In geotechnical and structural engineering, understanding soil pressure against retaining structures is fundamental to safe design. Among the three classical lateral earth pressure states (active, passive, and at-rest), the at-rest earth pressure condition represents the scenario where the retaining wall experiences no horizontal movement. This condition produces the highest lateral soil load among the three states, making it critical for many wall types. Engineers rely on the at-rest earth pressure coefficient (K0) to calculate these lateral forces, and the value of K0 depends primarily on soil type, stress history, and compaction level. This article explores the theory, calculation methods, and practical applications of at-rest earth pressure in retaining wall design. For a broader understanding of how lateral pressures compare across all soil states, see our guide on lateral earth pressure coefficients in retaining wall design.
Theory and Origin of At-Rest Earth Pressure
The at-rest condition occurs when a retaining wall is absolutely rigid and the soil behind it is not allowed to yield or move. Unlike active pressure (which develops when the wall tilts away from the soil) or passive pressure (which develops when the wall pushes into the soil), the at-rest condition represents the initial state of stress in the ground before any wall movement takes place.
The Coefficient of Earth Pressure at Rest (K0)
The coefficient of earth pressure at rest (K0) is defined as the ratio of horizontal effective stress to vertical effective stress in a soil mass under no lateral strain conditions:
K0 = σ’h / σ’v
Where σ’h is the horizontal effective stress and σ’v is the vertical effective stress at a given depth. Several theoretical and empirical relationships exist for estimating K0.
Jakys Formula for Normally Consolidated Soils
The most widely used relationship for estimating K0 in normally consolidated soils is Jakys formula:
K0 (NC) = 1 sin φ’
Where φ’ is the effective friction angle of the soil. This empirical relationship, developed by Jaky in 1944, has been verified through numerous laboratory and field studies for a wide range of soil types. For a typical sand with φ’ = 30 degrees, Jakys formula gives K0 = 1 sin 30 = 0.5. For a soft clay with φ’ = 20 degrees, K0 = 1 sin 20 = 0.66.
Overconsolidation Effects on K0
For overconsolidated soils, the at-rest earth pressure coefficient is higher due to stress history. Mayne and Kulhawy (1982) proposed:
K0 (OC) = K0 (NC) × OCRm
Where OCR is the overconsolidation ratio and m ranges from 0.4 to 0.5. A soil with OCR = 4 could have a K0 value nearly double that of its normally consolidated counterpart.
For comparison of how at-rest pressures relate to active and passive states, refer to the detailed article on active earth pressure theory using Rankine and Coulomb methods.
Calculating At-Rest Lateral Earth Pressure
The calculation of at-rest lateral earth pressure follows a systematic procedure that accounts for soil properties, groundwater conditions, and surcharge loads. The total lateral force on a retaining wall under at-rest conditions is determined by integrating the horizontal stress distribution over the wall height.
Vertical Stress Distribution
The vertical effective stress at any depth z below the ground surface is calculated as:
σ’v = γ × z (for dry soil above the water table)
σ’v = γ × z1 + γ’ × (z z1) (accounting for the water table at depth z1)
Where γ is the unit weight of soil above the water table and γ’ is the submerged unit weight below the water table.
Horizontal Stress and Lateral Force
The horizontal effective stress at depth z is:
σ’h = K0 × σ’v
The total lateral force per unit length of wall (P0) is the area of the horizontal stress diagram. For a homogeneous soil with no water table and a constant K0:
P0 = 0.5 × K0 × γ × H²
Where H is the total height of the wall. This force acts at a height of H/3 from the base of the wall for a triangular pressure distribution.
Worked Example: At-Rest Pressure on a 6m Retaining Wall
Consider a 6-meter-high retaining wall retaining dry sand with the following properties: γ = 18 kN/m³, φ’ = 32 degrees. No water table is present.
Step 1: Calculate K0 using Jakys formula
K0 = 1 sin 32 = 1 0.53 = 0.47
Step 2: Calculate vertical stress at the base
σ’v = 18 × 6 = 108 kPa
Step 3: Calculate horizontal stress at the base
σ’h = 0.47 × 108 = 50.76 kPa
Step 4: Calculate total lateral force
P0 = 0.5 × 50.76 × 6 = 152.28 kN/m length of wall
The force acts at 2m above the base of the wall.
Comparison of At-Rest, Active, and Passive Pressures
The table below compares the three earth pressure states for the same soil conditions (φ’ = 32 degrees, γ = 18 kN/m³, H = 6 m):
| Pressure State | Coefficient (K) | Formula | Lateral Force (kN/m) |
|---|---|---|---|
| At-Rest (K0) | 0.47 | 1 sin φ’ | 152.3 |
| Active (Ka) | 0.31 | (1 sin φ’)/(1 + sin φ’) | 100.4 |
| Passive (Kp) | 3.25 | (1 + sin φ’)/(1 sin φ’) | 1053.0 |
This comparison clearly shows that the at-rest condition produces significantly higher lateral forces than the active condition but far lower forces than the passive condition. For more details on passive resistance, see our article on passive earth pressure in retaining wall design.
Factors Influencing At-Rest Earth Pressure
Several factors can significantly affect the magnitude of at-rest earth pressure, and engineers must consider each one during design.
Soil Type and Density
Different soil types exhibit different K0 values due to variations in friction angle and stress-strain behavior:
- Loose sands: φ’ = 28-30 degrees, K0 ≈ 0.50-0.53
- Dense sands: φ’ = 35-40 degrees, K0 ≈ 0.36-0.43
- Soft clays: φ’ = 18-25 degrees, K0 ≈ 0.58-0.69
- Stiff clays: φ’ = 25-32 degrees (plus overconsolidation), K0 ≈ 0.5-1.5
Groundwater and Drainage Conditions
The presence of groundwater adds hydrostatic pressure to the lateral earth pressure. The total lateral pressure on the wall below the water table includes both the effective soil pressure (calculated using submerged unit weight) and the hydrostatic water pressure. Poor drainage behind a retaining wall can double or triple the total lateral load, which is why weep holes, drainage blankets, and granular backfill are essential design features. For practical drainage solutions, read our guide on why retaining wall drainage matters for long-lasting stability.
Surcharge Loads
Additional vertical loads on the soil surface behind the wall increase the lateral earth pressure proportionally. Common surcharge types include:
- Uniform surcharge (q): Adds a constant lateral pressure of K0 × q across the entire wall height
- Strip loads: From adjacent foundations or traffic lanes, analyzed using Boussinesq or elastic theory distribution methods
- Line loads: Parallel to the wall, such as from crane rails or conveyor belts
- Point loads: Localized loads that spread with depth
Compaction Effects
Compacted backfill develops significantly higher horizontal stresses than the at-rest condition calculated from Jakys formula. During compaction, the roller or plate compactor induces cyclic lateral stresses that can result in residual horizontal stresses several times higher than K0 conditions. These compaction-induced stresses are often largest near the top of the wall and decrease with depth. Design specifications should account for this by:
- Specifying light compaction equipment near the wall face
- Using a minimum 0.6 m buffer zone where hand-operated compactors are used
- Considering compaction pressure in the structural design of rigid walls
Practical Applications in Retaining Wall Design
The at-rest earth pressure condition governs the design of several types of retaining structures where even minimal wall movement cannot be tolerated.
Structures Designed for At-Rest Conditions
Basement walls of buildings are a classic example. These walls are typically restrained at the top by floor slabs and at the bottom by the foundation slab, preventing the rotation needed to mobilize active pressure conditions. Similarly, bridge abutments and tunnel walls often experience at-rest conditions due to their structural rigidity and restraint against lateral movement.
Types of Retaining Walls and Their Movement Potential
The amount of wall movement required to transition from at-rest to active conditions depends on the wall type:
- Gravity walls (stone, concrete, or crib): Require 0.5% to 1% of wall height movement to reach active state
- Cantilever walls (reinforced concrete): Require 0.1% to 0.5% of wall height movement
- Anchored walls or tieback walls: Movement depends on anchor stiffness, often very small
- Sheet pile walls: Can tolerate larger movements, often designed for active pressure
For walls that cannot tolerate movement, such as those supporting adjacent structures, the at-rest pressure must be used as the design load. For an in-depth look at another retaining wall configuration, see our article on buttressed retaining walls design and construction methods.
Construction Sequencing and At-Rest Pressures
Construction sequence plays a critical role in the actual earth pressure that develops against a wall. Key considerations include:
- Backfill placement: Compact backfill in lifts of 200-300 mm to prevent excessive lateral stresses
- Curing time: Allow concrete walls to achieve adequate strength before backfilling
- Temporary supports: Use temporary bracing for basement walls until floor slabs are cast
- Drainage installation: Install drainage systems concurrently with backfill placement
- Monitoring: Install inclinometers on critical projects to verify design assumptions
Seismic Considerations
Under earthquake loading, the at-rest condition is no longer valid as cyclic ground motions induce dynamic lateral pressures. The Mononobe-Okabe method extends Coulomb theory to account for seismic effects, adding both a dynamic pressure increment and a pseudo-static inertia force. In seismic design, the total lateral pressure is the sum of the static at-rest component and the dynamic increment, which can significantly exceed the static at-rest pressure alone.
Numerical Modeling for Complex Conditions
For complex soil profiles, irregular geometry, or interaction with adjacent structures, finite element analysis provides the most accurate assessment of at-rest earth pressures. Modern geotechnical software can model staged construction, soil-structure interaction, nonlinear soil behavior, and consolidation effects. While empirical formulas remain valuable for preliminary design, numerical modeling is recommended for high-risk projects where simplified methods may not capture critical behavior.
Understanding at-rest earth pressure is essential for safe retaining wall design, particularly for rigid structures where movement cannot occur. By applying correct K0 values, accounting for soil type and groundwater conditions, and following appropriate construction practices, engineers can ensure retaining walls perform safely throughout their design life. The at-rest condition often governs the most critical load case, and getting it right saves both time and cost in the long run.