410 Mechanics of Materials S15.8 (f)Mohr's modified shear stress theory (ay>oy) For the original formulation of the theory based on the results of pure tension,pure compression and pure shear tests the Mohr failure envelope is as indicated in Fig.15.10. In its simplified form,however,based on just the pure tension and pure compression results,the failure envelope becomes that of Fig.15.11. 402 ay, 9 Shear (a) (b) Fig.15.10.(a)Mohr theory on o-axes.(b)Mohr theory failure envelope on a-02 axes. C2 (a) (b) Fig.15.11.(a)Simplified Mohr theory ontaxes.(b)Failure envelope for simplified Mohr theory. 15.8.Graphical solution of two-dimensional theory of failure problems The graphical representations of the failure theories,or yield loci,may be combined onto a single set of o and o2 coordinate axes as shown in Fig.15.12.Inside any particular locus or failure envelope elastic conditions prevail whilst points outside the loci suggest that yielding or fracture will occur.It will be noted that in most cases the maximum shear stress criterion is the most conservative of the theories.The combined diagram is particularly useful since it allows experimental points to be plotted to give an immediate assessment of failure410 Mechanics of Materials 515.8 (f) Mohr’s modijied shear stress theory (cJ,,, > cy,) For the original formulation of the theory based on the results of pure tension, pure compression and pure shear tests the Mohr failure envelope is as indicated in Fig. 15.10. In its simplified form, however, based on just the pure tension and pure compression results, the failure envelope becomes that of Fig. 15.11. Fig. 15.10. (a) Mohr theory on u-T axes. (b) Mohr theory failure envelope on u,-u2 axes. Q2 Fig. 15.1 1. (a) Simplified Mohr theory on u-T axes. (b) Failure envelope for simplified Mohr theory. 15.8. Graphical solation of two-dimensional theory of failure problems The graphical representations of the failure theories, or yield loci, may be combined onto a single set of ol and o2 coordinate axes as shown in Fig. 15.12. Inside any particular locus or failure envelope elastic conditions prevail whilst points outside the loci suggest that yielding or fracture will occur. It will be noted that in most cases the maximum shear stress criterion is the most conservative of the theories. The combined diagram is particularly useful since it allows experimental points to be plotted to give an immediate assessment of failure