Cheatsheet: Earth Retaining Structures

1. Lateral Earth Pressure Theory

1.1 Types of Earth Pressure

1.2 Rankine Theory

1.2.1 Active Pressure Coefficient

• Kₐ = (1 - sin φ) / (1 + sin φ) = tan²(45° - φ/2)
• Active pressure: σₐ = Kₐ σᵥ - 2c√Kₐ
• For cohesionless soil: Pₐ = ½ Kₐ γ H²
• Acts at H/3 from base

1.2.2 Passive Pressure Coefficient

• Kₚ = (1 + sin φ) / (1 - sin φ) = tan²(45° + φ/2)
• Passive pressure: σₚ = Kₚ σᵥ + 2c√Kₚ
• For cohesionless soil: Pₚ = ½ Kₚ γ H²
• Acts at H/3 from base

1.2.3 At-Rest Coefficient

• K₀ = 1 - sin φ (Jaky's formula for normally consolidated)
• K₀ = (1 - sin φ) × OCR^(sin φ) (overconsolidated)
• P₀ = ½ K₀ γ H²

1.3 Coulomb Theory

1.3.1 Active Pressure

• Kₐ = cos²(φ - α) / [cos²α × cos(δ + α) × (1 + √[sin(φ + δ)sin(φ - β) / cos(δ + α)cos(β - α)])²]
• α = wall batter angle from vertical, β = backfill slope angle, δ = wall friction angle
• Pₐ = ½ Kₐ γ H²
• Acts at H/3 from base at angle δ to normal

1.3.2 Passive Pressure

• Kₚ = cos²(φ + α) / [cos²α × cos(δ - α) × (1 - √[sin(φ + δ)sin(φ + β) / cos(δ - α)cos(β - α)])²]
• For vertical wall (α = 0), horizontal backfill (β = 0): simplified form applies

1.4 Wall Friction and Soil-Wall Interface

2. Retaining Wall Types and Components

2.1 Gravity and Semi-Gravity Walls

2.2 Cantilever Retaining Walls

• Stem acts as vertical cantilever; base acts as horizontal cantilever
• Base width: B = 0.4H to 0.6H
• Heel length: 2/3 to 3/4 of total base width
• Toe length: 1/3 to 1/4 of total base width
• Economical for heights H = 10 to 25 ft
• Stem thickness at base: minimum 12 in or H/10

2.3 Counterfort and Buttress Walls

2.4 MSE Walls and Reinforced Earth

• Mechanically Stabilized Earth with reinforcing elements (strips, grids, geotextiles)
• Reinforcement length: L = 0.7H to 1.0H (minimum 8 ft)
• Active zone: extends 0.3H from face
• Resistant zone: provides anchorage beyond active zone
• Vertical spacing of reinforcement: 2 to 3 ft
• Maximum height: 40+ ft with proper design

2.5 Sheet Pile Walls

2.6 Basement and Underground Walls

• Consider full hydrostatic pressure if drainage fails
• Use at-rest pressure (K₀) for unyielding conditions
• Provide drainage system: gravel backfill + perforated drain pipe
• Waterproofing required below grade

3. Stability Analysis

3.1 Overturning Stability

• Factor of Safety: FS_overturning = ΣM_resisting / ΣM_overturning
• Minimum FS = 1.5 (2.0 for seismic)
• Resisting moments: weight of wall, weight of soil on heel
• Overturning moments: active earth pressure, water pressure, surcharge
• Moments taken about toe

3.2 Sliding Stability

• FS_sliding = (ΣV × tan δ_base + c_base × B + P_passive) / ΣH
• Minimum FS = 1.5 (2.0 for seismic)
• δ_base = friction angle at base (= φ for soil, reduce for concrete on soil)
• ΣV = total vertical forces (weight)
• ΣH = total horizontal forces (active pressure)
• Passive pressure in front of toe often neglected for conservatism
• Key may be added to increase sliding resistance

3.3 Bearing Capacity

• Check: q_max ≤ q_allowable
• Eccentricity: e = (B/2) - (ΣM_resisting - ΣM_overturning) / ΣV
• For e ≤ B/6: q_max = (ΣV/B) × (1 + 6e/B), q_min = (ΣV/B) × (1 - 6e/B)
• For e > B/6: loss of contact; q_max = 2ΣV / [3(B/2 - e)]
• Require e ≤ B/6 (middle third rule) to avoid tension
• Check against bearing capacity failure: apply FS = 2.5 to 3.0

3.4 Settlement Analysis

• Immediate settlement: elastic compression
• Consolidation settlement: for clayey soils, time-dependent
• Differential settlement: critical for continuous walls
• Allowable total settlement: 1 to 2 inches
• Allowable differential settlement: 0.5 inches per 20 ft

3.5 Global Stability

• Slope stability analysis including wall and retained soil mass
• Use circular arc method (Bishop, Spencer) or wedge method
• Minimum FS = 1.3 to 1.5 for static conditions
• Check deep-seated failure surfaces passing beneath wall

4. Design Loads and Load Combinations

4.1 Permanent Loads

4.2 Transient Loads

4.3 AASHTO Load Combinations for Service Limit State

• Service I: 1.0 DC + 1.0 EH + 1.0 ES + 1.0 WA + 1.0 LS
• Service II (overturning/sliding): 1.0 DC + 1.0 EH + 1.0 ES + 1.0 WA
• Service III (tension check): 1.0 DC + 0.8 EH + 1.0 ES + 1.0 WA

4.4 AASHTO Load Combinations for Strength Limit State

• Strength I: 1.25 DC + 1.5 EH + 1.5 ES + 1.75 LS
• Strength II: 1.25 DC + 1.5 EH + 1.5 ES
• Extreme Event I (seismic): 1.0 DC + 1.0 EH + 1.0 ES + 1.0 EQ
• Apply resistance factors: φ for bearing = 0.55, φ for sliding = 1.0

4.5 Surcharge Pressures

• Uniform surcharge: Δp = q × K (q = surcharge pressure in psf)
• Line load: use Boussinesq elastic theory for distribution
• Strip load: trapezoidal distribution, use 2V:1H spread
• Highway live load surcharge: equivalent 2 ft soil height or q = 240 psf

5. Drainage and Waterproofing

5.1 Drainage Requirements

• Prevent hydrostatic pressure buildup behind wall
• Perforated pipe at base: minimum 6 in diameter, slope 0.5% minimum
• Gravel drainage blanket: minimum 12 in thick, k ≥ 100 ft/day
• Geotextile filter fabric: prevent soil migration into gravel
• Weep holes: 4 in diameter at 5 to 10 ft spacing, 1 to 2 ft above grade

5.2 Drainage Materials

5.3 Waterproofing Systems

• Damp-proofing: asphalt coating, parging for non-critical walls
• Waterproofing: membrane systems for habitable basements
• Sheet membranes: bituminous, rubberized asphalt, PVC
• Liquid membranes: spray or trowel-applied
• Bentonite panels: self-healing clay barrier
• Apply to exterior face of wall before backfilling

5.4 Hydrostatic Pressure

• Full hydrostatic: p = γ_w × h (γ_w = 62.4 pcf)
• Resultant: P = ½ γ_w h²; acts at h/3 from base
• Design for full hydrostatic if drainage system reliability uncertain
• Combined earth and water: add pressures, check buoyancy

6. Seismic Design Considerations

6.1 Mononobe-Okabe Method

• Dynamic active earth pressure coefficient: K_AE
• K_AE = cos²(φ - θ - α) / [cos θ × cos²α × cos(δ + α + θ) × (1 + √[sin(φ + δ)sin(φ - β - θ) / cos(δ + α + θ)cos(β - α)])²]
• θ = tan⁻¹(k_h / (1 - k_v)); seismic inertia angle
• k_h = horizontal seismic coefficient = (1 - 1/FS) × PGA
• k_v = vertical seismic coefficient = 0 (conservative) or ± k_h/2
• PGA = peak ground acceleration from seismic maps

6.2 Seismic Earth Pressure Distribution

• Total active force: P_AE = ½ K_AE γ H²
• Increment: ΔP_AE = P_AE - P_A (static active pressure)
• Static component acts at H/3; dynamic increment acts at 0.6H from base
• Alternative: total P_AE acts at 0.5H to 0.6H

6.3 Seismic Design Parameters

6.4 Seismic Performance Considerations

• Yielding walls tolerate permanent displacement; non-yielding require elastic design
• MSE walls: generally flexible, perform well under seismic loading
• Free-draining backfill critical to prevent liquefaction and excess pore pressure
• Avoid saturated cohesionless backfill in seismic zones

7. Construction Considerations

7.1 Backfill Materials

7.2 Compaction Requirements

• Lift thickness: 8 to 12 inches loose measure
• Compaction within 3 ft of wall: use light equipment to avoid damage
• Minimum distance from wall for heavy compaction: 3 ft
• Behind MSE wall facing: hand-operated equipment within 3 ft
• Moisture content: within 2% of optimum for cohesive soils

7.3 Construction Sequence

• Excavate and prepare foundation; compact subgrade to ≥ 95% density
• Place drainage system and granular bedding layer
• Construct wall in stages; allow concrete to cure before backfilling
• Backfill in layers; compact each lift before next
• Install drainage features during backfilling
• Complete surface drainage and erosion protection

7.4 Quality Control

7.5 Common Construction Issues

• Inadequate compaction behind wall: leads to settlement and increased pressure
• Overcompaction too close to wall: induces high lateral stress
• Poor drainage installation: results in hydrostatic pressure buildup
• Premature backfilling: concrete not at design strength
• Frozen backfill: unacceptable; thaw and recompact

8. Reinforced Concrete Design

8.1 Stem Design

• Design as vertical cantilever for active earth pressure
• Critical section for moment: at base of stem (top of footing)
• M_u = ½ K_a γ H² × (H/3) for triangular pressure distribution
• Minimum reinforcement ratio: ρ_min = 0.0018 for Grade 60
• Temperature and shrinkage steel: #4 bars @ 12 in or #5 bars @ 18 in
• Cover: 2 in for earth face, 1.5 in for unexposed face

8.2 Heel Design

• Design as cantilever slab from stem
• Loads: weight of soil on heel, live load surcharge, self-weight
• Critical section: at back face of stem
• Downward loads cause tension at top surface
• Main reinforcement at top; distribution steel at bottom

8.3 Toe Design

• Design as cantilever slab from stem
• Load: upward soil bearing pressure
• Critical section: at front face of stem
• Upward pressure causes tension at bottom surface
• Main reinforcement at bottom; distribution steel at top

8.4 Key Design

• Projection below base slab to increase sliding resistance
• Width: 0.5 to 1.0 times depth
• Depth: determined by required passive resistance
• Design for shear and moment from lateral earth pressure
• Minimum reinforcement: 4 bars continuous

8.5 ACI 318 Requirements

9. Special Conditions

9.1 Walls with Broken Slopes

• Use Coulomb theory with actual backslope angle β
• For complex slopes: divide into segments and superpose pressures
• Equivalent fluid pressure method: conservative approximation
• Benching may reduce lateral pressure on wall

9.2 Walls with Batter

• Batter reduces overturning moment arm
• Use Coulomb theory with wall angle α from vertical
• Practical batter: 1:12 to 1:6 (horizontal:vertical)
• Increased formwork cost vs. structural benefit

9.3 Tiered or Terraced Walls

• Lower wall setback ≥ 2 × height of lower wall: design independently
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• Upper wall contributes surcharge load on lower wall
• Global stability analysis required

9.4 Walls on Slopes

• Foundation slope affects bearing capacity reduction
• Increased overturning potential on downward slopes
• Passive resistance reduced for sloping ground in front
• Consider slope stability with wall in place

9.5 Expansive and Collapsible Soils

9.6 High Water Table Conditions

• Use submerged unit weight γ' = γ_sat - γ_w for soil below water table
• Add full hydrostatic pressure from water
• Total pressure = σ'K + u (effective stress approach)
• Check uplift on base slab; drainage critical
• Consider tidal fluctuations for waterfront structures

10. Design Summary and Checks

10.1 Design Procedure

• 1. Establish design parameters: H, soil properties (γ, φ, c), loading
• 2. Select wall type and estimate preliminary dimensions
• 3. Calculate earth pressures (active, passive, at-rest as applicable)
• 4. Analyze overturning stability: FS ≥ 1.5
• 5. Analyze sliding stability: FS ≥ 1.5
• 6. Check bearing pressure: e ≤ B/6, q_max ≤ q_allowable
• 7. Design structural components (stem, toe, heel, key)
• 8. Check global stability: FS ≥ 1.3
• 9. Detail drainage system and reinforcement
• 10. Prepare construction drawings and specifications

10.2 Critical Design Checks

10.3 Common Design Errors

• Using active pressure for unyielding walls (should use at-rest)
• Neglecting water pressure or assuming drainage always functions
• Incorrect load factors or resistance factors
• Ignoring seismic loads in seismic zones
• Inadequate reinforcement development length
• Underestimating surcharge loads
• Relying on passive pressure in front of toe without certainty
• Insufficient global stability analysis

10.4 Key Design Values