Geotechnical Engineering

Geotechnical engineering applies soil mechanics, rock mechanics, and geology to the design and construction of foundations, earth structures, and underground facilities. Understanding soil behavior is essential for ensuring that structures remain stable and safe throughout their service life.

Soil Composition and Classification

Soil Phases

Soil is a three-phase material consisting of:

• Solids: Mineral particles (sand, silt, clay)
• Water: Pore water
• Air: Pore air (in unsaturated soil)

Phase Relationships

Void Ratio:

Porosity:

Degree of Saturation:

Water Content (Moisture Content):

Unit Weight Relationships:

Dry unit weight:

Saturated unit weight:

Moist unit weight:

Submerged (buoyant) unit weight:

Where $G_s$ is specific gravity of solids (typically 2.65-2.70) and $\gamma_w = 9.81$ kN/m$^3$.

Soil Classification

Unified Soil Classification System (USCS):

Atterberg Limits

Liquid Limit (LL): Water content at which soil transitions from plastic to liquid state

Plastic Limit (PL): Water content at which soil transitions from semi-solid to plastic state

Plasticity Index:

Liquidity Index:

Soil Stresses

Effective Stress Principle (Terzaghi)

The effective stress controls soil behavior:

Where:

• $\sigma'$ = effective stress
• $\sigma$ = total stress
• $u$ = pore water pressure

Vertical Stress in Soil

At depth $z$ below ground surface:

For layered soil:

Below water table:

Stress Distribution

Boussinesq's Equation (point load on elastic half-space):

Where $\beta$ is the angle from vertical.

2:1 Approximation for stress distribution:

For square footing:

Shear Strength

Mohr-Coulomb Failure Criterion

Where:

• $c$ = cohesion
• $\phi$ = angle of internal friction
• $\sigma'$ = effective normal stress on failure plane

Drained vs. Undrained Strength

Drained conditions (long-term): Use effective stress parameters $c'$, $\phi'$

Undrained conditions (short-term): Use total stress parameters $c_u$, $\phi_u = 0$ for saturated clay

Typical Strength Parameters

Bearing Capacity

Terzaghi's Bearing Capacity Equation

For strip footing:

Where:

• $q_u$ = ultimate bearing capacity
• $c$ = cohesion
• $q = \gamma D_f$ = overburden pressure at foundation level
• $B$ = foundation width
• $N_c$, $N_q$, $N_\gamma$ = bearing capacity factors (functions of $\phi$)

Bearing Capacity Factors

Shape Factors (Meyerhof)

For rectangular footings, multiply by shape factors:

Factor of Safety

Typical FS = 2.5 to 3.0 for foundations.

Settlement Analysis

Immediate Settlement

For flexible footing on elastic soil:

Where $I_s$ is influence factor (depends on shape and rigidity).

Consolidation Settlement

For normally consolidated clay:

For overconsolidated clay (if $\sigma'_0 + \Delta\sigma < \sigma'_c$):

Where:

• $C_c$ = compression index
• $C_r$ = recompression index
• $H$ = thickness of clay layer
• $e_0$ = initial void ratio
• $\sigma'_c$ = preconsolidation pressure

Time Rate of Consolidation

Where:

• $U$ = degree of consolidation
• $T_v = \frac{c_v t}{H_{dr}^2}$ = time factor
• $c_v$ = coefficient of consolidation
• $H_{dr}$ = drainage path length

Approximate relationships:

• For $U < 60\%$: $T_v \approx \frac{\pi}{4} U^2$
• For $U > 60\%$: $T_v \approx -0.933 \log(1-U) - 0.085$

Slope Stability

Factor of Safety

Infinite Slope Analysis

For cohesionless soil:

For cohesive soil with seepage parallel to slope:

Circular Slip Surface Methods

Ordinary Method of Slices (Fellenius):

Bishop's Simplified Method:

Where:

This requires iteration to solve.

Lateral Earth Pressure

At-Rest Pressure

Rankine Theory

Active pressure (wall moves away from soil):

Passive pressure (wall moves toward soil):

For cohesive soil:

Real-World Application: Foundation Design

Designing a shallow foundation for a building column.

Foundation Analysis Example

import math

# Site and loading conditions
foundation_params = {
    'column_load': 1200,      # kN (service load)
    'soil_type': 'medium_clay',
    'cohesion': 50,           # kPa (undrained)
    'phi': 0,                 # degrees (undrained, phi_u = 0)
    'gamma': 18,              # kN/m^3
    'depth': 1.5,             # m (foundation depth)
    'water_table': 4.0,       # m below ground surface
    'FS_required': 3.0,       # factor of safety
}

# Bearing capacity factors for phi = 0
Nc = 5.14  # For phi = 0
Nq = 1.0
Ngamma = 0

c = foundation_params['cohesion']
gamma = foundation_params['gamma']
Df = foundation_params['depth']
q = gamma * Df  # Overburden pressure

# Assume square footing, iterate for size
def calculate_bearing_capacity(B, c, q, gamma, Nc, Nq, Ngamma):
    """Calculate ultimate bearing capacity for square footing"""
    # Shape factors for square footing
    sc = 1.3
    sq = 1.0
    sgamma = 0.8

    # Terzaghi equation with shape factors
    qu = c * Nc * sc + q * Nq * sq + 0.5 * gamma * B * Ngamma * sgamma
    return qu

# Iterate to find required footing size
for B in [1.5, 2.0, 2.5, 3.0, 3.5, 4.0]:
    qu = calculate_bearing_capacity(B, c, q, gamma, Nc, Nq, Ngamma)
    q_allowable = qu / foundation_params['FS_required']
    q_applied = foundation_params['column_load'] / (B * B)

    if q_applied <= q_allowable:
        selected_B = B
        break

print(f"Foundation Design Results")
print(f"=" * 40)
print(f"\nSite Conditions:")
print(f"  Soil: {foundation_params['soil_type']}")
print(f"  Undrained cohesion: {c} kPa")
print(f"  Unit weight: {gamma} kN/m^3")
print(f"  Foundation depth: {Df} m")

print(f"\nLoading:")
print(f"  Column load: {foundation_params['column_load']} kN")

qu = calculate_bearing_capacity(selected_B, c, q, gamma, Nc, Nq, Ngamma)
q_allowable = qu / foundation_params['FS_required']
q_applied = foundation_params['column_load'] / (selected_B * selected_B)

print(f"\nBearing Capacity Analysis:")
print(f"  Ultimate bearing capacity: {qu:.1f} kPa")
print(f"  Allowable bearing capacity: {q_allowable:.1f} kPa")
print(f"  Applied bearing pressure: {q_applied:.1f} kPa")

print(f"\nSelected Foundation:")
print(f"  Size: {selected_B} m x {selected_B} m")
print(f"  Depth: {Df} m")
print(f"  Factor of Safety: {qu/q_applied:.2f}")

Your Challenge: Slope Stability Analysis

Analyze the stability of a slope using the method of slices.

Goal: Determine the factor of safety for a slope with specified geometry and soil properties.

Problem Setup

import math

# Slope geometry and soil properties
slope_config = {
    'height': 8.0,            # meters
    'angle': 30,              # degrees
    'soil_cohesion': 15,      # kPa (effective)
    'friction_angle': 25,     # degrees (effective)
    'unit_weight': 19,        # kN/m^3
    'water_table': 'at_toe',  # water table condition
}

# Convert angles to radians
beta = math.radians(slope_config['angle'])
phi = math.radians(slope_config['friction_angle'])

# Infinite slope analysis (simplified)
H = slope_config['height']
c = slope_config['soil_cohesion']
gamma = slope_config['unit_weight']

# For dry slope (no seepage)
# Critical depth for cohesive soil
z_critical = 4 * c / (gamma * (1 - math.cos(2 * beta)))

# Factor of safety using infinite slope method
FS_friction = math.tan(phi) / math.tan(beta)
FS_cohesion = c / (gamma * H * math.cos(beta) * math.sin(beta))
FS_total = FS_friction + FS_cohesion

print(f"Slope Stability Analysis")
print(f"=" * 40)
print(f"\nSlope Geometry:")
print(f"  Height: {H} m")
print(f"  Angle: {slope_config['angle']} degrees")

print(f"\nSoil Properties:")
print(f"  Cohesion: {c} kPa")
print(f"  Friction angle: {slope_config['friction_angle']} degrees")
print(f"  Unit weight: {gamma} kN/m^3")

print(f"\nInfinite Slope Analysis:")
print(f"  Frictional component: {FS_friction:.3f}")
print(f"  Cohesive component: {FS_cohesion:.3f}")
print(f"  Total Factor of Safety: {FS_total:.3f}")

if FS_total >= 1.5:
    status = "STABLE (FS >= 1.5)"
elif FS_total >= 1.0:
    status = "MARGINALLY STABLE (1.0 <= FS < 1.5)"
else:
    status = "UNSTABLE (FS < 1.0)"

print(f"\nStability Assessment: {status}")

# Sensitivity analysis
print(f"\nSensitivity to Water Table:")
# With full saturation and seepage
gamma_sat = gamma + 2  # Approximate saturated unit weight
gamma_sub = gamma_sat - 9.81  # Submerged unit weight
FS_with_seepage = (c / (gamma_sat * H * math.cos(beta) * math.sin(beta)) +
                   gamma_sub / gamma_sat * math.tan(phi) / math.tan(beta))
print(f"  FS with parallel seepage: {FS_with_seepage:.3f}")

How would you modify the analysis to account for earthquake loading using a pseudo-static approach?