720 



NATURE 



[February 24, igi6 



square inch will burst it, but when the ring D is 

 mounted upon it, even a pressure of 2000 lb. per sq. in. 

 may be applied with safety. In the experiments de- 

 scribed above, the ring had a thickness equal to the 

 interior breadth of the cup-leather, but a small per- 

 centage of the total pressure is absorbed by this 

 leather, and is not exerted upon the ring-. The 

 experimental results must therefore be slightly lower 

 than the calculated values, and, if the correction is 

 made, there is a very good agreement, as the table 

 shows. In more complicated cases the stresses, are 

 less amenable to calculation, as in cylinders provided 

 with ribs for aeroplane engines, or tubes constructed 

 in sectors bolted together, as in underground railways, 

 but it would not be difficult to determine experiment- 

 ally the stresses produced . under working conditions, 

 although not in so simple a manner. 



So far our measurements have only utilised a pro- 

 perty which enables us to measure the difference of 

 the "principal stresses at a point in a plate subjected 

 to plane stress, but by far the greater number of 

 problems require a knowledge of the magnitudes of 

 each stress, separately, as well as their directions, and 

 it is to these determinations which we must now 



To PUMP- 



Cross-Section. 



Cross-Section. 



Fig 3. Pressure chambers for applyine fluid p essuie to the i .ternal and 

 external boundaries of rings;. 



address ourselves. First, as to the determination of 

 the magnitude of the principal stresses : — • 



Principal Stresses. 

 A measure of the sum of the principal stresses at 

 a point can be obtained, as Mesnager suggested, if 

 advantage be taken of the fact that the stress causes 

 a change in the thickness of a plate of material pro- 

 portional to the sum (p + q) of the principal stresses in 

 its own plane. If, for example, both stresses are 

 tensions, there will be a lateral contraction of 

 {p + q)/mE, where E is the modulus of direct elasticity, 

 and m is Poisson's ratio. Both these latter quantities 

 can be determined, and the sum of the stresses can be 

 measured, if an extensometer is used of sufficient 

 accuracy to measure the lateral contraction. For each 

 1000 lb. of stress intensity, the corresponding lateral 

 contraction for plates of the usual thickness of J in. is 

 1/3000 of an inch, and to measure such a quantity to 

 an accuracy of within i or 2 per cent., it is advisable 

 to use an instrument capable of indicating a change 

 of at least one-hundredth of this quantity; such 

 changes have been measured with fair accuracy by 

 using a lateral extensometer capable of detecting a 

 change of about half a millionth of an inch. An in- 

 NO. 2417, VOL. 96] 



strument of this kind has been employed by Mr. 

 Scoble and myself for an investigation of the stress 

 produced by a rivet in a plate, and a photograph of 

 one form of this apparatus is now shown on the 

 screen. 



For investigating cases of plane stress in general 

 the combination of the optical and mechanical methods 

 described here is chosen, in which the sum of the two 

 principal stresses at a point is found by a lateral 

 extensometer, and the difference by an optical measure- 

 ment, since both can be made to depend upon mechan- 

 ical measurements only, and are therefore particularly 

 adapted for engineering- work. In some cases it 

 requires considerable care to obtain accurate values of 

 each quantity separately, especially if one stress is very 

 much smaller than the other, as then minute errors of 

 observation become a large percentage of the value of 

 the lesser stress ; but possibly this difficult}' would be 

 met with in any other method. 



Lines of. Principal Stress. 



Reference has already been made to the fact thai 

 any state of stress at a point in a plane may be repre- 

 sented by a pair of stresses at right angles through 

 the point. 



Between crossed Nicol's prisms a loaded plate 

 shows, in general, dark bands, which mark the i^osi- 

 tions of all points where the directions of principal 

 stress correspond to the axes of the polariser and 

 analyser, and by varying the angular positions of 

 these latter a series of bands is obtained, each corre- 

 sponding to definite, directions of the axes of stress. 



If, for example, the case of a simple tension mem- 

 ber is taken, with notches in it on each side as shown, 

 dark bands are observed, and these change their posi- 

 tions as the axes of the optical apparatus are rotated. 

 A diagram may be constructed which shows the centn- 

 lines of a number of these curves, with the directions 

 of the axes of stress marked on them. Other lines 

 of principal stress at right angles to the first set an- 

 also indicated by the measurements, and the twc) 

 systems give a kind of framework diagram whicli 

 shows the direction of the principal stresses at any 

 point, and therefore completes the experimental solu- 

 tion of the problem. The stress distribution in a 

 plate cut to a required form, and stressed in an 

 arbitrary manner by forces in its own plane, is there- 

 fore capable of solution experimentally. 



Complete Solution. 



The complete experimental solution of the stress 

 distribution in a plate stressed by forces in its own 

 plane, may be illustrated by an investigation 

 mentioned above of the action of a rivet near 

 the edge of a riveted joint, since we can deter- 

 mine the sum {p + q) of the principal stresses, their 

 difference (p — q), and their directions. In this problem 

 we can no longer neglect either principal stress, and 

 it is in general necessary to determine both their 

 directions and magnitudes. If the uniform tension 

 stress in the full section of a plate is represented by 

 equally-spaced lines in the direction of stress, we 

 maj' expect to find alterations in their directions and 

 distances apart as they draw near to the discontinuity 

 produced by the rivet, and an optical examination 

 shows that the lines of stress approach one another 

 very closely as they pass around the rivet, and after- 

 wards diverge again if the overlap of the plate i-^ 

 sufficient to permit this. It is not difficult to explore 

 the whole of a plate stressed in this way, by deter- 

 I mining both the sum and difference of the stresses at 

 j a sufficient number of points on the lines of stress 

 I ,so found, and some of the measurements for the cross 



