Terzaghi’s 1943 bearing capacity equation was a landmark, but it had significant practical limitations: it was formulated only for strip, square, and circular footings, it ignored the shear resistance of soil above the foundation level, and it included no factors for load inclination or depth effects. The Meyerhof, Hansen, and Vesic methods addressed all of these gaps, each in slightly different ways. Understanding where the three methods agree, where they diverge, and why — is essential for selecting the right approach for a given design situation.
The shared framework #
All three methods use the same general bearing capacity equation structure:
qu = c · Nc · (shape · depth · inclination factors) + q · Nq · (shape · depth · inclination factors) + 0.5 · γ · B · Nγ · (shape · depth · inclination factors)
where c is effective cohesion, q is effective overburden at foundation level, γ is soil unit weight below the foundation, B is foundation width, and Nc, Nq, Nγ are the bearing capacity factors. The three methods differ in their expressions for the bearing capacity factors — particularly Nγ — and in the correction factor formulas they apply for shape, depth, inclination, and (for Hansen) base and ground inclination.
For the bearing capacity factor values at common friction angles, see Bearing capacity factors Nc, Nq, Nγ — values and how to use them.
Meyerhof (1963) #
Meyerhof extended Terzaghi’s work by accounting for the shear resistance of soil above the foundation level — Terzaghi had treated this soil simply as a surcharge — and introduced three sets of correction factors not present in Terzaghi’s formulation: shape factors, depth factors, and inclination factors.
Shape factors adjust for the three-dimensional failure mechanism beneath square, circular, and rectangular footings relative to the strip footing baseline. They always increase bearing capacity above the strip case because a wider failure wedge engages more soil resistance.
Depth factors account for the additional shear resistance mobilised in the soil above foundation level as the embedment depth Df increases. Meyerhof’s depth factors increase with Df/B, producing higher allowable bearing capacities for more deeply embedded foundations.
Inclination factors reduce bearing capacity when the resultant load is inclined to the vertical. An inclined load reduces the effective horizontal component of the failure wedge resistance. Meyerhof’s inclination factors are expressed as functions of the load inclination angle β in degrees.
Meyerhof’s Nγ expression produces values that are moderately higher than Terzaghi’s original and slightly lower than Vesic’s at friction angles above 30°.
Hansen (1970) #
Hansen’s method includes everything in Meyerhof’s framework and adds two further sets of factors that neither Terzaghi nor Meyerhof covered:
Base inclination factors (bc, bq, bγ) reduce bearing capacity when the foundation base is tilted — for example, a footing bearing on sloped bedrock or a foundation with a deliberately inclined base to resist horizontal thrust.
Ground inclination factors (gc, gq, gγ) reduce bearing capacity when the ground surface slopes away from the foundation — for example, a footing near the crest of an embankment or slope. Meyerhof included ground slope factors in later work, but Hansen’s are more systematically formulated and integrated into the standard method.
Hansen’s inclination factors are more conservative than Meyerhof’s for heavily inclined loads, and his depth factors produce somewhat smaller corrections at shallow embedment ratios (Df/B < 1). For vertical concentric loading on level ground with a horizontal foundation base, Meyerhof and Hansen give results within 5–15% of each other in most practical cases.
Vesic (1973) #
Vesic proposed modifications to several correction factors, but his most significant contribution was a new Nγ expression:
Nγ = 2(Nq + 1) · tan(φ’)
This is now the most widely used modern expression for Nγ, having largely replaced earlier tabulated values. It gives the highest Nγ values of the three methods — particularly at friction angles above 30° — so Vesic typically produces the highest ultimate bearing capacity of the three for the same input parameters, most noticeably for foundations on dense sands and gravels.
Vesic’s shape and depth factors are slightly different from Hansen’s, generally producing somewhat higher corrections and therefore slightly higher qu values. For vertical concentric loading, Vesic and Hansen typically differ by 5–20% depending on friction angle and footing geometry.
Where the methods differ #
| Feature | Meyerhof | Hansen | Vesic |
|---|---|---|---|
| Nγ expression | Meyerhof (1963) tabulated | Hansen (1970) tabulated | 2(Nq+1)tanφ’ — highest values |
| Shape factors | Moderate correction | Moderate correction | Slightly higher correction |
| Depth factors | Larger at shallow Df/B | Smaller at shallow Df/B | Similar to Hansen |
| Inclination factors | Less conservative for large β | More conservative for large β | Intermediate |
| Base inclination factors | Not standard | Included | Included |
| Ground inclination factors | Limited | Fully included | Included |
When to use each method #
| Situation | Recommended method |
|---|---|
| North American practice, AASHTO, or US Army Corps of Engineers specifications | Meyerhof or Vesic |
| Eurocode 7 or European practice | Hansen |
| Inclined or eccentric loading | Hansen (most conservative inclination factors) |
| Foundation on sloped ground or sloped base | Hansen (base and ground inclination factors) |
| Vertical concentric load, level ground | Any — run all three and compare |
| Dense sand or gravel (φ’ > 30°) | Note: Vesic gives highest qu; Hansen gives lowest |
The single most reliable approach is to run all three methods simultaneously and compare. Where results agree within 10–15%, any method is adequate. Where results diverge significantly — which typically happens only at large inclination angles or high friction angles — apply engineering judgement informed by the design standard in use and the specific loading conditions. For allowable bearing capacity and factor of safety selection, see Allowable bearing capacity — factor of safety and design approach.
How Dartis Foundation and DartiGeo run all methods simultaneously #
Dartis Foundation calculates bearing capacity of shallow foundations using Meyerhof, Vesic, Hansen, and other well-known methods for shear failure determination. Both Dartis Foundation and DartiGeo run all three methods from a single set of input parameters — soil properties, foundation geometry, groundwater depth, and loading conditions. The fine calculations report shows every intermediate value: each bearing capacity factor, every shape, depth, inclination, base, and ground correction factor with its numerical value, and the final qu and qa for each method. Reports are exported as PDF or Word.
Download a free 14-day trial of DartiGeo or Dartis Foundation →
Frequently asked questions #
Which method gives the highest bearing capacity? #
For vertical, concentric loading on level ground, Vesic typically gives the highest ultimate bearing capacity — primarily because its Nγ expression is larger than Meyerhof’s or Hansen’s at friction angles above about 25°. The difference becomes more pronounced at higher friction angles: at φ’ = 35°, Vesic’s Nγ is noticeably higher than Hansen’s. For conservative design, Hansen is generally the lower bound; for typical North American and Middle Eastern practice, Meyerhof or Vesic are standard.
Why do Meyerhof and Hansen give different results for inclined loads? #
Their inclination factor formulas are different. Meyerhof’s inclination factors are expressed as (1 − β°/90°)² for the cohesion and surcharge terms. Hansen’s factors are derived from a different theoretical model and are more conservative for large inclination angles — reducing bearing capacity more aggressively as β increases. For structural loads with significant horizontal components — retaining walls, bridge abutments, machine foundations with lateral thrust — Hansen’s more conservative inclination factors are generally preferred. For the eccentric loading treatment, see Eccentric and inclined loads on shallow foundations.
Do I need to apply all correction factors every time? #
Every factor should be evaluated for every calculation, even if the result is 1.0 (no correction). For a vertical, concentric load on level ground with a horizontal foundation base: inclination factors = 1.0; base inclination factors = 1.0; ground inclination factors = 1.0. These still need to be confirmed as 1.0 rather than skipped. The shape factors and depth factors always differ from 1.0 for real foundations and must always be applied — omitting shape factors for a square footing is one of the most common errors in manual bearing capacity calculations.
Are these methods applicable to clay (undrained loading)? #
Yes. For undrained loading of saturated clay (φ = 0, total stress analysis), all three methods reduce to the same classical result: qu = su · Nc · (shape and depth corrections) + q. At φ = 0: Nc = 5.14, Nq = 1.0, Nγ = 0 for all three methods — so the Nγ differences between methods vanish entirely. Differences between Meyerhof, Hansen, and Vesic in clay are limited to their shape and depth correction factors, which are small in magnitude. The undrained shear strength su should be derived from laboratory UCS or triaxial testing — see the laboratory testing guide.
Related articles #
- Foundation design — bearing capacity and settlement guide
- Bearing capacity factors Nc, Nq, Nγ — values and how to use them
- Allowable bearing capacity — factor of safety and design approach
- Effect of groundwater on bearing capacity
- Eccentric and inclined loads on shallow foundations
- Estimating bearing capacity from SPT N-values
- Bearing capacity and settlement from CPT
- Geotechnical laboratory testing — complete guide
