Plotting Stress Paths Using Two-Dimensional Stress Parameters
For two-dimensional stresses, we can use an alternative stress path presentation based on Mohr’s circle. We can define
where t and s are the radius and center of Mohr’s circle, respectively, and represent the maximum shear stress and mean stress, respectively. This representation of stress neglects the effects of the intermediate principal stresses and is appropriate for plane stress condition. However, some geotechnical engineers use s9 or s and t for convenience, especially for plane strain condition, because we often do not know the value of the intermediate stress mobilized from conventional laboratory and fi eld test equipment. Recall that in a plane strain test the intermediate principal stress is not zero. So, by using s9 or s and t space, we are setting the intermediate principal stress to zero or a constant value. The s9 or s and t space is best used for plane stress condition (one principal stress equals zero). But plane stress condition rarely, if at all, represents conditions in the field.
You should be aware that the predicted changes in excess porewater pressure, which depend on mean stress p, would be different for stress path representations in (p, q) space and (s, t) space. For example, let us consider the triaxial compression test (axial stress increases and radial stress remains constant) for a linear, isotropic, elastic soil for which the TSP is represented by AB and the ESP is represented by AB9 (Figure. The predicted change in excess porewater pressure for (p, q) space (Figure ) is
Total and effective stress path in (p, q)
and (s, t) spaces.
For the (s, t) space
Thus, interpreting the excess porewater pressure from the stress path in (s, t) space would lead to a 50% greater excess porewater pressure than from the stress path in (p, q) space, because the intermediate stress is not accounted for in the (s, t) space. The slope of the TSP is also different for the two-stress-path space.
For the (p, q) space, the TSP for triaxial compression (TC) iswhile in (s, t) space it isIn the literature, p or p9 and q are sometimes used to denote the stress state characterized by s or s9 and t.
THE ESSENTIAL POINTS ARE:
1. A stress path is a graphical representation of stresses in stress space. For convenience, stress paths are plotted as deviatoric stress (q) on the ordinate versus mean effective stress (p9) and/or mean total stress (p) on the abscissa.
2. The effective stress path for a linear, elastic soil under the undrained condition is vertical; that is, Dp9 5 0 or Ds9 5 0.
3. The mean stress difference between the total stress path and the effective stress path is the excess porewater pressure.
4. The response, stability, and failure of soils depend on stress paths.