Bearing capacity and foundation settlements

A foundation must satisfy two separate criteria: safety against shear failure (bearing capacity) and acceptable settlement. The allowable pressure is the smaller of the two values. Terzaghi's bearing capacity theory gives the ultimate load a footing can take; consolidation theory predicts how much and how fast clay layers settle. These are among the most frequently examined geotechnical topics.

Key concepts

Bearing capacity terms

Term	Meaning
Ultimate bearing capacity $q_u$	Gross pressure at which soil fails in shear
Net ultimate $q_{nu}$	$q_u - \gamma D_f$
Net safe $q_{ns}$	$q_{nu}/F$ (F usually 2.5–3)
Safe bearing capacity $q_s$	$q_{ns} + \gamma D_f$
Allowable bearing pressure $q_a$	Lesser of net safe capacity and pressure giving permissible settlement

Factors affecting bearing capacity

Soil type and strength ($c$, $\phi$), unit weight, depth of foundation (more depth → more capacity), width and shape of footing, water table (submergence roughly halves the $\gamma$ terms), load eccentricity and inclination, compressibility, and method of construction.

Modes of foundation failure

Mode	Soil	Features
General shear	Dense sand, stiff clay	Well-defined slip surface to ground, heaving, sudden tilting failure; clear peak on load–settlement curve
Local shear	Medium dense sand, medium clay	Slip surface not reaching ground fully; some heave; no clear peak
Punching shear	Loose sand, soft clay, or deep footings	Vertical shearing around footing, large settlement, no heave

Terzaghi's local shear: use reduced $c' = \tfrac23 c$ and $\tan\phi' = \tfrac23\tan\phi$.

Terzaghi's general bearing capacity equation

Assumptions: shallow strip footing ($D_f \le B$), rough base, homogeneous soil, general shear failure, soil above base replaced by surcharge $\gamma D_f$. $q_u = cN_c + \gamma D_f N_q + 0.5\gamma B N_\gamma\quad\text{(strip)}$ $q_u = 1.3cN_c + \gamma D_fN_q + 0.4\gamma BN_\gamma\quad\text{(square)}$ $q_u = 1.3cN_c + \gamma D_fN_q + 0.3\gamma BN_\gamma\quad\text{(circular, } B = \text{diameter)}$

• $N_c, N_q, N_\gamma$ = bearing capacity factors depending only on $\phi$. For $\phi = 0$: $N_c = 5.7$, $N_q = 1$, $N_\gamma = 0$ (Terzaghi). Skempton's $N_c$ for $\phi = 0$ is 5 (strip) and 6 (square/circle) at the surface, rising with depth to 7.5 and 9; Prandtl's strip value is 5.14.
• Typical Terzaghi values: $\phi = 30^\circ$: $N_c = 37.2$, $N_q = 22.5$, $N_\gamma = 19.7$.

Cohesionless vs cohesive soils

• Cohesionless ($c = 0$): $q_u = \gamma D_fN_q + 0.5\gamma BN_\gamma$; capacity increases with both depth and width.
• Cohesive, undrained ($\phi = 0$): $q_u = 5.7c_u + \gamma D_f$ (strip); $q_{nu} = 5.7c_u$ — independent of width.
• Water table effect: correction factors $R_w$ (0.5 when water table at the relevant level, 1 when deep).

Consolidation

• Concept: gradual volume decrease of saturated fine-grained soil due to expulsion of pore water under sustained load; excess pore pressure dissipates with time (Terzaghi's spring-piston analogy).
• Types:
  • Primary consolidation – due to dissipation of excess pore pressure.
  • Secondary compression – creep of soil skeleton after pore pressure dissipates (important in organic soils).
  • Normally consolidated (present stress = max past stress) vs over-consolidated clays (OCR > 1).
• Test: oedometer — sample 60–75 mm diameter, ~20 mm thick, load doubled each 24 h; gives $e$–$\log\sigma'$ curve ($C_c$, $\sigma'_p$ by Casagrande's method), and time–settlement curve ($c_v$ by Casagrande log-time or Taylor square-root-of-time method).
• Time factor: $T_v = \dfrac{c_vt}{H_d^2}$; $H_d$ = drainage path (half the thickness for double drainage). $T_{50} = 0.197$, $T_{90} = 0.848$. For $U < 60\%$: $T_v = \dfrac{\pi}{4}U^2$.

Settlement

• Types: immediate (elastic) settlement $S_i$, primary consolidation $S_c$, secondary $S_s$. Total $S = S_i + S_c + S_s$.
• Nature: uniform, tilting, differential (most damaging — causes cracks).
• Effects: cracking of walls, tilt, damage to services, functional problems. Codes such as IS 1904 limit total settlement (of the order of 50 mm for isolated footings on sand, higher for clays and rafts) and angular distortion (about 1/300 is commonly quoted for onset of cracking in panel walls).

High-yield points

• Terzaghi strip: $q_u = cN_c + \gamma D_fN_q + 0.5\gamma BN_\gamma$; square uses $1.3c$ and $0.4\gamma B$; circle $1.3c$ and $0.3\gamma B$.
• $\phi = 0$: $N_c = 5.7$, $N_q = 1$, $N_\gamma = 0$, so $q_{nu} = 5.7c_u$.
• Local shear: $c' = \tfrac23c$, $\tan\phi' = \tfrac23\tan\phi$.
• $T_{50} = 0.197$, $T_{90} = 0.848$; consolidation time ∝ $H_d^2$.
• General shear in dense/stiff soils; punching shear in loose/soft soils.

Formulas

Immediate settlement: $S_i = qB\dfrac{(1-\mu^2)}{E}I_f$.

Primary consolidation (NC clay): $S_c = \frac{C_cH}{1+e_0}\log_{10}\frac{\sigma'_0 + \Delta\sigma}{\sigma'_0}$ or $S_c = m_vH\Delta\sigma$. $H$ = clay thickness, $e_0$ = initial void ratio, $\sigma'_0$ = initial effective stress at mid-layer, $\Delta\sigma$ = stress increase.

Worked example

A 4 m thick NC clay layer has $C_c = 0.3$, $e_0 = 1.0$, $\sigma'_0 = 80$ kPa at mid-depth. A building adds $\Delta\sigma = 40$ kPa. Find consolidation settlement and time for 90% consolidation (double drainage, $c_v = 2$ m²/year).

• $S_c = \dfrac{0.3\times4}{2}\log_{10}(120/80) = 0.6\times0.1761 = 0.106$ m ≈ 106 mm
• $H_d = 2$ m; $t_{90} = 0.848\times2^2/2 = 1.70$ years.

Exam tips

• For saturated clay ($\phi = 0$), net ultimate bearing capacity = $5.7c_u$ (Terzaghi) — independent of footing width.
• General shear → dense/stiff soils; punching shear → loose/soft soils.
• Rising water table can roughly halve the $\gamma$-related terms.
• Time for consolidation ∝ $H_d^2$: double drainage gives one-quarter the time of single drainage.
• $T_{50} = 0.197$, $T_{90} = 0.848$; differential settlement causes most damage.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. A 2 m wide strip footing rests at 1 m depth in dry sand with $\gamma = 18$ kN/m³ and $\phi = 30^\circ$ (Terzaghi's $N_q = 22.5$, $N_\gamma = 19.7$, $c = 0$). The ultimate bearing capacity is about

• A. 355 kPa
• B. 405 kPa
• C. 1115 kPa
• D. 760 kPa

Answer: D. $q_u = \gamma D_f N_q + 0.5\gamma B N_\gamma = 18 \times 1 \times 22.5 + 0.5 \times 18 \times 2 \times 19.7 = 405 + 354.6 \approx 760$ kPa.

2. Punching shear failure of a foundation is most likely to occur in

• A. dense sand
• B. loose sand or soft clay
• C. hard rock
• D. stiff overconsolidated clay

Answer: B. Punching shear occurs in highly compressible (loose or soft) soils, where the footing sinks without a well-defined failure surface reaching the ground.

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