4.4 Pseudostatic Analysis & Seismic Geotechnical Design

4.4 Pseudostatic Analysis & Seismic Geotechnical Design

1. Principles of Pseudostatic Slope Stability Analysis

Earthquake ground motions induce transient, cyclic accelerations in earth slopes and retaining structures. In engineering practice, the complex dynamic response is simplified using pseudostatic limit equilibrium analysis. The dynamic earthquake inertia force is represented by static force vectors applied at the centroid of a potential failure mass:

$$F_h = k_h \cdot W = \left(\frac{a_h}{g}\right) W$$ $$F_v = k_v \cdot W = \left(\frac{a_v}{g}\right) W$$

where:

• $W$ = weight of the potential sliding block.
• $k_h$ = horizontal pseudostatic seismic coefficient (dimensionless).
• $k_v$ = vertical pseudostatic seismic coefficient (dimensionless, often neglected or set to $k_v = \pm 0.5 k_h$).

The horizontal pseudostatic force $F_h$ destabilizes the slope by increasing driving overturning moments and reducing effective normal stresses along frictional slip surfaces.

Selection of Horizontal Seismic Coefficient ($k_h$)

Setting $k_h = PGA/g$ is overly conservative because peak ground accelerations act only for transient split-second intervals and do not act simultaneously across an entire slope mass. Recommended practice guidelines (Hynes-Griffin & Franklin 1984; Bray & Travasarou 2009):

• Standard Slope Design: $k_h = 0.33 \text{ to } 0.50 \times \left(\frac{PGA}{g}\right)$
• If the slope maintains a pseudostatic factor of safety $FS_{ps} \ge 1.10 - 1.15$ under $k_h = 0.50 (PGA/g)$, permanent slope deformations are generally small ($< 5\text{ cm} \approx 2\text{ inches}$), and slope performance is deemed acceptable.

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2. Yield Acceleration ($k_y$) and Newmark Permanent Displacement

When a slope has a pseudostatic $FS_{ps} < 1.0$, it does not automatically undergo catastrophic collapse. Instead, slope sliding occurs incrementally during transient time intervals when earthquake acceleration exceeds a critical threshold called the Yield Acceleration ($k_y$).

• Yield Acceleration ($k_y$): The horizontal seismic coefficient $k_h$ that produces a pseudostatic factor of safety equal to exactly $1.0$ ($FS_{ps} = 1.0$).
• Newmark Sliding Block Analysis (Newmark 1965): Models the sliding slope mass as a rigid block resting on an inclined plane. When ground acceleration $a(t)$ exceeds $k_y g$, the block breaks free and slides down slope. Permanent downslope displacement $u_d$ is obtained by double integrating the relative acceleration over time: $$u_d = \int \int \left[ a(t) - k_y g \right] dt^2 \quad \text{for } a(t) > k_y g$$

Empirical equations derived from strong-motion databases estimate Newmark displacement $u_d$ (Jibson 2007; Bray & Travasarou 2007):

$$\log u_d = -0.22 - 2.83 \log(k_y) - 0.333 (\log k_y)^2 + 0.566 \log(PGA) + 0.444 M_w$$

where $u_d$ is in centimeters.

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3. Mononobe-Okabe (M-O) Dynamic Earth Pressure Theory

For design of retaining walls subjected to earthquake shaking, the Mononobe-Okabe (M-O) method (Mononobe & Matsuo 1929; Okabe 1926) extends Coulomb's active earth pressure theory by incorporating horizontal ($k_h W$) and vertical ($k_v W$) static pseudostatic inertia forces into the soil wedge force equilibrium.

M-O Dynamic Active Earth Pressure Coefficient ($K_{AE}$)

$$K_{AE} = \frac{\cos^2(\phi - \theta - \beta)}{\cos\theta \cos^2\beta \cos(\delta + \beta + \theta) \left[ 1 + \sqrt{\frac{\sin(\phi + \delta) \sin(\phi - \theta - i)}{\cos(\delta + \beta + \theta) \cos(i - \beta)}} \right]^2}$$

where:

• $\phi$ = internal friction angle of backfill soil.
• $\delta$ = wall-soil friction angle (typically $\frac{1}{2} \phi$ to $\frac{2}{3} \phi$).
• $\beta$ = slope angle of wall face relative to vertical.
• $i$ = backfill slope inclination angle relative to horizontal.
• $\theta$ = seismic inertia angle, defined as: $$\theta = \tan^{-1}\left( \frac{k_h}{1 - k_v} \right)$$

Total dynamic active earth thrust $P_{AE}$ acting on a wall of height $H$ is:

$$P_{AE} = \frac{1}{2} \gamma H^2 (1 - k_v) K_{AE}$$

Division into Static and Dynamic Components:

Total active dynamic thrust $P_{AE}$ is divided into static active force ($P_A$) and dynamic active increment ($\Delta P_{AE}$):

$$P_{AE} = P_A + \Delta P_{AE}$$

• Static Active Force ($P_A$): $P_A = \frac{1}{2} \gamma H^2 K_A$, acting at a height $h_1 = \frac{H}{3}$ above the wall base.
• Dynamic Active Increment ($\Delta P_{AE}$): $\Delta P_{AE} = P_{AE} - P_A$, acting higher on the wall at $h_2 = 0.60 H$ above the base due to inverted dynamic inverted wedge mass distribution.

The resultant line of action $h_{res}$ above the base for $P_{AE}$ is:

$$h_{res} = \frac{P_A \left(\frac{H}{3}\right) + \Delta P_{AE} \left(0.60 H\right)}{P_{AE}}$$

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4. Seed-Whitman Simplified Dynamic Thrust Approximation

For vertical wall faces ($\beta = 0$) and horizontal backfills ($i = 0$), Seed & Whitman (1970) demonstrated that the dynamic active thrust increment $\Delta P_{AE}$ can be accurately approximated without solving the full M-O equation:

$$\Delta P_{AE} \approx \frac{3}{8} k_h \gamma H^2$$

This simple rule-of-thumb allows rapid verification of dynamic retaining wall loads on PE exams.

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5. Seismic Foundation Design Considerations

A. Shallow Foundations

Dynamic foundation bearing capacity under seismic loading is degraded by two factors:

1. Inclined and eccentric structural loads caused by lateral shear forces and overturning moments.
2. Soil inertia forces within the foundation supporting wedge (Richards et al. 1993 dynamic bearing capacity factors $N_{\gamma e}, N_{qe}, N_{ce}$).

B. Deep Foundations (Piles and Drilled Shafts)

Piles subjected to earthquake ground motions face critical failure modes:

1. Kinematic Bending: Caused by high strain gradients occurring at material interfaces between stiff and soft soil layers.
2. Inertial Bending: Structure superstructure acceleration transmits large dynamic shear and overturning moments to the pile head.
3. Liquefaction Degradation & Downdrag: Liquefied layers lose lateral resistance (modelled using $p-y$ multipliers $p_{mult} = 0.1 - 0.3$). Subsequent post-liquefaction settlement causes heavy negative skin friction (downdrag) loads on pile shafts.

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6. Comprehensive Step-by-Step Worked Example

Problem Statement:

A vertical retaining wall ($H = 6.0\text{ m}$) supports a horizontal granular backfill ($i = 0$, $\beta = 0$).

• Backfill unit weight: $\gamma = 18.5\text{ kN/m}^3$
• Friction angle: $\phi = 34^\circ$
• Wall friction angle: $\delta = 0^\circ$ (smooth wall)
• Design earthquake parameters: $PGA = 0.30g$, horizontal seismic coefficient $k_h = 0.50 \times (PGA/g) = 0.15$, $k_v = 0$.

Calculate:

1. Static Rankine active earth pressure coefficient $K_A$ and static active force $P_A$.
2. Seismic inertia angle $\theta$.
3. Mononobe-Okabe dynamic active earth pressure coefficient $K_{AE}$ and total dynamic thrust $P_{AE}$.
4. Dynamic thrust increment $\Delta P_{AE}$ using both M-O theory and the Seed-Whitman approximation.
5. Line of action $h_{res}$ of the total dynamic resultant thrust above the wall base.

Solution:

Step 1: Static Active Thrust ($P_A$)
Rankine static active coefficient:

$$K_A = \tan^2\left(45^\circ - \frac{\phi}{2}\right) = \tan^2\left(45^\circ - 17^\circ\right) = \tan^2(28^\circ) = 0.5317^2 = \mathbf{0.2827}$$

Static active force per unit length:

$$P_A = \frac{1}{2} \gamma H^2 K_A = \frac{1}{2} \times 18.5 \times (6.0)^2 \times 0.2827 = 0.5 \times 18.5 \times 36 \times 0.2827 = \mathbf{94.14\text{ kN/m}}$$

Point of application: $h_1 = \frac{H}{3} = \frac{6.0}{3} = \mathbf{2.0\text{ m}}$ above base.

Step 2: Seismic Inertia Angle ($\theta$)

$$\theta = \tan^{-1}\left(\frac{k_h}{1 - k_v}\right) = \tan^{-1}(0.15) = \mathbf{8.531^\circ}$$

Step 3: Mononobe-Okabe Dynamic Active Coefficient ($K_{AE}$) & Thrust ($P_{AE}$)
For $\beta = 0$, $i = 0$, $\delta = 0$:

$$K_{AE} = \frac{\cos^2(\phi - \theta)}{\cos\theta \left[ 1 + \sqrt{\frac{\sin\phi \sin(\phi - \theta)}{\cos\theta}} \right]^2}$$ $$\phi - \theta = 34^\circ - 8.531^\circ = 25.469^\circ$$ $$\cos^2(25.469^\circ) = (0.9028)^2 = 0.8151$$ $$\cos\theta = \cos(8.531^\circ) = 0.9889$$ $$\sin(34^\circ) = 0.5592, \quad \sin(25.469^\circ) = 0.4300$$ $$\sqrt{\frac{0.5592 \times 0.4300}{0.9889}} = \sqrt{\frac{0.2405}{0.9889}} = \sqrt{0.2432} = 0.4931$$ $$K_{AE} = \frac{0.8151}{0.9889 \times [1 + 0.4931]^2} = \frac{0.8151}{0.9889 \times 2.2294} = \frac{0.8151}{2.2046} = \mathbf{0.3697}$$

Total dynamic active thrust:

$$P_{AE} = \frac{1}{2} \gamma H^2 K_{AE} = \frac{1}{2} \times 18.5 \times 36 \times 0.3697 = \mathbf{123.11\text{ kN/m}}$$

Step 4: Dynamic Increment ($\Delta P_{AE}$)

• Via Mononobe-Okabe: $$\Delta P_{AE} = P_{AE} - P_A = 123.11 - 94.14 = \mathbf{28.97\text{ kN/m}}$$
• Via Seed-Whitman Approximation: $$\Delta P_{AE,SW} \approx \frac{3}{8} k_h \gamma H^2 = 0.375 \times 0.15 \times 18.5 \times (6.0)^2 = 0.375 \times 0.15 \times 18.5 \times 36 = \mathbf{37.46\text{ kN/m}}$$ (Seed-Whitman gives a slightly conservative upper bound).

Step 5: Line of Action ($h_{res}$)
Using the M-O dynamic increment ($\Delta P_{AE} = 28.97\text{ kN/m}$) acting at $0.60 H = 0.60 \times 6.0\text{ m} = 3.60\text{ m}$:

$$h_{res} = \frac{P_A (2.0\text{ m}) + \Delta P_{AE} (3.60\text{ m})}{P_{AE}}$$ $$h_{res} = \frac{(94.14 \times 2.0) + (28.97 \times 3.60)}{123.11} = \frac{188.28 + 104.29}{123.11} = \frac{292.57}{123.11} = \mathbf{2.377\text{ m}}$$

(The resultant dynamic thrust acts at $2.38\text{ m}$ above the base, compared to $2.0\text{ m}$ for static conditions).