CELE Geotechnical Engineering — Lateral Earth Pressure and Retaining StructuresSummary

A soil mass can exist in three distinct states with respect to lateral pressure: (1) At-rest (K₀), when the wall is immovable and stress has been stable; (2) Active (Ka), when the wall moves away from the soil or the soil is allowed to expand laterally, reducing pressure to its minimum value; (3) Passive (Kp), when the wall moves into the soil, compressing it and increasing pressure to its maximum value. The transition between states is continuous, but design typically assumes either purely active or purely passive conditions for safety. For a vertical wall with horizontal cohesionless backfill, the Rankine coefficients are Ka = tan²(45° − φ/2) and Kp = tan²(45° + φ/2), where φ is the friction angle. The at-rest coefficient K₀ = 1 − sin φ applies to normally consolidated soil in its initial state. In the Philippines, tropical and residual soils may have non-standard stress histories, so field testing is often necessary to confirm earth-pressure behavior.

The earth-pressure coefficient scales the vertical effective stress to give the lateral (horizontal) effective stress. For Rankine theory (smooth wall, horizontal backfill), Ka is always less than K₀, which is less than Kp. The relationship Kp = 1/Ka holds for cohesionless soil. For example, with φ = 30°: Ka = tan²(45° − 15°) = tan²(30°) ≈ 0.333, and Kp = tan²(45° + 15°) = tan²(60°) ≈ 3.0. The at-rest coefficient K₀ = 1 − sin(30°) = 0.5 falls between them. These coefficients are dimensionless and depend only on the friction angle and wall geometry. Coulomb theory extends this to inclined walls and sloping backfills, introducing a wall-friction angle δ that increases Ka slightly (reducing active thrust) and increasing Kp substantially. In NSCP 2015, the values must be verified against actual soil properties obtained from site investigation.

For a dry cohesionless backfill of height H against a smooth vertical wall, the lateral effective stress σ increases linearly with depth: σ(z) = Ka·γ·z, where γ is the unit weight. The pressure diagram is triangular, with zero at the top and maximum Ka·γ·H at the base. The resultant thrust Pa = (1/2)·Ka·γ·H² acts at the centroid of the pressure triangle, i.e., at height H/3 from the base. This is a key point—the thrust acts below the mid-height, concentrating force near the toe. When cohesion c is present, the active pressure is reduced: σₐ(z) = Ka·γ·z − 2c·√Ka, and a tension crack of depth zc = 2c/(γ√Ka) may form near the surface where pressure would otherwise be negative. A uniform surcharge q on the backfill surface adds a rectangular pressure block of height Ka·q, which contributes thrust q·Ka·H acting at H/2. Groundwater creates additional hydrostatic pressure (triangular distribution) and reduces effective stress in saturated soil, increasing the design thrust significantly. All these components must be superposed to find the total lateral thrust.

In cohesive (undrained or drained) backfill, the active pressure envelope is not purely triangular. Near the surface, cohesion can resist tension, so the pressure may reach zero (or become negative, signifying tension) at a certain depth zc below the surface. The critical depth for a tension crack is zc = 2c/(γ·√Ka). Above this depth, the wall and soil naturally separate; water can infiltrate this crack, adding a hydrostatic component that is often neglected conservatively (by assuming the wall 'drains freely'). Some design codes require the tension crack to be included in the thrust calculation by superposing a hydrostatic force. For design purposes, the safest approach is to assume the crack extends full height and is filled with water, adding (1/2)·γw·H² to the thrust. This is particularly important for Philippine clays with high cohesion and for structures in monsoon regions with prolonged saturation.

Lateral earth pressure must be analyzed in terms of **effective stress** (stress between soil particles) or **total stress** (all stress), depending on the soil drainage condition and design approach. In the short term (immediately after loading, undrained condition), use undrained shear strength su and assume zero dilatancy; in the long term (drained condition), use drained friction angle φ' and effective stress paths. For saturated soil below the water table, the effective unit weight γ' = γsat − γw ≈ γsat − 9.81 kN/m³ must be used in thrust calculations. The hydrostatic water pressure σw = γw·h (h is depth below water surface) acts independently and is added algebraically. In Philippines, where seasonal fluctuations and tropical rainfall are common, the water table often fluctuates; design should account for the highest probable water table. Using total stress (φu, su) simplifies analysis for saturated clay but is less accurate over long time scales; effective stress analysis is preferred for permanent structures.

Rankine theory assumes a smooth wall (δ = 0°). In reality, friction between wall and soil (friction angle δ) reduces the active pressure and increases the passive pressure. Coulomb's wedge theory treats the soil as a rigid wedge in limiting equilibrium and solves for the critical slip surface and corresponding thrust. The thrust depends on the inclination of the wall, the slope angle of the backfill, and the wall-friction angle δ. For a vertical wall with horizontal backfill and wall friction δ, the Coulomb active coefficient becomes slightly less than Rankine (reducing thrust), and the passive coefficient becomes much larger (increasing passive resistance at the toe). In practice, δ is taken as a fraction of φ (e.g., δ = (2/3)φ for sand on concrete) or estimated from friction tests. In the Philippines, where many walls are built with relatively rough surfaces (plastered concrete or brick), δ ≈ 0.75φ to φ is reasonable. However, Coulomb coefficients are cumbersome to calculate by hand; tables or specialized software are often used in practice.

Common retaining-wall types include: (1) **Gravity walls**, relying on their massive weight (stone, concrete) to resist thrust; typically trapezoidal in cross-section, simple but costly. (2) **Cantilever walls**, with a thin vertical stem and a base slab; the stem acts as a cantilevered beam, and bending stresses must be checked. (3) **Counterfort walls**, adding vertical ribs between the stem and base to increase bending capacity and reduce thickness. (4) **Sheet piles** (steel or concrete), used for temporary or narrow excavations; very thin and flexible. (5) **Soldier piles and lagging**, combining stiff vertical piles with flexible horizontal boards. Geometry profoundly affects thrust distribution: a sloping or battered (inward-tilted) wall reduces the height of backfill acting on the wall and may slightly reduce active pressure. The orientation of the base (vertical, inclined, or stepped) affects the distribution of resisting moment and the bearing-pressure profile. In Philippine practice, cantilever walls in reinforced concrete are the most common for small to medium heights (up to 6–8 m), often with a sloped backfill for drainage.

A retaining wall must satisfy three external stability conditions: (1) **Overturning (FS_OT)**: The sum of moments resisting rotation about the toe must exceed the overturning moment from active thrust by a factor of 1.5–2.0 (NSCP 2015 recommends ≥ 1.5 for static loads). FS_OT = Σ(M_resisting) / Σ(M_overturning). (2) **Sliding (FS_slide)**: The sum of vertical loads (weight of wall and soil above the base) times the coefficient of friction at the base, plus any passive resistance at the toe, must exceed the horizontal component of active thrust. FS_slide = [μ·Σ(W) + Pp] / Pa,H ≥ 1.5. Passive resistance Pp is often neglected for conservatism unless the toe is deeply embedded or the backfill is very well-compacted. (3) **Bearing capacity**: The contact pressure between wall base and foundation soil must not exceed the allowable bearing capacity. To ensure no tension at the toe, the resultant of all vertical and horizontal forces should lie within the **middle third** of the base (kern), i.e., the eccentricity e ≤ B/6 (B is base width). NSCP 2015 Section 212 governs foundations and requires verification of these conditions under factored or service loads depending on the design method used.

A surcharge is any additional vertical load on the backfill surface (e.g., a building foundation, stockpile, or traffic load). For a uniform surcharge q acting over the full width of the backfill, the lateral pressure increases by a constant Ka·q at all depths, as if the surcharge were a layer of soil of height q/γ. The corresponding thrust from the surcharge is Pq = Ka·q·H, acting at height H/2 (mid-height of the wall), adding to the thrust from self-weight of the soil. A **point load** or **line load** (e.g., from a nearby building) creates a non-uniform pressure increase that depends on the distance from the wall and the depth; the influence zone typically extends to a depth of 6B (B is the distance from the load to the wall). Calculating the effect of point loads requires the use of Boussinesq's equations or influence charts, which is laborious; in practice, point loads are either treated conservatively as equivalent surcharges or the zone of influence is excluded from the retaining-wall design. In the Philippines, many walls are built in urban areas where adjacent structures or parking impose surcharges; careful investigation of existing and future loads is essential.

When groundwater is present (water table at depth hw below the surface), the soil below the water table becomes saturated, and its effective unit weight is γ' = γsat − γw ≈ γsat − 9.81 kN/m³ (typically 9–10 kN/m³ less than the dry unit weight). The lateral effective stress at depth z below the water table is σ'h = Ka·γ'·z + Ka·γ·(hw), where the first term uses the reduced effective weight below the table and the second term accounts for the stress from saturated soil above. Additionally, water pressure σw = γw·(z − hw) acts perpendicular to the wall and is independent of soil friction. The total lateral stress is σh,total = σ'h + σw. Seepage through or around the wall can create additional pressures (seepage forces) that increase the effective unit weight and the lateral stress. To prevent seepage-induced failure, retaining walls must include **drainage provisions**: weep holes (small openings to allow drainage), a **drainage layer** (gravel or geotextile) behind the wall to intercept water, and a **sump** or **drainage line** at the toe to collect and remove seeping water. In the Philippines, monsoonal rainfall and high groundwater in coastal and lowland areas make drainage a critical design consideration; walls built without proper drainage frequently fail or develop substantial movements.

Beyond external stability (overturning, sliding, bearing), the **internal strength** of the wall must be verified. For concrete cantilever walls, the vertical stem is subject to a distributed lateral load (active pressure) and must be checked for bending moment and shear using reinforced concrete design methods (ACI 318). The base slab is subject to the concentrated reaction from the stem (which transmits the lateral thrust) and the distributed bearing pressure from the foundation, creating both hogging and sagging moments; these must be designed as a two-way slab or beam depending on geometry. For gravity walls, internal stresses are generally low and cracking is avoided, but material quality (stone, concrete) must be adequate. **Reinforced earth walls** (geosynthetic reinforcement) and **segmental retaining walls** (modular blocks with internal reinforcement) are increasingly used in Philippines; their design follows specific proprietary guidelines but still requires external stability checks. Material selection is crucial: in the Philippines, high-sulfate groundwater in some regions requires sulfate-resistant concrete (NSCP 2015 Table 3.1.1.2); corrosion protection of steel reinforcement is essential in coastal areas due to chloride exposure.

The choice of pressure state (active, at-rest, passive) directly determines the coefficient K used in the thrust formula P = (1/2)K·γ·H². Active (smallest K) gives minimum thrust for a wall that moves outward or a retaining wall in service. Passive (largest K) is used at the toe if the wall is pushed inward (rare) or in anchored excavations where the wall restrains the soil.

Higher friction angle → smaller Ka (less active pressure) and larger Kp (more passive resistance). The relationship Ka = tan²(45° − φ/2) shows that even a 10° increase in φ can reduce Ka by ~50%. This is why soil improvement (e.g., compaction, stabilization) that increases φ can significantly reduce wall thickness and cost.

Overturning moment M_OT = P·d, where d is the distance from the toe (pivot point). Since P ∝ H² and d ∝ H/3, the moment grows as H^2 × H = H^3. This cubic relationship means tall walls are exponentially harder to hold against overturning; at some height, overturning becomes the governing criterion.

Heavier walls (from gravity mass or backfill soil on a sloped base slab) generate larger resisting moments and higher normal stress at the base, improving both FS_OT and FS_slide. However, excessive weight increases bearing pressure and cost; optimal design balances these competing effects.

Cohesion reduces active pressure near the surface, potentially eliminating the pressure over a depth zc (tension-crack depth). However, the crack usually fills with water (if drainage fails), so the benefit is lost and additional hydrostatic pressure is added. This paradoxical effect is why cohesion is often neglected in active-pressure calculations for design.

A surcharge at the wall (close to the toe) creates a point load with concentrated pressure; a distant surcharge creates a more distributed influence. For uniform surcharges over the entire backfill, the added pressure is constant (Ka·q) and contributes thrust at mid-height, simplifying the calculation. Off-center or partial surcharges require influence charts or numerical integration.

A higher water table (shallower saturation) increases the height of saturated soil, reducing the effective unit weight and increasing absolute water pressure; the net effect is usually an increase in total lateral stress. Proper drainage (weep holes, drainage layer) prevents water from accumulating and effectively lowers the water table, reducing pressure. This demonstrates the critical importance of drainage in wall design.

For a given thrust and wall weight, a wider base reduces bearing pressure (distributed over larger area) and lowers the eccentricity (e = M/N), keeping the resultant within the middle third and avoiding tension. A narrower base increases pressure and eccentricity, risking uplift at the toe and uneven settlement. Base width is a key design variable for economic and safety optimization.

Rough walls (higher δ) and sloped backfills reduce Ka and increase Kp, but the calculations are complex. Rankine theory (δ = 0°, horizontal backfill) is a simplified but safe lower bound for active pressure and upper bound for passive; using Rankine avoids the need for Coulomb calculations unless high precision is required.

The lateral pressure on the stem of a cantilever wall creates a distributed load that increases with depth (triangular). The maximum bending moment occurs at the base and is proportional to P·(H/3). This moment must be resisted by flexural reinforcement (rebar) designed per ACI 318, and shear must be checked to ensure the concrete (or composite section) is adequate.

In highway cut-and-fill projects across the Philippines, retaining walls are built to create stable embankments. A 6 m cut on a hillside near Tagaytay is made stable with a reinforced concrete cantilever wall. Site soil is volcanic ash (φ = 32°, γ = 18 kN/m³, no cohesion). A nearby building foundation imposes a 20 kPa surcharge. Engineers calculate Ka = 0.307, the active thrust from self-weight is (1/2)(0.307)(18)(6²) = 99 kN/m, and from surcharge is (0.307)(20)(6) = 37 kN/m, for a total of 136 kN/m. The wall design must accommodate this thrust, satisfy overturning and sliding checks, and include drainage to prevent water buildup from seasonal rains. The wall's base is typically 40–50% of the height (2.4–3.0 m) to achieve adequate factors of safety. This is a common design task in Philippine infrastructure.

In Metro Manila, many commercial and residential buildings require basement excavations in reclaimed land, which is often loose sand and silt with high water tables (1–2 m below surface). A 5 m deep basement wall must retain a saturated fill (φ = 28°, γsat = 19 kN/m³, γ' = 10 kN/m³, γw = 9.81 kN/m³). Above the water table (1 m), dry fill applies; below, saturated conditions with effective stress and water pressure both act. The active thrust is split: (1) from 0–1 m, Ka·γ·h²/2 = 0.361 × 18 × 1²/2 = 3.25 kN/m; (2) from 1–5 m, Ka·γ'·(4)·2 + Ka·γ'·4²/2 = 0.361 × 10 × 4 × 2 + 0.361 × 10 × 8 = 28.9 + 28.9 = 57.8 kN/m from effective stress; (3) plus hydrostatic from 1–5 m: γw × 4²/2 = 9.81 × 8 = 78.5 kN/m. Total ≈ 139.6 kN/m. The wall must be deeply anchored or braced because both gravity and overturning stability are insufficient for a free-standing wall at 5 m. Drainage and water management are critical to prevent base heave and internal erosion.

Port and waterfront development often uses sheet pile walls to retain soft bay mud and allow land reclamation or dredging. A 4 m high sheet pile bulkhead is installed along the Pasig River (water on one side, backfill on the other). The bay mud has φ = 18°, c = 25 kPa, γsat = 17 kN/m³. Active pressure is reduced by cohesion: Ka = 0.528 at φ = 18°. The tension-crack depth is zc = 2(25) / [17 × √0.528] = 3.0 m, nearly to the surface, meaning cohesion is ineffective over most of the height and the wall must resist the full (reduced) pressure. Including water on both sides (river side at 0 m, backfill side at some depth) adds complexity. The thin sheet pile section must be adequate in lateral bending and corrosion-resistant (stainless steel or coated) due to the marine environment. Passive resistance from embedded length is critical for stability; without sufficient embedment, the wall fails by rotating.

In mountainous provinces (Benguet, Cordillera, etc.), highway cuts through steep terrain often require slope stabilization. One method is a **soil-nailed wall**, where a thin shotcrete (reinforced concrete) facing is placed on a cut slope, and grouted steel nails are drilled into the slope to anchor and reinforce it. The nails reduce lateral pressure on the facing by transferring load directly into the stable soil mass. The design uses the active earth-pressure coefficient to calculate the initial load on the facing (before nails are effective); as the nails are installed, they redistribute stress and reduce the pressure. The facing must resist the pressure until the nails are activated. For a 10 m cut in weathered rock (φ = 35°, γ = 19 kN/m³, small cohesion), Ka = 0.271. The initial thrust is (1/2)(0.271)(19)(10²) = 257 kN/m. The facing (20 cm thick shotcrete) spans between nails (1.5 m spacing) and is designed as a continuous beam with point loads from the nails, similar to an anchored wall problem.

Bridge abutments at river crossings or highway interchanges function as retaining walls, holding back embankment soil on one or both sides. For a 4 m high abutment with approach embankment (φ = 30°, γ = 18 kN/m³, no cohesion), Ka = 0.333. The active thrust is (1/2)(0.333)(18)(16) = 48 kN/m (per meter of width). The abutment stem is typically a reinforced concrete wall, and the base slab sits on a spread foundation. The abutment must resist not only the earth pressure but also live loads from bridge traffic (surcharges on the approach), thermal expansion/contraction, and earthquake acceleration (important for Philippine Seismic Design Code). The bearing capacity of the foundation soil below the abutment must accommodate the combined load of the abutment weight and the reaction from the bridge superstructure. The design also accounts for water in the embankment (drainage or perched water table) and potential scour at river crossings.

Modern quay walls at Philippine ports (Port of Manila, Subic Bay, Cebu) are massive structures built on deep foundations (pilings or caissons) to support cargo-handling equipment and hold back dredged seabed. A quay wall in reclaimed land with soft clay (φ = 15°, c = 30 kPa, γsat = 18 kN/m³) must resist active pressure from a 12 m depth, plus surcharge from stacked containers (equivalent to 50 kPa). Ka = tan²(45° − 7.5°) = 0.589. Cohesion slightly reduces pressure, but the tension-crack depth zc = 2(30) / [18 × √0.589] ≈ 4.3 m, allowing cohesion to be effective only in the upper zone. The wall is designed with robust reinforcement and anchorage, often using sheet piles or diaphragm-wall construction to reach depths of 20+ m. Seepage and groundwater management (sumps, dewatering systems) are essential for long-term stability.

In residential subdivisions on sloped terrain (common in Quezon City, Caloocan, Cavite), multiple retaining walls are built in series to create level building lots. Each wall retains the slope above it. For a 3 m wall on residual volcanic soil (φ = 30°, γ = 18 kN/m³, small cohesion), Ka = 0.333, and active thrust is (1/2)(0.333)(18)(9) = 27 kN/m. To minimize cost, simple gravity walls (masonry or small concrete blocks) or short cantilever walls are used. If the wall is not properly drained (weep holes, drainage layer), water accumulates during monsoon season, increasing pressure and risking failure—a common cause of wall collapse in Philippine residential areas. Site investigations must confirm soil properties (especially liquidity index and cohesion of clays); many failures stem from underestimating the strength or overestimating the drainage capacity of the backfill.

In dense urban areas like Makati or Manila, basement construction requires temporary support of adjacent properties during deep excavation. A **soldier pile and lagging system** (steel H-piles with wooden boards) is used to retain soil and prevent damage to nearby structures. The active pressure from the retained soil is transmitted to the piles via the lagging. The piles are braced at multiple levels to control deflection and prevent wall collapse. For a 7 m excavation in loose sand (φ = 32°, γ = 17 kN/m³), Ka = 0.307, and the lateral thrust is (1/2)(0.307)(17)(49) ≈ 128 kN/m. The soldier piles are spaced 2–3 m apart and designed as cantilevers or struts (with internal bracing) to carry the load. The lagging must be strong enough to bridge the span between piles without excessive deflection. After the basement structure is built, the temporary bracing and lagging are removed, and the permanent basement wall becomes the support. This is a temporary design, but stability and safety are critical to avoid injury or property damage.