9.1 Ultimate & Allowable Bearing Capacity Theories

Introduction to Shallow Foundations

A shallow foundation is defined geotechnical engineering-wise as a structural element whose depth of embedment $D_f$ is less than or equal to its minimum width $B$ ($D_f / B \le 1$), though some modern codes extend this boundary to $D_f / B \le 4 \text{ to } 8$. Shallow foundations transfer structural column and wall loads to upper soil strata through direct base contact pressure. The fundamental design requirements demand that:

1. The ultimate shear strength of the supporting soil mass is not exceeded (ultimate limit state for bearing capacity with an adequate Factor of Safety, $FS \ge 3.0$).
2. Immediate elastic and long-term consolidation settlements remain within tolerable structural limits (serviceability limit state).

---

Shear Failure Modes Beneath Shallow Foundations

Depending on soil relative density $D_r$, soil compressibility, and foundation embedment depth, three distinct shear failure modes can develop beneath loaded footings:

                      General Shear Failure Geometry
                      
                             Applied Load P
                                   │
                       ┌───────────┴───────────┐
                       │   Footing (B x L)     │
         Ground Surface └───┬───────────────┬───┘ Ground Surface
        ───┬───────────────┘               └───────────────┬───
          ╱ │ Zone I:                      Zone I: │ ╲
        ╱   │ Active Wedge               Active Wedge │   ╲  Zone III: Passive
      ╱     │ (Elastic)                     (Elastic) │     ╲ Rankine Zone
    ╱       └─────────────────┬───────────────────────┘       ╲
  ╱     Zone II: Radial       │      Zone II: Radial            ╲
 └────── Shear Zone ──────────┴────── Shear Zone ────────────────┘

For local and punching shear failures, Vesic and Terzaghi recommended adjusting soil shear strength parameters before applying standard bearing capacity equations:

$$c'^* = \frac{2}{3} c'$$ $$\tan\phi'^* = \frac{2}{3} \tan\phi'$$

---

Terzaghi's Ultimate Bearing Capacity Theory

Karl Terzaghi (1943) developed the first comprehensive theory for evaluating ultimate bearing capacity beneath a continuous strip footing of infinite length ($L \gg B$) under drained conditions. Terzaghi assumed a rigid active wedge (Zone I) directly beneath the footing base inclined at angle $\alpha = \phi'$ to the horizontal, flanked by radial shear zones (Zone II) and passive Rankine zones (Zone III).

Terzaghi's Equation for Continuous Strip Footings

$$q_{ult} = c' N_c + q N_q + \frac{1}{2} \gamma B N_\gamma$$

Where:

• $q_{ult} =$ Gross ultimate bearing capacity (kPa or psf)
• $c' =$ Effective soil cohesion (kPa or psf)
• $q = \gamma D_f =$ Effective vertical surcharge stress at footing base level (kPa or psf)
• $\gamma =$ Total/moist unit weight of soil supporting the wedge (kN/m³ or pcf)
• $B =$ Footing width (m or ft)
• $N_c, N_q, N_\gamma =$ Terzaghi non-dimensional bearing capacity factors (functions strictly of friction angle $\phi'$)

Mathematical Formulation of Bearing Capacity Factors

$$N_q = \frac{e^{2(3\pi/4 - \phi'/2)\tan\phi'}}{2 \cos^2(45^\circ + \phi'/2)}$$ $$N_c = (N_q - 1) \cot\phi'$$ $$N_\gamma = \frac{1}{2} \left( \frac{K_{p\gamma}}{\cos^2\phi'} - 1 \right) \tan\phi'$$

Undrained Cohesive Soils (Saturated Clay, $\phi_u = 0$)

Under quick, undrained loading in saturated clay ($\phi_u = 0, c = c_u = s_u$):

$$N_q = 1.0, \quad N_c = 5.7 \text{ (Terzaghi)} \text{ or } 5.14 \text{ (Prandtl/Vesic)}, \quad N_\gamma = 0$$ $$q_{ult} = c_u N_c + q = 5.14 s_u + \gamma D_f$$

---

General Bearing Capacity Equation (Meyerhof, Hansen, Vesic)

Terzaghi's original formulation was restricted to continuous strip footings loaded vertically. Meyerhof (1963), Hansen (1970), and Vesic (1973) expanded the equation to accommodate non-strip footprints, shallow-to-deep embedment, and inclined loadings:

$$q_{ult} = c' N_c s_c d_c i_c + q N_q s_q d_q i_q + \frac{1}{2} \gamma B N_\gamma s_\gamma d_\gamma i_\gamma$$

Where $s_i$ are shape factors, $d_i$ are depth factors, and $i_i$ are load inclination factors.

Terzaghi Empirical Shape Modifications

For square and circular footings, Terzaghi simplified shape effects as follows:

• Square Footing ($B \times B$): $$q_{ult} = 1.3 c' N_c + q N_q + 0.4 \gamma B N_\gamma$$
• Circular Footing (Diameter $B$): $$q_{ult} = 1.3 c' N_c + q N_q + 0.3 \gamma B N_\gamma$$

Meyerhof / Vesic General Shape Factors

For rectangular footings ($B \times L$ where $B \le L$):

$$s_c = 1 + \left(\frac{B}{L}\right)\left(\frac{N_q}{N_c}\right)$$ $$s_q = 1 + \left(\frac{B}{L}\right) \tan\phi'$$ $$s_\gamma = 1 - 0.4 \left(\frac{B}{L}\right)$$

Vesic Bearing Capacity Factor $N_\gamma$

In modern PE exam specifications, Vesic's $N_\gamma$ equation is widely accepted:

$$N_\gamma = 2 (N_q + 1) \tan\phi'$$

---

Groundwater Table (GWT) Corrections

The presence of a groundwater table alters effective vertical stresses and unit weights within the shear failure zone. The correction depends on the depth $d_w$ of the water table relative to ground surface and footing base $D_f$.

               Case 1: GWT Above Footing Base (0 <= dw <= Df)
               
               Ground Surface ───────────────────────────── dw = 0 to Df
                              │                        │
                              ▼                        ▼
               GWT ═════════════════════════════════════════
                              │ (Df - dw)
               Footing Base   └───► ┌───────────┐
                                    │ B x L     │
                                    └───────────┘

---

Factors of Safety and Allowable Bearing Capacity

To prevent shear failure with an adequate margin of safety, ultimate bearing capacity values are converted to allowable design pressures:

Gross Allowable Bearing Capacity ($q_{all}$)

$$q_{all} = \frac{q_{ult}}{FS}$$

Where $FS$ is typically $3.0$ for primary static building loads ($FS = 2.0 \text{ to } 2.5$ for temporary or seismic loading).

Net Ultimate Bearing Capacity ($q_{net,ult}$)

Net ultimate bearing capacity represents the net stress increase above existing overburden pressure:

$$q_{net,ult} = q_{ult} - q = q_{ult} - \gamma D_f$$

Net Allowable Bearing Capacity ($q_{net,all}$)

$$q_{net,all} = \frac{q_{net,ult}}{FS} = \frac{q_{ult} - \gamma D_f}{FS}$$

Maximum allowable column force $P_{all}$ for a footing area $A = B \times L$:

$$P_{all} = q_{net,all} \cdot A$$

---

Comprehensive Worked Calculation Example

Problem Statement

A square footing measuring $B = 2.0\text{ m} \times 2.0\text{ m}$ is founded at depth $D_f = 1.5\text{ m}$ in a sandy soil. The soil properties are:

• Soil friction angle $\phi' = 32^\circ$
• Effective cohesion $c' = 0\text{ kPa}$
• Moist soil unit weight above GWT $\gamma = 18.0\text{ kN/m}^3$
• Saturated soil unit weight $\gamma_{sat} = 19.5\text{ kN/m}^3$
• Water unit weight $\gamma_w = 9.81\text{ kN/m}^3$
• Groundwater table is located at $d_w = 0.5\text{ m}$ below ground surface.

Using Terzaghi bearing capacity factors for $\phi' = 32^\circ$ ($N_q = 23.2, N_\gamma = 22.0$), compute:

1. The gross ultimate bearing capacity $q_{ult}$
2. The net ultimate bearing capacity $q_{net,ult}$
3. The allowable net column load $P_{all}$ using $FS = 3.0$

Solution Steps

Step 1: Calculate Effective Surcharge $q$ (Case 1 GWT)

Since $d_w = 0.5\text{ m} < D_f = 1.5\text{ m}$, Case 1 applies:

$$\gamma' = \gamma_{sat} - \gamma_w = 19.5 - 9.81 = 9.69\text{ kN/m}^3$$ $$q = \gamma d_w + \gamma' (D_f - d_w) = (18.0 \times 0.5) + (9.69 \times (1.5 - 0.5)) = 9.0 + 9.69 = 18.69\text{ kPa}$$

Step 2: Compute Terzaghi Gross Ultimate Capacity $q_{ult}$ for Square Footing

For square footing with $c' = 0$:

$$q_{ult} = 1.3 c' N_c + q N_q + 0.4 \gamma' B N_\gamma$$ $$q_{ult} = 0 + (18.69 \times 23.2) + (0.4 \times 9.69 \times 2.0 \times 22.0)$$ $$q_{ult} = 433.61 + 170.54 = 604.15\text{ kPa}$$

Step 3: Compute Net Ultimate Bearing Capacity $q_{net,ult}$

$$q_{net,ult} = q_{ult} - q = 604.15 - 18.69 = 585.46\text{ kPa}$$

Step 4: Compute Net Allowable Bearing Pressure and Column Load $P_{all}$

$$q_{net,all} = \frac{q_{net,ult}}{FS} = \frac{585.46}{3.0} = 195.15\text{ kPa}$$ $$P_{all} = q_{net,all} \cdot (B \times B) = 195.15 \times (2.0 \times 2.0) = 780.6\text{ kN}$$