Ductile Materials under Combined Stress. 121 



The stresses at the elastic limit have been considered in 

 several cases, and have been found to follow the same varia- 

 tions approximately. 



53. Graphical presentation of Variations. — The tables and 

 figures may be held to prove, iu the case of the materials 

 experimented upon — steel, copper and brass — that to a first 

 approximation the shearing-stress is constant whenever the 

 third principal stress is zero, or whenever the lesser principal 

 stress is zero or small. 



It can readily be found by trial that no other simple rela- 

 tion between the stresses or strains is approximately constant. 



The results of the series of tests upon steel, with the excep- 

 tion of the unannealed specimens, are shown in fig. 19, which 

 has been constructed as follows : — Ox is the axis of principal 

 stress most nearly coinciding with the generators, and the 

 abscissae represent the relative values of the stresses in the 

 various expei'iments, a mean axial yield-point stress being- 

 selected as the basis of the scale and marked 10. Relative 

 values of the other principal stresses were then set off parallel 

 to Oy and Oz, those parallel to Oy being the mean, and those 

 parallel to Oz being the minimum principal stress. The 

 paper should be now considered as folded at a right-angle 

 along the line Ox; and the observations recorded in the 

 quadrant zox to lie actually on the paper, while those plotted 

 in the quadrant xoy, though nearly in the plane of the paper, 

 should correctly be considered as projected by lines parallel 

 to 0,2 on to a plane through Ox inclined to the plane xoy at 

 an angle whose tangent is half the ratio of the internal pressure 

 to the circumferential stress it produces. The value of the 

 angle is about tan -1 O0225, so that the points may be regarded 

 as practically in the plane of the paper. 



On the paper are drawn the lines BA(J, BD representing 

 the locus of the points had the maximum stress been true ; 

 the lines GAH, KAL, MAN, representing the locus had the 

 maximum strain- law been true, and a had the values 05, 

 0*25, and 356 respectively, the latter being the mean ex- 

 perimental value. The constant shearing-stress law locus is 

 the line EFABD, and it will be seen to what extent this 

 agrees with the mean distribution of the observations. No 

 movement of the other lines parallel to themselves can pro- 

 duce any approximation to the mean of the actual distribution. 



It must, however, be remembered, while inspecting this 

 figure, that this method of setting out the points was adopted 

 as being just to any theory of strength ; had a mean basis 

 been adopted the points could have followed the shearing- 

 stress law much more closely. 



