Soil compressibility governs how much a foundation will settle under load.
When a load is applied to soil, the soil skeleton compresses, water is expelled, and settlement occurs. The amount and rate of settlement depend on:
- Soil type: granular vs. cohesive
- Stress history: normally consolidated vs. overconsolidated
- Loading magnitude: net stress increase
- Drainage conditions: drainage path length and permeability
Types of Settlement
| Settlement Type | Soil Type | Time Frame | Mechanism |
|---|---|---|---|
| Immediate (elastic) settlement | All soils | During construction | Elastic deformation of soil skeleton without volume change |
| Primary consolidation | Fine-grained soils | Months to years | Gradual expulsion of pore water under increased load |
| Secondary compression (creep) | All soils, especially organic | Years to decades | Viscous adjustment of soil skeleton under constant effective stress |
The Consolidation Process
Terzaghi's Theory of One-Dimensional Consolidation
Terzaghi's theory describes the time-dependent settlement of saturated clay layers under increased load.
Fundamental equation:
$$ c_v \frac{\partial^2 u}{\partial z^2} = \frac{\partial u}{\partial t} $$Where:
- $c_v$ = coefficient of consolidation
- $u$ = excess pore water pressure
- $z$ = depth
- $t$ = time
The Consolidation Analogy
Load (Δσ)
↓
┌───────────┐
│ Sand │ Drainage layer
├───────────┤
│ Clay │ ← Excess pore pressure = Δσ at t=0
│ Layer │
│ │ → Pore pressure dissipates over time
├───────────┤
│ Sand │ Drainage layer
└───────────┘
At t=0: All load is carried by pore water ($u = \Delta\sigma$)
At t=∞: All load is carried by soil skeleton ($\sigma' = \sigma_0 + \Delta\sigma$)
The Oedometer Test (AS 1289.6.3.1)
The one-dimensional consolidation test (oedometer test) determines:
| Parameter | Symbol | Determination Method |
|---|---|---|
| Compression index | $C_c$ | Slope of virgin compression line |
| Swell index | $C_s$ | Slope of unloading/reloading line |
| Preconsolidation pressure | $\sigma'_p$ | Casagrande construction |
| Coefficient of consolidation | $c_v$ | Casagrande log-time or Taylor √t method |
| Coefficient of volume compressibility | $m_v$ | $m_v = \Delta\epsilon / \Delta\sigma'$ |
| Coefficient of permeability | $k$ | $k = c_v m_v \gamma_w$ |
The e-log σ' Curve
The void ratio vs. log effective stress plot is the primary output of an oedometer test:
e
│
│ ┌───────────────────┐
│ │ Recompression │ ── Swelling line (C_s)
│ │ │
│ │● σ'_p (Preconsolidation Pressure)
│ │ │
│ │ Virgin Compression Line (C_c)
│ │ │
│ └───────────────────┘
│
└─────────────────────────── log σ'
Key Parameters from the e-log σ' Curve
Compression Index ($C_c$):
$$ C_c = \frac{e_1 - e_2}{\log(\sigma'_2/\sigma'_1)} $$Correlations for $C_c$:
| Formula | Source | Applicability |
|---|---|---|
| $C_c = 0.009(LL - 10)$ | Terzaghi & Peck | Inorganic clays |
| $C_c = 0.007(LL - 7)$ | Skempton | Remoulded clays |
| $C_c = 0.01(w_n - 5)$ | — | CL clays |
| $C_c = 0.0046LL + 0.014$ | — | General |
| $C_c = 1.15(e_0 - 0.27)$ | — | All clays |
Swell Index ($C_s$):
$$ C_s \approx 0.1\ \text{to}\ 0.2 \times C_c $$Preconsolidation Pressure
The Casagrande construction determines $\sigma'_p$:
- Identify the point of maximum curvature on the e-log σ' curve
- Draw the tangent and horizontal lines at this point
- Bisect the angle between them
- Extend the straight-line portion of the virgin compression curve
- The intersection of the bisector and the virgin line = $\sigma'_p$
Overconsolidation Ratio (OCR)
$$ OCR = \frac{\sigma'_p}{\sigma'_{v0}} $$| OCR | Stress History |
|---|---|
| OCR = 1 | Normally consolidated (NC) |
| OCR > 1 | Overconsolidated (OC) |
| OCR < 1 | Underconsolidated (unlikely unless fill recently placed) |
| OCR | Degree of Overconsolidation |
|---|---|
| 1 | Normally consolidated |
| 1–2 | Slightly overconsolidated |
| 2–4 | Moderately overconsolidated |
| 4–10 | Highly overconsolidated |
| > 10 | Heavily overconsolidated |
Settlement Calculations
Primary Consolidation Settlement
For normally consolidated clay ($\sigma'_{v0} + \Delta\sigma' \leq \sigma'_p$):
$$ S_c = \frac{C_c H}{1+e_0} \log\left(\frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_{v0}}\right) $$For overconsolidated clay ($\sigma'_{v0} + \Delta\sigma' > \sigma'_p$):
$$ S_c = \frac{C_s H}{1+e_0} \log\left(\frac{\sigma'_p}{\sigma'_{v0}}\right) + \frac{C_c H}{1+e_0} \log\left(\frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_p}\right) $$For overconsolidated clay ($\sigma'_{v0} + \Delta\sigma' \leq \sigma'_p$):
$$ S_c = \frac{C_s H}{1+e_0} \log\left(\frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_{v0}}\right) $$Where:
- $H$ = thickness of clay layer
- $e_0$ = initial void ratio
- $\sigma'_{v0}$ = initial effective overburden stress
- $\Delta\sigma'$ = stress increase at mid-depth of layer
Immediate Settlement (Elastic)
For flexible foundations on clay:
$$ S_i = \frac{qB(1-\mu^2)}{E_s} I_f $$Where:
- $q$ = applied pressure
- $B$ = foundation width
- $\mu$ = Poisson's ratio (0.3–0.5 for clay)
- $E_s$ = Young's modulus
- $I_f$ = influence factor (depends on shape and rigidity)
Typical elastic modulus values:
| Soil Type | $E_s$ (MPa) |
|---|---|
| Very soft clay | 2–15 |
| Soft clay | 5–25 |
| Medium clay | 15–40 |
| Stiff clay | 30–100 |
| Very stiff clay | 80–200 |
| Loose sand | 10–25 |
| Medium dense sand | 25–50 |
| Dense sand | 50–80 |
| Dense sand and gravel | 80–200 |
Secondary Compression (Creep)
$$ S_s = \frac{C_\alpha H}{1+e_p} \log\left(\frac{t_2}{t_1}\right) $$Where:
- $C_\alpha$ = coefficient of secondary compression
- $e_p$ = void ratio at end of primary consolidation
- $t_1$ = time at end of primary consolidation
- $t_2$ = design life
| Soil Type | $C_\alpha / C_c$ |
|---|---|
| Clays | 0.04 ± 0.01 |
| Silts | 0.03 ± 0.01 |
| Organic clays | 0.05 ± 0.02 |
| Peat | 0.06–0.10 |
Rate of Consolidation
Degree of Consolidation
$$ U = \frac{S_t}{S_c} \times 100\% $$Where $S_t$ = settlement at time $t$, $S_c$ = final primary consolidation settlement.
Time Factor
$$ T_v = \frac{c_v t}{d^2} $$Where:
- $d$ = **drainage path length** (half the layer thickness for double drainage; full thickness for single drainage)
- $c_v$ = coefficient of consolidation
Relationship between $U$ and $T_v$:
| $U$ (%) | $T_v$ |
|---|---|
| 0 | 0 |
| 10 | 0.008 |
| 20 | 0.031 |
| 30 | 0.071 |
| 40 | 0.126 |
| 50 | 0.197 |
| 60 | 0.287 |
| 70 | 0.403 |
| 80 | 0.567 |
| 90 | 0.848 |
| 95 | 1.163 |
| 99 | 1.815 |
Approximate formulas:
$$ T_v = \frac{\pi}{4}\left(\frac{U}{100}\right)^2 \quad \text{for} \ U < 60\% $$ $$ T_v = 1.781 - 0.933\log(100 - U) \quad \text{for} \ U > 60\% $$Time to Reach a Given Settlement
$$ t = \frac{T_v d^2}{c_v} $$Coefficient of Consolidation Determination
Casagrande's Log-Time Method
Plot dial reading vs. log time:
- Identify the theoretical zero point (final parabolic portion)
- Identify the theoretical 100% primary consolidation (intersection of tangents)
- The time at 50% consolidation ($t_{50}$) gives $c_v$:
Taylor's √t Method
Plot dial reading vs. square root of time:
- Extend the initial straight-line portion
- Draw a second line at 1.15 times the slope
- The intersection with the curve gives $\sqrt{t_{90}}$
- Calculate $c_v$:
Stress Distribution in the Ground
Boussinesq's Solution
The vertical stress increase at depth $z$ below a point load $Q$:
$$ \Delta\sigma_z = \frac{3Q}{2\pi z^2} \left[ \frac{1}{1+(r/z)^2} \right]^{5/2} $$Approximate Methods
2:1 Method (simplified):
$$ \Delta\sigma_z = \frac{Q}{(B+z)(L+z)} $$Where $B$ = foundation width, $L$ = foundation length
Influence Factors for Common Shapes
| Shape | Stress at depth $z$ below centre |
|---|---|
| Square ($B \times B$) | $\Delta\sigma_z = q \times I$ |
| Strip (width $B$, infinite length) | $\Delta\sigma_z = \frac{q}{\pi}(\alpha + \sin\alpha\cos(\alpha+2\beta))$ |
| Circular (diameter $B$) | $\Delta\sigma_z = q\left[1 - \frac{1}{(1+(B/2z)^2)^{3/2}}\right]$ |
Practical Considerations
Allowable Settlements
| Structure Type | Maximum Total Settlement | Maximum Differential Settlement |
|---|---|---|
| Isolated footings — clay | 65 mm | 1:300 |
| Isolated footings — sand | 50 mm | 1:300 |
| Raft foundations | 50–75 mm | 1:500 |
| Frame structures | 50 mm | 1:300 |
| Steel structures | 100 mm | 1:200 |
| Bridges | 50 mm | 1:300 |
| Machinery foundations | 25 mm | 1:1000 |
| Warehouses | 150 mm | 1:150 |
Australian Standards
| Standard | Relevance |
|---|---|
| AS 2870 | Residential slabs (differential settlement limits) |
| AS 2159 | Piling (settlement of piled foundations) |
| AS 5100 | Bridge design (foundation settlement limits) |
| AS 4678 | Earth retaining structures (settlement behind walls) |
9.3 Methods to Reduce Settlement
| Method | How It Works | Applicable Soil |
|---|---|---|
| Preloading | Apply surcharge before construction | Soft clays, silts |
| Vertical drains (PVDs) | Shorten drainage path | Soft clays |
| Deep compaction | Increase density | Granular soils |
| Stone columns | Reinforce + drain | Soft clays, silts |
| Deep mixing | Improve soil stiffness | Soft clays, organic |
| Grouting | Fill voids, improve stiffness | All soils |
| Pile foundations | Transfer load to competent strata | All weak soils |
Worked Example
Problem: A 3 m × 3 m square footing carries a load of 900 kN on a 5 m thick clay layer. The clay has $e_0 = 1.0$, $LL = 45\%$, $\sigma'_p = 150$ kPa, $\sigma'_{v0} = 80$ kPa, $C_c = 0.32$, $C_s = 0.05$, $c_v = 5$ m²/year. Water table at surface.
Step 1 — Stress increase at mid-depth (2.5 m):
Using 2:1 method:
$\Delta\sigma = \frac{900}{(3+2.5)(3+2.5)} = \frac{900}{30.25} = 29.8$ kPaStep 2 — Check OCR at mid-depth:
$\sigma'_{v0} = 80$ kPa, $\sigma'_p = 150$ kPa $OCR = 150/80 = 1.88$ → OverconsolidatedStep 3 — Settlement calculation:
$\sigma'_{v0} + \Delta\sigma = 80 + 29.8 = 109.8$ kPa < $\sigma'_p = 150$ kPaUse OC equation:
$$ S_c = \frac{0.05 \times 5}{1+1.0} \log\left(\frac{109.8}{80}\right) = 0.125 \times \log(1.373) = 0.125 \times 0.138 = 0.017\ \text{m} = 17\ \text{mm} $$Step 4 — Time to 90% consolidation:
$d = 2.5$ m (double drainage, clay on sand) $T_{90} = 0.848$ $$ t = \frac{0.848 \times 2.5^2}{5} = 1.06\ \text{years} $$