§15.9 Theories of Elastic Failure 411 2 Load line for equal biaxiol tension or compression 6小 Load iine for torsion a)Principal stress Max.shear stress Max.principal strain d】 Total strain energy Shear strain energy Modified Mohr Fig.15.12.Combined yield loci for the various failure theories. probability according to the various theories.In the case of equal biaxial tension or compression for example o1/a2=1 and a so-called load line may be drawn through the origin with a slope of unity to represent this loading case.This line cuts the yield loci in the order of theoriesd;(a,b,e,f);and c.In the case of pure torsion,however,o=rand 2=-t, i.e.a/2=-1.This load line will therefore have a slope of-1 and the order of yield according to the various theories is now changed considerably to(b;e,f,d,c,a).The load line procedure may be used to produce rapid solutions of failure problems as shown in Example 15.2. 15.9.Graphical representation of the failure theories for three-dimensional stress systems 15.9.1.Ductile materials (a)Maximum shear strain energy or distortion energy (von Mises)theory It has been stated earlier that the failure of most ductile materials is most accurately governed by the distortion energy criterion which states that,at failure, (G1-02)2+(02-03)2+(3-01)2=203=constant In the special case where 3=0,this has been shown to give a yield locus which is an ellipse symmetrical about the shear diagonal.For a three-dimensional stress system the above equation defines the surface of a regular prism having a circular cross-section,i.e.a cylinder with its central axis along the line o=o2=o3.The axis thus passes through the origin of the principal stress coordinate system shown in Fig.15.13 and is inclined at equal angles to each515.9 Theories of Elastic Failure 41 1 Fig. 15.12. Combined yield loci for the various failure theories. probability according to the various theories. In the case of equal biaxial tension or compression for example al/uz = 1 and a so-called load line may be drawn through the origin with a slope of unity to represent this loading case. This line cuts the yield loci in the order of theories d; (a, b, e, f ); and c. In the case of pure torsion, however, u1 = z and uz = - z, i.e. al/az = - 1. This load line will therefore have a slope of - 1 and the order of yield according to the various theories is now changed considerably to (b; e, f, d, c, a). The load line procedure may be used to produce rapid solutions of failure problems as shown in Example 15.2. 15.9. Graphical representation of the failure theories for threedimensional stress systems 15.9.1. Ductile materials (a) Maximum shear strain energy or distortion energy (uon Mises) theory It has been stated earlier that the failure of most ductile materials is most accurately governed by the distortion energy criterion which states that, at failure, (al - az)’ + (az - a3)’ + (a3 - al)’ = 2a,Z = constant In the special case where u3 = 0, this has been shown to give a yield locus which is an ellipse symmetrical about the shear diagonal. For a three-dimensional stress system the above equation defines the surface of a regular prism having a circular cross-section, i.e. a cylinder with its central axis along the line u1 = uz = u3. The axis thus passes through the origin of the principal stress coordinate system shown in Fig. 15.13 and is inclined at equal angles to each