Equation for Shear Strength of Unsaturated Soil

Terzaghi (1943) presented the Mohr-Coulomb shear strength theory in terms of effective stresses, (σ-u_w) for a saturated soil 

Saturated soil (after Terzhagi, 1943)

Where:

τ  = shear strength of soil (kPa)

c’  = effective cohesion (kPa)

(σ-u_w) = effectives stress (kPa)

ϕ’  = effective friction angle (^o)

Fredlund and Morgenstern (1977) proposed constitutive equations for unsaturated soils that can be formulated as an extension of the constitutive equations for saturated soils. Constitutive equations describe the relationships between shear strength or volume change with two independent stress state variables. The constitutive equations for unsaturated soils show a smooth transition to the constitutive equations for saturated soils when degree of saturation approaches 100 % or matric suction goes to zero.

Unsaturated soil (after Fredlund et al, 1978) 

Where:

(σ – u_w) = net normal stress(kPa)

(u_a – u_w) = matric suction (kPa)

c = total cohesion

ϕ^b = angle indicating the rate of increase in shear strength with respect to a change in matric suction (^o)

Equation for estimating the shear strength of unsaturated soil

Goh et al (2010) developed equation to estimate the drying and wetting shear strength of an unsaturated soil. They assumed the shear strength of soil to be linearly proportional to matric suction, where tan ϕ^b is equal to tan ϕ’, at matric suctions lower than air-entry value. At matric suctions higher than air-entry value, the relationship between shear strength and matric suction is non-linear where the calculated logarithm difference of matric suction and the air-entry value of soil increases nonlinearly when matric suction increases. 

Schnellmann et al (2015) developed equation to estimate the shear strength of an unsaturated soil. They observed that the effective degree of saturation play important role in controlling parameter for unsaturated soil property functions. They also concluded that the shear strength due to soil suction is interpreted to become zero at residual state.

Extended Mohr–Coulomb failure envelope for coarse-grained soils proposed by Schnellmann et al (2015)

References

1. Schnellmann, R., Rahardjo, H. and Schneider, H. R. (2015). “Controlling parameter for unsaturated soil property functions: validated on the unsaturated shear strength.” Canadian Geotechnical Journal, August, 52(3), 374-381. doi
2. Fredlund, D. G., Rahardjo, H. and Fredlund, M. D. (2012). Unsaturated soil mechanics in engineering practice, John Wiley & Sons, Inc., New York, 926 pages (ISBN 978-1-118-13359-0). doi
3. Goh, S. G., Rahardjo, H. and Leong, E. C. (2010). “Shear strength equations for unsaturated soil under drying and wetting.” Journal of Geotechnical and Geoenvironmental Engineering. April, 136(4), 594 –606. doi | handle
4. Terzaghi, K. (1943). Theoretical Soil Mechanics, Wiley, New York. doi
5. Fredlund, D. G. and Morgenstern, N. R. (1977), “Stress State Variables for Unsaturated Soils”, Journal of Geotechnical Engineering, 103(5), 447-466.