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measuring the various curvatures and changes of curvatures of such a sample, it would be possible to construct a strain stress diagram with
the help of the following formula :
It must not be forgotten that p, the radius of curvature, grows smaller with
de increasing x, and that is negative.
As in the case of torsion experiments, *---*
the second term in the bracket is very
small when the point of rupture is Fig. 124
reached, and to estimate the ultimate strength of a plastic material, including cast iron, but not glass or other brittle substances, it is sufficiently accurate to use the formula
This is 33 % less than obtained by the generally accepted elastic formula, and accounts for the fact that beams are apparently so much stronger than they should be.
Compound Stresses. It has already been pointed out that the elastic limit and ultimate strength of a material are very much less in shear than in tension, or in compression. But it is well known that a shearing stress is composed of a tension and a compression stress, each of equal intensity, as shown in fig. 125, and it is therefore of importance to ascertain what other compound stresses exist.
There are, firstly, the simple stresses, (I.) tension, and (II.) compression.
A combination of two of these acting at right angles produces a (III.) shearing stress (fig. 125).
IV. Two equal compression stresses are met with chiefly in railway axles when the wheel boss has been shrunk on them. This combination might be called a shrinking or strangling stress, to distinguish it from a compression stress, which acts only in one direction. (See fig. 126.)
V. The reverse of this stress might be called drum tension (fig. 127), as it is best represented by that case; it is also met with in thin spherical shells subjected to internal pressure.
By adding stresses at right angles to the planes in which the last two are acting four others are obtained.
VI. Fluid Pressure (fig. 128).—In this case there are only compression stresses. If they are all changed into tension stresses we get à combination which might be called (VII.) negative fluid pressure, or solid tension (fig. 129).
VIII. When two of the stresses are compression, and the other one tension, we have the case of wire-drawing. This might be called a draw stress (fig. 130). (See p. 148.)
IX. By combining two tension stresses with one compression we reproduce a condition which is found on the inner spherical surfaces of very thick-walled exploding shells. This might be called a bomb stress (figs. 131 and 181). Of the last six combinations there is only one
about which anything definite is known, and that is that no material, however weak it may otherwise be, has been destroyed by fluid pressure, however great.
Solid tension (or negative fluid pressure) does not occur in prac tice, but the following two cases are an approach to this condition. If a solid sphere is heated to redness, and then plunged into cold water, the outer surface solidifies while the centre is still red hot. The external diameter will be somewhat larger than it would have been if cooled slowly; and when the centre has grown cold, a tension will be found
there acting in every direction. In large masses of steel this tension even comes into existence while the centre is still red hot, and cavities are formed, to prevent which ingots are never cast circular, but square or of polygonal shape, so that the sides may collapse. Of course it it impossible to estimate the stress which has produced these holes.
A somewhat similar stress is found at the point of contraction of test pieces (fig. 132) when of a circular section. The sample is being stretched in the direction of the arrows, 1, 1, and the lines of force, s, s, will adapt themselves to the fibres, at any rate at the circumference, and there their curved shape will produce radial tensions, as indicated by the looped arrows (fig. 133), so that at the centre of the smallest section there exists a tension in every direction.
This might explain why with mild steel, where there is considerable contraction of area, the fracture starts at the centre, the material being less able to withstand a solid tension than a simple one, as at the circumference of the fracture. A careful analysis of the distribution of stresses produced by the load at the instant of rupture will
perhaps enable one to obtain numerical values for different materials. Thus, it is not unusual for the contraction to exceed 50 %, and if the load at rupture was 80 % of the maximum, or, say, 24 tons per square in. instead of 30, the average stress in the reduced section must have been 48 tons; and when this is combined with the drum tension, due to the shape, which also exists there, it is not unreasonable to assume that the sample only gave way to a solid stress whose components amounted to from 96 to 144 tons. Hard cast steel will not resist these compound stresses without rupture, and therefore does not contract; and it might even be questioned whether for compound stresses this material is as strong as the milder qualities.
Recently Mr. J. J. Guest (Phil. Mag.,' July 1900, p. 70) has investigated this subject by experimenting on steel, brass, and copper tubes. These were placed in a tensile testing machine and had attached to them other appliances, so that stresses by tension, torsion, or fluid pressure could be produced both singly and combined. Each tube was tested over and over again, but only up to its elastic limit. A summary of some of the results is contained in the following table :-
It will be seen that a material which stands a stress of 40,000 lbs. when there is no cross stress (1) will stand a greater stress, viz. 45,000 lbs. (2), when there is also a circumferential stress of 30,000 lbs., and it is only when the latter stress is raised to 48,000 lbs. (3) that the material again gives way at the original stress of 40,000 lbs. It would have been very interesting if the test pieces could have been subjected to an axial compression stress, and to an axial compression combined with internal fluid pressure; then we would have obtained the limit of elasticity for (a) axial compression, (b) for circumferential tension, and (c) for circumferential tension combined with various intensities of axial compression. This latter information would have been very valuable, because this combination is a shearing stress, which, according to (4) of the table, causes the material to break down when the individual stresses are a little more than half the limit for the axial tension stress. The material dealt with in these experiments seems to have been fairly hard; it is therefore not desirable to draw definite conclusions from them as regards mild steel, though seeing how well crank shafts stand the most complicated combinations of stresses, it is perhaps not unreasonable to expect mild steel to offer more resistance when subjected to two right-angled tension stresses, than to one only, and that it is perhaps better able to resist a single tension than tension combined with compression, from which one could argue that this material will break down most easily if two right-angled compression stresses (strangling stress, p. 166) act on it.
These surmises appear to be confirmed by some very interesting experiments on flat unstayed plates which were subjected to various pressures and very carefully gauged by Professor C. Bach (* Deut. Ing.,' 1897, p. 1158, &c.).
These plates, five in number, were made of mild steel of 24.5 tons tenacity, with a limit of elasticity of 10.5 tons in tension and limit of plasticity of 15 tons. They ranged from 3 to 3 in. in thickness. In each plate the first permanent bulging occurred when the tension and compression stresses in the centres of the plates, as calculated by the author from the acquired curvatures, had reached from 7,000 to 8,000 lbs. (7,650 lbs. mean). This is almost exactly 30 % of the limit of elasticity of tension. Now these experiments do not show whether this giving way was due to V. p. 167, drum tension’on the outer surfaces of the plates, or to IV., 'strangling stresses' on the inner surfaces. Nor do these experiments show the drum tension due to the expansion of the shell. This shell plate was s in. thick and 28 ins. diameter. Its enlargement would tend to increase the drum tensions (see p. 166) and reduce the strangling stresses, but this influence would hardly affect the stresses by more than 10 %. Should it be shown that this steel gave way at 7650 + 765 = 8415 lbs. drum tension, then the result is in direct conflict with Mr. Guest's experiments, which show that with his harder steel the limit of elasticity for .drum tension’ may be in excess of that for simple tension. Possibly it will one day be shown that the plates gave way under a strangling stress of 7650–765=6885 lbs.i.e. at one-third of the elastic limit for tension, and that therefore a strangling stress is severer on mild steel than a shearing stress.
The nature of fractures of different materials indicates that they do not all behave in the same way under compound stresses. Thus mild steel tension fractures are always much inclined to the axis except near the middle, whereas for hard steel the fractured surfaces are normal to it and crystalline. Under torsion stresses mild steel fractures are normal to the axis, while hard steel fractures show a helical surface. Apparently mild steel yields with comparatively greater ease to shear than to tension, whereas hard steel is more easily torn than sheared. For this reason it is useless to speculate on the discrepancies between Platt and Hargraves and J. J. Guest's ratio of tensile and shearing strengths.
Further experiments dealing with this very interesting subject are urgently required.
Until experiments have been made to determine the permissible compound stresses, it would be useless to speculate as to their action, but enough has been said to show that we are as yet groping in the dark. It is even impossible to say whether a spherical boiler end may be made half as thick as the cylindrical shell; for although theory shows that the stress is only one-half, it also shows that there are two equal stresses, which have been called drum tension.