Bearing capacity is the ability of soil to support the loads applied by a foundation without shear failure or excessive settlement
Bearing capacity analysis determines the maximum load a soil can safely support. Two key concepts:
- Ultimate Bearing Capacity ($q_{ult}$): The maximum pressure the soil can sustain before shear failure
- Allowable Bearing Capacity ($q_{all}$): The maximum safe pressure considering a factor of safety
Typical Factors of Safety:
| Loading Condition | FS |
|---|---|
| Normal static loading | 2.5–3.0 |
| Maximum design load | 2.0 |
| Including wind/seismic | 1.5–2.0 |
| Temporary loading | 1.5–2.0 |
Modes of Bearing Capacity Failure
| Failure Mode | Description | Typical Soil |
|---|---|---|
| General shear failure | Continuous failure surface from footing edge to ground surface; sudden failure with tilting | Dense sand, stiff clay |
| Local shear failure | Significant compression before failure; failure surface does not fully develop | Medium sand, firm clay |
| Punching shear failure | Vertical shear around footing perimeter; soil compresses beneath footing | Loose sand, soft clay |
Terzaghi's Bearing Capacity Theory
Terzaghi (1943) developed the first comprehensive bearing capacity theory, still widely used:
General Equation (Strip Footing)
$$ q_{ult} = cN_c + \gamma D_f N_q + 0.5\gamma B N_\gamma $$Where:
- $c$ = cohesion of soil
- $\gamma$ = unit weight of soil
- $D_f$ = depth of foundation
- $B$ = width of foundation
- $N_c, N_q, N_\gamma$ = bearing capacity factors (function of $\phi$)
Terzaghi's Bearing Capacity Factors
| $\phi$ (°) | $N_c$ | $N_q$ | $N_\gamma$ |
|---|---|---|---|
| 0 | 5.7 | 1.0 | 0.0 |
| 5 | 7.3 | 1.6 | 0.5 |
| 10 | 9.6 | 2.7 | 1.2 |
| 15 | 12.9 | 4.4 | 2.5 |
| 20 | 17.7 | 7.4 | 5.0 |
| 25 | 25.1 | 12.7 | 9.7 |
| 30 | 37.2 | 22.5 | 19.7 |
| 32 | 44.4 | 30.0 | 27.8 |
| 34 | 53.0 | 40.0 | 39.5 |
| 36 | 63.6 | 54.0 | 56.5 |
| 38 | 76.0 | 73.0 | 82.0 |
| 40 | 95.7 | 100.0 | 120.0 |
| 45 | 172.0 | 280.0 | 380.0 |
Formulas for bearing capacity factors:
$$ N_q = \frac{a^2}{2\cos^2(45° + \phi/2)} \quad \text{where} \quad a = e^{(0.75\pi - \phi/2)\tan\phi} $$ $$ N_c = (N_q - 1)\cot\phi $$ $$ N_\gamma = \frac{1}{2} \left(\frac{K_{p\gamma}}{\cos^2\phi} - 1\right)\tan\phi $$Shape and Depth Corrections (Terzaghi)
Shape factors (for non-strip footings):
| Shape | $s_c$ | $s_q$ | $s_\gamma$ |
|---|---|---|---|
| Strip | 1.0 | 1.0 | 1.0 |
| Square | 1.3 | 1.2 | 0.8 |
| Circle | 1.3 | 1.2 | 0.6 |
| Rectangle ($B/L$) | $1 + 0.3(B/L)$ | $1 + 0.2(B/L)$ | $1 - 0.4(B/L)$ |
Terzaghi — Special Cases
Undrained condition ($\phi = 0$, $s_u$):
$$ q_{ult} = 5.7s_u + \gamma D_f $$Square footing, undrained:
$$ q_{ult} = 1.3 \times 5.7s_u + \gamma D_f = 7.4s_u + \gamma D_f $$Meyerhof's Bearing Capacity Theory
Meyerhof (1963) extended Terzaghi's work with improved factors for depth, shape, and inclination:
General Equation
$$ q_{ult} = cN_c s_c d_c i_c + \gamma D_f N_q s_q d_q i_q + 0.5\gamma B N_\gamma s_\gamma d_\gamma i_\gamma $$Meyerhof's Bearing Capacity Factors
| $\phi$ (°) | $N_c$ | $N_q$ | $N_\gamma$ |
|---|---|---|---|
| 0 | 5.14 | 1.0 | 0.0 |
| 10 | 8.35 | 2.47 | 0.47 |
| 20 | 14.83 | 6.40 | 2.87 |
| 25 | 20.72 | 10.66 | 6.77 |
| 30 | 30.14 | 18.40 | 15.67 |
| 32 | 35.49 | 23.18 | 22.36 |
| 34 | 42.16 | 29.44 | 32.50 |
| 36 | 50.59 | 37.75 | 48.03 |
| 38 | 61.35 | 48.93 | 72.35 |
| 40 | 75.31 | 64.20 | 111.40 |
| 45 | 133.88 | 134.88 | 328.70 |
Formulas:
$$ N_q = e^{\pi\tan\phi}\tan^2(45° + \phi/2) $$ $$ N_c = (N_q - 1)\cot\phi $$ $$ N_\gamma = (N_q - 1)\tan(1.4\phi) $$Meyerhof's Shape and Depth Factors
Shape factors:
| Factor | Formula |
|---|---|
| $s_c$ | $1 + 0.2K_p(B/L)$ |
| $s_q$ | $1 + 0.1K_p(B/L)$ |
| $s_\gamma$ | $1 + 0.1K_p(B/L)$ |
Where $K_p = \tan^2(45° + \phi/2)$
Depth factors:
| Factor | For $D_f/B \leq 1$ | For $D_f/B > 1$ |
|---|---|---|
| $d_c$ | $1 + 0.2\sqrt{K_p}(D_f/B)$ | $1 + 0.2\sqrt{K_p}\tan^{-1}(D_f/B)$ |
| $d_q$ | $1 + 0.1\sqrt{K_p}(D_f/B)$ | $1 + 0.1\sqrt{K_p}\tan^{-1}(D_f/B)$ |
| $d_\gamma$ | 1.0 | 1.0 |
Hansen's Bearing Capacity Theory
Hansen (1970) provides the most comprehensive framework, including factors for:
- Base inclination ($b$)
- Ground inclination ($g$)
- Load inclination ($i$)
General Equation
$$ q_{ult} = cN_c s_c d_c i_c b_c g_c + \gamma D_f N_q s_q d_q i_q b_q g_q + 0.5\gamma B N_\gamma s_\gamma d_\gamma i_\gamma b_\gamma g_\gamma $$Hansen's Bearing Capacity Factors
$N_q$ and $N_c$ follow the same formulas as Meyerhof.For $N_\gamma$, Hansen proposed:
$$ N_\gamma = 1.5(N_q - 1)\tan\phi $$Hansen's Load Inclination Factors
For a horizontal load $H$ and vertical load $V$:
$$ i_q = \left[1 - \frac{0.5H}{V + A_f c\cot\phi}\right]^5 $$ $$ i_c = i_q - \frac{1 - i_q}{N_q - 1} \quad \text{(for $\phi > 0$)} $$ $$ i_\gamma = \left[1 - \frac{0.7H}{V + A_f c\cot\phi}\right]^5 $$Bearing Capacity from In-Situ Tests
SPT Correlation (Bowles)
$$ q_{ult} = \frac{N}{F_1} \times F_2 \times F_3 \quad \text{(kPa)} $$Where:
- $N$ = SPT blow count (corrected)
- $F_1$ = 0.08 for SI units
- $F_2$ = shape factor
- $F_3$ = depth factor
Simplified (for $B \leq 1.2$ m):
| Allowable $q_{all}$ (kPa) | SPT N-value | Soil Description |
|---|---|---|
| 75–100 | 4–8 | Loose sand |
| 100–200 | 8–15 | Medium sand |
| 200–300 | 15–30 | Dense sand |
| 300–450 | 30–50 | Very dense sand |
CPT Correlation
$$ q_{ult} = \frac{q_c}{F} $$Where $q_c$ is the average cone resistance over the bearing influence zone, and $F$ is a factor typically 10–20.
For strip footings on sand:
$$ q_{ult} = 0.2 q_c \left(\frac{B}{0.3}\right) \quad \text{(for B ≤ 1.2 m)} $$Eccentric and Inclined Loading
Meyerhof's Effective Width Method
$$ B' = B - 2e $$Where $e$ = eccentricity of the resultant load.
Effective area: $A' = B' \times L'$
Bearing capacity calculated using: $q_{ult} = f(B', L')$
Check: $q_{max} = \frac{V}{A'} \leq q_{all}$
7.2 Footings on Slopes
For footings near slope crests, bearing capacity is reduced:
$$ q_{ult(slope)} \approx q_{ult(flat)} \times \text{reduction factor} $$| $b/B$ | Slope 1V:2H | Slope 1V:3H | Slope 1V:5H |
|---|---|---|---|
| 0 | 0.50 | 0.55 | 0.80 |
| 1 | 0.60 | 0.75 | 0.85 |
| 2 | 0.75 | 0.85 | 0.90 |
| 4 | 0.90 | 0.95 | 0.95 |
Where $b$ is the distance from slope crest to footing edge.
Allowable Bearing Capacity in Australian Codes
AS 2870 — Residential Slabs and Footings
| Site Class | Soil Type | Allowable Bearing Pressure (kPa) |
|---|---|---|
| A | Rock/sand | 100–300 (engineering assessment) |
| S | Slightly reactive | 75–150 |
| M | Moderately reactive | 50–100 |
| H | Highly reactive | 30–50 |
| E | Extremely reactive | < 30 |
AS 2159 — Piling
Bearing capacity for piles is determined by:
- Static analysis (shaft friction + end bearing)
- Dynamic analysis (wave equation)
- Static load testing (maintained load test, rapid load test)
- Dynamic load testing (PDA)
Typical Presumptive Values (Various Australian Codes)
| Soil/Rock Type | Allowable Bearing Capacity (kPa) |
|---|---|
| Sound hard rock | 2,000–5,000+ |
| Shale/mudstone | 500–2,000 |
| Sandstone (sound) | 1,000–3,000 |
| Stiff clay | 150–300 |
| Firm clay | 75–150 |
| Soft clay | < 75 |
| Dense sand/gravel | 300–600 |
| Medium dense sand | 150–300 |
| Loose sand | 50–150 |
| Well-compacted fill | 100–200 |
Groundwater Effects
Water table reduces bearing capacity by reducing effective stress:
Correction factor (Terzaghi & Peck):
If water table is within $B$ of the base of footing:
$$ q_{ult(WT)} = q_{ult(dry)} - 0.5\gamma_w N_\gamma $$More precise correction (Meyerhof):
$$ R_{w1} = 0.5\left(1 + \frac{z_w}{D_f + B}\right) \quad \text{(for $N_q$ term)} $$ $$ R_{w2} = 0.5\left(1 + \frac{z_w}{B}\right) \quad \text{(for $N_\gamma$ term)} $$Where $z_w$ = depth from base of footing to water table.
Worked Examples
Example 1 — Strip Footing on Sand
Given: $B = 1.5$ m, $D_f = 1.0$ m, $\gamma = 18$ kN/m³, $\phi' = 34°$, $c' = 0$, water table > 3 m deep.
Terzaghi:
$N_q = 40.0$, $N_\gamma = 39.5$ (from table for $\phi = 34°$) $q_{ult} = 0 + 18 \times 1.0 \times 40.0 + 0.5 \times 18 \times 1.5 \times 39.5$ $q_{ult} = 720 + 533 = 1,253$ kPa $q_{all} = 1253/3.0 = 418$ kPaExample 2 — Square Footing on Clay
Given: $B = 2.0$ m, $D_f = 1.5$ m, $\gamma = 19$ kN/m³, $s_u = 80$ kPa, $\phi = 0$
Terzaghi (undrained, square):
$q_{ult} = 1.3 \times 5.7 \times 80 + 19 \times 1.5$ $q_{ult} = 593 + 29 = 622$ kPa $q_{all} = 622/2.5 = 249$ kPaSettlement Check
After the bearing capacity is satisfied, check settlement:
$$ S_{total} = S_i + S_c $$| Allowable Settlement | Structure Type |
|---|---|
| 50 mm | Frame structures |
| 65 mm | Isolated footings on clay |
| 100 mm | Steel structures |
| 25 mm | Machinery foundations |
| 1:300 | Differential settlement limit |
Net allowable bearing pressure:
The lower value from:
- Ultimate bearing capacity / FS
- Settlement-limited pressure