As it was mentioned earlier, Bishop (1959) was the first one to suggest two independent stress state variables viz. net normal stress and matric suction. He proposed an expression for shear strength in terms of these two independent stress state variables and a material variable constant, χf in a lumped form. The effective stress parameter χf , is a function of negative pore water pressure or volumetric water content of the soil. However, controlling or defining this parameter is not possible. Therefore, Bishop had proposed an indirect procedure to obtain effective stress parameter, by the use of traditional M-C criterion. The traditional M-C criterion, after substituting Bishop's effective stress definition for unsaturated soils, is given by
...........................................................(6.5)
where τf is the shear strength of the unsaturated soil; and c' and φ' are the effective cohesion and internal friction at full saturation.
The normal stresses and matric suction are known in a modified triaxial shear test. Thus, the M-C criterion can be written in terms of principle stresses using the geometrical analysis of the following Mohr's circle as
Fig. 6.19 M-C criterion for unsaturated soils by Bishop's approach
.................................................(6.6)