Foundation Engineering / The bearing capacity equation
The factor nobody agrees on
Nγ, and what to do when the references disagree.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A geotechnical reviewer must confirm drainage and stress basis, bearing-factor formulation, correlations and settlement influence factors. Partial factors and project-specific ground conditions are outside these teaching checks.
- Shape/depth/inclination methods and loading direction need exact provenance and compatible assumptions. Current Eurocode 7/UK NA conformity and numerical model limits are not certified.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
Open three textbooks at the bearing capacity tables. Nc and Nq will match to the last digit in all three. Nγ will not. This is not a misprint, and it is not one author being careless — and if nobody tells you, you will waste an afternoon looking for the mistake.
Nc and Nq come from a closed-form solution. Prandtl and Reissner solved the mechanism for a weightless soil, which is what makes the mathematics tractable, and the answer is exact within that idealisation.
The self-weight term is the one that breaks the idealisation, because it exists only by virtue of the weight that was assumed away. There is no closed-form solution for Nγ. Every published value comes from an approximate analysis, a numerical solution, or a fit to model tests — and reasonable people making reasonable choices have arrived at noticeably different answers.
At φ′ = 35°, three widely used forms give:
- Vesic: Nγ ≈ 48.0
- Meyerhof: Nγ ≈ 37.1
- EC7 Annex D: Nγ ≈ 45.2
The spread between the highest and lowest is about 29%. In a sand-dominated design where the self-weight term is the larger one, that is a 20% swing in calculated capacity depending on nothing but which reference is open on the desk.
Check yourself
Why do published values of Nγ differ between authorities while Nc and Nq do not?
Try it
The three terms
A strip footing, no corrections. Change the geometry and the soil, and see which term is doing the work.
Nγ form
- c·Nc0 kPa
- q·Nq633 kPa
- ½γB·Nγ913 kPa
- Nc
- 46.1
- Nq
- 33.3
- Nγ
- 48.0
- Effective ultimate q′_ult
- 1545 kPa
- Net ultimate qnu
- 1526 kPa
- Nγ spread across the three forms
- 29 %
The three published forms of Nγ disagree by 29% at this friction angle. Switch between them and watch the self-weight term move — that disagreement is real, not a rounding difference.
Summary
- Nc and Nq are exact for a weightless soil; Nγ describes the weight that assumption removed
- Published Nγ values differ by roughly 30% at typical friction angles
- Use the form your design code assumes, and record which it was
- Never combine a resistance model from one source with partial factors calibrated against another