9.4 Consolidation Settlement & Mat/Raft Foundation Design

Primary Consolidation Settlement ($S_c$) Formulation

Primary consolidation settlement occurs in saturated fine-grained cohesive soils (clays and silty clays) as excess pore water pressure $\Delta u$ induced by foundation loads dissipates, transferring load to the soil skeleton over time.

                   1D Consolidation e - log(sigma_v') Curve
                   
       Void Ratio e
           │
        e0 ├───────────┐ Recompression Path (Cr)
           │           └───┐
        ep ├───────────────┼───────────┐ Preconsolidation Pressure sigma_p'
           │               │           └───┐ Virgin Compression Path (Cc)
           │               │               └───┐
           └───────────────┴───────────────────┴─────────────► log(sigma_v')
                         sigma_v0'           sigma_f'

Stress History States and Governing Equations

Let:

• $H_c =$ Thickness of compressible clay layer
• $e_0 =$ Initial void ratio of clay
• $C_c =$ Virgin compression index
• $C_r =$ Recompression index ($C_r \approx 0.10 C_c \text{ to } 0.20 C_c$)
• $\sigma_{v0}' =$ Initial vertical effective stress at mid-height of clay layer
• $\sigma_p' =$ Preconsolidation pressure (maximum past effective vertical stress)
• $\Delta \sigma_{avg} =$ Average vertical stress increment across clay layer

Weighted Average Stress Increment Calculation (Simpson's Rule):

$$\Delta \sigma_{avg} = \frac{\Delta \sigma_{top} + 4 \Delta \sigma_{mid} + \Delta \sigma_{bot}}{6}$$

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1. Normally Consolidated Clay ($NC$: $\sigma_{v0}' \approx \sigma_p'$)

$$\sigma_f' = \sigma_{v0}' + \Delta \sigma_{avg}$$ $$S_c = \frac{C_c H_c}{1 + e_0} \log_{10}\left( \frac{\sigma_{v0}' + \Delta \sigma_{avg}}{\sigma_{v0}'} \right)$$

2. Overconsolidated Clay ($OC$ Case A: $\sigma_{v0}' + \Delta \sigma_{avg} \le \sigma_p'$)

The final stress state remains entirely on the elastic recompression curve:

$$S_c = \frac{C_r H_c}{1 + e_0} \log_{10}\left( \frac{\sigma_{v0}' + \Delta \sigma_{avg}}{\sigma_{v0}'} \right)$$

3. Overconsolidated Clay ($OC$ Case B: $\sigma_{v0}' + \Delta \sigma_{avg} > \sigma_p'$)

The stress path traverses recompression up to $\sigma_p'$, then transitions onto the virgin compression line:

$$S_c = \frac{C_r H_c}{1 + e_0} \log_{10}\left( \frac{\sigma_p'}{\sigma_{v0}'} \right) + \frac{C_c H_c}{1 + e_0} \log_{10}\left( \frac{\sigma_{v0}' + \Delta \sigma_{avg}}{\sigma_p'} \right)$$

Secondary Compression (Creep) Settlement ($S_s$)

After primary consolidation completes ($U = 100\%$):

$$S_s = \frac{C\alpha H_c}{1 + e_p} \log_{10}\left( \frac{t_2}{t_1} \right)$$

Where $C_\alpha$ is secondary compression index, $e_p$ is void ratio at end of primary consolidation, and $t_1, t_2$ represent reference times.

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Mat (Raft) Foundations

A mat foundation is a continuous reinforced concrete slab supporting all columns and walls of a structure. Mat foundations are selected when:

1. Individual spread footings would cover more than $50\%$ of the building footprint area.
2. Soil bearing capacity is low, or soil strata exhibit erratic local soft pockets.
3. Structural loads are extremely heavy (e.g., high-rise buildings, silos, power plants).
4. Differential settlement must be minimized across sensitive equipment footprints.

                   Compensated (Floating) Mat Foundation
                   
             Building Load Q_total
             ═════════╤═════════
                      │
          ┌───────────┴───────────┐ Basement Structure
          │                       │ Excavation Depth Df
    ──────┴───────────────────────┴────── Ground Surface
    ░░░░░░│                       │░░░░░░
    ░░░░░░└───────────────────────┘░░░░░░ Mat Slab (Area A)
    ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ Uplift Net Stress Delta_q_net

Undrained Bearing Capacity of Mat Foundations on Clay ($\phi_u = 0$)

Skempton (1951) net ultimate bearing capacity for mats in clay:

$$q_{net,ult} = 5.14 s_u \left( 1 + 0.2 \frac{B}{L} \right) \left( 1 + 0.2 \frac{D_f}{B} \right)$$

Where maximum depth factor term $\left(1 + 0.2 \frac{D_f}{B}\right) \le 1.5$.

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Compensated (Floating) Foundation Design Principle

A compensated foundation reduces net soil stress by excavating soil weight equal to all or part of the total structural weight.

$$\Delta q_{net} = \frac{Q_{total}}{A_{mat}} - \gamma D_f$$

• Fully Compensated (Floating) Mat ($\Delta q_{net} = 0$): $$D_f = \frac{Q_{total}}{\gamma A_{mat}}$$ When $\Delta q_{net} = 0$, primary consolidation settlement $S_c$ in underlying soft clay layers is theoretically reduced to near zero (except for small rebound/recompression strain).
• Partially Compensated Mat ($0 < \Delta q_{net} < \frac{Q_{total}}{A_{mat}}$): Reduces consolidation settlement significantly compared to surface loading.

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Modulus of Subgrade Reaction ($k_s$)

In mat structural analysis (Winkler spring models), soil support is represented by vertical springs with subgrade modulus $k_s = q / \delta$ (units of $\text{kN/m}^3$ or $\text{lb/in}^3$).

Size Scaling Equations (Terzaghi 1955)

Given test plate subgrade modulus $k_1$ from a $0.3\text{ m} \times 0.3\text{ m}$ ($1\text{ ft} \times 1\text{ ft}$) plate:

• For Cohesionless Sand:

  $$k_s = k_1 \left( \frac{B + 0.3}{2 B} \right)^2 \quad \text{(B in meters)}$$ $$k_s = k_1 \left( \frac{B + 1}{2 B} \right)^2 \quad \text{(B in feet)}$$

• For Cohesive Clay:

  $$k_s = k_1 \left( \frac{0.3}{B} \right) \quad \text{(B in meters)}$$ $$k_s = k_1 \left( \frac{1}{B} \right) \quad \text{(B in feet)}$$

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Tolerable Settlement and Angular Distortion ($\beta$)

Foundation design must limit differential settlement $\delta_{diff}$ between adjacent columns spaced distance $L_{span}$ apart.

$$\text{Angular Distortion } \beta = \frac{\delta_{diff}}{L_{span}}$$

                      Angular Distortion Definition
                      
             Column i                  Column i+1
                 │                          │
                 ▼                          ▼
            ┌────────┐                 ┌────────┐
            │        │                 │        │
            └────┬───┘                 └───┬────┘
                 │                         │
           ──────┼─────────────────────────┼────── Original Grade
                 │ delta_i                 │
                 ▼                         │ delta_i+1
               ─────                       ▼
                                         ─────
                 ◄──────── L_span ────────►
                 Angular Distortion beta = (delta_i+1 - delta_i) / L_span

Bjerrum (1963) Serviceability Damage Criteria:

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Comprehensive Worked Calculation Example

Problem Statement

A building load $Q_{total} = 54,000\text{ kN}$ is supported by a mat foundation measuring $18\text{ m} \times 25\text{ m}$ embedded at $D_f = 3.0\text{ m}$ in sand ($\gamma = 18.0\text{ kN/m}^3$). Directly beneath the sand is a $4.0\text{ m}$ thick clay layer ($H_c = 4.0\text{ m}$) with its midpoint located at $z = 7.0\text{ m}$ below the mat base.

Clay layer properties:

• Initial vertical effective stress at midpoint $\sigma_{v0}' = 110\text{ kPa}$
• Preconsolidation pressure $\sigma_p' = 150\text{ kPa}$
• Initial void ratio $e_0 = 0.85$
• Compression index $C_c = 0.32$
• Recompression index $C_r = 0.04$

Calculate:

1. The net foundation stress increment $\Delta q_{net}$ at mat level
2. The stress increment $\Delta \sigma_{avg}$ at the midpoint of the clay layer using the 2:1 method
3. The primary consolidation settlement $S_c$ of the clay layer

Solution Steps

Step 1: Calculate Net Applied Foundation Pressure $\Delta q_{net}$

$$A_{mat} = 18 \times 25 = 450\text{ m}^2$$ $$q_{gross} = \frac{54,000}{450} = 120\text{ kPa}$$ $$\sigma_{v0,exc}' = \gamma D_f = 18.0 \times 3.0 = 54\text{ kPa}$$ $$\Delta q_{net} = q_{gross} - \sigma_{v0,exc}' = 120 - 54 = 66\text{ kPa}$$

Step 2: Compute Stress Increment $\Delta \sigma_{avg}$ at Mid-Clay Depth ($z = 7.0\text{ m}$)

Using 2:1 stress distribution:

$$B_z = B + z = 18 + 7.0 = 25.0\text{ m}$$ $$L_z = L + z = 25 + 7.0 = 32.0\text{ m}$$ $$\Delta \sigma_{avg} = \frac{\Delta q_{net} \cdot (B \times L)}{(B + z)(L + z)} = \frac{66 \times 450}{25.0 \times 32.0} = \frac{29,700}{800} = 37.125\text{ kPa}$$

Step 3: Evaluate Overconsolidation Stress State

$$\sigma_{f}' = \sigma_{v0}' + \Delta \sigma_{avg} = 110 + 37.125 = 147.125\text{ kPa}$$

Compare $\sigma_f'$ to preconsolidation pressure $\sigma_p' = 150\text{ kPa}$:

$$\sigma_f' = 147.125\text{ kPa} \le \sigma_p' = 150\text{ kPa}$$

Since $\sigma_f' \le \sigma_p'$, the soil remains overconsolidated ($OC$ Case A), deforming strictly along the recompression curve governed by $C_r$.

Step 4: Calculate Consolidation Settlement $S_c$

$$S_c = \frac{C_r H_c}{1 + e_0} \log_{10}\left( \frac{\sigma_{v0}' + \Delta \sigma_{avg}}{\sigma_{v0}'} \right)$$ $$S_c = \frac{0.04 \times 4.0}{1 + 0.85} \log_{10}\left( \frac{147.125}{110} \right)$$ $$S_c = \left( \frac{0.16}{1.85} \right) \log_{10}(1.3375) = 0.086486 \times 0.126297 = 0.01092\text{ m} = 10.92\text{ mm}$$