Table of Contents
1. Introduction to Bearing Capacity Theories
The ultimate bearing capacity (qu) of a shallow foundation is the maximum pressure the foundation can exert on the soil without causing shear failure. The most widely used theoretical frameworks for determining qu are those developed by Terzaghi (1943), Meyerhof (1963), and Hansen (1970). Each successive theory refined and generalized the earlier work by adding correction factors for foundation shape, depth, load inclination, base tilt, and ground slope.
All three methods share the same fundamental form: qu = c × Nc + gamma × Df × Nq + 0.5 × gamma × B × Ngamma, where c is cohesion, gamma is unit weight, Df is foundation depth, B is foundation width, and Nc, Nq, Ngamma are dimensionless bearing capacity factors dependent on the soil friction angle phi. The differences lie in the values of the bearing capacity factors and the inclusion of various correction factors.
The choice of method depends on soil type, foundation geometry, and loading conditions. Terzaghi's method is adequate for simple strip footings on homogeneous soil under vertical centric loads. Meyerhof's method handles inclined loads and rectangular foundations. Hansen's method is the most comprehensive, accounting for all geometric and loading conditions including base tilt and ground slope. The Soil Bearing Capacity Calculator implements all three methods for rapid comparison.
2. Terzaghi's Bearing Capacity Theory (1943)
Karl Terzaghi's 1943 theory was the first comprehensive analytical solution for bearing capacity. He assumed a general shear failure mechanism comprising three zones beneath the footing: an elastic wedge (Zone I) that moves downward with the footing, a radial shear zone (Zone II) bounded by a log-spiral failure surface, and a passive Rankine zone (Zone III) extending to the ground surface.
Terzaghi's bearing capacity factors are computed from:
Key assumptions: (a) homogeneous soil, (b) horizontal ground surface, (c) vertical centric loading, (d) footing base is rough (full friction develops at soil-footing interface), (e) slip surface does not extend above the base level—overburden above base level is treated as a surcharge only, (f) general shear failure mechanism. Terzaghi's theory does not include depth factors, inclination factors, or base tilt factors. The shape factors (1.3 and 0.4/0.3) are empirical.
For local shear failure (loose sands, soft clays), Terzaghi recommended reduced strength parameters: c' = (2/3)c and phi' = arctan((2/3)tan(phi)). The bearing capacity factors are then computed using the reduced phi. This accounts for the progressive nature of failure in compressible soils. The Learn Geotechnical Engineering section provides detailed derivations.
3. Meyerhof's General Bearing Capacity Theory (1963)
Meyerhof (1963) extended Terzaghi's work to consider the shearing resistance along the failure surface above the base level and included depth factors. His general equation includes shape factors (sc, sq, sgamma), depth factors (dc, dq, dgamma), and inclination factors (ic, iq, igamma):
Meyerhof's bearing capacity factors use the following relationships (computed using phi in radians for the exponential):
Shape factors (Meyerhof): sc = 1 + 0.2 × Kp × B/L for phi > 10°, else sc = 1 + 0.2 × B/L. sq = sgamma = 1 + 0.1 × Kp × B/L for phi > 10°, else sq = sgamma = 1.0. Here Kp = tan²(45 + phi/2). For strip footings (B/L = 0), all shape factors = 1.0.
Depth factors (Meyerhof): dc = 1 + 0.2 × sqrt(Kp) × Df/B for Df/B <= 1.0, and dc = 1 + 0.2 × sqrt(Kp) × arctan(Df/B) for Df/B > 1.0. dq = dgamma = 1 + 0.1 × sqrt(Kp) × Df/B for Df/B <= 1.0, and dq = dgamma = 1 + 0.1 × sqrt(Kp) × arctan(Df/B) for Df/B > 1.0.
Inclination factors (Meyerhof): ic = iq = (1 - theta/90°)², where theta is the angle of load inclination from vertical (in degrees). igamma = (1 - theta/phi)² for theta <= phi, and 0 for theta > phi. Meyerhof's inclination factors are simpler than Hansen's and are most accurate for vertical or moderately inclined loads. The Engineering Standards Reference provides complete factor tables.
4. Hansen's Extended Theory (1970)
Hansen (1970) further generalized bearing capacity analysis by adding base tilt factors (bc, bq, bgamma) and ground slope factors (gc, gq, ggamma), making his theory applicable to foundations on sloping ground and with inclined bases. He also refined the shape, depth, and inclination factors using extensive experimental data:
Hansen's Nq and Nc factors are identical to Meyerhof's. Hansen's Ngamma factor is:
Hansen's shape factors: sc = 1 + 0.2 × (B/L) × Nq/Nc (for phi > 0) or sc = 1 + 0.2 × B/L (for phi = 0). sq = 1 + (B/L) × sin(phi). sgamma = 1 - 0.4 × B/L. Note that sgamma decreases with increasing B/L ratio, reflecting reduced Ngamma contribution for rectangular footings.
Hansen's depth factors: dc = 1 + 0.4 × k (where k = Df/B for Df/B <= 1 or k = arctan(Df/B) in radians for Df/B > 1). dq = 1 + 2 × tan(phi) × (1 - sin(phi))² × k. dgamma = 1.0 for all cases.
Hansen's inclination factors: ic = iq - (1 - iq) / (Nc × tan(phi)) for phi > 0, and ic = 0.5 - 0.5 × sqrt(1 - H/(B×L×c×Nc)) for phi = 0. iq = (1 - 0.5 × H / (V + B×L×c × cot(phi))) to the power of 5, where H = horizontal load component and V = vertical load component. igamma = same as iq but with exponent 5.
Base tilt and ground slope factors (Hansen): bc = 1 - (eta/147°), where eta is base inclination from horizontal in degrees. bq = e^(-0.035 × eta × tan(phi)). gc = gq = (1 - beta/147°)², where beta is ground slope. ggamma = (1 - beta/147°)². These factors make Hansen's theory the most suitable for foundations on sloping terrain, such as hillside buildings and bridge abutments. The Soil Bearing Capacity Calculator implements all Hansen factors.
5. Bearing Capacity Factors Comparison Table
The following table compares Nc, Nq, and Ngamma values for Terzaghi, Meyerhof, and Hansen at various friction angles:
| phi (deg) | Method | Nc | Nq | Ngamma |
|---|---|---|---|---|
| 0 | Terzaghi | 5.70 | 1.00 | 0.00 |
| Meyerhof | 5.14 | 1.00 | 0.00 | |
| Hansen | 5.14 | 1.00 | 0.00 | |
| 10 | Terzaghi | 9.60 | 2.70 | 1.20 |
| Meyerhof | 8.34 | 2.47 | 1.12 | |
| Hansen | 8.34 | 2.47 | 1.47 | |
| 20 | Terzaghi | 17.70 | 7.40 | 5.00 |
| Meyerhof | 14.83 | 6.40 | 5.39 | |
| Hansen | 14.83 | 6.40 | 5.39 | |
| 30 | Terzaghi | 37.20 | 22.50 | 19.70 |
| Meyerhof | 30.14 | 18.40 | 22.40 | |
| Hansen | 30.14 | 18.40 | 15.07 | |
| 35 | Terzaghi | 57.80 | 41.40 | 42.40 |
| Meyerhof | 46.12 | 33.30 | 45.40 | |
| Hansen | 46.12 | 33.30 | 32.67 | |
| 40 | Terzaghi | 95.70 | 81.30 | 100.40 |
| Meyerhof | 75.31 | 64.20 | 109.40 | |
| Hansen | 75.31 | 64.20 | 87.42 |
Key observations: (1) Terzaghi's Nc values are consistently higher than Meyerhof/Hansen because of the different assumed failure mechanism (Prandtl-type vs log-spiral). (2) Terzaghi's Ngamma is between Meyerhof and Hansen for most phi values. (3) Meyerhof and Hansen share identical Nc and Nq factors, differing only in Ngamma. (4) At phi = 0 (undrained clay), Terzaghi gives Nc = 5.70 vs Meyerhof/Hansen Nc = 5.14 (the difference is due to the assumed wedge angle). The factor of safety is typically applied as FoS = 3.0 for general shear. The Engineering Formula Library provides exact computation functions for all factor sets.
6. Eurocode 7 and IS 6403 Provisions
Eurocode 7 (EN 1997-1:2004) does not prescribe a specific bearing capacity formula but provides design approaches (DA1, DA2, DA3) that apply partial safety factors to actions (loads), soil parameters, and resistances. The characteristic bearing resistance is typically computed using Hansen's or Vesic's method, then divided by the resistance factor gamma_Rv (typically 1.0 for DA1 and DA3, 1.4 for DA2). The design bearing resistance Rd must satisfy Vd <= Rd for the ultimate limit state.
The partial factors on soil parameters in EC7 are: tan(phi_k)/gamma_phi (gamma_phi = 1.0-1.25), c_k/gamma_c (gamma_c = 1.0-1.25), and cu_k/gamma_cu (gamma_cu = 1.0-1.4). These factored parameters are used in the bearing capacity equation. EC7 also requires verification of the serviceability limit state (settlement) using the characteristic values of soil stiffness (constrained modulus M or Young's modulus E).
IS 6403:1981 is the Indian standard for bearing capacity of shallow foundations. It adopts Meyerhof's bearing capacity factors with modifications for local shear failure. The standard distinguishes between general shear failure (dense soils, phi >= 36°) and local shear failure (loose soils, phi < 36°). For local shear, reduced strength parameters c' = 0.67c and tan(phi') = 0.67 tan(phi) are used.
IS 6403 provides tables of bearing capacity factors Nc, Nq, and Ngamma for both general and local shear conditions. The allowable bearing pressure qa = qu / FoS, where FoS = 2.5 for general shear and 3.0 for local shear. Settlement analysis is required per IS 8009 to verify that the allowable bearing pressure is not governed by settlement limits. The standard also specifies minimum depth of foundation (Df_min = P/(gamma × (1-sin(phi)/(1+sin(phi)))^2) per Rankine's theory.
Bearing Capacity Failure Mechanism Diagram
[SVG Diagram: Cross-section of a strip footing on soil showing three failure zones: Zone I (elastic wedge) directly beneath the footing moving downward, Zone II (radial shear zone) with log-spiral failure surfaces radiating from the footing edges at angle phi to horizontal, Zone III (passive Rankine zone) extending outward and upward to the ground surface. Labels show the applied load q, the overburden pressure gamma*Df, footing width B, and depth Df. For Terzaghi the failure surface terminates at base level; for Meyerhof it extends to the ground surface.]
7. Worked Example — Same Problem Solved with All Three Methods
Determine ultimate bearing capacity for a strip footing using Terzaghi, Meyerhof, and Hansen
Given: Strip footing (B = 2.0 m, L >> B), depth Df = 1.5 m. Soil: c' = 10 kPa, phi = 30°, gamma = 18 kN/m³ (above water table), gamma_sat = 20 kN/m³ (below water table). Groundwater table at 3.0 m below ground surface (below footing base by 1.5 m). Load: vertical centric only. Use water table correction factor per ACI 336.2R for water table below base by distance dw = 1.5 m.
Water table correction: Since dw > B (1.5 m > 2.0 m), no correction for Nq term. For Ngamma term, gamma_e = gamma + (dw/B) × (gamma_sat - gamma) = 18 + (1.5/2.0) × (20 - 18) = 18 + 1.5 = 19.5 kN/m³. (When dw > B, gamma_e = gamma. Here dw = 1.5 m < B = 2.0 m, so use weighted average.)
Method 1 — Terzaghi (strip footing):
Nc = 37.2, Nq = 22.5, Ngamma = 19.7 (from Terzaghi table for phi = 30°). qu = c×Nc + gamma×Df×Nq + 0.5×gamma_e×B×Ngamma = 10×37.2 + 18×1.5×22.5 + 0.5×19.5×2.0×19.7 = 372 + 607.5 + 384.2 = 1363.7 kPa.
Method 2 — Meyerhof (strip footing, sc = sq = sgamma = 1.0, vertical load):
Nc = 30.14, Nq = 18.40, Ngamma = 22.40 (Meyerhof for phi = 30°). Depth factors: Df/B = 1.5/2.0 = 0.75 (< 1.0). Kp = tan²(45+30/2) = tan²(60) = 3.0. dc = 1 + 0.2 × sqrt(Kp) × Df/B = 1 + 0.2 × 1.732 × 0.75 = 1.260. dq = dgamma = 1 + 0.1 × 1.732 × 0.75 = 1.130. Inclination factors = 1.0 (vertical load). qu = 10×30.14×1.0×1.260 + 18×1.5×18.40×1.0×1.130 + 0.5×19.5×2.0×22.40×1.0×1.130 = 379.8 + 561.4 + 493.4 = 1434.6 kPa.
Method 3 — Hansen (strip footing, vertical load, horizontal ground, level base):
Nc = 30.14, Nq = 18.40, Ngamma = 15.07 (Hansen for phi = 30°). Depth factors: k = Df/B = 0.75. dc = 1 + 0.4 × 0.75 = 1.300. dq = 1 + 2 × tan(30) × (1 - sin(30))² × k = 1 + 2 × 0.577 × (0.5)² × 0.75 = 1 + 0.217 = 1.217. dgamma = 1.0. Shape factors = 1.0 for strip. Base tilt/ground slope factors = 1.0. qu = 10×30.14×1.0×1.300 + 18×1.5×18.40×1.0×1.217 + 0.5×19.5×2.0×15.07×1.0×1.0 = 391.8 + 604.4 + 293.9 = 1290.1 kPa.
Comparison and discussion:
| Method | qu (kPa) | qa (FoS=3) (kPa) | Key Difference |
|---|---|---|---|
| Terzaghi | 1364 | 455 | Higher Nc, no depth factors |
| Meyerhof | 1435 | 478 | Highest, depth factors increase qu |
| Hansen | 1290 | 430 | Lower Ngamma reduces qu |
Meyerhof gives the highest qu because the depth factors increase the Nq and Ngamma contributions while using the same Nc and Nq as Hansen but with a larger Ngamma. Hansen gives the lowest because Ngamma is smaller than both Terzaghi and Meyerhof for phi >= 30°. The difference of about 10% between methods is typical and within the accuracy of bearing capacity estimation given natural soil variability. Always use the method most appropriate for the specific problem and apply an adequate factor of safety. Verify with the Soil Bearing Capacity Calculator for rapid cross-checking.
Method Selection Guidelines
| Condition | Recommended Method | Reason |
|---|---|---|
| Strip footing, vertical load, phi < 35 | Terzaghi | Simple, conservative for shallow Df |
| Rectangular/square footing, any load | Meyerhof | Includes shape and depth factors |
| Inclined or eccentric load | Meyerhof or Hansen | Both include inclination factors |
| Sloping ground or tilted base | Hansen | Only method with base/ground factors |
| EC7 or Indian code design | Hansen / Meyerhof | Most codes reference these methods |
| Undrained clay (phi = 0) | Hansen (Nc = 5.14) | Prantdl solution, shape factors verified |
The Engineering Standards Reference provides detailed selection guidance for each code jurisdiction (ACI, EC7, IS, BS).
8. Frequently Asked Questions
What is the difference between Terzaghi, Meyerhof, and Hansen bearing capacity theories?
Terzaghi (1943) developed the first comprehensive theory for strip footings with vertical centric loads. Meyerhof (1963) added depth, shape, and inclination factors and extended the failure surface above the base level. Hansen (1970) added base tilt and ground slope factors and refined all correction factors. Each successive theory is more general and applicable to a wider range of foundation geometries and loading conditions.
Which bearing capacity method gives the most conservative result?
Terzaghi's method tends to give the most conservative (lowest) qu for shallow strip footings with depth factors not considered, because it ignores shearing resistance above the base level. However, for square footings, Terzaghi's shape factors (1.3 for cohesion, 0.4 for Ngamma) combined with higher Nc can give higher qu than Meyerhof. The relative conservatism depends on phi, B/L ratio, and Df/B ratio.
How does the water table affect bearing capacity calculations?
When the water table is above the foundation base, use the submerged unit weight (gamma' = gamma_sat - gamma_water) for the Ngamma term and for the soil above the base in the Nq term. When the water table is below the base by distance dw, apply correction: for dw >= B, no correction needed. For dw < B, use weighted average gamma for Ngamma: gamma_e = gamma + (dw/B)(gamma_sat - gamma).
What factor of safety should I apply to bearing capacity?
Typical factors of safety: 3.0 for general shear failure (cohesive soils, dense sands), 2.5 for local shear failure (loose sands, soft clays), 2.0 when bearing capacity is verified by field plate load tests. For seismic loading, the FoS may be reduced to 1.5-2.0. EC7 uses partial factors on actions and resistance instead of a global FoS.
When should I use Vesic's bearing capacity theory instead of Hansen's?
Vesic (1973) uses a different Ngamma expression: Ngamma = 2(Nq+1)tan(phi)/(1+sin(phi)). Vesic's Ngamma is slightly larger than Hansen's for low phi and slightly smaller for high phi. Vesic also provides modified compressibility factors for soils with high compressibility. Use Vesic when dealing with loose sands or soft clays where soil compressibility affects the failure mechanism.
How do I calculate bearing capacity for eccentric loads?
For eccentric loads, use Meyerhof's effective width method: B' = B - 2e, where e is the load eccentricity. The ultimate bearing capacity is then calculated using the effective footing dimensions B' and L'. The contact pressure is assumed uniform over the effective area. Hansen also provides eccentricity factors that reduce bearing capacity directly.
What is the Ngamma factor and why do different methods give different values?
Ngamma is the bearing capacity factor for the soil weight term (0.5gamma B Ngamma). The variation between methods arises from different assumptions about the failure surface shape (log-spiral radius ratio) and the passive earth pressure coefficient in the weight term. Terzaghi used Kp_gamma from graphical solutions, Meyerhof used Ngamma = (Nq-1)tan(1.4phi), and Hansen used Ngamma = 1.5(Nq-1)tan(phi). These differ by up to 30% at high phi values.
How is bearing capacity affected by soil layering?
For layered soils, punching shear analysis may be required. If a strong layer overlies a weak layer, failure may occur by punching through the strong layer followed by general shear in the weak layer (Hanna and Meyerhof method). If a weak layer overlies a strong layer, the failure surface may be contained within the weak layer. Weighted average strength parameters based on the extent of the failure zone provide an approximate solution.
What is the bearing capacity of clay (phi = 0) per these methods?
For undrained clay (phi = 0): Terzaghi gives qu = 5.70cu + gamma Df, Meyerhof gives qu = 5.14cu(1+0.2B/L)(1+0.2Df/B) + gamma Df, Hansen gives qu = 5.14cu(1+0.2B/L)(1+0.4Df/B) + gamma Df. For a strip footing at surface: qu = 5.14cu (Meyerhof/Hansen) vs 5.70cu (Terzaghi). Skempton's formula provides intermediate values based on Df/B ratio.
How do I select between general shear and local shear failure?
General shear failure occurs in dense soils (relative density > 70%, SPT N > 30) and stiff clays (cu > 100 kPa). Local shear failure occurs in loose sands (N < 10) and soft clays (cu < 50 kPa). Intermediate soils may experience punching shear. IS 6403 uses phi = 36° as the boundary: phi >= 36° for general shear, phi < 36° for local shear with reduced parameters.
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References & Standards
- Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons, New York.
- Meyerhof, G.G. (1963). "Some Recent Research on the Bearing Capacity of Foundations." Canadian Geotechnical Journal, 1(1): 16-26.
- Hansen, J.B. (1970). "A Revised and Extended Formula for Bearing Capacity." Danish Geotechnical Institute Bulletin, No. 28.
- Vesic, A.S. (1973). "Analysis of Ultimate Loads of Shallow Foundations." Journal of the Soil Mechanics and Foundations Division, ASCE, 99(SM1): 45-73.
- EN 1997-1:2004. Eurocode 7: Geotechnical Design — General Rules. CEN, 2004.
- IS 6403:1981. Code of Practice for Determination of Bearing Capacity of Shallow Foundations. BIS, 1981.
- Bowles, J.E. (1996). Foundation Analysis and Design. 5th ed., McGraw-Hill.
- Coduto, D.P. (2016). Foundation Design: Principles and Practices. 3rd ed., Pearson.
- Civil Engineering Handbook — Geotechnical Engineering chapter.
- Engineering Formula Library — Bearing capacity factor formulas and correction factors.
- Eurocode 7 Standard, IS 6403 Standard references.
- Engineering Glossary — Bearing capacity and foundation terms.