Shear strength is the fundamental engineering property that governs the stability of slopes, foundations, retaining walls, and excavations.
Shear strength is the maximum shear stress a soil can sustain before failure. Unlike many engineering materials, the shear strength of soil depends on:
- Effective normal stress — the stress carried by the soil skeleton
- Drainage conditions — drained vs. undrained behaviour
- Soil type and density — granular vs. cohesive
- Stress history — normally consolidated vs. overconsolidated
- Rate of loading — static vs. dynamic/cyclic
Mohr-Coulomb Failure Criterion
The Fundamental Equation
$$ \tau_f = c' + \sigma'_n \tan\phi' $$Where:
- $\tau_f$ = shear stress at failure
- $c'$ = effective cohesion (kPa)
- $\sigma'_n$ = effective normal stress on the failure plane
- $\phi'$ = effective friction angle (degrees)
Total stress form:
$$ \tau_f = c_u + \sigma_n \tan\phi_u $$(Used for undrained conditions, typically $\phi_u = 0$ for saturated clays)
Mohr's Circle Representation
τ
│
│ ┌──────────────────────────┐
│ │ Failure envelope │
│ │ τ_f = c' + σ'_n tan ϕ' │
│ │ │
│ │ ┌───────┐ │
│ │ / Circle \ │
│ c'──/ at \─── │
│ │ │ failure │ │ │
│ │ \ / │
│ │ └───────┘ │
│ │σ'_3 σ'_1 │
└────┴─────────────────────────── σ'
Major principal stress at failure:
$$ \sigma'_1 = \sigma'_3 \tan^2\left(45° + \frac{\phi'}{2}\right) + 2c' \tan\left(45° + \frac{\phi'}{2}\right) $$Failure plane orientation:
$$ \theta = 45° + \frac{\phi'}{2} $$Where $\theta$ is the angle between the failure plane and the major principal plane.
Drained vs Undrained Behaviour
Drained Conditions
- All excess pore pressures have dissipated
- Loading rate is slow relative to soil permeability
- Effective stresses govern behaviour
- Parameters: $c'$ and $\phi'$
Undrained Conditions
- No pore water drainage occurs during loading
- Loading rate is faster than drainage
- Total stresses govern short-term behaviour
- For saturated clays: $\phi_u = 0$ concept, $s_u = c_u$
When to use each:
| Condition | Drained | Undrained |
|---|---|---|
| Sand/granular — long-term | ✓ | — |
| Sand/granular — short-term | ✓ | — |
| Clay — short-term (construction) | — | ✓ |
| Clay — long-term (operational) | ✓ | — |
| Rapid loading (earthquake, storm) | — | ✓ |
| Slow loading (fill placement) | ✓ | — |
Laboratory Shear Strength Tests
Triaxial Compression Test (AS 1289.6.4.x)
The triaxial test is the most versatile and reliable laboratory shear strength test.
Test types:
| Test Type | Drainage Phase | Loading Phase | Parameters Obtained |
|---|---|---|---|
| UU (Unconsolidated Undrained) | No drainage | No drainage | $s_u$ (total stress) |
| CIU (Consolidated Isotropic Undrained) | Drainage allowed | No drainage | $c'$, $\phi'$ (with pore pressure measurement) |
| CID (Consolidated Isotropic Drained) | Drainage allowed | Drainage allowed | $c'$, $\phi'$ (effective stress) |
Typical test procedure (CIU):
- Saturation: Back pressure saturation (B-check, $B \geq 0.95$)
- Consolidation: Drainage allowed under cell pressure
- Shearing: Axial compression at constant strain rate (0.05–1.0 mm/min)
- Measurement: Deviator stress, pore pressure, axial strain
Typical strain rates:
| Soil Type | UU | CIU | CID |
|---|---|---|---|
| Clay | 1–2%/min | 0.1–0.5%/min | 0.01–0.05%/min |
| Silt | 1–2%/min | 0.1–0.5%/min | 0.01–0.05%/min |
| Sand (saturated) | — | — | 0.1–0.5%/min |
Direct Shear Test (AS 1289.6.2.2)
A simpler, lower-cost alternative to the triaxial test.
Procedure:
- Soil sample placed in a split box
- Normal load applied
- Shear force applied to the upper half of the box
- Test repeated at 3–4 different normal stresses
Advantages: Simple, quick, inexpensive
Limitations: Forced failure plane, non-uniform stress distribution, drainage control difficult
Unconfined Compression Test (AS 1289.6.1.1)
For cohesive soils only:
$$ q_u = \frac{P_{max}}{A} $$ $$ s_u = \frac{q_u}{2} $$| Consistency | $q_u$ (kPa) | $s_u$ (kPa) |
|---|---|---|
| Very soft | < 25 | < 12 |
| Soft | 25–50 | 12–25 |
| Firm | 50–100 | 25–50 |
| Stiff | 100–200 | 50–100 |
| Very stiff | 200–400 | 100–200 |
| Hard | > 400 | > 200 |
Vane Shear Test (AS 1289.6.2.1)
In-situ or laboratory test for soft clays:
$$ s_u = \frac{T}{\pi D^2 (H/2 + D/6)} $$Where $T$ = torque at failure, $D$ = vane diameter, $H$ = vane height.
Correction factors (Bjerrum):
$$ s_u(\text{design}) = \mu \times s_u(\text{measured}) $$| PI | Correction Factor $\mu$ |
|---|---|
| 20 | 1.0 |
| 40 | 0.85 |
| 60 | 0.75 |
| 80 | 0.70 |
| 100 | 0.65 |
Field Shear Strength Tests
Standard Penetration Test (SPT)
N-value correlations for friction angle ($\phi'$) in sands:
| SPT N-value | Relative Density | $\phi'$ (degrees) |
|---|---|---|
| 0–4 | Very loose | 25–28 |
| 4–10 | Loose | 28–32 |
| 10–30 | Medium dense | 32–36 |
| 30–50 | Dense | 36–40 |
| > 50 | Very dense | 40–45 |
Peck, Hanson & Thornburn correlation:
$$ \phi' = 0.5N + 27 \quad (\text{for } N \leq 30) $$Cone Penetration Test (CPT)
Undrained shear strength from CPT:
$$ s_u = \frac{q_t - \sigma_{v0}}{N_{kt}} $$Where $N_{kt}$ = cone factor (typically 12–20, average ≈ 15)
Friction angle from CPT (Robertson & Campanella):
$$ \tan\phi' = 0.1 + 0.38\log\left(\frac{q_c}{\sigma'_{v0}}\right) $$Factors Affecting Shear Strength
Effect of Drainage
| Condition | Sand | Normally Consolidated Clay | Overconsolidated Clay |
|---|---|---|---|
| Undrained | $s_u$ depends on density | $s_u/\sigma'_{v0} \approx 0.22$ | $s_u/\sigma'_{v0}$ increases with OCR |
| Drained | $\phi' = 30-40°$ | $\phi' = 22-28°$ | $\phi' = 24-32°$ |
Effect of Overconsolidation
Undrained strength ratio:
$$ \left(\frac{s_u}{\sigma'_{v0}}\right)_{OC} = \left(\frac{s_u}{\sigma'_{v0}}\right)_{NC} \times OCR^{0.8} $$Effect of Strain Rate
Typically, shear strength increases by approximately 5–15% per log cycle of strain rate.
Effect of Structure and Bonding
Natural clays often have a "structured" component — additional strength from cementation/bonding that is destroyed upon disturbance.
Peak, Critical State, and Residual Strengths
Dense Sands and Overconsolidated Clays
τ
│
│ ┌───── Peak
│ /│\
│ / │ \ ─── Critical State
│ / │ \
│/ │ \ ──── Residual
└────┴────┴───── Strain
| Strength | Sand | Overconsolidated Clay |
|---|---|---|
| Peak ($\phi'_p$) | 38–45° | 25–35° |
| Critical state ($\phi'_{cs}$) | 30–35° | 20–30° |
| Residual ($\phi'_r$) | — | 8–20° (clay with clay minerals) |
Sensitivity of Clays
$$ S_t = \frac{s_u(\text{undisturbed})}{s_u(\text{remoulded})} $$| Sensitivity | Classification |
|---|---|
| 1–2 | Insensitive |
| 2–4 | Medium sensitivity |
| 4–8 | Sensitive |
| 8–16 | Extra sensitive |
| > 16 | Quick clay |
Typical Strength Parameters
Effective Stress Parameters
| Soil Type | $c'$ (kPa) | $\phi'$ (°) |
|---|---|---|
| Clean sand (loose) | 0 | 30–33 |
| Clean sand (dense) | 0 | 35–40 |
| Silty sand | 0–5 | 28–35 |
| Sandy clay | 5–20 | 25–32 |
| Silty clay | 5–25 | 20–28 |
| Clay (low plasticity, CL) | 5–15 | 25–30 |
| Clay (high plasticity, CH) | 5–20 | 15–25 |
| Overconsolidated clay | 10–50 | 20–30 |
| Peat / organic | 0–10 | 15–30 |
Undrained Shear Strength
| Soil | $s_u$ (kPa) | $s_u/\sigma'_{v0}$ |
|---|---|---|
| Normally consolidated clay | 10–40 | 0.20–0.30 |
| Overconsolidated clay | 40–200 | 0.5–2.0 |
| Stiff fissured clay | 80–250 | 1.0–3.0 |
| Soft marine clay | 5–20 | 0.15–0.25 |
Residual Strength of Clays
| Mineral | $\phi'_r$ (°) |
|---|---|
| Quartz (silt/sand) | 25–35 |
| Kaolinite | 10–18 |
| Illite | 8–15 |
| Montmorillonite (Ca) | 5–12 |
| Montmorillonite (Na) | 3–8 |
Applications in Australian Design
Bearing Capacity
$$ q_{ult} = c'N_c + \gamma D_f N_q + 0.5\gamma B N_\gamma $$Using $c'$ and $\phi'$ for drained (long-term) analysis, or $s_u$ for undrained (short-term) analysis.
Slope Stability
Factor of safety:
$$ FS = \frac{\text{Resisting force}}{\text{Driving force}} = \frac{\int(c' + \sigma'_n \tan\phi')dl}{\int W\sin\alpha} $$Earth Pressure
| Condition | Active $K_a$ | Passive $K_p$ | At-Rest $K_0$ |
|---|---|---|---|
| Drained | $\tan^2(45-\phi'/2)$ | $\tan^2(45+\phi'/2)$ | $1-\sin\phi'$ |
| Undrained | $1 - \frac{2s_u}{\sigma_v}$ | $1 + \frac{2s_u}{\sigma_v}$ | 1.0 |
Australian Standards
| Standard | Application |
|---|---|
| AS 5100 | Bridge foundations — bearing capacity, friction piles |
| AS 2159 | Piling — shaft friction, end bearing |
| AS 4678 | Retaining structures — drained shear parameters |
| AS 2870 | Residential slabs — undrained shear strength for site class |
| TfNSW QA Specification R10 | Road earthworks — fill shear strength |