## Stress paths in threedimensional stress space

We have considered two different forms of two-dimensional stress paths in Chapters 3 and 9 and we must now examine the form of these paths if they were plotted in three-dimensional space defined by p', q and v.

### Undrained tests

If we consider the plane q-p' then we can plot the effective stress paths for undrained shear in a manner similar to the previous two-dimensional stress paths. Remember that q = — 03 and that

The resulting diagram is shown in Fig. 13.4a. The points A,, A2 and A3 lie on the isotropic normal consolidation line and their respective stress paths reach the failure boundary at points B,, B2 and B3. As the tests are undrained the values of void ratio at points B[, B2, B3 are the same as they were when the soil was at the stress states Ab A2 and A3 respectively. Knowing the e values we can determine the values of specific volume and prepare the corresponding plot on the v-p' plane (Fig. 13.4b).

It is seen that the failure points B], B2 and B3 lie on a straight line in the q-p' plane and on a curve, similar to the normal consolidation curve, in the v-p' plane.

### Drained tests

The effective stress paths for drained shear are shown in Fig. 13.5. For the q-p' plane the plot consists of straight lines which are inclined to the horizontal at tan-1 3. The reason why is illustrated in Fig. 13.5.

Stress paths for drained shear.

Stress paths for drained shear.

The points Ct, C2 and C3 represent the failure points after drained shear, so the void ratio values at these points are less than those at the corresponding A points.

The stress paths in the v-p' plane are illustrated in Fig. 13.5b. As with the undrained case, the failure points Q, C2 and C3 lie on a curved line similar to the normal consolidation line.

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