Shear strength of soil and the Mohr–Coulomb equation

Soil fails in shear: slopes slide, foundations punch in and retaining walls tilt because shear stress along some surface exceeds what the soil can carry.

Mohr–Coulomb strength

τ_f = c + σ tan φ

c is cohesion and φ the angle of internal friction. Clean sands have c ≈ 0. Soft saturated clays loaded quickly behave as φ ≈ 0 with all their strength in c.

Effective stress

Water in the pores carries part of the normal stress but no shear. Strength depends only on the stress carried by the grains, Terzaghi's effective stress:

σ' = σ − u

τ_f = c' + σ' tan φ'

That is why slopes fail after heavy rain: pore pressure u rises, σ' falls and so does the strength.

Which test?

Test	Good for	Notes
Direct shear	Sands, quick c and φ	Failure plane is forced; no pore pressure control
Triaxial (UU, CU, CD)	All soils	Controls drainage and measures pore pressure
Unconfined compression	Saturated clays	cu = qu / 2
Vane shear	Soft clays in the field	Gives undrained strength in place

Angle of the failure plane

In a triaxial test the failure plane makes this angle with the major principal plane:

θ = 45° + φ2

Drainage conditions

• UU (unconsolidated undrained): short-term stability, such as just after building an embankment on soft clay.
• CU (consolidated undrained): staged construction, sudden drawdown.
• CD (consolidated drained): long-term stability of slopes and foundations.

See Mohr's circle step by step in our Mohr's circle tool.