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the Week of Proper 28 / Ordinary 33
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Strength of Materials
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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.
Stress is the mutual action between two bodies, or between two 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 place where it is about to give way the portion of metal on either side of the plane of shear is tending to drag the portion on the other side with a force in that plane.
1 Normal and Tangential Stress
2 Simple Longitudinal Stress
3 Compound Stress
4 Equality of Shearing Stress in Two Directions
5 Limits of Elasticity
6 Young's Modulus
7 Modulus of Cubic Compressibility
8 Ultimate Strength
9 Permissible Working Stress
10 Factor of Safety
11 Tests of Strength
12 Single-lever Testing Maclaine
13 Diaphragm Testing Machines
14 Autographic Recorders
15 Hardening Effect of Permanent Set
16 Annealing
17 Extensometers
18 Contraction of Section at Rupture
19 Influence of Local Stretching on Total Elongation
20 Influence on Strength
21 Plane of Shear
22 Liiders's Lines
23 Yielding under Compound Stress
24 Experiments on Compound Stress
25 Fatigue of Metals
26 Fracture by Compression
27 Resilience
28 Microscopic Examination
29 Influence of Foreign Matter
30 Data as to Strength of Steel
31 Bending beyond Elastic Limits
32 Strain produced by Bending
33 Diagrams of Bending Moment and Shearing Force
34 Distribution of Shearing Stress
35 Principal Stresses in a Beam
36 Long Columns and Struts: Compression and Bending
Normal and Tangential Stress
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 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 SP/SS, where SP is the amount of stress on an indefinitely small area SS 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.
Simple Longitudinal Stress
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.
Compound Stress
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 (see Elasticity) 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.
Resolution of Stress. - Returning now to the case of a single simple longitudinal stress, let AB (fig. i) be a portion of a tie or a strut A 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 F EF whose normal is inclined to the axis at an angle 0 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 0. 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., fig. 2) is P cos 0; the tangential component (P t ) is P sin 0.0. Hence the intensity of normal pull or push on EF, or pn, is p cos t 0, and the 8 intensity of shearing stress EF, or pt, FIG. I. is p sin 0 cos 0. This expression makes p t a maximum when 0=45°: surfaces inclined at 45° to the axis are called surfaces of maximum shearing stress; the intensity of shearing stress on F IG. 2. them is 2p.
Combination of Two Simple Pull or Push Stresses at Right Angles to One Another.--Suppose next that there are two principal stresses: in other words that in addition to the simple pull or push stress of fig. i there is a second pull or push stress acting at right angles to it as in fig. 3. Call these P x and P, E respectively. On any inclined surface EF there will be an intensity of stress whose normal component p n and tangential component p t are found by summing up the effects due to P x and P, F separately. Let p x and p y be the intensities of stress produced by P x and P u respectively on planes perpendicular to their own direc tions. Then p„= (P x cos', B +A, sin e 9, pt = (px - py ) sin B cos 0, 0 being the angle which the normal to the surface makes with the direction of Px.
The tangential stress p t becomes a maximum when 0 is 45°, and its value then is Max. p t =2 (p0 - p,,). If in addition there is a third principal stress P z, it will not produce any tangential component on planes perpendicular to the plane of the figure. Hence the above expression for the maximum tangential stress will still apply, and it is easy to extend this result so as to reach the important general proposition that in any condition of stress whatever the maximum intensity of shearing stress is equal to onehalf the difference between the greatest and least principal stresses and occurs on surfaces inclined at 45° to them.
Q State of Simple Shear. - A special case of great importance occurs when there are two principal stresses only, equal in magnitude and opposite in sign; in other words, when one is a simple push and the other a simple pull. Then on surfaces inclined at 45° to the axes of pull and push there is nothing but tangential stress, for p„,= o; and this intensity of tangential stress is numerically equal to px or to pa,. This condition of stress is called a state of simple shear.
The state of simple shear may FIG. 4. also be arrived at in another way. Let an elementary cubical part of any solid body (fig. 4) have tangential stresses QQ applied to one pair of opposite faces, A and B, and equal tangential stresses applied to a second pair of faces C and D, Figs. 3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 23, 24 and 25 are from Ewing's Strength of Materials, by permission of the Cambridge University Press.
as in the figure. The effect is to set up a state of simple shear. On all planes parallel to A and B there is nothing but tangential stress, and the same is true of all planes parallel to C and D. The intensity of the stress on both systems of planes is equal throughout to the intensity of the stress which was applied to the face of the block.
To see the connexion between these two ways of specifying a state of simple shear consider the equilibrium of the parts into which the block may be divided by ideal diagonal planes of section. To balance the forces QQ (fig. 5), there must be normal pull on the diagonal plane, the amount of which is P = Q1/2. But the area of the surface over which P acts is greater than that of the surface over which Q acts in the proportion which P bears to Q, and hence the intensity of P is the same as the intensity of Q Q FIG. 5
Again, taking the other diagonal plane (fig. 6), the same argument applies except that here the normal force P required for equilibrium is a push instead of a pull. Thus the state of stress represented in fig. 4 ?Q admits of analysis into two equal principal stresses, one of push and one of pull, acting in directions at right angles to one another and inclined at 45° to the planes of shear stress.
Equality of Shearing Stress in Two Directions
No tangential stress can exist in one direction without an equal intensity of tangential stress existing in another direction at right angles to the first. To prove this it is sufficient to consider the equilibrium of the elementary cube of fig. 4. The tangential forces acting on two sides A and B produce 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 equal tangential forces on C and D.
Fluid Stress. - Another important case occurs when there are three principal stresses all of the same sign and of equal intensity p. The tangential components on any planes cancel each other: the stress on every plane is wholly normal and its intensity is p. This is the only state of stress that can exist in a fluid at rest because a fluid exerts no statical resistance to shear. For this reason the state is, often spoken of as a fluid stress.
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.
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| P y | Limits of Elasticity 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 most of it is found, when the stress is removed, to consist of permanent set. The Q FIG. 3. Y B C D A Q Q Q. Missing image Strengthofmaterials-4.jpg | Q | FIG. | 6. 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. Hooke's Law. - 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 und^rgoes when stressed. Let the original length be 1, and let the whole change of length be Sl when a stress is applied whose intensity p is within the elastic limit. Then the strain is measured by Sill, and this by Hooke's law is proportional to p. This may be written Sl/l = pÆ, 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. Young's Modulus 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 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 moduli applicable to other modes of stress. Since E = pl/Sl, the modulus may be defined as the ratio of the intensity of stress p to the longitudinal strain Sl/l. Modulus of Rigidity. - 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 4) in radians; then by Hooke's law p/o = 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. Modulus of Cubic Compressibility 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 SV, the ratio of the stress p to the strain SV/V is called the modulus of cubic compressibility, and will be denoted by K. Of these three moduli 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 pÆ. This is accompanied by a lateral contraction or expansion, in each transverse direction, whose amount may be written p/0-E, where v is the ratio of longitudinal to lateral strain. It is shown in the article Elasticity, that for an isotropic material E= 9CK an 3K-2C 3K+C 3K-2C Plastic Strain. - 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 (perhaps 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. Ultimate Strength 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 crosssection is 2 sq. in. 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 be 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 that their compressive strength may be determined with some precision. Some of the materials used in engineering, notably timber and wrought iron, are 1 P o 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, provided the process of rolling is completed at a temperature high enough to allow recrystallization to take place in cooling. Cold-rolled or cold-drawn metal is not isotropic because the crystals of which it is made up have been elongated in one direction by the process: but isotropy may be restored by heating the piece sufficiently to allow the crystals to re-form. Permissible Working Stress In applying a knowledge of the strength of materials to determine the proper sizes of parts in an engineering structure we have to estimate a permissible working stress. This is based partly on special tests and partly on experience of the behaviour of the material when used in similar structures. The working stress is rarely so much as one-third of the ultimate strength; it is more commonly one-fourth or one-fifth and in some cases, especially where the loads to be* borne are liable to reversal or to much change, it may be prudent to make the working stress even less than this. Factor of Safety The ratio of the ultimate strength to the working stress is called the factor of safety. The factor should in general be such as to bring the working stress within the limit of elasticity and even to leave within that limit a margin which will be ample enough to cover such contingencies as imperfection in the theory on which the calculation of the working stress is founded, lack of uniformity in the material itself, uncertainty in the estimation of loads, imperfections of workmanship which may cause the actual dimensions to fall short of those that have been specified, alterations arising from wear, rust and so forth. An important distinction has to be drawn in this connexion between steady or " dead " loads and loads which are subject to variation and especially to reversal. With the former the working stress may reach or pass the elastic limit without destroying the structure; but in a piece subject to reversals a stress of the same magnitude would lead inevitably to rupture, and hence a larger margin should be left to ensure that in the latter case the elastic limit shall not even be approached. It is in fact the elastic limit rather than the ultimate strength of the material on which the question mainly depends of how high the working stress may safely be allowed to rise in any particular conditions as to mode of loading, and accordingly it becomes a matter of much practical importance to determine by tests the amount of stress which can be borne without permanent strain. From an engineering point of view the structural merit of a material, especially when variable loads and possible shocks have to be sustained, depends not only on the 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 crosssection 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. Missing image Strengthofmaterials-5.jpg Missing image Strengthofmaterials-6.jpg Tests of Strength 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 lever. The lower holder is jointed to a cross-head C, which is connected by two vertical screws to a lower cross-head B, upon which the hydraulic plunger shown in section in fig. 7 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 the plunger 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 cross-heads C and B, these screws being turned by a handle applied at F. The knife edges are made long enough to prevent the load on them from ever exceeding 5 tons to the linear inch. To adapt a machine of this class for tests in compression, a small platform is suspended like a stirrup by four rods from the weigh-beam, and hangs below the cross-head, which is pulled FIG. 8. | A | Scale Fee: 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. Single-lever Testing Maclaine Figs. 7 and 8 show an excellent form of single-lever testing machine designed by J. H. Wicksteed ( Proc. Inst. Mech. Eng., August 1882) 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 are in common use which exert a force of Too tons or more. AA is the lever, on which there is a graduated scale. The stress on the testpiece 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. 8.) The holder for the upper end of the sample hangs from a knife-edge 3 in. from the fulcrum of the down when the hydraulic cylinder is put in action. The arrangement is that of two stirrups linked with one another, so that when the two pull against each other a block of material placed between them becomes compressed. For tests in bending one of the stirrups, namely, the platform which hangs from the weigh-beam, is made some 4 or 5 ft. long, and carries two knifeedge supports at its ends on which the ends of the piece that is to be bent rest, while the cross-head presses down upon the middle of the piece. In both cases the force which is exerted is measured by means of the weigh-beam and travelling weight, just as in the tension tests. To apply the force continuously, without shock, and either quickly or slowly at will, a very convenient plan is to use an hydraulic intensifier, consisting of a large hydraulic piston operating a much smaller one. By gradually admitting water to the large piston from any convenient source under moderate pressure, such as the town water mains, a progressively increased pressure is produced in the fluid on which the small piston acts, and this fluid is admitted to the straining cylinder of the machine. One of the advantages of the vertical type of machine, with its single horizontal lever, is the facility with which the correctness of its readings may be verified. The two things to be tested are (1) the distance between the knife-edges, one of which forms the fulcrum of the weigh-beam and from the other of which the shackle holding the upper end of the specimen is hung, and ( 2) the weight of the travelling poise. The weight of the poise is readily ascertained by using a supplementary known weight to apply a known moment to the beam, and measuring how far the poise has to be moved to restore equilibrium. The distance between the knife-edges is then found by hanging a known heavy weight from the shackle, and again observing how far the poise has to be moved. 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.1 Multiple-lever Testing Machines. - 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. In most cases the fulcrums are fixed, and the stress is applied to one end of the specimen by hydraulic power or by screw gearing, which takes up the stretch, as in the single-lever machines already described. David Kirkaldy, who was one of the earliest as well as one of the most assiduous workers in this field, applied in his 1,000,000 lb machine a horizontal hydraulic press directly to one end of the horizontal test-piece. The other end of the piece was connected to the short vertical arm of a bellcrank lever; the long arm of this lever was horizontal, and was connected to a second lever to which weights were applied. Machines have been employed 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. Diaphragm Testing Machines In another class of testing machines the stress (applied as before to one end of the piece, by gearing or by hydraulic pressure) is measured by connecting the other end to a flexible diaphragm, on which a liquid acts whose pressure is determined by a gauge. Fig. 9 shows FIG. 9. 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 applied in the remarkable testing machine of Watertown arsenal, built in 1879 by the U.S. government to the designs of A. H. Emery. This is a horizontal machine, taking specimens of any length up 1 Maschine zum Priifen d. Festigkeit d. Materialen, &c. (Munich, 1882). | | ? | | | | | | | | | | | | | | | | | | ild p | | | Bessemer | Steel | | | | | | | | | | | | | | | | | t ?r0n? | ?nr? | e For | ing) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | to 30 ft., 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 420,000.2 Stress-strain Diagrams. - 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. To, the data for which are taken from tests of long rods by Kirkaldy.3 Up to the elastic limit these diagrams show sensibly the same rate of extension for all the materials to which they refer. Soon after the limit of elasticity is passed, a point, which has been called by Sir A. B. W. Kennedy the yield-point, is reached, 6 8 10 12 14 16 18 20 Extension, per cent FIG. 10. 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 o as the end of the test is approached. Fig. II is a stress-strain diagram for cast iron in extension and compression, taken from Eaton Hodgkinson's experiments. 4 The extension was measured on a rod 50 ft. long; the compression was also measured on a long rod, which 2 See Report of the U.S. Board appointed to test Iron, Steel and other Metals (2 vols., 1881). For full details of the Emery machine, see Report of the U.S. Chief of Ordnance (1883), app. 24. 3 Experiments on the Mechanical Properties of Steel by a Committee of Civil Engineers (London, 1868 and 1870). 4 Report of the Commissioners on the Application of Iron to Railway Structures (1849). | S: | | | | | | Ccm.ression .er | | | cent | | | | I | ' p 10 | | v n | | | c | | | | | | | / | | | | j ? | J | | | | | 20 | 45 40 35 0 30 w 25 Q o 20 "Z 15 10 4 FIG. II.
2 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. Autographic Recorders Testing machines are sometimes 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.' In 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 Kennedy's apparatus autographic diagrams are drawn by applying the stress to the test-piece through an elastic masterbar of larger section. The master-bar is never strained beyond its elastic limit, and within that limit its extension furnishes an accurage measure of the stress; this gives motion to a pencil, which writes on a paper moved by the extension of the testpiece. 3 In R. H. 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.4 Strain beyond the Elastic Limit: Influence of Time. - 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 1 For descriptions of these and other types of autographic recorder, see a paper by Professor W. C. Unwin, " On the Employment of Autographic Records in Testing Materials," Journ. Soc. Arts (Feb., 1886); also Sir A. B. W. Kennedy's paper, " On the Use and Equipment of Engineering Laboratories," Proc. Inst. Civ. Eng. (1886), which contains much valuable information on the whole subject of testing and testing machines. On the general subject of tests see also Adolf Martens's Handbook of Testing Materials, trans. by G. C. Henning. 2 Proc. Inst. Mech. Eng. (1886). An interesting feature of this apparatus is a device for preventing error in the diagram through motion of the test-piece as a whole. Proc. Inst. Mech. Eng. (1886); also Proc. Inst. Civ. Eng. vol. lxxxviii. pl. 1 (1886). 4 Thurston's Materials of Engineering, pt. ii. For accounts of work done with this machine, see Trans. Amer. Soc. Civ. Eng. (from 1876); also, Report of the American Board, cited above. 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. 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 be fore, 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. Hardening Effect of Permanent Set 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. 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 ( e, fig. 13). A similar and 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 Extension FIG. 12.
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 0 5 10 15 Extension,per cent FIG. 14. load of 232 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 451 hours, the stress of 231 tons being maintained during the interval in both cases.1 It may be concluded that, when a piece of metal has in any way been overstrained 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. 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. Annealing The hardening effect of overstrain is removed by the process of annealing, that is, by heating to redness and cooling slowly. 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. Extensometers Much attention has been paid to the design of extensometers, or apparatus for observing the small deformation which a test-piece in tension or compression undergoes before its limit of elasticity is reached. Such observations afford the most direct means of measuring the modulus of longitudinal elasticity of the material, and they serve also to determine the limits within which the material is elastic. In such a material 1 J. A. Ewing, Proc. Roy. Soc. (June, 1880). as wrought iron the elastic extension is only about of the length for each ton per square inch of load, and the whole amount up to the elastic limit is perhaps T A T, of the length; with a length of 8 in., which is usual in tensile tests, it is desirable to read the extension to, say, 5 0 in. if the modulus of elasticity is to be found with fair accuracy, or if the limits of proportionality between strain to stress are under examination. Measurements taken between marks on one side of the bar only are liable to be affected by bending of the piece, and it is essential either to make independent measurements on both sides or to measure the displacement between two pieces which are attached to the bar in such a manner as to share equally the strain on both sides. In experiments carried out by Bauschinger, independent measurements of the strains on both sides of the bar were made by using mirror micrometers of the type illustrated diagrammatically in fig. 15. Two clips a and b clasp the test-piece at the place between FIG. 15. which the extension is to be measured. The clip b carries two small rollers d 1 d 2 which are free to rotate on centres fixed in the clip. These rollers press on two plane strips c i cz attached to the other clip. When the specimen is stretched the rollers consequently turn through angles proportional to the strain, and the amount of turning is read by means of small mirrors g i and g 2, fixed to the rollers, which reflect the divisions of a fixed scale f into the reading telescopes e 1 ez. In Martens's extensometer each of the rollers is replaced by a rhombic piece of steel with sharp edges, one of which bears against the test-piece, while the other rests in a groove formed in the spring projecting parallel to the test-piece from the distant clip. Much FIG. 16.. excellent work has been done by extensometers of this class, but in point of convenience of manipulation it is of great advantage to have the apparatus self-contained. J. A. Ewing has introduced a microscope extensometer of the self-contained type which is shown in fig. 16; its action will be seen by reference to the diagram fig. 17. Two clips B and C are secured on the bar, each by means of a pair of opposed set-screws. Between the two is a rod B' which is hinged to B and has a blunt pointed upper end which makes a ball-andsocket joint with C at P. Another bar R hangs from C, and carries a mark which is read by a microscope attached to B. Hence, when the specimen stretches, the length of B' being fixed, the bar R is pulled up relatively to the microscope, and the amount of the movement is measured by a micrometer scale in the eyepiece. A screw at P serves to bring d, --e a b 5 1.0 15 Extension,per cent FIG. 13. Missing image Strengthofmaterials-7.jpg Missing image Strengthofmaterials-8.jpg so 25 c 120 C 3 a Q15 _ l0 a 5 0 FIG. 17.
on R into mark the the field of view, and also to calibrate the readings of the micrometer scale. The scale allows readings to be taken to 564,0 in., by estimating tenths of the actual divisions. The arms CP and CQ are equal, and hence the movement of Q represents twice the extension of the bar under test. In another form of the instrument adapted to measure the elastic compression of short blocks the arm CQ is four times the length of CP, and consequently there is a mechanical magnification of five besides the magnification afforded by the microscope. When the behaviour of specimens of iron, steel, or other materials possessing plasticity, is watched by means of a sensitive extensometer during the progress of a tensile test, it is in general observed that a very close proportionality between the load and the extension holds during the first stages of the loading, and that during these stages there is little or no " creeping " or supplementary extension when any particular load is left in action for a long time. The strain is a linear function of the stress, almost exactly, and disappears when the stress is removed. In other words, the material obeys Hooke's law. This is the stage of approximately perfect elasticity, and the elastic limit is the point rather vaguely defined by observations of the strain, at which a tendency to creep is first seen, or a want of proportionality between strain and stress. " Creeping " is usually the first indication that it has been reached. As the load is further augmented, there is in general a clearly marked yield-point, at which a sudden large extension ensues. In metals which have been annealed or in any way brought into a condition which is independent of the effects of earlier applications of stress, this elastic stage is well marked, and the limit of elasticity is as a rule sharply defined. But if the metal has been previously overstrained, without having had its elasticity restored by annealing or other appropriate treatment, a very different C FIG. 18. behaviour is exhibited. The yield-point may be raised, as, for instance, in wire which has been hardened by stretching, but the elasticity is much impaired, and it is only within very narrow limits, if at all, that proportionality between stress and strain is found. Subsequent prolonged rest gradually restores the elasticity, and after a sufficient number of weeks or months the metal is found to be elastic up to a point which may be much higher than the original elastic limit.' It has been shown by ' See experiments by Johann Bauschinger, Mitt. aus dem meth-tech. Lab. in Munchen (1886), and by the writer, Proc. Roy. Soc., vol. xlviii. (1895). A summary of Bauschinger's conclusions will be found in Martens's book, cited above, and in Unwin's Testing of Materials. J. Muir 2 that the rate at which this recovery of elasticity occurs depends on the temperature at which the piece is kept, and that complete recovery may be produced in iron or steel by exposure of the overstrained specimen for a few minutes to the temperature of boiling water. Figs. 18 and 19 illustrate interesting points in Muir's experiments. In these figures the geometrical device is adopted of shearing back the curves which show extension in relation to load by reducing each of the observed extensions. by an amount proportional to the load, namely, by one unit of extension for each 4 tons per square inch of load. The effect is to contract the width of the diagrams, and to make any want of straightness in the curves more evident than it would otherwise be. To escape confusion, curves showing successive operations. are drawn from separate origins. In the experiment of figs. 18 and 19 the material under test was a medium steel, containing. about 0.4% of carbon, which when tested in the usual way showed a breaking strength of 39 tons per square inch with a well-marked elastic limit at about 22 tons. In fig. 18 the line A relates to a test of this material in its primitive condition; the loading was raised to 35 tons so as to produce a condition of severe overstrain. The load was then removed, and in a few minutes it was reapplied. The line B exhibits. the effect of this application. Its curved form shows plainly that all approach to perfect elasticity has disappeared, as a consequence of the overstraining. There is now no elastic limit, no range of stress within which Hooke's law applies. With the lapse of time the curve gradually recovers its straightness, and the material, if kept at ordinary atmospheric temperature, would show almost complete recovery in a month or two. But in this instance the recovery was hastened by immersing the piece for four minutes in boiling water, and line C shows that this treatment restored practically perfect elasticity up to a limit as high as the load by which the previous overstraining had been effected. The loading in C was continued past a new yield-point;. this made the elasticity again disappear, but it was restored in the same way as before, namely, by a few minutes' exposure to ioo C., and the line D shows the final test, in which the elastic limit has been raised in this manner to 45 tons. Other tests have shown that a temperature of even 50° C. has a considerable influence in hastening the recovery of elasticity after overstrain. In the non-elastic condition which follows immediately on overstrain the metal shows much hysteresis in the relation of r 2 3 a FIG. 19. strain to stress during any cyclic repetition of a process of loading. This is illustrated in fig. 19, where the arrows indicate the sequence of the operations. When a piece of iron or steel which has been overstrained in tension is submitted to compression, it shows, as might Muir, " On the Recovery of Iron from Overstrain," Phil. Trans.. A, vol. 193 (1900). Missing image Strengthofmaterials-9.jpg Missing image Strengthofmaterials-10.jpg
-40 Tons per sq.inch -35 -30 -25 -20 15 -45 -40 -35 -30 -25 -20 - 15 -50 Tons per sq. inch A. Primary test B.10 minutes after A C. After exposure for 4 minutes to 100°Cent. D.A fter further straining and exposure to 100`Cent. 3 4 be expected, no approach to conformity with Hooke's law until recovery has been brought about either by prolonged rest at ordinary temperature or by exposure for a short time to some higher temperature. After recovery has taken place the elastic limit in compression is found to have been lowered; that is to say, it occurs at a lower load than in a normal piece of the same metal. But it appears from Muir's experiments that the amount of this lowering is not at all equal to the amount by which the elastic limit has been raised in tension. In other words, the general effect of hardening by overstrain, followed by recovery of elasticity, is to widen the range within which a practically complete proportionality between strain and stress holds good. Contraction of Section at Rupture 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 FIG. 20. 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 Copyright Statement
These files are public domain. Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Strength of Materials'. 1911 Encyclopedia Britanica. https://www.studylight.org/encyclopedias/eng/bri/s/strength-of-materials.html. 1910.
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