Foundation Engineering / How a footing fails

The mechanism beneath the footing

A wedge, a fan and a passive zone — and where the exponential comes from.

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.

Corrections checked in this pass

Removed the hidden cap from the friction-angle sensitivity comparison.

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.
Friction-angle selection and general/local failure assumptions require ground-specific assessment.

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.

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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 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 highly sensitive to φ′.

Surcharge bearing capacity factor

N_q = e^{\pi \tan \phi'} \tan^2\!\left(45^\circ + \frac{\phi'}{2}\right)

What it calculates: How effectively surcharge beside the footing resists the failure mechanism

N_q: Surcharge bearing capacity factor (—)
$\phi'$: 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%.

Check the sensitivity to φ′ alongside the uncertainty in the ground model and other design parameters.

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

N_q is exponential in tan φ′. Move the friction angle and watch how fast the capacity factor runs away.

Friction angle φ′32.0 °

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 N_q.
In undrained clay φ′ = 0, N_q = 1, and capacity is independent of width entirely.

N_q: 23.2
N_c: 35.5
N_γ (Vesic): 30.2
N_q at φ′ + 5°: 42.9
Increase for +5°: 85 %

Five degrees of friction angle — well inside the scatter of any correlation — changes N_q 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
N_q 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