In saturated soft clay under total stress conditions (φu = 0), what is the theoretical depth of a tension crack expressed in terms of undrained cohesion c_u and unit weight γ?

Introduction / Context:Tension cracks can develop near the crest of slopes or excavations in purely cohesive soils (φu = 0) because the soil has finite tensile capacity related to its undrained shear strength. Estimating the crack depth informs safe excavation and stability checks, including the possibility of block toppling or piping paths for water.

Given Data / Assumptions:

• Undrained condition: φu = 0 (total stress analysis).
• Soil has undrained cohesion c_u and total unit weight γ.
• No surcharge and a vertical face idealization near the crest.

Concept / Approach:For φu = 0 soils, the maximum tensile stress that can be mobilized at the ground surface is related to c_u. Equilibrium of a thin vertical slice leads to a limiting crack depth proportional to c_u/γ. Classical analysis gives z_tc = 2 c_u / γ. Beyond this depth, vertical total stress exceeds the tensile capacity and crack propagation ceases.

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