CELE Geotechnical Engineering — Bearing Capacity of SoilsSummary

The ultimate bearing capacity is expressed as qu = cNc + γDfNq + 0.5γBNγ for a strip footing. This three-term decomposition represents: (1) **cohesion term** cNc — shear strength from soil cohesion; (2) **surcharge term** γDfNq — contribution from overburden pressure at footing depth; (3) **width term** 0.5γBNγ — contribution proportional to footing width and self-weight of soil. The bearing-capacity factors Nc, Nq, Nγ depend only on the soil friction angle φ and are tabulated or calculated using closed-form equations (e.g., Nc = (Nq − 1) cot φ). For strip footings, the coefficients 1.0, 1.0, 0.5 apply; these change for square and circular shapes.

These dimensionless factors quantify how soil friction angle φ affects load-carrying ability. Typical values: φ = 0° (clay): Nc = 5.7, Nq = 1.0, Nγ = 0; φ = 20°: Nc ≈ 17.7, Nq ≈ 7.4, Nγ ≈ 5.1; φ = 30°: Nc ≈ 37.2, Nq ≈ 22.5, Nγ ≈ 19.1; φ = 40°: Nc ≈ 95.7, Nq ≈ 81.3, Nγ ≈ 100+. The factors increase sharply with φ because higher friction angle means greater interlocking and resistance to shear. For φ = 0 (undrained clay), Nγ = 0 because there is no dilation; all strength comes from cohesion and surcharge. Engineers reference standardized tables (textbooks, design codes) or use Terzaghi/Meyerhof correlations.

Terzaghi's equation applies strictly to strip (infinite length) footings. Real footings are often square or rectangular. Shape factors (sc, sq, sγ) modify each term: for a square footing, typically sc = 1.3, sq = 1.0, sγ = 0.4; for circular, sc = 1.3, sq = 1.0, sγ = 0.3. The corrected equation becomes qu = sc·cNc + sq·γDf·Nq + 0.5·sγ·γ·B·Nγ. Intuitively, compact shapes (square, circular) mobilize more shear strength than elongated strips, particularly in the width term (smaller sγ reflects this). For rectangular footings with length L and width B, intermediate corrections interpolate between L/B = 1 (square) and L/B → ∞ (strip).

The presence of groundwater affects the effective stress and unit weight of soil. When the water table is at or above the footing base, the soil below is saturated (submerged). In the bearing-capacity equation, the unit weight γ in the width term should be replaced with the **submerged unit weight** γ' = γsat − γw, where γsat is the total saturated weight (≈ 19–20 kN/m³ for most soils) and γw = 9.81 kN/m³. If the water table is within depth B below the footing base, a weighted interpolation is used. If it is deeper than B, no correction is needed. The surcharge term γDf uses the in-situ unit weight above the footing (typically γ = 17–18 kN/m³ in unsaturated conditions). This distinction is critical: only the soil self-weight contributing to the passive resistance zone (depth B) is modified by saturation.

The **allowable bearing capacity** (qa) is the design pressure applied to a footing, calculated by dividing the ultimate bearing capacity by a **factor of safety** (FS): qa = qu / FS. Common FS values range from 2.5 to 3.5 depending on soil variability, load predictability, and project importance (NSCP 2015 and PRC guidelines recommend FS ≈ 3 for typical building foundations). **Net allowable bearing capacity** (qa,net) subtracts the overburden: qa,net = (qu − γDf) / FS. This distinction matters when reporting pressures relative to the footing base: qa,net is the incremental pressure beyond the overburden at that depth, used to calculate the allowable column load Q = qa,net × A. For a rectangular footing with net allowable qa,net (kPa) and area A (m²), the allowable load is Q = qa,net × A (kN).

In dense, strong soils (e.g., dense sand, stiff clay, φ > 30°), a footing undergoes **general shear** failure: a well-defined failure surface develops from the footing edge into the soil, with clear bulging and heave around the footing. The Terzaghi equation (with Nc, Nq, Nγ values) applies directly. In loose, compressible soils (φ < 25°), a footing experiences **local shear** (or punching shear): the failure surface is less defined, and the footing settles significantly before catastrophic failure. For local shear, the effective cohesion and friction angle are reduced: c' = (2/3)c and tan φ' = (2/3) tan φ. These reduced values are substituted into the Terzaghi equation to yield a lower (conservative) qu. Recognizing soil type and mode is essential for accurate design: loose sand near the water table (φeff ≈ 20–25°) typically warrants local shear reduction.

While the bearing-capacity equation determines the pressure at which the soil shears, **settlement** — the vertical displacement of the footing — is often the controlling design criterion, particularly in sand and loose soils. A footing may shear at qu = 500 kPa (FS = 2 → qa = 250 kPa), but a settlement analysis (using elastic or Boussinesq methods) might show that qa = 150 kPa produces tolerable settlement (<25 mm for typical buildings per NSCP 2015). Many Philippine foundation design projects are settlement-governed rather than bearing-governed. The designer must calculate both qa (from shear) and then verify settlement under qa. If settlement exceeds code limits (often 20–50 mm depending on structure), qa is reduced or the footing is enlarged.

The depth Df of a footing (distance from ground surface to footing base) affects bearing capacity through two mechanisms. First, it increases the **surcharge term** q = γDf in the Nq component: deeper footings benefit from more overburden confinement, raising qu. Second, a depth factor dq (and dc, dγ) further refines this, particularly for very shallow or very deep footings. For typical shallow footings (Df/B ≤ 1–2), depth effects are modest; for Df/B > 4, they become significant. In bearing-capacity design, increasing Df is a strategy to raise allowable qa without enlarging the footing area. However, deeper footings are more expensive to construct and may encounter weaker soil strata. The optimization balance between cost and bearing capacity is common in practice and can appear in design-oriented PRC questions.

The friction angle φ uniquely determines the shape and magnitude of the failure surface and hence the bearing-capacity factors. Higher φ (dense sand, strong soil) yields much larger Nc, Nq, Nγ; lower φ (clay, loose sand) yields smaller factors. This relationship is tabulated and forms the lookup step in any bearing-capacity calculation.

Square and circular footings mobilize shear resistance more efficiently than strip footings. Shape factors (typically 1.0–1.3 for sc, sq; 0.3–0.4 for sγ) adjust the Terzaghi equation. The more compact the footing, the higher the bearing capacity for a given soil and depth.

Saturation reduces the effective unit weight from γ to γ' = γsat − γw. In the Terzaghi equation, the width term (which depends on soil self-weight in the passive zone) is reduced if that zone is saturated. This adjustment lowers qu and hence qa, often governing design in low-lying or coastal Philippine sites.

The allowable pressure is a direct fraction of ultimate: qa = qu / FS. The choice of FS depends on soil predictability, load uncertainty, and code requirements (NSCP 2015 recommends FS ≈ 3 for typical buildings). Higher FS implies more conservative design but higher cost.

Deeper footings experience higher confinement and surcharge, increasing the qu linearly (via the qNq term). This is a primary strategy to increase allowable pressure without enlarging the footing: deepen the foundation. However, cost and geotechnical investigation depth increase with Df.

In loose, fine-grained soils (low φ, silt, loose sand), settlement often governs design — qa is limited by tolerable vertical displacement, not shear. In dense coarse soils (high φ), shear governs. The designer must evaluate both and use the more restrictive qa.

For practical design, the total allowable column load is Q = qa,net × A, where qa,net is in kPa and A in m². Increasing footing area directly increases allowable load; this is a key lever for accommodating large column loads in soft soils.

In general shear (dense soil), failure is sudden and well-defined; FS provides a clear safety margin. In local shear (loose soil), failure is gradual and diffuse; the soil settles before shearing catastrophically. Local shear mode is accounted for by reducing c and φ (2/3 rule) and thus lowering calculated FS, reflecting the reduced strength margin.

A typical Philippine residential building on a 1.5 m square footing at depth Df = 1.5 m in silty clay (c = 20 kPa, φ = 18°, γ = 18 kN/m³) requires bearing-capacity calculation. Engineer obtains Nc, Nq, Nγ from a table, applies square shape factors, checks water table (often high in Metro Manila), calculates qu, and divides by FS = 3 to obtain qa. The allowable load is Q = qa,net × 2.25 m². This workflow is repeated for every footing size in the foundation plan. Variations in soil strata (encountered in boring logs) require recalculation for each zone or a conservative lower-bound qa across the site.

In coastal areas or reclaimed land (e.g., parts of Metro Manila and Quezon City), bearing capacity may be very low (soft clay, φ ≈ 0°, qu ≈ 50–100 kPa → qa ≈ 16–33 kPa). Rather than construct oversized surface footings, engineers deepen the foundation to stronger strata at Df = 3–5 m. The increased surcharge (q = γDf) and confinement raise qu substantially. Alternatively, piling is considered. Choosing between deep footings and piles is a cost–benefit decision informed by bearing-capacity calculations at various depths.

In Philippine tropical climates, the water table may rise during monsoon season (May–October) and drop during dry season. A bearing-capacity design must account for the **worst-case** (highest water table) scenario. For example, a footing designed with water table 2 m below base might experience water at the base during monsoon, changing γ to γ' in the width term and reducing qa by 10–20%. Engineering practice includes specifying a design water table and documenting the assumption in the foundation report. If in-situ conditions differ, bearing capacity must be recalculated.

After calculating qa (from shear), the engineer performs a settlement analysis using elastic theory or the Boussinesq/Newmark method. For a footing in sand (φ = 30°, relatively compressible), qa,shear might be 200 kPa, but settlement analysis shows that 150 kPa yields 30 mm settlement — within the 50 mm code limit for a low-rise building (NSCP 2015). The engineer adopts qa = 150 kPa, even though shear capacity allows 200 kPa. This settlement-governed design is common in Philippine sand and silt layers.

During construction, a plate-load test is sometimes performed to verify assumed bearing capacity. A rigid plate (typically 0.3–0.6 m square, much smaller than the actual footing) is loaded and settlement is measured. The test yields a load–settlement curve; the bearing capacity is inferred by fitting this to theoretical relations or using standard correlations. If the measured qa is lower than designed, remedial action (e.g., ground improvement, deeper footing, reduced column load) is necessary. This practical feedback loop is critical in the Philippines where soil variability is high.

Where in-situ soil is weak or compressible, engineers may improve it to increase qa. Common methods: (1) **Dense sand replacement** — excavate soft clay and backfill with compacted sand (increases φ, hence Nc, Nq, Nγ); (2) **Soil stabilization** — cement or fly ash addition increases c (particularly effective for clay); (3) **Vibro-compaction** — for sand, densifies soil in-situ, raising φ. After improvement, bearing-capacity calculation is repeated with improved soil parameters. Ground improvement is cost-effective for moderate buildings in weak soils, avoiding expensive piles.

If bearing capacity at economical depth is extremely low (soft clay layer with c = 10 kPa, φ ≈ 0°, qu < 50 kPa → qa < 20 kPa), a shallow footing becomes impractically large. Engineers switch to **pile foundations**, which bypass weak upper soil and transfer load to stronger strata at depth or develop side friction along the pile shaft. Pile design uses different methods (API, LCPC, etc.) rather than Terzaghi. In the Philippines, piling is common in Intramuros (historical soft clay), Makati (silt), and other areas with deep weak soil.

A foundation design report submitted to local building departments (per RA 544 engineering practice requirements) must document: (1) soil boring logs and lab results (c, φ, γ, Gs, w); (2) bearing-capacity calculation with cited method (Terzaghi), factors used, and all assumptions (water table, FS, failure mode); (3) allowable bearing pressure qa with unit; (4) settlement analysis results; (5) recommendations (footing depth, dimensions, quality control). The report must be sealed by a licensed Professional Engineer (PE) under RA 544. Exam candidates must be familiar with this format and documentation rigor.