Materials under Combined Stress. 125 



The maximum stress hypothesis assumes that F 1 — constant. 

 If the maximum strain theory be correct, then, for stresses in 

 two directions only, Px— ^P 3 = constant. The stress differ- 

 ence or shear-stress law states that P x — P 3 = constant. Since all 

 these equations are particular forms of Pj + mP 3 = constant, it 

 appears to be desirable to examine the results to find whether the 

 stress distribution at yield may be expressed in a similar form. 

 The maximum and minimum principal stresses for these 

 tests are plotted in fig. 4. The points for the copper tubes 

 lie very close to a straight line because the original results 

 were exceptionally good. The other points would have been 

 close to the straight lines if corrected values had been taken, 

 but, if the remarks against the points are considered, the 

 lines represent the relations between the principal stresses 

 fairly well. The letters against the points in fig. 4 indicate 

 the positions of the corresponding points in the former 

 diagrams when compared to the mean curves. The letter 

 indicates the error in the original result from all causes, the 

 experimental error, difference in material, and in the case of 

 the steel tubes, due to the effect of repeated loading. On the 

 diagram, fig. 4, A is the maximum and B is the minimum 

 principal stress. Using the original notation, the equations 

 which express the relations between the principal stresses 

 are 



Pi- 1-57 P 3 = 71000, 



P x - 1-37 P 3 = 62200, 



P x - 1-26 P 3 = 54500. 

 The constant varies because it indicates the strength of the 

 material, and is the tensile stress at yield. The values for " m " 

 are 1*57, 1'37, and l'2b', but the materials are different. 



It is now evident that the shear-stress law is most nearly 

 correct of those stated above, and it is intended to be applied 

 to all ductile materials. Although the deviation from the 

 shear-stress law is sometimes considerable (under combined 

 bending and twisting), if an equation be adopted to express 

 the conditions at yield more closely, it must contain a con- 

 stant which depends upon the material considered. This is 

 a very serious complication, and engineers will probably 

 prefer to assume a constant shear stress at failure, although 

 the results obtained will be slightly inaccurate. Also 

 combined bending and torsion is not the only example of 

 combined loading, and it is desirable for practical purposes 

 to adopt one rule to apply to all cases, if that be possible. 

 Tt is therefore proposed to further consider this matter by 

 examining the results obtained by other observers, who 

 employed different methods and combinations of loadino-. 



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