# Elasticity

### stress section load strength shearing steel piece iron material rupture

ELASTICITY, § 47, that E= 9C÷KC and 2(3K + C) Beyond the limits of elasticity the relation of strain to stress Plastic becomes very indefinite. Materials then exhibit, to a greater or strain. 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 icing) 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.

The ultimate strength of a material with regard to any stated Ultimode of stress is the stress required to produce rupture. In reckon- mate ing ultimate strength, however, engineers take, not the actual in- strength. tensity 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 strengths of the material is reckoned to be 30 toss per square inch, although theactual in tensity 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. Tho 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 ft, f,, and f„ 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 bo 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.

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 svill he 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).

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.

Ordinary tests of strength are made by submitting the piece Tests of to direct pull, direct compression, bending, or torsion. Testing strength, machines are frequently arranged so that they may apply any of these four modes of stress ; tests by direct tension are the most common,

Fig. 5. Fig. 6.

Wicksteed's Single-Lever Testing Machine.

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. Iii 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).

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 ; this screw-shaft is driveu 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 //4.' specimen, and the weight W is then run out along the lever so that the lever is just kept floating between the X*.

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- Fig. 7.

heads 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.

lever levers is employed to reduce the force between the specimen and testing the measuring weight. Probably the earliest machine of this class machines. was that of Major Wade,5 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 np the stretch, as in the single-lever machines already described. Mr Iiirkahly, who was one of the earliest as well as one of the most assiduous workers in this field, applies 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 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. lu 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.3 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 the 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 Min, but its inertia, when referred to the test-piece, is (Min) x n2 or Mn. The inertia which is effective for producing oscillation is thus increased is 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.

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.

In another important class of testing machines the stress Dia(applied as before to one end of the piece, by gearing or by phragm hydraulic pressure) testing is measure by con- machines.

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 bymeans of a screw at the top, and pulls up the lower side of a Wim hydrostatic bellows ; aaass■ water from the bel- , Ass.

Fla. 0. - Tilomasset's Testing Machine.

Darbel, Maillard,4 and Bailey. It has found its most important ap- Waterplication in the remarkable testing machine of Watertown arsenal, town built in 1879 by the U.S. Government to the designs of Mr A. H. machine. 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 420,000.5 The results of tests are very commonly exhibited by means Stress-of stress-strain diagrams, or diagrams showing the relation of strain strain to stress. A few typical diagrams for wrought-iron and diagrams. steel in tension are given in fig. 10, the data for which are taken from tests of long rods by Mr Kirkaldy.6 Up to the elastic bruit 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 Prof. Kennedy the yield-point, is reached, which is marked by a very sudden extension of and Lechler'. Memorial de r Artillerie et de fa Marine, 1883.

of Ordnance, 1883, appendix 24.

Experiments on the Mechanical Properties of Steel by a Committee of Civil Engineers, London, 1868 and 1870.

Masehine sum PrOfen d. Festigkeit d. Materialen, &c, Munich, 1882.

the specimen. After this the extension becomes less rapid ; then it another wire, fastened to a clip near one end of the specimen, and continues at a fairly regular and gradually increasing rate ; near 4 passing over a pulley near the other end, draws a pencil through the point of rupture the metal again begins to draw out rapidly.

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 Millh. ass dsm Mech.-Tech. Lab. in Munchen, heft 5.

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 Hodgkin son's experiments.1 The extension was mea sured on a rod 50 feet long ; the com- pression was also measured on a long rod, which was pre- vented from buckl- ing by being sup- ported in a trough with partitions. The full line gives the strain produced by loading ; it is continuous through the origin, showing that Young's mod- ulus is the same for pull and push. (Similar experi- ments on wrought- iron and steel in extension and com- pression have given the same result.) Fig IL 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.

Auto- 26. Testing machines are now frequently fitted with autographic wire or cord, so that the drum is made to revolve through angles proportional to the travel of the weight. At the same time Again, near the point of rapture, the flow again becomes specially rapid ; the weight on the lever has P 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 steal a m Harden- 30. But it is not only big effect through what we may call of per- the viscosity of materials manent that the time rate of load-set. ing affects their behaviour under test. In iron and steel, and probably in some other effect of a very re- . .

no. 12. - Autographie Diagram for a test of Influence markable kind. Let the mild steel.

of time. 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 he 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 The full line be in fig. 13 shows the sub- 1.,-€.

sequent behaviour of the piece. But now . .

, let the experiment be ‘; vat of time, of a few ff, hours or more, is al- ',,, 5 er z lowed to elapse after be found that a process r., r, of hardening has been ff going on during this `'' interval of rest; for, when the loading is continued, the new not at b as formerly, Extension, Per Cent.

change has taken place is afforded by the fact that the ultimate extension is reduced and the ultimate strength is increased (c, 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 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 i two experiments of this kind, made with annealed iron wire. A load of 231 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.

to rupture in 4 minutes and the other at a rate about 5000 times slower.

than it would have borne had the stress been uniformly distributed. 39. Local stretching causes the percentage of elongation which Influena This bad effect of punching is especially noticeable in thick plates a test-piece exhibits before rupture (an important quantity in en- of local of mild steel. It disappears when a narrow ring of material gineers' specifications) to vary greatly with the length and section stretch-surrounding the hole is removed by means of a rimer, so that of the piece tested. It is very usual to specify the length which ing on the material that is left is homogeneous. Another remarkable is to exhibit an assigned percentage of elongation. This, however, total instance of the same kind of action is seen when a mild-steel plate is not enough ; the percentage obviously depends on the relation elongawhich is to be tested by bending has a piece cut from its edge by of the transverse dimensions to the length. A fine wire of iron tion.

a shearing machine. The result of the shear is that the metal or steel, say 8 inches long, will stretch little more in proportion close to the edge is hardened, and, when the plate is bent, this part, to its length than a very long wire of the same quality. An being unable to stretch like the rest, starts a crack or tear which 8-inch bar, say 1 inch in diameter, will show something like twice quickly spreads across the plate on account of the fact that in as much the percentage of elongation as a very long rod. The ex-the metal at the end of the crack there is an enormously high periments of M. Barba , show that, iu material of uniform quality, local intensity of stress (see ELASTICITY, § 72). By the simple the percentage of extension is constant for test-pieces of similar expedient of planing off the hardened edge before bending the form, that is to say, for pieces of various size in which the plate homogeneity is restored, and the plate will then bend with- transverse dimensions are varied in the same proportion as the out damage. length. It is to be regretted that in ordinary testing it is not Anneal- 36. The hardening effect of strain is removed by the process of practicable to reduce the pieces to a standard form, with one ing. annealing, that is, by heating to redness and cooling slowly. In iron, proportion of transverse dimensions to length, since an arbitrary very mild steel, and most other metals the rate of cooling is a matter choice of length and cross-section gives results which are incapable of indifference; but in steel that contains more than about 0.2 per of direct comparison with one another.

cent. of carbon another kind of hardening is produced if the metal, 40. The form chosen for test-pieces iu tension tests affects not Influen after being heated to redness, is cooled suddenly. When the only the extension but also the ultimate strength. In the first on proportion of carbon is considerably greater than this, steel may place, if there is a sudden or rapid change in the area of cross strengt Harden- be rendered excessively hard and brittle (" glass-hard ") by sudden section at any part of the length under tension (as at AB, fig. 17), ing and cooling from a red heat. Further, by being subsequently heated the stress will not be uniformly distributed there. temper- to -a moderate temperature, it may be deprived of some of this The intensity will be greatest at the edges A and B, Mg of hardness and rendered elastic through a wide range of strain. and the piece will, in consequence, pass its elastic steel. This process is called the tempering of steel ; its effects depend on limit at a less value of the total load than would be the temperature to which the steel is heated after being hardened, the case if the change from the larger to the smaller and the grade of temper which is acquired is usually specified by section were gradual. In a non-ductile material, rupthe colour (blue, straw, &c.) which appears on a clean surface of tare will for the same reason take place at AB, with the metal during this heating, through the formation of a film of a less total load than would otherwise be borne. On oxide. iu the ordinary process of rolling plates or bars of iron or the other hand, with a sufficiently ductile material, mild steel the metal leaves the rolls at so high a temperature that although the section AB is the first to be permanently it is virtually annealed, or pretty nearly so.1 The case is different deformed, rupture will preferably take place at some with plates and bars that are rolled cold : they, like wire sup- section not near AB, because at and near AB the con-plied in the hard-drawn state (that is, without being annealed after traction of sectional area which precedes rupture is it leaves the draw-plate), exhibit the higher strength and greatly partly prevented by the presence of the projecting rig. 17 reduced plasticity which result from permanent set. portions C and D. Hence, too, with a ductile material Contrac- 37. The extension which occurs when a bar of uniform section is samples such as those of fig. 18, in which the part of smallest section tion of pulled is at first general, and is distributed with some approach between the shoulders or enlarged ends of the piece is short, will section to uniformity over the length of the bar. Before the bar breaks, break with a greater at nip- however, a large additional amount of local extension occurs at and load than could be tune. near the place of rupture. The material flows in that neighbour- borne by long unihood much more than in other parts of the bar, and the section is form rods of the much more contracted there than elsewhere. The contraction of same section. In .1 area at fracture is frequently stated as one of the results of a test, good wrought-iron and is a useful index to the quality of materials. If a flaw is pre- and mild steel the sent sufficient to determine the section at which rupture shall occur flow of metal prethe contraction of area will in general be distinctly diminished ceding rupture and as compared with the contraction in a specimen free from flaws, causing local conalthough little reduction may be noted in the total extension of traction of section Fig. 18.

difficult and the breaking load of the sample will be raised.' the mere interpretation of special tests. An important practical These considerations have of course a wider application than to 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.' In several of Mr Kirkaldy's papers, a comparison is given of the elastic first remarked by Mr Kirkaldy (Experiments on Wrought from and Steel, p. 74, limit, ultimate strength, and ultimate extension of samples which were annealed also Experiments on Fagersta Steel, p. 27). See also a paper by Mr E. Richards, before testing, and of samples which were tested In the commercial state ; in on testa of mild steel, Jour. Iron and Steel Inst., 1882.

the Steel Committee, part i. referred to in 135.

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.

sectional area and of different lengths, provided the length of both sequently greater than if the height were sufficient for shearing in were great enough to prevent the action described in § 40 from affect- a single plane.

ing the result. But, since no material is perfectly homogeneous, 44. The inclination of the surfaces of shear, when fracture takes Plane of the longer rod will in general be the weaker, offering as it does place by. shearing under a simple stress of pull or push, is a matter shear. more chances of a weak place ; and the probable defect strength of much interest, throwing some light on the question of how the in the lonon rod will depend on the degree of variability of the resistance which a material exerts to stress of one kind is affected by material. When this has been established by numerous tests of the presence of stress of another kind, - a question scarcely touched short samples, the strength which a rod of any assigned length may by direct experiment. At the shorn surface there is, in the ease be expected to possess can be calculated by an appli-of tension tests, a normal pull as well as a shearing stress, and in cation of the theory of probabilities. A theory of the the case of compression tests a normal push as well as shearing strength of long bars has been worked out on this stress. If this normal component were absent the material basis by Prof. Chaplin,1 and has been experimentally (assuming it to be isotropic) would shear in the surface of greatest confirmed by tests of long and short samples of wire. shearing stress, which, as we have seen in § 5, is a surface inclined The theory does not apply when the length is so small tj at 45° to the axis. In fact, however, it does not shear on this that the action of § 40 enters into the case, and the surface. Hodgkinson's experiments on the compression of cast-experimental data on which it is based must be takeniron give surfaces of shear whose normal is inclined at about 55° from tests of samples long enough to exclude that to the axis of stress,4 and Kirkaldy's, on the tension of steel, show action.that when rupture takes place by shear the normal to the surface Fracture 42. In tension tests, rupture may occur, as in fig. is inclined at about 25° to the axis.' These results show that Y 19, by direct separation over a surface which is nearly normal pull diminishes resistance to shearing and normal push ension. plane and normal to the line of stress. This is usual increases resistance to shearing. In the case of cast-iron under in hard steel and other comparatively non-ductile mate- compression, the material prefers to shear on a section where the rials. Or it may occur by shearing along an oblique intensity of shearing stress is only 0'94 of its value on the surface plane, as in fig. 20, which of maximum shearing stress (inclined at 45°), but where the shows the fracture of a piece - - normal push is reduced to 0.66 of its value on the surface of maxiof steel softer than the speci- mum shearing stress.

men of fig, 19. In very due-Fig. 45. Fatigue of Metals. - A matter of great practical as well as Fatigue Fig. 20. Fig. 21, ring is in two parts, one above and one millions of times in a year, and the spring works for years without below the surface of rupture of the central flat core. In other apparent injury. In such cases the stresses lie well within the instances, such as that of the sample shown in fig. 16, the shorn elastic limits. On the other hand, the toughest bar breaks after a by com- a process of flow may go on without limit ; the piece (which must normal ultimate strength. A laborious research by Wohler,5 ex- Wohler's pressinn- of course be short, to avoid buckling) shortens and bulges out in tending over twelve years, has given much important information experithe form of a cask. This is illustrated . regarding the effects on iron and steel of very numerous repeated ments.

by fig. 22 (from one of Fairbairn's exile- alternations of stress from positive to negative, or between a higher riments), which shows the compression and a lower value without change of sign. By means of ingeniously of a round block of steel (the original contrived machines lie submitted test-pieces to direct pull, alterheight and diameter of which are shown nated with complete or partial relaxation from pull, to repeated by the dotted lines) by a load equal tol'" bending in one direction and also in opposite directions, and to retional area. The surface over which the results show that a stress greatly less than the ultimate strength (as stress is distributed becomes enlarged, +1. 41 tested in the usual way by a single application.of load continued to and the total load must be increased in rupture) is sufficient to break a piece if it be often enough removed Fig, 22.

a corresponding degree to maintain the and restored, or even alternated with a less stress of the same process of flow.' The bulging often produces longitudinal cracks, kind. In that case, however, the variation of stress being less, as in the figure, especially when the material is fibrous as well as the number of repetitions required to produce rupture is greater.

plastic (as in the case of wrought-iron). A brittle material, such In general, the number of repetitions required to produce rupas cast-iron brick, or stone, yields by shearing on inclined planes ture is increased by reducing the range through which the stress as in figs. 23 and 24, which are taken from is varied, or by lowering the upper limit of that range. If the Hodgkinson's experiments on cast-iron.3 greatest stress be chosen small enough, it may be reduced, re-The simplest fracture of this kind is exem- moved, or even reversed many million times without destroying plified by fig. 23, where a single surface (ap- 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 ;6 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.

i..1 Stress. Number of Applications Stress. Number Number of Applications .1, Max. 31in causing Rupture.

10,141,645 106,901 __ 2,313,424 480,852 140 240 Not broken with 4 millions.

Fig. 23. Fig. 24.

proximately 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 I 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 COEran Nostrand's Engineering Magazine, Dec. 1880; Proc. Engineers' Club of Philadelphia, March, 1882.

Iron bar bent by transverse load :- Stress. Number of headings Stress. Number of Bendings Max. 11in, causing Rupture. Max. Min. causing Rupture.

Steel bar bent by transverse load: - Stress. Number of Bendings Stress. Number of Bending* Max. Situ. causing Rupture. Max. Min. causing Rupture.

I Die Festigkeits.Versuche mit Eisen and Stahl, Berlin, 1870, or Zeilschr. fur Bauwesen, 1860-70; see also Engineering, vol. xi., 1871. For early experiments by Fairbairn on the same subject, see Phil. Trans., 1864.

IV. Iron bar bent by supporting at one end, the other end being this strain falls within the elastic limit, the strain and the stress loaded ; alternations of stress from pull to push caused by rotating are twice as great as the same load would produce when in equine bar : - librium. Instances of load applied with complete suddenness, and Strain Strews. Number of Rotations Stress. Nnmber of Rotations yet without shock, are rare ; but it is a common case for loads to produe From + to - causing Rupture. From 4- to - causing Rupture. be applied so rapidly that the stress reaches a value intermediate by "lit 280183,145 180 19,186,791 same load applied. at once. Thus the Railway Commissioners found From these and other experiments Wohler concluded that fact that a "live" load produces greater stress than a dead. load is the wrought-iron to which the tests refer could probably bear an of course to be distinguished from the question Wohler's experiindefinite number of stress changes between the limits stated (in ments deal with - the greater destructiveness of the intermitted or round numbers) in the following table (the ultimate tensile strength varied stress which a live load causes. In many cases engineers was about 19i tons per square inch) : - allow in one operation for these quite independent influences of a Stress in Tons per Sq. Inch.

From poll to push +7 to - 7 live load by choosing a higher factor of safety for the live than for From pull to no stress 13 to 0 the dead part of the whole load on a structure, or (what is the From pull to less pull 19 to 10i same thing) by multiplying the live load by a coefficient (often 1i), Hence it appears that the actual strength of this material varies in adding the product to the dead load, and treating the sum as if a ratio which may be roughly given as 3 : 2 : 1 in the three cases of all were dead load.

(a) steady pull, (b) pull alternating with no stress, very many times 49. A useful application of diagrams showing the relation of Resilirepeated, and (c) pull alternating with push, very many times strain to stress is to determine the amount of work done in strain- once. repeated. Factors of safety applicable to the three cases might ing a piece in any assigned way. The term "resilience" is conventherefore rationally stand to one another in the ratio of 1 : 2 : 3. iently used to specify the amount of work done when the strain For steel Wohler obtained results of a generally similar kind. His just reaches the corresponding elastic limit. Thus a rod in simple experiments were repeated. by Spangenberg, who extended the tension or simple compression has a resilience per unit of volume inquiry to brass, gun-metal, and phosphor-bronze.' On the basis - f2/2E, where f is the greatest elastic pull or push. A blow whose of Wohler's results formulas have been devised by Launhardt, energy exceeds the resilience (reckoned for the kind of stress to Weyrauch, and others to express the probable actual strength of which the blow gives rise) must in the most favourable case pro-metals under assigned variations of stress ; these are, of course, of duce a permanent set ; in less favourable cases local permanent a merely empirical character, and the data are not yet extensive set will be produced although the energy of the blow is less than enough to give them much value.' the resilience, in consequence of the strain being unequally disThere are as yet no experiments showing how far fatigue of probably due to initial stress. In plastic metal a nearly complete strength is affected by the frequency, as distinguished from the state of ease is brought about by annealing ; even annealed pieces, mere number, of the stress-changes, nor whether a period of rest, however, sometimes show, in the first loading, small defects of under small loads which Hodgkinson discovered in cast-iron is gated by Thomson causes such stress-changes as occur rapidly to do the time the inside has become solid. The inside then contracts, and work on the material, and the destructive effect of repeated changes its contraction is resisted by the shell, which is thereby compressed may be due in great part to this cause. His experiments show that in a tangential direction, while the metal in the interior is pulled rapid stress-changes often repeated do produce a cumulative effect in the direction of the radius. Allusion has already been made to in reducing the modulus of elasticity ; and it is very probable that the fact, pointed out by J. Thomson, that the defect of elasticity after fatigue has been induced, restores strength. That it does so elasticity, which are probably due to initial stress, as they disappear may be conjectured from Thomson's discovery that rest restores when the load is reapplied.

elasticity after elastic fatigue. The conjecture is strengthened by 51. Little is exactly known with regard to the effect of tempera- Effect of Bauschinger's discovery that, after a permanent set has been pro- ture on the strength of materials. Sonic metals, notably iron or temperaduced and a period of rest follows, the apparent limit of elasticity steel containing much phosphorus, show a marked increase in brittle- ture on (in the strict sense of that term) rises slowly with the lapse of time. ness at low temperatures, or "cold shortness." Experiments on the strength. Both questions are of obvious practical interest.4 tensile strength of wrought-iron and steel show in general little When a strain is produced within the limits to which variation within the usual atmospheric range of heat and cold. Hooke's law applies, the work done in producing it is half the The tensile strength appears to be slightly reduced at very low product of the stress into the strain. A load applied to a piece temperatures, and to reach a maximum when the metal is warmed suddenly, but without impact, does an amount of work in straining to a temperature between 100° C. and 200° C. When the tempera-the piece which is measured by the weight of the load into the ture exceeds 300° C. the tensile strength begins to fall off rapidly, distance it sinks in consequence of the strain. Hence, provided and •at 1000° C. it is less than one-tenth of the normal value.?

I Ueber dos Verhalten der Metalle bei weiderholten A nstrengungen, Berlin, 1875. Reference may be made, in this connexion, to the effect which a 2 See Weyranch, "On the Calculation of Dimensions as depending on the "blue heat," or temperature short of red heat, is believed to have Ultimate Working Strength of Materials," Min. Proc. Inst. C.E„ vol. lxiii. p. 275; on the plasticity and strength of iron, and more especially of mild.

also a correspondence In Engineering, vol. xxix., and Linwin's Machine Design, chap. it. a Ewing, Phil. Trans•, 1885, 1880, Steel. It appears that steel- plates and bars bent or otherwise ac., and resnits of experiments showing reduced plasticity in fatigued metal, 5 Report of Commissioners on the Application of Iron to Railway Structures, the phenomenon is complicated by the occurrence of blows, or shocks whose energy a Comb. and Dub. Math. Journ., Nov., 1848.

is absorbed In producing strains often exceeding the elastic limits, sometimes 7 See Report of a Committee of the Franklin Institute, 1837 ; Fairbairn, Brit.

of a very local character In consequence of the inertia of the strained pieces. Ass. Rep., 1856; Styffe on Iron and Steel, trans. by C. P. Sandberg. Notices of Such shocks may cause an accumulation of set which finally leads to rupture in a these and other experiments will be found In Thurston's Materials of Engineering, way that Is not to be confused with ordinary fatigue of strength. It appears Si. chap. x., and in papers by J. J. Webster, Min. Proc. Inst. C.E., vol. lx,, and that the effects of fatigue may be removed by annealing. A. Martens, Zeitschr. des Ver. Deutsch. Ing., 1883.

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.1 ri- 52. The following table gives a few representative data regarding Graphic 53. Space admits of no more than a short and elementary of distri- The stress which acts on any plane surface AB (fig. 25), such as buted an imaginary cross-section of a strained stress, piece, may be represented by a figure formed by setting up ordinates Aa, Bb, 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 the stress figure, parallel to the ordinates Fig. 25.

Centre of Aa, &c., determines the point C, which is called the centre of stress, tress. 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 All. When a bar is subjected to simple pull applied axially - that Stromeyer, ” The Injurious Effect of a Blue heat on Steel and Iron," Min. Proc. Inst. C.E., vol. lxxxiv., 1886.

is to say, so that the resultant stress passes through the centre of gravity of every cross-section, - the stress maybe 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.

in a long strut or column where buckling 6 stress.

varying makes the stress become non-axial. In -- proportional to the -- . ----------- distance of P from . -gl.

a line M N, called _ Fig. 2G the neutral axis, which 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 vana- 4._.__,2,., C, the centre of gravity of the stir- Flg. 27.

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 dS be an indefinitely small part of the surface at a distance x from the neutral axis through C, and let p he the intensity of stress on dS. The moment of the stress on dS is xpdS. But p - pix/x1=p2x/x2 (see fig. 27). The whole moment of the stress on AB is fxpdS(pilxi)l'x2dS - pillxi or p2I/x-2, where I is the moment of inertia of the surface AB about - the neutral axis through C.

intensity at the centre of gravity of the surface C) and a stress of ' the kind shown in fig. 27. The resultant is NS, where S is the whole area of the surface, and it acts at a distance CD from C such that the moment poS . CD= (Ps - Pe)I/xi - (Pi +Po)i/x2. Hence 1/2 - p0(1 +x,S . CD/I), and pt - ingoecurs when a Fig. 23.

W,/i - W2/2, the portion of the beam lying between W1 and W5 is subjected to a simple • pull and push, and has for its resultant a couple whose moment 11FilgL 21;,/s - W2/2. This is called the bending moment at the section. If the stress be within non-elastic strain will begin as soon as either Let the bending moment now be increased; or , exceeds the corresponding limit of elasticity, 3 beyond /I and the distribution of stress will be changed in stance 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.

In consequence of the action which has been illustrated (in a somewhat crude fashion) by figs. 31 and 32, the moment required Bridge Iron 1 19 across „ 'about Bars, finest 27 to 29 1 off, Bessemer) ! average ( about 30 about Crucible tool steel 40 to 65 ...

Chrome steel SO Whitworth's fluid-compressed Cast-iron „ average American (ordnance) Zinc, cast „ rolled Pitch pine.

Sandstone Limestone.

to break the beam (Mr) cannot be calculated from the ultimate tensile or compressive strength of the material by using the for- mula Mi - LT/Yi, or Mi - f,Itya. 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 A of the modulus of trans- A verse rupture. Let the section be such that y ture. defined as f - Mq• This mode of stating t e results of experiment on transverse strength is unsatisfactory, inasmuch as the modulus of rupture thus determined will vary with different Fig. 31. Fig. 32.

forms of section. Thus a plastic material for which f, and f, 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 rupture sensibly equal to f, or fe. 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 ft, and the other half in uniform compression of the same intensity. The moment of stress is then fib/0, b being the breadth and h the depth of the section ; but by definition of the modulus of rupture!, /11,-.Ifbh2. 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.

Strain 59. The strain produced by bending stress in a bar or beam is, as produced regards any imaginary filament taken along the length by bend- of the piece, sensibly the same as if that filament were big. directly pulled or compressed by itself. The resulting ii deformation of the piece consists, in the first place and II chiefly, of curvature in the direction of the length, due ; to the longitudinal extension and compression of the I filaments, and, in the second place, of transverse flex- si ure, due to the lateral compression and extension which go along with their longitudinal extension and compression (see ELASTICITY, § 57). Let 4, fig. 33, be a I short portion of the length of a beam strained by a bend- II ing moment M (within the limits of elasticity). The I beam, which we assume to he originally straight, bends in the direction of its length to a curve of radius 13, f such that R/i - y,/8/, 81 being the change of 1 by exten.

sion or compression, at a distance yi from the neutral I axis. But 8/-4/01/E by § 10, and pi - Bin/1. 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 where a is the ratio of longitudinal extension to lateral contraction under simple pull. Fig. 33.

Ordinary 60. Bending combined with shearing is the mode of stress to bending which beams are ordinarily subject, the loads, or externally applied of beams. 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, which lies on one side of AB, is in equilibrium under the joint action of the external forces F1, F2, Fa, &c., and the forces which the other portion U exerts on V in consequence moments arc Fix„ Faxo, Fora, &c. Hence the stress at AB must equilibrate, first, a couple whose moment is and, second, a force whose valve is V, 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 F3 is opposite to the sign of F1 and F2. 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 (2Fx), and a shearing force equal to the suns of the forces about one side of the section (IF). 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 iu horizontal planes parallel to the length of the beam.

- 32:SI/r/63-8M/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 (Si and 82 respectively) should be proportioned to the value of the ultimate strengths!, and f„ so that Si! ,=--Sof,. Thus for cast-iron beams Hodgkinson has recommended that the tension flange should have six tunes the sectional area of tho compression flange. The intensity of longitudinal stress on the two flanges of an I beam is approximately MIS,h, and M/Salt, being the depth from centre to centre of the flanges.

In the examination of loaded beams it is convenient to re- Diagrams present graphically the bending moment and the shearing force at of bend-various sections by setting up ordinates to represent the values of ing mothese quantities. Curves of bending moment and shearing force ment and for a number of important practical cases of beams supported at shearing the ends will be found in the article BRIDGES, with expressions for force. 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 +8x, the bending moment changes to 117(x +8x), or SDI-8x2F. 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+ 8x, the moment changes by an additional amount which depends on (8x)2 and may therefore be neglected.

But 8N1/3x is the whole shearing force Q on the section of the beam. Hence Q frfr , ; i and this is also the intensityzo of vertical shearing stress at the distance yo from the neutral axis. This expression may conveniently be written g -Q412.01, 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. -IQ/bh. In other words, the maximum intensity of shearing stress ou any section is -I of the mean intensity. Similarly, in a beam of circular section the maximum is 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.

Principal 64. The foregoing analysis of the stresses in a beam, which stresses resolves them into -longitudinal pull and push, due to bending in a moment, along with shear in longitudinal and transverse planes, is beam. 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 or 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 tangential. Let AC (6g. 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 intensity 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 intensity of that stress. Let 0 be the angle (to be determined) which BC makes with AB. Then the equilibrium of the triangular wedge ABC requires that rBC cos 0 -p . AB+ q. AC , and rBC sin 0- q . AB ; or (r - p) cos q sin 8, and r sin 0 q cos .

Hence, q2 s. r(r - tan 20=2q1p , 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 0 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 ire+42+ iy2 (by § 5).

The equation for 0 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.

dz ; dx ' Integrating this for a beam of uniform section, of span L, supported at its ends and loaded with a weight IV at the centre, we have, for the greatest slope and greatest deflexion, respectively, i, = WL2/16EI, ul-IVL2/48E1. If the load W is uniformly distributed over L, 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.

the plane BB. As drawn at any dis- Fig. 40.

tance rfrom the axis, and originally straight, changes into the helix AD'. Let 0 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 Odx-rdi. Hence q, the intensity of shearing stress iu a plane of cross-section, varies as s', since q= CO - Cr . The resultant moment of the whole shearing stress on each plane of cross-section is equal to the twisting moment M. Thus ../.2ne-qdr .

Calling r1 the outside radius (where the shearing stress is greatest) and q1 its intensity there, we have q - rqilri, and hence, for a solid shaft, q,.--2M/rrr,3. For a hollow shaft with a central hole of radius r, the same reasoning applies : the limits of integration are now ri and r2, and 2Mr, qr ir(r14 -7.24) 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 -A) the stress is greatest at the middle of each side, and its intensity there' is M/0'28110.

For round sections the angle of twist per unit of length is 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 uniform value and the moment of torsion will be 41271-740r =Irrq'(r12- 7'52). In the ease 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 3.1 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 ft and f,. 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.

Torsion 70. Twisting combined with Bending. - This important practical of angles to the plane of the crank.

cranks. At any section of the shaft (between the crank and the bearing) there is a twisting moment M1-P . AB, and a bending moment M2 = P . BC. There is also a direct shearing force P, but this does not require to be taken into account in calculating _ In" the stress at points at the top or bottom of the tributed so that its intensity is zero at these points. The stress there is consequently made up of longitudinal normal stress (due to bending), h -4Mdwri3, and shearing stress (due to torsion), 2Nf1brr13. Combining these, as in § 64, we find for the principal stresses r -2(M,IVIS112+M22)/irri3, or r =2P(B0 AC)/irri3. The greatest shearing stress is 2P . AC/aril; and the axes of principal stress are inclined so that tan 20=.111/112= AB/BC. The axis of greater principal stress bisects the angle ACS.

The above theory, which is Euler's, assigns P, as a limit to the strength of a strut on account of flexural instability ; but a stress less than PI may cause direct crushing. Let S be the area of section, and f, 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,S is less than Pi, but by bending if f,S is greater than P5. 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 P1), 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,S or P1. 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 recouciled 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, or ft 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 tint the greatest deflexion consistent with stability will vary as L2/b, b being the least breadth, and the greatest and least stress, at opposite edges of the middle section, will consequently be -g- if -7-)r) , where a is a coefficient depending on the material and the form of the section. This gives, for the breaking load, P=Sf,/(1 +aL2/82) or - Sft/(1 - aL21b2), 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 off 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,S, and for very long struts it is T2EI/L2. If we write P =f,S/(1 +f,SL2/71-2E1), 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=fS/(1 +eSL2 /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.3 For values of the constants, see BRIDGES. In the case of fixed ends, c is to be divided by 4.

Bursting Strength of Circular Cylinders and Spheres. - Space Strength remains for the consideration of only one other mode of stress, of of shells great importance from its occurrence in boilers, to resist pipes, hydraulic and steam cylinders, and guns. T -r bursting.

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 So thickness. Let p be the intensity of fluid pressure 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 perpendicular to the direction of the axis, of length l and width RIO, 80 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=p/R80 and T =fit. But by the triangle of forces (fig. 43) P= T50. Hence f= pRit.

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 -r the ends, a transverse section, the area of which is 2,rRt, has to bear a total force pirR2. Hence, if f' be the intensity of Fig. 43.

longitudinal stress, f' =pRi2t=4f.

Thick 75. When the thickness is not small compared with the radius, cylinder. 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 1 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')/cE. 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 - r)-p'.

d r(p These two equations give by integration, p-C+Clr2, and p' =0 - OW°.

If 7.1 be the external and 7.2 the internal radius, and po the pressure on the inner surface, the conditions that p=p, when r = r2 and p = 0 when r =r1 give C /v2/(r12 - r22) and 0'= - Cr12. Hence the circumferential stress at any radius r is p'= - por22(1+ri2/72)/(ri2 - r22). At the inside, where this is greatest, its value is - po(r,2+r22)/(r,=- r22), - a quantity always greater than po, 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 aro shrunk on over the barrel or over the inner hoops. In Mr Long-ridge's system, now under trial, the gun is made by winding steel wire or ribbon, with suitable initial tension, on a central barrel.

-po(r13+ 2r23)/2(r,° -r23). (J. A. E.) This condition is realized in practice when the fluid causing internal pre,sure Is held in by a piston, and the stress between this piston and the other end of the cylinder Is taken by some other part of the structure than the cylinder sides. 2 Tho solution which follows In the text is applicable even when there is longitudinal stress, provided that the longitudinal stress is uniformly distributed over each transverse section. If we call this stress p", the longitudinal strain Is p"/E+(p-Fp')/0-E. Since the whole strain is uniform, and p" is uniform, the sum of p andp' Is constant at all points, as in the case where the ends are free.

gained the important victory usually named after the village of Hohenfriedberg (June 4, 1745).