Structural Geology

Stress, Strain & the Mohr Circle

Resolve normal and shear stress on any plane, read principal stresses off a stress block, and predict fault orientation with the Mohr–Coulomb criterion.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Stress at a point is a tensor: on any plane it resolves into a normal (

σ_{n}

) and shear (

τ

) component. The Mohr circle plots all

(σ_{n}, τ)

pairs as the plane orientation rotates — its centre is the mean stress

(σ_{1} + σ_{3}) /2

and its radius is the maximum shear

(σ_{1} - σ_{3}) /2

. Rock fails along planes where the Mohr circle touches the Coulomb failure envelope

τ = c + σ_{n} tan ϕ

; these conjugate shear planes lie at

4 5^{\circ} - ϕ /2

σ_{1}

The Core Formula Matrix

Normal stress on a plane (normal at

θ

σ_{1}

σ_{n} = \frac{σ _{1} + σ _{3}}{2} + \frac{σ _{1} - σ _{3}}{2} cos 2 θ

Shear stress:

τ = \frac{σ _{1} - σ _{3}}{2} sin 2 θ

**Principal stresses (from

σ_{x}, σ_{y}, τ_{x y}

)**:

σ_{1, 3} = \frac{σ _{x} + σ _{y}}{2} \pm (\frac{σ _{x} - σ _{y}}{2})^{2} + τ_{x y}^{2}

Maximum shear:

τ_{ma x} = \frac{σ _{1} - σ _{3}}{2}

Coulomb failure:

τ = c + σ_{n} tan ϕ

; conjugate shears at

4 5^{\circ} - ϕ /2

σ_{1}

Mohr circle with a Coulomb failure envelope (φ=30°, c=10).

The ‘IIT Traps’

⚠
** $2 θ$ , not $θ$ .** The angle on the Mohr circle is twice the physical plane angle.
⚠
Max shear ≠ on the σ₁ plane. Maximum shear acts on planes at $4 5^{\circ}$ to $σ_{1}$ , where $τ = (σ_{1} - σ_{3}) /2$ .
⚠
Sign of τ. Track the sense of shear consistently; magnitude is $\frac{σ _{1} - σ _{3}}{2} sin 2 θ$ .

📚 Standard references

Structural Geology — Haakon Fossen
Earth Structure — van der Pluijm & Marshak

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · NAT

With

σ_{1} = 120

MPa and

σ_{3} = 40

MPa, the maximum shear stress is _____ MPa.

σ₁=120, σ₃=40 MPa.

MPa

Q2MEDIUM2 Marks · NAT

With

σ_{1} = 120

and

σ_{3} = 40

MPa, the normal stress on a plane whose normal makes

3 0^{\circ}

with

σ_{1}

is _____ MPa.

Plane at 30° to σ₁.

MPa

Q3HARD2 Marks · NAT

At a point

σ_{x} = 100

σ_{y} = 20

τ_{x y} = 30

MPa. The major principal stress

σ_{1}

is _____ MPa.

σx=100, σy=20, τxy=30 MPa.

MPa