The ground is a mechanism

The bearing capacity of a footing is

$$q_{ult} = cN_c + qN_q + \tfrac{1}{2}\gamma BN_\gamma$$

and it is usually met as a formula with three terms and a table of coefficients. Read that way it is arbitrary: three products, three symbols nobody can derive, and a table to look them up in.

Read as a mechanism it is none of those things. Every coefficient is a property of one drawing, every angle in that drawing is a function of the friction angle alone, and the reason the three terms have different behaviours is that they are three different kinds of quantity.

Which free body produced the number #

Three of them, in sequence, and the sequence is the mechanism.

Directly under the footing is a wedge of soil that goes down with it, rigid, bounded by two planes at $45 + \phi/2$ to the horizontal. It is at rest relative to the footing and it acts as part of it.

Outward from each edge of that wedge is a fan of radial shear. The soil here is at failure everywhere, and the direction of the major principal stress rotates through exactly ninety degrees between the two wedges. Because the soil obeys a friction law, the boundary of the fan is a logarithmic spiral $r = r_0e^{\theta\tan\phi}$ — the curve whose tangent makes a constant angle $\phi$ with its own radius, which is precisely the condition for the resultant on it to pass through the spiral’s centre.

Beyond the fan is a passive wedge, bounded by planes at $45 - \phi/2$, which has to be pushed up and out to the surface for the footing to go down.

The whole thing is nine tenths geometry. The angles come from Mohr’s circle at failure and nothing else; the spiral’s growth rate is $\tan\phi$; and the ninety degrees is the difference between the two wedge angles, which is $(45 + \phi/2) - (-(45 - \phi/2)) = 90°$ at every friction angle. The fan is a quarter turn whatever the soil.

Where the coefficients come from #

Follow the stress round the spiral and the ratio of the passive side to the active side comes out as

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

The exponential is the fan. The squared tangent is the two Rankine wedges either end of it. At $\phi = 30°$ it is 18.4, at 35° it is 33.3, at 40° it is 64.2 — an exponential in a variable engineers quote to the nearest degree, which is the first practical consequence of the derivation and not a small one.

$N_c = (N_q - 1)/\tan\phi$ is the same result written for a cohesion instead of a surcharge, and its limit as $\phi \to 0$ is exactly $2 + \pi$.

$N_\gamma$ is the odd one out and it is worth saying so plainly: it has no closed form. The mechanism above assumes weightless soil; putting the soil’s own weight inside the failing region makes the problem statically indeterminate in a way the spiral construction cannot handle, and every expression in use — Vesic’s $2(N_q+1)\tan\phi$, Meyerhof’s, Hansen’s — is a fit to a numerical solution. At $\phi = 32°$ they give 30.2, 22.0 and 26.9. A number quoted to three figures inside a formula whose authors disagree by a third.

Three terms, three kinds of quantity #

The most useful thing the mechanism explains is why the three terms behave so differently, and the answer is dimensional.

The first two terms are stresses resisting stresses. A cohesion is a stress; a surcharge is a stress. Neither knows anything about how large the footing is, so neither term contains $B$.

The third is a stress resisting a weight, and a weight inside the mechanism grows with the mechanism, which grows with $B$. So the third term is proportional to the footing width — and the consequence is one of the strangest facts in the subject.

A footing on clean dry sand at the surface has a bearing capacity of zero at zero width. Not a small capacity: none. Set a grain of sand on a bed of sand and it carries nothing, because there is no cohesion to hold the mechanism together and no depth of soil above the failure surface to weigh it down. The capacity is entirely bought with the size of the failure region, which is bought with the size of the footing.

That is why a small pad on sand is founded at a depth, and why a raft on sand is very much more efficient per unit area than a set of pads. Doubling the width of a footing on sand doubles the pressure it can carry as well as doubling the area — so it carries four times the load, not two.

It is also why the capacity is almost never what governs. A footing on sand wide enough for its capacity to be large is a footing whose settlement is the design criterion, and settlement is a stiffness problem rather than a strength one.

The exponential, and what it does to a design #

There is a consequence of the derivation that is worth a section on its own, because it decides how much site investigation a project should buy.

$N_q$ is an exponential in $\tan\phi$. Between 30° and 35° it rises from 18.4 to 33.3 — an 81% increase in capacity for a 5° change in a quantity that is measured indirectly, from a penetration test correlated to a friction angle by an empirical relation with a scatter of several degrees of its own.

So a foundation designed on $\phi = 35°$ where the soil is at 32° has about half the capacity assumed, and there is no partial factor anywhere in the calculation with a value of two. The response the profession has settled on is to apply the factor to $\tan\phi$ rather than to the capacity — which is the only place it does any good, because it is the only place where a small change is amplified.

What the exponential is doing is visible in the drawing rather than only in the number, because the friction angle sets every angle in it and therefore its size.

A longer surface with more shear stress along all of it is where the exponential comes from: the fan turns the stress through the same ninety degrees, and the spiral it turns on grows at tanφ, so a small change in the angle compounds all the way round. That is a general property worth carrying beyond soil. Whenever a design quantity appears inside an exponential, the factor of safety belongs on the exponent and not on the answer, and a sensitivity study on the answer will systematically understate the risk. It is the same reasoning the load that never came near failing anything applies to a fatigue calculation, where the life goes as the cube of a stress range and the factor belongs on the range.

The bounds, which close #

The undrained case is worth its own section, because it is one of a handful of problems in plasticity where the two bound theorems meet.

Set $\phi = 0$. The soil has a shear strength $c_u$ and no friction; the spiral becomes a circle; the two wedges are at 45°.

The shrinkage is the reason the undrained problem is tractable. A mechanism this compact, with circular arcs and two 45° wedges, is one whose work equation can be written out and whose matching stress field can be constructed by hand.

Two ways of being wrong sets out the theorems: a mechanism gives an upper bound, because a mechanism that satisfies compatibility overestimates the collapse load, and a stress field in equilibrium that nowhere breaks the yield condition gives a lower bound.

Four calculations, all of them classic:

A two-zone stress field, with a vertical stress under the footing and a horizontal one outside it, satisfies equilibrium everywhere and gives $N_c = 4$. The footing certainly carries that.

A circular slip through the footing edge, centred on the edge, is a mechanism and gives $N_c = 2\pi = 6.28$.

Optimising the circle brings the mechanism down to 5.52.

Prandtl’s mechanism gives $2 + \pi = 5.1416$, and a stress field that matches it can be constructed. The mechanism cannot be improved and neither can the stress field, so the bounds have closed and the collapse load is not estimated but known.

That is why the undrained bearing capacity factor is quoted to four figures and the drained one is quoted three different ways by three authors. It is not that one soil is better understood; it is that one problem has been solved and the other has not.

There is a second reason the bounds are worth knowing about, and it is practical rather than aesthetic. Bounds tell a designer which way an approximation errs, and that is often more useful than the approximation itself. A circular slip is easy to compute, easy to check by hand, and easy to extend to a slope, a layered soil or an eccentric load — and because it is a mechanism it is always unsafe, so an answer from it is an upper bound and has to be treated as one. A stress field is harder to construct and always safe. Knowing which of the two a method is stops it being a number of unknown reliability and makes it a number with a direction.

The point the mechanism turns about is the same reasoning applied to a rigid-body collapse, and the same discipline: a mechanism gives a load, the load is an upper bound, and the way to improve it is to find a mechanism that gives a lower one.

Why they close for one case and not the other #

There is a deeper reason the undrained problem is solved and the drained one is not, and it is not that anybody has tried harder. The bound theorems have a hypothesis, and frictional soil does not satisfy it.

Both theorems require an associated flow rule: the plastic strain increment must be normal to the yield surface. For a Mohr–Coulomb material that condition has a physical reading — it says the soil must dilate as it shears, at a rate set by the friction angle, so that a material failing at $\phi = 35°$ expands with a dilation angle of 35° as well.

Real sand does nothing of the kind. A dense sand dilates at perhaps 10 to 15 degrees and a loose one contracts. The flow rule is non-associated, by a wide margin, and the theorems that make an upper bound an upper bound simply do not apply. What can still be said is weaker: the true collapse load of a non-associated material is bounded above by that of the associated material with the same yield surface, and the gap between them is not small.

Now set $\phi = 0$. The yield surface is a pair of horizontal lines, the normal to it is a pure shear with no volume change, and the dilation angle is zero — which is the friction angle. The flow rule is associated exactly, with no approximation and no argument, and both theorems are available in full strength.

So the reason $2 + \pi$ is exact and $N_\gamma$ is quoted three ways is not a difference in effort. One of the two problems is a genuine limit-analysis problem and the other only looks like one, and the exponential coefficients of the drained case are properly understood as the answer for a fictitious soil that dilates far more than any real one — which is one more reason the factor belongs on $\tan\phi$ rather than on the capacity.

Where the mechanism reaches, and what that means #

The drawing has a width as well as angles. For $\phi = 32°$ and a 3 m footing the toe of the mechanism is 15.9 metres from the centre — 10.6 times the footing’s half width, on each side.

Two consequences follow directly.

It also fixes the depth of ground that has to be investigated. A borehole taken to one footing width below founding level has not been taken through the mechanism, whose deepest point on the drawing is about a width below the underside and whose passive wedges reach four widths sideways. The free body is a choice decides, here, how deep to drill.

Two footings closer together than about four widths do not have separate bearing capacities. Their mechanisms overlap, and the honest calculation is one mechanism for both — which is generally better than two separate ones, because the passive wedges interfere, but which no formula covers.

A footing beside an excavation has lost most of it. The passive wedge on the open side has been dug away, and the loss is not proportional to how much was removed — the mechanism reorganises into a one-sided one whose capacity can be a third of the two-sided value. That is the arithmetic behind the rule that a trench must not be dug beside an existing foundation without checking, and it is the same reasoning as the load that depends on what carries it, where the pressure a wall receives depends on how far it has let the soil move.

And a footing near a slope has lost part of its mechanism. The passive wedge on the downhill side has to be pushed up into ground that is not there, so the capacity falls sharply — by half or more within a few widths of a crest, and by an amount that depends on the slope rather than on the soil.

Push that far enough and the footing stops being the problem. The ground on its own has a mechanism, and finding it is the same exercise with the formula removed entirely: there is no closed form, no coefficient and no table, only a family of candidate surfaces and the requirement to find the worst.

That is the bound theorems’ discipline arriving as a working procedure rather than as a theorem. Every circle on the figure is a mechanism, every mechanism is an upper bound, and the only way to improve an upper bound is to find a worse mechanism — so a slope analysis that reports one circle has reported nothing at all, and a bearing capacity taken from a table has reported one circle.

Where the model stops #

The soil was treated as rigid-plastic. It is not: it deforms a great deal before it reaches the mechanism, and the settlement that accompanies the last twenty per cent of the load is usually intolerable long before collapse. The mechanism is a real limit and is very rarely the design criterion.

The load was assumed vertical and central. An inclined or eccentric load changes the mechanism completely: the wedge is no longer symmetric, part of the footing lifts off, and the capacity is computed on an effective area centred on the resultant — which is the middle third applied to soil rather than to masonry.

Water was ignored. A submerged soil has an effective unit weight of about half its bulk one, so the width term halves; and a rapidly loaded clay is undrained, which is a completely different calculation on the same ground.

The footing was assumed rigid and the pressure under it uniform. Neither is true: a rigid footing on clay has a pressure distribution peaked at its edges and a flexible one on sand has it peaked at the centre, and the difference decides the bending moment the footing itself has to carry. The beam that sits on the ground is that calculation, and it has nothing to do with the collapse mechanism above.

And the ground was assumed uniform. A stiff crust over soft clay, or a sand layer over peat, produces a mechanism that punches through rather than rotating — a different drawing with different arithmetic and no coefficients at all.

Why the mechanism is drawn at all #

It is fair to ask why any of this matters, given that the mechanism is almost never the design criterion and the formula is available whether or not anybody understands it.

The answer is that the mechanism is what makes the formula extensible, and every real foundation is outside the case the formula covers. A footing beside a slope, a footing on a layered profile, a pair of footings close together, a footing carrying an inclined load, a footing on the crest of an excavation, a raft with a soft patch under one corner — none of them is in the table, all of them are ordinary, and each is a small modification to a drawing that anybody who has seen the drawing can make.

The formula, on its own, offers a set of correction factors: for shape, for depth, for inclination, for base tilt, for ground slope. Each is a number multiplying a term, each was derived from a modified mechanism by somebody else, and used without the picture they are a second table on top of the first. Drawing as calculation is the alternative, and here it is not a stylistic preference — it is the difference between a method that covers one case and a method that covers the cases that occur.

The generalisation #

The idea to carry away is that a tabulated coefficient is usually a drawing that somebody has already made.

Three numbers looked up from a table are three numbers. The same three understood as properties of a wedge, a spiral and a passive block are a picture, and a picture can be interrogated: what happens near a slope, what happens with two footings, what happens if the soil is layered, what happens if the load is inclined. Every one of those questions is unanswerable from the table and straightforward from the drawing.

The second is about the bound theorems, and it is a rare piece of good news. Two ways of being wrong presents them as a pair of one-sided answers, which is how they usually behave. Here they close, and when bounds close the result stops being an engineering estimate and becomes a theorem — which is why $2 + \pi$ appears in a geotechnical formula at all, and why it is the only number in this essay that will still be exactly right in a hundred years.