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ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 493 



probable limit in torsion was 6^ per cent, greater than half the elastic 

 limit in tension. Several comparative tests by Scoble^o are also represented 

 in the same diagram. These were carried out on solid unannealed test- 

 pieces subjected to combined bending and torsion, and the yield-points 

 were quoted as the bases of comparison. The limits under combined 

 stress appear unusually low in this series, lying well within the straight 

 line HG. 



Experiments ivith Combinatimi of three Finite Principal Stresses. 



For combinations of three finite principal stresses, the above method 

 of comparison is less convenient than that adopted by Cook and Robertson" 

 in their investigation of the elastic-limit of thick-walled tubes subjected 

 to internal fluid -pressure. In this investigation, a number of tubes were 

 tested under gradually increasing pressure, producing hoop-tension, axial- 

 tension and radial pressure in the material of the inner layer. By varying 

 the ratio, k, between the external and internal diameters of the tubes, the 

 relative magnitudes of the three principal stresses were varied widely, 

 affording a direct comparison of the effects of difierent complex stresses. 

 The elastic-limit of the material under simple tension, F, was also measured 

 and the limiting internal pressure, P, was expressed as a fraction of this. 

 Since the lengths of the tubes were great enough to render the influence 

 of the ends negligible. Lame's method of calculating the principal stresses 

 could be applied with confidence. These stresses are : 



X = hoop tension = 7 {k^ -{- 1) -^ {k^ — 1) 

 Y = axial tension = P -h (F — 1) 

 Z = radial pressure = P 



Cook and Robertson showed that, for each of the current hypotheses, 

 the limiting internal pressure, P, could be expressed in terms of the elastic- 

 limit, F ; thus : — 



On Rankine's hypothesis P/F = (jfc^ _ i) _j. (j^s + i) 



On Guest's hypothesis P/F = (P — 1) ^ 2k^ 



On de Saint Tenant's (with m = 4) P/F = 4 (P — 1) h- (5P + 2). 



Figure 20 shows these expressions plotted, as functions of k, for tubes 

 of different thicknesses. Cook and Robertson's experimental values are 

 also plotted and lie well below graphs I and II, representing the Rankine 

 and St. Venant hypotheses, but above the third graph. III, which repre- 

 sents Guest's hypothesis of constant maximum tangential-stress. The 

 author has added two graphs, IV, giving the values of the ratio, P/F, 

 corresponding to the hypothesis of constant strain-energy ; and it is 

 observed that these follow the experimental points very closely. 



According to the hypothesis of constant strain-energy the relation 



>o W. A. Scoble, Phil. 3Iag. ii. 1906. 

 " Cook and Robertson, Engineering, December 15, 1911. 



