10.4 Shallow and Deep Foundation Bearing Capacity and Slope Stability

10.4 Shallow and Deep Foundation Bearing Capacity and Slope Stability

Foundation design bridges structural engineering and soil mechanics. Structural engineers must proportion structural footings and deep piles to transmit superstructure column loads into underlying soil strata without exceeding the shear strength of the supporting ground or causing excessive differential settlement. Furthermore, natural and engineered slopes must be stabilized against catastrophic gravitational mass wasting.

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Terzaghi's Bearing Capacity Theory for Shallow Foundations

A shallow foundation is defined as one where the embedment depth is less than or equal to its width ($D_f \le B$). In 1943, Karl Terzaghi published the classical ultimate bearing capacity equation for a continuous strip footing based on general shear failure:

                    Column Load P
                         |
                     +---+---+
     Ground Surface  |       |        Ground Surface
    -----------------+   Df  +-----------------
                     |       |
                     +-------+ Footing Base (Width B)
                    /|   I   |\
                   / |       | \
                  /  +-------+  \
                 /   /       \   \
                / II/         \II \
               /   /           \   \
              +---+             +---+
              <------- Zone III ------->

General shear failure divides the underlying soil mass into three distinct behavioral zones:

• Zone I (Triangular Elastic Wedge): Directly under the base, pushes downward into the soil as a rigid body.
• Zone II (Radial Shear Zone): Plastic shear zones along log-spiral boundaries.
• Zone III (Rankine Passive Linear Wedge): Pushed upward against the surcharge overburden.

General Form for Continuous Strip Footings

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

• $c$ = soil cohesion ($\text{kPa}$)
• $q = \gamma D_f$ = effective surcharge at foundation base level ($\text{kPa}$)
• $\gamma$ = unit weight of soil supporting the base wedge ($\text{kN/m}^3$)
• $B$ = footing width (shortest lateral dimension, $\text{m}$)
• $N_c, N_q, N_\gamma$ = dimensionless Terzaghi bearing capacity factors, which depend purely on the internal friction angle $\phi$: $N_q = \frac{e^{2(3\pi/4 - \phi/2)\tan \phi}}{2 \cos^2(45^\circ + \phi/2)}$ $N_c = (N_q - 1) \cot \phi \quad (\text{for } \phi = 0^\circ, N_c = 5.7)$ $N_\gamma \approx \frac{1}{2} \left( \frac{K_{p\gamma}}{\cos^2\phi} - 1 \right) \tan \phi$

For saturated clays under undrained conditions ($\phi = 0^\circ$): $N_c = 5.7, \quad N_q = 1.0, \quad N_\gamma = 0$ $q_{\text{ult}} = 5.7 c_u + q$

Footing Shape Modifications

To adapt the 2D strip footing equation to 3D isolated footings, empirical shape coefficients are applied:

General Bearing Capacity Equation (Meyerhof, Hansen, Vesic)

For footings subjected to inclined loads, deep embedment, or eccentricities, the General Bearing Capacity Equation incorporates shape ($s$), depth ($d$), and load inclination ($i$) factors: $q_{\text{ult}} = c N_c s_c d_c i_c + q N_q s_q d_q i_q + \frac{1}{2} \gamma B N_\gamma s_\gamma d_\gamma i_\gamma$

Groundwater Table Modifications

The presence of a high water table reduces effective stresses and diminishes bearing capacity:

   Case 1: Water at depth d_w <= D_f
   -------------------------- GWT (d_w)
   /////////// Submerged Overburden
   ----------- Footing Base (D_f)
   
   Case 2: Water at depth D_f < d_w <= D_f + B
   ----------- Footing Base (D_f)
   ........... Moist Wedge
   -------------------------- GWT (d_w)
   /////////// Submerged Wedge
   -------------------------- Depth D_f + B

1. Case 1: Groundwater Table Above Footing Base ($d_w \le D_f$):
   • Surcharge $q$ becomes: $q = \gamma d_w + \gamma' (D_f - d_w)$
   • Unit weight in third term becomes buoyant: $\gamma = \gamma' = \gamma_{\text{sat}} - \gamma_w$
2. Case 2: Groundwater Table Within Wedge Depth ($D_f < d_w \le D_f + B$):
   • Surcharge is unaffected: $q = \gamma D_f$
   • Unit weight in third term is an effective weighted average: $\bar{\gamma} = \gamma' + \left( \frac{d_w - D_f}{B} \right) (\gamma - \gamma')$
3. Case 3: Groundwater Table Deep Below Base ($d_w > D_f + B$):
   • Water table has no physical effect on bearing capacity; use moist/dry $\gamma$ in both terms.

Allowable Bearing Capacity & Factor of Safety

$q_{\text{all}} = \frac{q_{\text{ult}}}{FS} \quad (\text{standard } FS = 3.0)$ $\text{Net Allowable Bearing Capacity: } q_{\text{all,net}} = \frac{q_{\text{ult}} - q}{FS}$ $\text{Allowable Column Load: } P_{\text{all}} = q_{\text{all}} \times A_{\text{footing}}$

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Deep Foundations: Piles and Drilled Shafts

When surficial soils are too weak or compressible to support shallow footings within tolerable settlement limits, structural loads are transferred to deeper, competent strata via deep foundations.

Ultimate Axial Pile Capacity

The ultimate compressive load capacity $Q_{\text{ult}}$ of an isolated single pile is the sum of its base end-bearing resistance ($Q_p$) and shaft skin friction resistance ($Q_s$): $Q_{\text{ult}} = Q_p + Q_s = q_p A_p + \sum f_s A_s$ Where:

• $A_p$ = cross-sectional area of the pile tip
• $q_p$ = unit point bearing capacity
• $A_s = p \Delta L$ = surface area of pile shaft perimeter over length segment $\Delta L$
• $f_s$ = unit skin friction (shaft resistance)

Cohesive Soils (Clays)

1. End-Bearing ($Q_p$): For saturated clays under undrained conditions, $N_c^* = 9.0$: $q_p = 9 c_u \implies Q_p = 9 c_u A_p$
2. Skin Friction ($Q_s$) — The $\alpha$-Method: $f_s = \alpha c_u \implies Q_s = \sum \alpha c_u p \Delta L$ Where $\alpha$ is the empirical adhesion factor (ranging from $\approx 1.0$ for soft clays down to $0.4$ for stiff clays).

Cohesionless Soils (Sands)

1. End-Bearing ($Q_p$): $q_p = q' N_q^* \le q_{\text{limit}}$ Where $q'$ is the effective vertical stress at the pile tip (often capped at a critical depth $L_c \approx 10D \text{ to } 20D$).
2. Skin Friction ($Q_s$) — The $\beta$-Method: $f_s = \beta \sigma'_v = K \tan \delta \sigma'_v$ Where $K$ is the lateral earth pressure coefficient and $\delta$ is the interface friction angle.

Pile Group Efficiency & Block Failure

Piles are typically installed in clusters connected by a reinforced concrete pile cap. The ultimate capacity of a pile group containing $n$ piles is governed by the lesser of:

1. Sum of Individual Capacities (with efficiency $\eta$): $Q_{\text{group}} = \eta \cdot n \cdot Q_{\text{ult,single}}$ Where the Converse-Labarre efficiency equation models group interaction: $\eta = 1 - \frac{\theta}{90^\circ} \left[ \frac{(n-1)m + (m-1)n}{m \cdot n} \right]$ Where $m$ is number of rows, $n$ is number of columns, and $\theta = \arctan(d/s)$ in degrees ($d = \text{diameter}, s = \text{spacing}$).
2. Block Failure Capacity ($Q_{\text{block}}$): The pile cluster and enclosed soil mass act as a single large monolithic pier of dimensions $B_g \times L_g$ and depth $L$: $Q_{\text{block}} = 9 c_u (B_g L_g) + 2(B_g + L_g) L c_u$ (Note: For block failure, soil shears along soil, so $\alpha = 1.0$.)

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Slope Stability Analysis

Slope failures occur when gravity-induced shear stresses along a potential sliding surface exceed the available shear strength of the soil.

1. Infinite Slopes (Translational Failure)

Applies to long, planar natural slopes where the depth of the active sliding layer $H$ is small compared to slope length.

• Cohesionless Soil ($c' = 0$), Dry or No Seepage: $FS = \frac{\text{Resisting Shear Stress}}{\text{Driving Shear Stress}} = \frac{\sigma' \tan \phi'}{\tau} = \frac{\gamma H \cos \beta \tan \phi'}{\gamma H \sin \beta} = \frac{\tan \phi'}{\tan \beta}$ The slope is stable against failure as long as slope angle $\beta \le \phi'$.

• Cohesionless Soil with Steady Seepage Parallel to Slope (Water at Surface): $FS = \frac{\gamma'}{\gamma_{\text{sat}}} \frac{\tan \phi'}{\tan \beta} \approx \frac{1}{2} \frac{\tan \phi'}{\tan \beta}$

  Important

  Steady seepage parallel to the slope face with water at the surface cuts the factor of safety roughly in half because buoyant unit weight $\gamma'$ is approximately half of saturated unit weight $\gamma_{\text{sat}}$.

• Cohesive-Frictional Soil ($c' - \phi'$): $FS = \frac{c'}{\gamma H \sin \beta \cos \beta} + \frac{\tan \phi'}{\tan \beta}$

2. Finite Slopes (Rotational Circular Arc Failure)

Applies to engineered embankments, cuts, and retaining berms.

• Swedish Circle / Mass Method ($\phi = 0^\circ$ Saturated Clay): $FS = \frac{\text{Resisting Moment}}{\text{Overturning Moment}} = \frac{c_u L_a R}{W \cdot d} = \frac{c_u (R \theta) R}{W \cdot d}$ Where $L_a = R \theta$ is the circular arc length, $R$ is radius of the slip circle, $W$ is total weight of sliding mass, and $d$ is horizontal distance from circle center to mass centroid.

• Ordinary Method of Slices (Fellenius): The sliding soil mass is subdivided into vertical slices. Assumes interslice forces are zero or equal and opposite: $FS = \frac{\sum [ c' \Delta l + (W_i \cos \alpha_i - u_i \Delta l) \tan \phi' ]}{\sum W_i \sin \alpha_i}$

• Bishop's Simplified Method: Considers normal interslice forces while assuming interslice shear forces are zero. Solved iteratively: $FS = \frac{\sum \frac{c' b_i + (W_i - u_i b_i) \tan \phi'}{m_\alpha}}{\sum W_i \sin \alpha_i}, \quad m_\alpha = \cos \alpha_i \left( 1 + \frac{\tan \alpha_i \tan \phi'}{FS} \right)$ Bishop's method is standard practice, typically providing $5\% \text{ to } 15\%$ higher and more realistic factors of safety than Fellenius.

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Step-by-Step Worked Problem Examples

Worked Example: Square Footing with Shallow Water Table

Problem: A reinforced concrete square column footing $B \times B = 2.0\text{ m} \times 2.0\text{ m}$ is founded at depth $D_f = 1.5\text{ m}$ below the ground surface. Soil properties are: moist unit weight $\gamma_{\text{moist}} = 18.5\text{ kN/m}^3$, saturated unit weight $\gamma_{\text{sat}} = 20.0\text{ kN/m}^3$ (giving $\gamma' = 20.0 - 9.81 = 10.19\text{ kN/m}^3$), cohesion $c = 20.0\text{ kPa}$, and $\phi = 25^\circ$. Terzaghi bearing capacity factors for $\phi = 25^\circ$ are $N_c = 25.1$, $N_q = 12.7$, $N_\gamma = 9.7$. The groundwater table is located at depth $d_w = 2.5\text{ m}$ below the ground surface. Utilizing a factor of safety of $FS = 3.0$:

1. Compute the modified effective unit weight $\bar{\gamma}$ for the soil wedge below the base.
2. Determine the ultimate bearing capacity $q_{\text{ult}}$.
3. Calculate the maximum allowable column load $P_{\text{all}}$.

Solution:

Step 1: Water Table Position Analysis $D_f = 1.5\text{ m}, \quad d_w = 2.5\text{ m}, \quad D_f + B = 1.5 + 2.0 = 3.5\text{ m}$ Since $D_f < d_w \le D_f + B$ ($1.5 < 2.5 \le 3.5\text{ m}$), this is Case 2: the water table lies within the failure wedge depth $B$ below the base.

• Surcharge $q$ (above footing base) is entirely dry/moist: $q = \gamma_{\text{moist}} D_f = 18.5\text{ kN/m}^3 \times 1.5\text{ m} = 27.75\text{ kPa}$
• Effective unit weight $\bar{\gamma}$ for the third term is weighted over depth $B$: $\bar{\gamma} = \gamma' + \left( \frac{d_w - D_f}{B} \right) (\gamma_{\text{moist}} - \gamma')$ $\bar{\gamma} = 10.19 + \left( \frac{2.5 - 1.5}{2.0} \right) (18.5 - 10.19) = 10.19 + (0.50)(8.31) = 10.19 + 4.155 = 14.345\text{ kN/m}^3$

Step 2: Calculate Ultimate Bearing Capacity ($q_{\text{ult}}$) Using Terzaghi's square footing equation: $q_{\text{ult}} = 1.3 c N_c + q N_q + 0.4 \bar{\gamma} B N_\gamma$ Evaluate each term:

• Cohesion term: $1.3 \times 20.0 \times 25.1 = 652.60\text{ kPa}$
• Surcharge term: $27.75 \times 12.7 = 352.43\text{ kPa}$
• Soil wedge term: $0.4 \times 14.345 \times 2.0 \times 9.7 = 111.32\text{ kPa}$ $q_{\text{ult}} = 652.60 + 352.43 + 111.32 = 1,116.35\text{ kPa} \approx 1,116.4\text{ kPa}$

Step 3: Calculate Allowable Column Load ($P_{\text{all}}$) $q_{\text{all}} = \frac{q_{\text{ult}}}{FS} = \frac{1,116.35}{3.0} = 372.12\text{ kPa}$ $P_{\text{all}} = q_{\text{all}} \times B^2 = 372.12\text{ kPa} \times (2.0\text{ m})^2 = 372.12 \times 4.0 = 1,488.48\text{ kN} \approx 1,488.5\text{ kN}$

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