6.3 Uplift Pressure, Piping & Heave Safety

6.3 Uplift Pressure, Piping & Heave Safety

Seepage Force & Quick Condition Mechanics

Water seeping through soil exerts a drag force on individual soil grains known as seepage force. Per unit volume of soil mass ($V = 1.0\text{ m}^3$), the seepage force vector $\mathbf{j}$ acts parallel to the hydraulic gradient vector $\mathbf{i}$:

$$j = i \cdot \gamma_w$$

When seepage occurs vertically upwards, the seepage drag force acts directly counter to the effective gravitational weight of the saturated soil mass. The vertical effective stress $\sigma'$ at depth $z$ subject to upward seepage becomes:

$$\sigma'(z) = \sigma_{total} - u = \gamma_{sat} z - (\gamma_w z + i \gamma_w z) = (\gamma_{sat} - \gamma_w) z - i \gamma_w z = \gamma' z - i \gamma_w z$$

Where $\gamma' = \gamma_{sat} - \gamma_w$ is the submerged (buoyant) unit weight. As the upward hydraulic gradient $i$ increases, effective stress $\sigma'$ diminishes. At the Critical Hydraulic Gradient ($i_{cr}$), effective stress reaches zero:

$$\sigma' = 0 \implies \gamma' z - i_{cr} \gamma_w z = 0$$ $$\ni_{cr} = \frac{\gamma'}{\gamma_w} = \frac{\gamma_{sat} - \gamma_w}{\gamma_w}$$

Expressing $i_{cr}$ in terms of specific gravity of solids $G_s$ and void ratio $e$:

$$\gamma' = \frac{(G_s - 1)\gamma_w}{1 + e} \implies i_{cr} = \frac{G_s - 1}{1 + e}$$

When $i \ge i_{cr}$, cohesionless soils (sands and silts) completely lose shear strength ($\tau_f = \sigma' \tan\phi' = 0$) and behave like a heavy liquid, creating a Quick Condition (quicksand). For typical soils ($G_s \approx 2.65-2.70, e \approx 0.5-0.8$), $i_{cr}$ ranges from 0.90 to 1.10 (frequently approximated as $i_{cr} \approx 1.0$).

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Terzaghi Exit Prism Heave Analysis

In sheeted excavations in sand where water table differential exists across embedded sheet piles, upward seepage occurs at the bottom of the excavation adjacent to the wall. Karl Terzaghi demonstrated that heave failure occurs within a finite prism of soil of width $D/2$ and depth $D$ (where $D$ is the sheet pile embedment depth below excavation level).

  SHEET PILE WALL EXCAVATION BOTTOM HEAVE (TERZAGHI PRISM)
  
             Sheet Pile Wall
                   ||
   High Water      ||
   Side            ||   Excavation Bottom
  ~~~~~~~~~~~~~~~~~||~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                   ||  |                        |
                   ||  |   Terzaghi Heave       | Depth D
                   ||  |   Prism (D x D/2)      |
                   ||  |   Submerged W'         |
                   ||  v                        |
                   ||============================
                   ||  ^ Upward Seepage Force U
                   ||  | (h_p_avg * gamma_w * D/2)

Stability Factors for Heave Prism:

1. Submerged Weight of Soil Prism ($W'$): Per unit length of wall:

   $$W' = \gamma' \cdot D \cdot \left(\frac{D}{2}\right) = \frac{1}{2} \gamma' D^2$$

2. Upward Seepage Force on Prism Base ($U$):

   $$U = \bar{h}_p \cdot \gamma_w \cdot \left(\frac{D}{2}\right) = \frac{1}{2} \bar{h}_p \gamma_w D$$

   Where $\bar{h}_p$ is the average excess piezometric pressure head along the base of the prism ($z = -D$).

3. Factor of Safety against Heave ($FS_{heave}$):

   $$FS_{heave} = \frac{W'}{U} = \frac{\frac{1}{2} \gamma' D^2}{\frac{1}{2} \bar{h}_p \gamma_w D} = \frac{\gamma' D}{\bar{h}_p \gamma_w}$$

Standard geotechnical engineering specifications require $FS_{heave} \ge 1.50 \text{ to } 2.00$ for temporary excavations, and $FS_{heave} \ge 2.00 \text{ to } 3.00$ for permanent earth retention systems.

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Base Uplift Pressure & Structural Flotation Safety

Subsurface structures such as concrete dams, weir aprons, dry docks, and deep basement slabs subjected to differential head experience hydraulic uplift forces $U_{uplift}$.

Calculation of Base Uplift Pressure Profile:

Along the base of a concrete dam or mat slab of width $B$, pore water pressure $u(x)$ at distance $x$ from upstream toe is:

$$\nu(x) = \gamma_w \cdot h_p(x) = \gamma_w \left[ H_1 - n(x) \cdot \Delta h - z_{base} \right]$$

Where $n(x)$ is the cumulative equipotential drops at position $x$. The Total Uplift Force ($U$) per unit length is obtained by integrating $u(x)$ across base length $B$:

$$U = \int_{0}^{B} u(x) dx$$

  UPLIFT PRESSURE DISTRIBUTION UNDER CONCRETE DAM BASE
  
        Upstream Head H1                          Downstream Head H2
      ~~~~~~~~~~~~~~~~~~~~                    ~~~~~~~~~~~~~~~~~~~~~
             |    |                                  |    |
             |    |________ Dam Structure ___________|    |
             |   /                                   \    |
             |  /                                     \   |
             | /                                       \  |
             |/=========================================\|
   u_upstream |\                                          | u_downstream
   = gamma_w*H1 \                                        / = gamma_w*H2
                 \_____ Linear/Flow Net Uplift Profile _/

Factor of Safety against Structural Flotation ($FS_{flotation}$):

To prevent buoyant uplift (flotation) of basements or empty structures:

$$FS_{flotation} = \frac{\sum W_{dead}}{U_{uplift}} = \frac{W_{concrete} + W_{soil\_cover}}{U_{uplift}} \ge 1.50$$

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Piping Mechanics & Lane's Weighted Creep Ratio

Piping (backward erosion piping) is progressive internal erosion where seepage velocities at exit points remove soil grains, creating hollow erosion pipes that track backwards upstream toward the reservoir. To prevent piping failure, the exit hydraulic gradient $i_{exit}$ must satisfy:

$$FS_{piping} = \frac{i_{cr}}{i_{exit}} \ge 3.0 - 4.0$$

Lane's Weighted Creep Theory ($C_w$):

E.W. Lane developed an empirical piping risk check evaluating seepage flow path length along structural contact surfaces. Recognizing vertical contacts ($>45^\circ$) resist piping three times more effectively than horizontal contacts ($<45^\circ$):

$$L_{weighted} = L_V + \frac{1}{3} L_H$$ $$C_w = \frac{L_{weighted}}{\Delta H} = \frac{L_V + \frac{1}{3} L_H}{\Delta H}$$

Where $L_V$ is total vertical contact path length, $L_H$ is total horizontal contact path length, and $\Delta H$ is net differential head. Minimum safe $C_w$ values specified by Lane:

• Fine Sand: $C_w \ge 7.0$
• Medium Sand: $C_w \ge 6.0$
• Coarse Sand: $C_w \ge 5.0$
• Gravel / Clay: $C_w \ge 3.0 - 4.0$

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Engineering Countermeasures for Heave and Piping

1. Vertical Cutoff Walls: Driving sheet piles or slurry walls below excavation bottom increases path length $L_V$, reducing exit gradient $i_{exit}$ and excess head $\bar{h}_p$.
2. Downstream Weighted Filters (Inverted Filters): Placing coarse sand/gravel filter layers over downstream exit zones provides overburden surcharge $W_{filter}'$ while permitting free water drainage without soil particle loss.
3. Relief Wells: Drilled perforated wells placed downstream relieve high sub-artesian pressures in underlying permeable layers.
4. Upstream Impermeable Blankets: Extended clay layers upstream lengthen horizontal seepage path $L_H$, dissipating head prior to structure arrival.

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Worked Numerical Examples

Worked Example 1: Critical Gradient & Heave Safety Factor

Problem Statement: An excavation in saturated sand ($\gamma_{sat} = 19.8\text{ kN/m}^3$, $G_s = 2.65$) is supported by sheet piling driven to embedment depth $D = 5.0\text{ m}$ below the pit floor. Hydraulic analysis of the flow net reveals that average excess pressure head along the base of Terzaghi's heave prism is $\bar{h}_p = 2.0\text{ m}$. Calculate:

1. Critical hydraulic gradient $i_{cr}$.
2. Terzaghi heave factor of safety $FS_{heave}$.

Solution:

1. Calculate submerged unit weight $\gamma'$ and critical gradient $i_{cr}$ (taking $\gamma_w = 9.81\text{ kN/m}^3$):

   $$\gamma' = \gamma_{sat} - \gamma_w = 19.8 - 9.81 = 9.99\text{ kN/m}^3$$ $$i_{cr} = \frac{\gamma'}{\gamma_w} = \frac{9.99}{9.81} = 1.018$$

2. Calculate Terzaghi heave factor of safety $FS_{heave}$:

   $$FS_{heave} = \frac{\gamma' \cdot D}{\bar{h}_p \cdot \gamma_w} = \frac{9.99\text{ kN/m}^3 \times 5.0\text{ m}}{2.0\text{ m} \times 9.81\text{ kN/m}^3} = \frac{49.95}{19.62} = 2.546 \approx 2.55$$

   Since $FS_{heave} = 2.55 > 1.50$, the excavation base is safe against bottom heave.