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Terzaghi Bearing Capacity

Calculate the ultimate bearing capacity of a shallow continuous strip footing using Terzaghi's equation. Mathematically determine the maximum load soil can support before catastrophic shear failure occurs.

Terzaghi Bearing Capacity Calculator

Terzaghi's 1943 general bearing capacity theory is the bedrock of shallow foundation design. It models ultimate soil failure as three simultaneous mechanisms: cohesion shear (soil stickiness), surcharge pressure (overburden above the footing), and soil wedge friction (the lateral soil mass under the footing). All three must be summed to find the true failure load.

Friction Angle Presets (auto-fill Nc, Nq, Nγ)

0 for cohesionless (sand)

q_u = (c × Nc) + (γ × Df × Nq) + (0.5 × γ × B × Nγ)
Cohesion term: 200.00 × 17.7 = 3540.00 psf
Surcharge term: 110.00 × 3.00 × 7.4 = 2442.00 psf
Density term: 0.5 × 110.00 × 4.00 × 5.0 = 1100.00 psf
q_u = 3540.00 + 2442.00 + 1100.00 = 7082.00 psf
q_safe = q_u / 3.0 = 2360.67 psf
Ultimate Bearing Capacity (q_u)
7082
psf
Very High Capacity
Safe Design Capacity (q_safe, FS=3)
2361
psf
This is what the structural engineer uses for footing design
Term Contribution to q_u
Cohesion (c·Nc)
3540.0 psf (50.0%)
Surcharge (γ·Df·Nq)
2442.0 psf (34.5%)
Density (½γ·B·Nγ)
1100.0 psf (15.5%)
q_u vs. Footing Width (c=200, γ=110, Df=3)
2 ft
6532 psf
3 ft
6807 psf
4 ft
7082 psf
6 ft
7632 psf
8 ft
8182 psf

Practical Example

A geotechnical engineer designs a strip footing for a load-bearing wall on a medium-density silt-clay soil (φ = 20°). Soil properties: c = 200 psf, γ = 110 pcf. Footing geometry: B = 4 ft wide, Df = 3 ft deep. Bearing capacity factors from Terzaghi's tables: Nc = 17.7, Nq = 7.4, Nγ = 5.0.

Cohesion term: 200 × 17.7 = 3,540 psf
Surcharge term: 110 × 3 × 7.4 = 2,442 psf
Density term: 0.5 × 110 × 4 × 5.0 = 1,100 psf
q_u = 3,540 + 2,442 + 1,100 = 7,082 psf
q_safe = 7,082 / 3 = 2,361 psf.

If the wall imposes a column load of 80,000 lbs on a 4 ft × 20 ft footing (80 ft²), the contact pressure = 80,000 / 80 = 1,000 psf — well below q_safe. The footing passes. If it failed, the engineer would widen B to reduce contact pressure.

💡 Field Notes

  • Terzaghi vs. Meyerhof/Hansen: Terzaghi's equation is for continuous (strip) footings only and is considered conservative. For square/rectangular footings, apply Meyerhof's shape factors or use the general bearing capacity equation with inclination and depth factors (Nc·Fcs·Fcd·Fci + …).
  • Factor of Safety: The standard FS = 3 for bearing capacity is not arbitrary — it accounts for site variability, load eccentricity, construction tolerance, and soil disturbance during excavation. For well-characterized soils with careful construction, some codes permit FS = 2.5.
  • Groundwater table: If the water table rises to footing level, the effective unit weight drops to approximately γ − γ_water ≈ 110 − 62.4 = 47.6 pcf. This can reduce q_u by 30–40% — always adjust for water table depth in your geotech report.
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What is Geotechnical Soil Mechanics & Shallow Foundation Design?

Terzaghi's bearing capacity theory, published in 1943, remains the undisputedly most widely used method globally for designing shallow footings. It models raw soil failure as a simultaneous macroscopic collapse of three wedges: an active wedge plunging directly below the footing, a radial shear transition zone, and a massive passive wedge violently shoving upward at the outer edge. The master equation captures this resistance as three additive terms — cohesion, surcharge, and soil density — each amplified by an empirically derived exponential bearing capacity factor.

Mathematical Foundation

Ultimate Bearing Capacity (Terzaghi)
qu=cNc+γDfNq+0.5γBNγq_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma
cc= Soil cohesion (psf or kPa) — the 'stickiness' of the cohesive soil. Zero for clean sands.
γ\gamma= Effective soil unit weight (pcf or kN/m³) — average structural density of the soil mass.
DfD_f= Depth of the footing bottom below final grade. Deeper footings benefit from massive surcharge confinement.
BB= Width of the continuous strip footing. Wider footings mobilize exponentially more soil density resistance.
Nc,Nq,NγN_c, N_q, N_\gamma= Non-dimensional bearing capacity factors — functions entirely of soil friction angle φ, tabulated from Terzaghi's theory of general shear failure.
Safe Bearing Capacity (Design)
qsafe=quFSq_{safe} = \frac{q_u}{FS}
FSFS= Factor of Safety — universally 3.0 for foundation bearing capacity. Accounts for lethal site variability, invisible load uncertainty, and severe construction disturbance.

Laws & Principles

  • Cohesion Term (c·Nc): Represents the shear strength of strictly cohesive soil particles sticking together. In heavily saturated clays where the friction angle (φ) drops near 0°, this is the only term that matters—the soil fails in rapid undrained shear. In clean river sands, cohesion c = 0 and this entire term mathematically vanishes.
  • Surcharge Term (γ·Df·Nq): The physical weight of the dirt sitting directly above the footing elevation acts like a giant, heavy blanket creating confining pressure that ruthlessly resists the passive wedge pushing upward. Deeper footings dramatically improve capacity — burying a footing twice as deep roughly doubles this defensive term.
  • Density Term (0.5·γ·B·Nγ): The sheer internal friction contribution from the actual mass of the soil wedge trapped beneath the footing. Wider footings mobilize a vastly wider, deeper frictional failure arc. This term grows linearly with footing width (B), which is why massive monolithic mat foundations can achieve extraordinary capacities in deep granular soils.
  • The Global Factor of Safety: FS = 3.0 is the unforgiving industry standard for bearing capacity. It is NOT applied to settlement predictions — a totally separate deformation analysis (e.g., Terzaghi primary consolidation or Boussinesq elastic settlement) is legally required alongside the initial bearing capacity check.
  • Groundwater Table Destruction: If the unseen groundwater table rises up to the footing elevation, the soil becomes instantly buoyant. You must replace the dry unit weight with the submerged effective unit weight (γ' ≈ γ_sat - 62.4). This single water event instantly reduces the foundation capacity by an overwhelming 40%.

Step-by-Step Example Walkthrough

" A structural engineer is designing a load-bearing wall strip footing in medium-density silt-clay. Laboratory borings show soil cohesion c = 200 psf, unit weight γ = 110 pcf. The architect sets the footing depth Df = 3 ft and footing width B = 4 ft. The Triaxial test friction angle φ = 20°, which references chart factors: Nc = 17.7, Nq = 7.4, Nγ = 5.0. "

  1. 1. Calculate Cohesion term: 200 psf × 17.7 (Nc) = 3,540 psf of resistance.
  2. 2. Calculate Surcharge term: 110 pcf × 3 ft deep × 7.4 (Nq) = 2,442 psf of resistance.
  3. 3. Calculate Density term: 0.5 × 110 pcf × 4 ft wide × 5.0 (Nγ) = 1,100 psf of resistance.
  4. 4. Calculate Ultimate Failure (q_u): 3,540 + 2,442 + 1,100 = 7,082 psf ultimate capacity.
  5. 5. Apply Life Safety Limit: q_safe = 7,082 / 3.0 (Factor of Safety) = 2,361 psf usable capacity.
Final Result: The ultimate bearing capacity before catastrophic failure is 7,082 psf. The legal design bearing capacity (with FS=3) applied to the blueprints is 2,361 psf. A standard timber residential wall load of 1,200 lbs per foot on a 4-foot wide footing produces only 300 psf contact pressure, meaning this foundation is massively over-engineered and profoundly safe.

Quick Answer: What is the Terzaghi bearing capacity equation?

The Terzaghi equation (q_u = c·Nc + γ·Df·Nq + 0.5·γ·B·Nγ) calculates the ultimate weight a soil foundation can hold before catastrophically failing. It sums three distinct sources of soil strength: the sticky cohesion of the clay (c), the heavy surcharge dirt burying the footing (Df), and the sheer frictional density and width of the soil wedge underneath the load (B). The result is always aggressively divided by a Factor of Safety of 3.0 for construction.

The Tri-Part Terzaghi Theory

Part 1 (Cohesion) = Cohesion value (c) × Nc

Part 2 (Burial/Surcharge) = Soil Density (γ) × Depth × Nq

Part 3 (Footing Width) = 0.5 × Soil Density (γ) × Width × Nγ

Note: The N-factors (Nc, Nq, Nγ) are dimensionless coefficients determined strictly from the soil's internal angle of friction. They rise exponentially as friction increases.

Common N-Factors by Soil Friction Angle (φ)

Friction Angle (φ) Typical Soil Type Nc Factor Nq Factor
0° Pure Saturated Clay 5.7 1.0
20° Silt / Silty-Clay blend 17.7 7.4
30° Loose Clean Sand 37.2 22.5
40° Dense Gravel / Stone 95.7 81.3
Note: Notice how an increase in friction from 20 to 40 degrees creates a massive 10x multiplier explosion in the surcharge bearing capacity factor (Nq).

Geotechnical Bearing Failures

The Water Table Wipeout

An engineer designs a massive silo foundation on extremely dense sand assuming 120 pcf soil density. The math works perfectly, and the silo is constructed. Five years later, an unprecedented heavy flood season raises the invisible water table 15 feet until it touches the bottom of the silo footing. Because the soil is now physically submerged under water, it gains buoyancy (Archimedes' principle). The 120 pcf density is suddenly cut in half to an "effective" density of ~58 pcf. The footing loses 50% of its bearing capacity overnight and catastrophic rapid shear settlement instantly tilts the silo.

Punching Shear in Clay

A contractor pours a small 1x1 foot square column pad onto incredibly soft, wet clay. Because clay has a friction angle of virtually zero (φ=0), the N_gamma factor mathematically vanishes, meaning widened footings don't help much. Instead of a massive soil failure wedge violently pushing up the adjacent dirt (General Shear), the small footing simply acts like a giant needle and smoothly plunges straight down into the unresisting wet clay (Local/Punching Shear). For soft clays, Terzaghi's standard factors must be heavily modified downwards to account for punching.

Professional Geotechnical Strategies

Do This

  • ✓Check Settlement independently. A footing may technically be able to successfully hold 5,000 psf without shear-collapsing, but the soil might squish and settle 6 inches vertically to do it. Modern building finishes crack after just 1 inch of settlement. Terzaghi Bearing Capacity ONLY checks failure; it does NOT check settlement.
  • ✓Always trust the lowest borings. If a soil boring shows stiff clay for 3 feet, but abruptly turns to soft plunging mud 5 feet down, your failure wedge will intercept the mud. You must use the weakest shear metrics inside the B-width zone beneath the footing.

Avoid This

  • ✗Do not apply Terzaghi to Deep Foundations. Terzaghi's strict formula is exclusively derived for shallow footings where Depth (Df) is roughly less than or equal to Width (B). If you apply this formula to a 40-foot deep drilled concrete pier, the surcharge math will generate a fake billion-pound capacity result.
  • ✗Don't guess soil friction. Guessing a soil has a 34-degree friction angle instead of 28 degrees might seem like a minor 15% error, but because Terzaghi's N-factors are highly exponential, that guess will aggressively double or triple your spreadsheet capacity, leading to a disastrously undersized footing.

Frequently Asked Questions

What does "Ultimate Capacity" vs "Safe Capacity" mean?

Ultimate Capacity is the exact mathematical breaking point where the soil structure literally ruptures and the building violently caves in. Safe Capacity divides that breaking limit by an extreme safety margin (usually 3.0x), which ensures construction anomalies can't trigger a failure.

Why do deeper footings hold more weight?

When a footing fails, the soil directly under it wants to violently push out and sideways, forcing the soil next to the footing upwards. If the footing is buried 6 feet deep, the failing wedge physically has to lift 6 feet of heavy dirt. That added massive defensive "surcharge" weight locks the footing in place.

Why don't sands use the Cohesion part of the formula?

Cohesion represents electrical and molecular 'stickiness' between ultra-fine clay particles. Sand grains are too massive for these tiny electrical forces to matter; sand solely relies on jagged particle friction. Therefore, cohesion (c) in clean sand is mathematically 0.

What are the N_c, N_q, and N_gamma values?

They are empirically derived, non-dimensional multipliers published by Karl von Terzaghi. Because calculating exact physics for trillions of soil particles is impossible, these tabulated factors exponentially scale up bearing strength based solely on the soil's measured internal friction angle.

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