Geomechanics & Ground Control

Mohr–Coulomb Failure

The shear-strength envelope τ = c + σ tanφ and the principal-stress failure criterion — the backbone of rock and soil strength.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Rock fails in shear when the shear stress on a plane reaches the material's resistance, which has two parts: a cohesion

c

(the strength at zero normal stress) and a frictional part that grows with the normal stress through the angle of internal friction

ϕ

. Plotted as

τ = c + σ tan ϕ

, this is the Mohr–Coulomb envelope; a stress state is safe while its Mohr circle stays below the line and fails when the circle just touches it. Re-cast in principal stresses, the criterion predicts the major principal stress

σ_{1}

a sample can carry for a given confinement

σ_{3}

The Core Formula Matrix

Shear-strength envelope:

τ = c + σ_{n} tan ϕ

Principal-stress form (at failure):

σ_{1} = σ_{3} N_{ϕ} + 2 c N_{ϕ}, N_{ϕ} = \frac{1 + sin ϕ}{1 - sin ϕ} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})

Uniaxial compressive strength (

σ_{3} = 0

σ_{c} = 2 c N_{ϕ} = 2 c tan (4 5^{\circ} + \frac{ϕ}{2})

c

= cohesion,

ϕ

= friction angle,

σ_{n}

= normal stress.

Mohr circle for σ₃ = 5, σ₁ = 49.64 MPa just touching the c = 10 MPa, φ = 30° envelope.

The ‘IIT Traps’

⚠
** $N_{ϕ}$ uses $4 5^{\circ} + ϕ /2$ , not $4 5^{\circ} - ϕ /2$ .** The half-angle is added for the strength (active-failure) direction.
⚠
Cohesion is the intercept, friction is the slope. Confusing $c$ (a stress) with $tan ϕ$ (dimensionless) breaks the units.
⚠
**The envelope is a straight line in $σ$ – $τ$ space, not the Mohr circle.** Failure is the circle *tangent* to the line, not where the circle peaks.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

A rock has cohesion

c = 20 kPa

and friction angle

ϕ = 3 0^{\circ}

. The shear strength on a plane carrying a normal stress of

100 kPa

is:

Q2MEDIUM2 Marks · NAT

A rock has cohesion

c = 10 MPa

and friction angle

ϕ = 3 0^{\circ}

. Its uniaxial compressive strength is ______ MPa. (Round off to two decimal places.)

MPa

Q3HARD2 Marks · NAT

For

c = 10 MPa

and

ϕ = 3 0^{\circ}

, the major principal stress at failure under a confining stress

σ_{3} = 5 MPa

is ______ MPa. (Round off to two decimal places.)

MPa