Bearing Capacity Calculator (Terzaghi equation with Vesic factors, strip footing)

Understanding the bearing capacity calculator — concept and principle

A footing must not load the ground beyond the pressure at which it fails in shear. This calculator uses Terzaghi's three-term bearing capacity equation for a long strip footing, with bearing capacity factors in the form popularised by Vesic, to find the ultimate bearing capacity and an allowable value after a safety factor.

How it is calculated

q_u = c N_c + γ D_f N_q + ½ γ B N_γ

• N_q = e^(π tan φ) × tan²(45° + φ/2)
• N_c = (N_q − 1) cot φ (5.14 when φ = 0)
• N_γ = 2(N_q + 1) tan φ
• q_allow = q_u / FS

c is cohesion (kPa), φ the friction angle, γ the soil unit weight (kN/m³), D_f the footing depth and B the footing width (m).

The principle

At failure a wedge of soil under the footing pushes sideways, shearing the soil along curved surfaces and heaving it up beside the footing. Resistance comes from three sources: cohesion along the slip surface (first term), the weight of soil above footing level acting as a surcharge that holds the wedge down (second term), and the weight and friction of the soil in the failure zone itself (third term, which grows with width). The N factors depend only on φ and come from plasticity solutions: Prandtl for N_c, Reissner for N_q, and Vesic's 1973 expression for N_γ. Terzaghi's own tabulated factors differ somewhat; at φ = 30° his N_c and N_q are higher and N_γ lower.

Worked example

1. Inputs: c = 10 kPa, φ = 30°, γ = 18 kN/m³, D_f = 1 m, B = 2 m, FS = 3.
2. N_q = e^(π × 0.5774) × tan²60° = 6.133 × 3 = 18.40.
3. N_c = (18.40 − 1)/0.5774 = 30.14; N_γ = 2 × 19.40 × 0.5774 = 22.40.
4. Cohesion term = 10 × 30.14 = 301.40 kPa.
5. Surcharge term = 18 × 1 × 18.40 = 331.22 kPa.
6. Width term = 0.5 × 18 × 2 × 22.40 = 403.24 kPa.
7. q_u = 1,035.86 kPa; q_allow = 1,035.86/3 = 345.29 kPa.

How to read the result

Compare the allowable value with the service footing pressure. A safety factor of about 3 is traditional for shear failure; settlement often governs before it, especially on wide footings and soft clay. This is a gross value; the net capacity subtracts the existing overburden γD_f (18 kPa here). Square and circular footings need shape factors, inclined loads need inclination factors, and a water table near the footing requires the submerged unit weight in the affected terms.

Common mistakes

• Mixing drained and undrained strength parameters.
• Applying strip-footing results to square footings.
• Ignoring groundwater.
• Using general shear factors for loose sand or soft clay, where local or punching shear may control.

Frequently asked questions

Why does the result rise so fast with φ?

N_q contains e^(π tan φ), so a few degrees of friction angle change the factors by large amounts; careful testing of φ is essential.

What is the net bearing capacity?

The gross value minus the existing overburden γD_f.

What about purely cohesive clay?

With φ = 0, N_c = 5.14, N_q = 1 and N_γ = 0, so q_u ≈ 5.14c + γD_f for a strip footing.

Sources: Vesic, A. S. (1973), Analysis of ultimate loads of shallow foundations, Journal of the Soil Mechanics and Foundations Division, ASCE, 99(SM1) · Das, B. M., Principles of Foundation Engineering, Cengage