C'∕ r 2 .
If r 1 be the external and r 2 the internal radius, and p 0 the pressure on the inner surface, the conditions that p=p 0 when r = r 2 and p = 0 when r=r 1 give C = - p 0 r 2 2 ∕(r 1 2 - r 2 2 ) and C' = -C r 1 2 . Hence the circumferential stress at any radius r is p' = p 0 r 2 2 (l+ r 1 2 / r 2 )∕(r 1 2 - r 2 2 ). At the inside, where this is greatest, its value is - p 0 (r 1 2 + r 2 2 )∕(r 1 2 - r 2 2 ),—a quantity always greater than p 0 , however thick the cylinder is.
In the construction of guns various devices have been used to equalize the circumferential tension. With cast guns a chilled core has been employed to make the inner layers solidify and cool first, so that they are afterwards compressed by the later contraction of the outer layers. In guns built up of wrought-iron or steel hoops the hoops are bored small by a regulated amount and are shrunk on over the barrel or over the inner hoops. In Mr Longridge’s system, now under trial, the gun is made by winding steel wire or ribbon, with suitable initial tension, on a central barrel.
76. The circumferential stress at any point of a thick hollow sphere exposed to internal fluid pressure is found, by a process like that of the last paragraph, to be - p 0 r 2 3 (1 + r 1 3 ∕2 r 3 )∕(r 1 3 - r 2 3 ), which gives, for the greatest tension, the value
- p 0 (r 1 3 + 2 r 2 3 )∕2(r 1 3 - r 2 3 ). (J. A. E. )
