When designing earth retention structures, engineers must consider the lateral pressure exerted by soil under various movement conditions. Among the three classical earth pressure states (active, passive, and at-rest), the at-rest earth pressure condition represents the case where a retaining wall experiences zero lateral movement. This condition governs the design of rigid structures such as basement walls, bridge abutments, and braced excavations, where even slight displacement could cause serviceability or stability problems. Understanding at-rest earth pressure is fundamental to retaining wall engineering types earth pressure analysis sheet pile walls and drainage systems for earth retention, as it often produces the largest lateral forces among the three states.
Defining the At-Rest Earth Pressure Condition
The at-rest state occurs when a soil mass is subjected to no lateral strain. In practical terms, this means the retaining wall does not move at all, neither away from nor into the soil mass. This zero-displacement condition creates a specific stress state that differs significantly from the active or passive states described by Rankine’s theory.
In geotechnical engineering, the at-rest condition is represented by the coefficient of lateral earth pressure at rest, denoted as K0. This coefficient is defined as the ratio of horizontal effective stress to vertical effective stress:
K0 = σ’h / σ’v
Where σ’h is the horizontal effective stress and σ’v is the vertical effective stress at a given point in the soil mass. Unlike active and passive coefficients, which are derived from Mohr-Coulomb failure theory, K0 represents a stress state before any shear failure occurs in the soil. For a deeper understanding of how this relates to overall soil mechanics and foundation engineering classification shear strength consolidation and earth pressure principles, it is important to study these interconnected topics together.
The at-rest condition applies in several common situations:
- Basement walls cast directly against soil, where the wall is rigid and no movement is permitted
- Bridge abutments that are restrained at the top by the bridge deck
- Braced excavations where struts or tiebacks prevent lateral wall movement
- Sheet pile walls supported by multiple levels of anchors or struts
- Retaining walls founded on competent rock where sliding is not possible
The Coefficient K0 and Jaky’s Formula
The most widely used method for estimating the coefficient of earth pressure at rest for normally consolidated soils is Jaky’s empirical formula. János Jaky, a Hungarian engineer, derived this relationship in the 1940s based on theoretical considerations of the stress state in an unloaded soil prism:
K0 (NC) = 1 − sin φ’
Where φ’ is the effective internal friction angle of the soil. This simple yet remarkably accurate formula has been validated by numerous laboratory and field studies over the decades. For typical sandy soils with a friction angle of 30°, Jaky’s formula gives K0 = 0.50. For denser sands with φ’ = 40°, K0 drops to approximately 0.36. Understanding the active earth pressure retaining mechanisms provides useful context since the at-rest condition represents the upper bound of lateral forces before wall movement initiates active conditions.
For overconsolidated soils, the at-rest coefficient increases significantly due to the stress history of the deposit. The following relationship accounts for overconsolidation effects:
K0 (OC) = K0 (NC) × (OCR)m
Where OCR is the overconsolidation ratio and m is an exponent typically taken as sin φ’ (Mayne and Kulhawy, 1982). For heavily overconsolidated clays, K0 can approach or even exceed 1.0, meaning the horizontal stress equals or exceeds the vertical stress.
| Soil Type | Friction Angle φ’ (°) | K0 (NC) | Typical K0 (OC) Range |
|---|---|---|---|
| Loose sand | 28–30 | 0.47–0.53 | 0.55–0.75 |
| Medium dense sand | 32–36 | 0.41–0.47 | 0.50–0.70 |
| Dense sand | 38–42 | 0.33–0.38 | 0.45–0.65 |
| Soft clay (normally consolidated) | 22–26 | 0.56–0.63 | 0.60–0.80 |
| Stiff clay (overconsolidated) | 25–30 | 0.50–0.58 | 0.80–1.20 |
| Dense gravel | 40–45 | 0.29–0.36 | 0.40–0.60 |
Calculating At-Rest Lateral Earth Pressure Distribution
Once K0 is determined, the lateral earth pressure at any depth can be calculated. For a homogeneous soil deposit with a horizontal ground surface and no surcharge loading, the horizontal stress at depth z is given by:
σ’h = K0 × γ × z
Where γ is the unit weight of the soil and z is the depth below the ground surface. The lateral pressure distribution is triangular, increasing linearly with depth. When groundwater is present, the calculation must separate effective stress and pore water pressure components:
- Above the water table: use the moist unit weight (γm) for vertical stress calculation
- Below the water table: use the buoyant unit weight (γ’ = γsat − γw) for effective stress, then add hydrostatic water pressure separately
- The total lateral pressure = effective lateral earth pressure + pore water pressure
The total resultant horizontal force acting on a wall of height H is obtained by integrating the pressure distribution over the wall height. For a simple triangular distribution with no water table or surcharge, the resultant force acts at H/3 from the base of the wall and is calculated as:
P0 = 0.5 × K0 × γ × H2
When surcharge loads, layered soil profiles, or sloping backfills are present, the calculation requires superposition of individual pressure contributions. Understanding the lateral pressure of fresh concrete on formwork sides follows similar principles of pressure distribution, though with different material properties and time-dependent behavior.
Comparing At-Rest with Active and Passive States
The three classical earth pressure states represent different wall movement scenarios. The at-rest condition produces the highest lateral pressure among the three when the wall is prevented from moving at all. Understanding these distinctions is essential for selecting the appropriate design condition.
Active state: Occurs when the wall moves away from the soil (e.g., a cantilever retaining wall tilting forward). The soil mobilizes its shear strength, reducing lateral pressure to a minimum value. The active coefficient is Ka = (1 − sin φ’) / (1 + sin φ’) for Rankine theory. Typical wall movements required to reach active conditions are 0.001H to 0.004H for dense sand and 0.01H to 0.04H for clay.
Passive state: Occurs when the wall is pushed into the soil (e.g., a massive gravity wall resisting overturning). The soil provides increasing resistance up to a maximum value defined by Kp = (1 + sin φ’) / (1 − sin φ’). Passive conditions require much larger movements, typically 0.01H to 0.05H for dense sand and 0.02H to 0.10H for clay.
At-rest state: The wall does not move at all. K0 typically falls between Ka and Kp, being higher than Ka but lower than Kp. For a soil with φ’ = 30°, the three coefficients are: Ka = 0.33, K0 = 0.50, and Kp = 3.00. The concept of how lateral forces distribute through structural elements relates to the anatomy of a toilet how gravity flow and pressure assisted toilets work, though the applications are entirely different.
A comparison table of the three states is shown below:
| Parameter | Active (Ka) | At-Rest (K0) | Passive (Kp) |
|---|---|---|---|
| Wall movement | Away from soil | Zero movement | Toward soil |
| Lateral pressure magnitude | Minimum | Intermediate | Maximum |
| Movement required | Small | None | Large |
| Typical application | Cantilever walls | Basement walls | Anchor blocks |
| K value (φ=30°) | 0.33 | 0.50 | 3.00 |
| Shear failure | Yes (extension) | No | Yes (compression) |
Practical Design Considerations and Special Cases
Several factors influence the magnitude of at-rest earth pressure in real-world design scenarios. Engineers must account for these variables to produce safe and economical designs.
Compacted Backfill Effects
When backfill is compacted in layers behind a wall, the compaction process generates residual horizontal stresses that exceed the theoretical at-rest pressure. Compaction-induced stresses can be estimated using the vertical stress from compaction equipment (typically equivalent to a uniform surcharge) combined with the at-rest coefficient. For light compaction equipment, an additional surcharge of 10 to 15 kPa is commonly assumed. Heavy compaction rollers may induce equivalent surcharges of 20 to 30 kPa.
Drained versus Undrained Conditions
In fine-grained soils such as clays and silts, the rate of wall construction relative to soil drainage determines whether total stress or effective stress analysis applies. For short-term conditions in low-permeability soils, undrained analysis using total stress parameters (cu) is appropriate. For long-term conditions, drained analysis using effective stress parameters (c’, φ’) governs. In undrained conditions, K0 for normally consolidated clay can be approximated as 0.6 to 0.8, while the undrained shear strength approach uses σh = σv − 2cu for active-type failures.
Seismic Considerations
During earthquake loading, the at-rest condition is no longer valid because the dynamic ground motion induces cyclic wall movements that mobilize both active and passive resistance. The pseudostatic Mononobe-Okabe method is generally used for seismic earth pressure calculations, where the total lateral force includes both static and dynamic components. The dynamic increment is calculated using horizontal and vertical seismic coefficients (kh and kv) derived from the design ground acceleration. The concept of pressure bulb or stress isobar concept helps visualize how stresses spread through soil under various loading conditions.
Summing Up At-Rest Earth Pressure Design
At-rest earth pressure represents the most conservative design condition for rigid earth retention structures. The key design steps include determining the appropriate K0 value based on soil type and stress history, calculating the lateral pressure distribution considering groundwater and surcharge effects, and verifying that the structural system can resist the resulting lateral forces without excessive movement. Modern design codes such as Eurocode 7 and AASHTO provide specific guidance on when at-rest conditions should be assumed versus active conditions.
The choice between at-rest and active conditions has significant economic implications. Designing a basement wall for at-rest pressure may require thicker wall sections and more reinforcement compared to an active pressure design. However, the cost savings from a thinner wall must be weighed against the risk of wall movement, which could cause cracking in the superstructure, damage to adjacent buildings, or water leakage through the wall. In urban environments where neighboring structures and utilities are present, the at-rest assumption is typically the safest engineering choice.
Understanding the link between earth pressure and what is pressure head in fluid mechanics provides useful parallels, as both concepts involve distributed forces on retaining boundaries. The triangular distribution in earth retention mirrors hydrostatic pressure in fluids, except that K0 modifies lateral stress magnitude based on soil properties rather than assuming equal pressure in all directions.
At-rest earth pressure governs the design of rigid retaining structures where displacement cannot be tolerated. Jaky’s formula provides a reliable K0 estimate for normally consolidated soils, while overconsolidation increases K0 significantly. Proper consideration of groundwater, compaction, and seismic effects enables safe and economical earth retention design.