Module 3 · Lesson 3.2
The mechanism beneath the footing
A wedge, a fan and a passive zone — and where the exponential comes from.
Why this matters
The bearing capacity equation has three terms and three factors with unhelpful names. Learned as symbols it is forgettable and easy to misapply. Learned as a picture it is neither, and the picture also explains the equation's most important property: its violent sensitivity to the friction angle.
Picture a strip footing pressed into the ground. Beneath it, three zones form.
The wedge. Directly under the footing, a triangular block of soil is trapped. It cannot escape sideways because of friction on the underside of the footing, so it moves down with the foundation as a rigid body, acting like a wedge being driven into the ground.
The shear fan. Either side, the wedge pushes soil outwards through a fan-shaped zone of radial shearing. This is where most of the resistance is generated.
The passive zone. Beyond the fan, a block of soil must be pushed up and outwards to make room. It resists exactly as the passive wedge behind a retaining wall does — and this is the zone that produces the visible heave in a general shear failure.
The footing cannot go down unless all three move. Its capacity is the load that forces the whole mechanism at once.
The exponential comes from the fan. Along a logarithmic spiral, the radius grows as
radius = r₀ · e^(θ tan φ′)
where θ is the angle turned through the fan. The fan sweeps roughly 90°, so the resistance mobilised across it is scaled by e^(π tan φ′). That single factor is the reason the surcharge term's bearing capacity factor takes the form it does, and it is why bearing capacity is not merely sensitive to φ′ but explosively so.
What it calculates: How effectively surcharge beside the footing resists the failure mechanism
- Nq
- Surcharge bearing capacity factor (—)
- Effective angle of shearing resistance (degrees)
This assumes
- General shear failure — a continuous slip surface reaching the ground surface
- Weightless soil in the fan, which is what makes a closed solution possible
In plain terms: The first factor is the log-spiral fan; the second is the passive zone. Both grow with φ′, which is why the product rises so steeply. At φ′ = 0 it is exactly 1: a frictionless soil offers no amplification, and the surcharge simply resists as itself.
The numbers make the point better than the algebra. At φ′ = 30° the factor is about 18.4. At 35° it is about 33.3, and at 40° about 64.2.
So a 10° increase in friction angle multiplies this term by roughly 3.5. Put the other way round: a 5° error in φ′ — well within the scatter of an SPT correlation — changes the calculated bearing capacity by something like 80%.
That is the single most important practical fact in shallow foundation design, and it is why the effort spent establishing φ′ properly is never wasted.
Check yourself
Two engineers derive φ′ for the same dense sand from the same SPT data. One gets 35°, the other 38°. Roughly what does that do to the calculated surcharge term of the bearing capacity?
Try it
Why φ′ matters so much
Nq is exponential in tan φ′. Move the friction angle and watch how fast the capacity factor runs away.
Reading it
- The vertical axis is logarithmic — a straight line here is exponential growth.
- A five-degree error in φ′ is ordinary. Look at what it does to Nq.
- In undrained clay φ′ = 0, Nq = 1, and capacity is independent of width entirely.
- Nq
- 23.2
- Nc
- 35.5
- Nγ (Vesic)
- 30.2
- Nq at φ′ + 5°
- 42.9
- Increase for +5°
- 85 %
Five degrees of friction angle — well inside the scatter of any correlation — changes Nq by 85%. This is why a confident φ′ is worth more than a refined bearing capacity equation.
Summary
- The mechanism is a rigid wedge, a log-spiral shear fan and a passive zone that must be lifted
- The fan's exponential geometry is where e^(π tan φ′) comes from
- Nq is about 18 at 30°, 33 at 35° and 64 at 40° — a 10° change multiplies it by roughly 3.5
- Because the input is uncertain by ±3°, a sensitivity check is part of the calculation, not an extra
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint