Terzaghi and Vesic Bearing Capacity Calculator
The ultimate bearing capacity qu and allowable bearing capacity qadm define the pressure that a footing can transmit to the ground without shear failure or excessive settlement. This calculator applies the general Terzaghi (1943) formula with extensions by Vesic (1973) and Meyerhof, incorporating shape, depth, and load inclination factors. Compatible with the design criteria of BS EN 1997-1 (Eurocode 7) and common practice in building construction.
What is it and when is it applied?
Bearing capacity applies to shallow foundations (isolated, strip, combined) and, with adaptations, to raft foundations. It defines the maximum load per unit area before general shear failure of the soil. In practice, qu is divided by a factor of safety (typically 3.0 static and 2.0 seismic according to BS EN 1997-1) to obtain qadm. The calculation is sensitive to c, φ, γ, embedment depth, geometry, and the presence of a water table.
Applied Formulas
Terzaghi (strip footing, ideal soil):
qu = c·Nc + q·Nq + 0.5·γ·B·Nγ
Generalised Vesic: qu = c·Nc·sc·dc·ic + q·Nq·sq·dq·iq + 0.5·γ·B·Nγ·sγ·dγ·iγ
Capacity factors (Vesic):
Nq = e^(π·tan φ) · tan²(45 + φ/2)
Nc = (Nq − 1) / tan φ (or 5.14 if φ = 0)
Nγ = 2 · (Nq + 1) · tan φ
Shape factors (Vesic): sc = 1 + (B/L)(Nq/Nc), sq = 1 + (B/L) tan φ, sγ = 1 − 0.4 (B/L)
Depth factors: dq = 1 + 2 tan φ (1 − sin φ)² · (Df/B); dc = dq − (1 − dq)/(Nc tan φ); dγ = 1.
Allowable capacity: qadm = (qu − γ·Df) / FS + γ·Df (net allowable load)
Calculation example
| Parameter | Value |
|---|---|
| B × L (width × length) | 2.0 × 2.0 m |
| Embedment depth Df | 1.5 m |
| Cohesion c | 10 kPa |
| Friction angle φ | 30° |
| Unit weight γ | 18 kN/m³ |
| Water table | Not detected within 5 m |
| Static FS | 3.0 |
Factors with φ = 30°: Nq = e^(π·0.577) · tan²60° = 6.40 · 3.0 = 18.4; Nc = (18.4 − 1) / 0.577 = 30.1; Nγ = 2 · 19.4 · 0.577 = 22.4. Overburden q = γ · Df = 18 · 1.5 = 27 kPa. Shape factors (B/L = 1): sc = 1 + 1·(18.4/30.1) = 1.61; sq = 1 + 1·0.577 = 1.58; sγ = 1 − 0.4 = 0.6. Depth factors (Df/B = 0.75): dq = 1 + 2·0.577·(1 − 0.5)² · 0.75 = 1 + 0.217 = 1.22; dc ≈ 1.23; dγ = 1. qu = 10·30.1·1.61·1.23 + 27·18.4·1.58·1.22 + 0.5·18·2·22.4·0.6·1 = 596 + 957 + 242 = 1 795 kPa. qadm = (1 795 − 27) / 3 + 27 = 616 kPa.
Result: qu = 1 795 kPa · qadm = 616 kPa (FS = 3, static).
Interpretation of results
An allowable capacity of 616 kPa is high and typical of dense sandy soil with apparent cohesion. For actual design, settlements must also be verified (Schmertmann or elastic) because they often govern over bearing capacity. In the presence of a water table below the footing, γ must be replaced by γ' = γsat − γw in the affected terms, which can reduce qadm by 30-50 %. In seismic zones, BS EN 1997-1 allows FS = 2 for combinations with earthquake.
Reference standards
- BS EN 1997-1:2004 — Geotechnical design — Part 1: General rules (Eurocode 7)
- BS EN 1997-1:2004 — Chapter on foundations and retaining walls
- Terzaghi, K. (1943). Theoretical Soil Mechanics
- Vesic, A.S. (1973). Analysis of ultimate loads of shallow foundations
- Meyerhof, G.G. (1963). Some recent research on bearing capacity
- BS EN 1997-1:2004 and BS EN 1998-5:2004 — Seismic combinations
Frequently asked questions
When do I use Terzaghi and when Vesic?
Classic Terzaghi is useful for quick manual verification with strip footings without load inclination or water table. Vesic is the general formulation and is recommended in current practice because it incorporates shape, depth, inclination, and inclined base factors. BS EN 1997-1 accepts both as long as the assumptions are documented.
What factor of safety should I use?
For static combinations, FS = 3.0 is the traditional minimum. BS EN 1997-1 allows FS = 2.0 for combinations with severe earthquake. If there are cyclic loads without earthquake (machinery), it is increased to FS = 4.0. LRFD designs use resistance factors (0.45-0.60) instead of a global FS.
How is the water table treated?
If the water table is above the footing base, all γ in the affected terms are replaced by γ'. If it is between the base and Df, it is weighted according to depth. If it is below the footing at more than 1·B, it does not significantly affect. It is a frequent error to forget this adjustment: it can overestimate qu by 30-50 %.
Does bearing capacity or settlement govern the design?
In dense granular soils and rocks, bearing capacity is usually conservative and settlement governs. In soft clays and compressible silts, the opposite is true. Designing only for bearing capacity without verifying settlement is poor practice that can lead to building collapse due to excessive differential settlements.