Whether it tips or slides

Weight is the only thing resisting overturning, and that essay checked a body against one failure. A free body pushed sideways has two, and nothing in the overturning calculation says which of them is nearer.

It can rotate about its leeward edge, or it can translate along its base. Both are collapses of the same free body, both are decided without any material strength, and they are usually written up as separate checks with separate factors — which conceals the fact that one comparison settles both.

The two checks, written so that they can be compared #

Take the body as one free body, cut on the plane of its base, with a horizontal force $F$ applied at height $H/2$ and a weight $W$ acting through the centre of a base of width $B$.

Overturning. Moments about the leeward toe. The wind’s moment is $F H/2$ and the weight’s is $W B / 2$, so

$$\text{FoS}_{\text{ot}} = \frac{W B / 2}{F H / 2} = \frac{W B}{F H}$$

Sliding. Horizontal forces on the same free body. The contact can supply at most $\mu W$, so

$$\text{FoS}_{\text{sl}} = \frac{\mu W}{F}$$

Both contain $W$ and both contain $F$, and that is the observation the rest of this page is about. Divide one by the other:

$$\frac{\text{FoS}_{\text{ot}}}{\text{FoS}_{\text{sl}}} = \frac{B}{\mu H}$$

The weight has gone, the wind force has gone, and the depth of the body has gone with them. What is left is a ratio of the body’s own shape to the coefficient of friction under it. Overturning is the smaller factor — and therefore the failure that arrives first — exactly when

$$\frac{B}{H} < \mu$$

The hoarding above is 1.6 m over 4.5 m, a ratio of 0.356, against a coefficient of 0.5. It tips.

The crossing height, which is one number #

Because the ratio of the two factors does not depend on the load, the height at which they change places is a property of the body alone. Setting them equal gives

$$H^{*} = \frac{B}{\mu}$$

For a 1.6 m base on a contact of $\mu = 0.5$ that is 3.2 m. Anything taller tips; anything shorter slides.

The two curves fall at different rates, and that is the whole mechanism. Raising a body by a factor of two doubles the wind it collects and doubles the arm that wind acts on, so the overturning factor falls by four. The sliding factor falls by two, because the contact’s capacity has not changed at all.

A structure therefore changes which failure it is prone to as it gets taller, without anything about it getting worse in the ordinary sense. A 3 m hoarding and a 5 m hoarding of the same base are not the same design problem, and the second is not simply a worse version of the first.

What a different contact does to the same body #

Everything above holds the coefficient at 0.5. It is the least reliable number in the calculation, and moving it moves the crossing rather than the answer.

Halving $\mu$ moved $H^{*}$ from 3.2 m to 6.4 m and turned the same object from one that tips into one that slides. Nothing about the hoarding changed; the floor it stands on did.

That is worth stating carefully because it inverts a common habit. A stability calculation is usually presented as a property of the structure, with the ground as a passive supplier of whatever reaction is required. Here half of the answer belongs to the surface, and the coefficient is the number in the whole exercise that nobody has measured — a property of a pair of surfaces rather than of either material, quoted to two figures and known to about one.

Which free body produced the number #

One free body produced both, and the cut is the reason the two checks share their terms.

Cut the hoarding free on the plane of its base and hold it in equilibrium. Three things cross the cut: a distributed normal pressure, a distributed shear, and nothing else. The two checks are the two ways that traction can run out.

Sliding is the shear resultant reaching $\mu$ times the normal resultant — a statement about the direction of the contact reaction, which must lie within $\arctan\mu$ of the vertical.

Overturning is the normal pressure reaching the leeward edge and having nowhere further to go — a statement about the position of the contact reaction, which must lie within the base.

Both are constraints on one resultant: sliding bounds its inclination and overturning bounds its point of application. Neither is a strength, and the free body is the same in both cases; only the question asked of the resultant changes.

That also explains why the two factors share a denominator. The wind force sets both the shear that must be carried and the moment that must be balanced, so it divides out of the comparison and cannot influence which failure comes first.

What “overturning” looks like before it happens #

Overturning does not arrive as an event. It arrives as the resultant walking outward across the base, and the interesting thing happens well before it reaches the edge.

Once the resultant leaves the middle third the windward edge lifts, the contact shortens, and the pressure at the leeward toe rises much faster than the load does — 34 kPa to 92 kPa here for an eccentricity of a quarter of the width. The overturning factor is still 2.0 at that point and the bearing pressure has nearly tripled, which is the sense in which the ground usually decides the matter before the moment does.

Sliding has no equivalent warning. The friction force rises in exact proportion to the demand and gives no sign at all until it is exhausted, at which point the body moves. The bound is a bound and nothing about the contact says how near it is — a property this collection’s other capacities do not share, since a member at 95 per cent of its strength is usually visibly working.

The taller version, and what changes #

Take the same base and make the body half again as tall, with the weight rising in proportion.

The overturning factor has fallen from 4.35 to 2.78 and the sliding factor is essentially unchanged at 6.07, because both the weight and the wind force grew in proportion to the height and their ratio did not. The gap between the two failures has widened, and the body has moved further into the regime where only one of them is worth calculating.

That is the practical use of the crossing height. A body well clear of $H^{*}$ on either side has one governing failure and the other is a formality; a body near $H^{*}$ has two, at similar factors, and both need the arithmetic done properly. On this base with this contact the awkward height is 3.2 m — which is roughly the height of every temporary hoarding and site fence ever erected.

There is one more consequence of the two curves falling at different rates, and it decides how a temporary structure is propped. A prop placed near the top of a hoarding removes almost all of the overturning demand, because it takes the wind force out at a height where its lever arm is large; the same prop does almost nothing for sliding, because the horizontal force it removes from the base is exactly the force it applies to whatever it leans on. A kentledge block placed on the base does the opposite. Neither is a general answer, and the choice between them is the crossing height again, applied to the structure as it will actually be built rather than as it is drawn.

The same condition, seen as an angle #

There is a second way to write $B/H < \mu$ that makes the comparison look like what it is.

The line from the base’s centre to the body’s centre of mass leans back from the vertical by $\alpha = \arctan(B/H)$ — half the base width over half the height. The contact’s cone of admissible reactions has half-angle $\phi = \arctan\mu$. The body tips first when $\alpha < \phi$: when the geometry runs out before the friction does.

Read that way the check is a comparison of two angles, one belonging to the shape and one to the surface, and both independent of everything else in the problem. It is also the reason a tilting table works as a test: raise a plane under a body until it moves, and whether it slides or topples tells which of $\alpha$ and $\phi$ is smaller without measuring either.

Where this reappears, and it is not a hoarding #

The same two angles decide a problem that looks nothing like this one.

A block that is rocking rather than standing has the same $\alpha$ in it, and the ground acceleration at which uplift begins is $\tan\alpha = 0.356\,g$ for this body. If the friction available is less than that, the body slides across the floor instead of rocking on it — and sliding, in an earthquake, is the benign outcome. The condition that decides which happens is the one derived above, with the wind replaced by an inertia force that scales with the same weight and therefore cancels in the same way.

The same pair of angles also settles a question about erection that looks like a different subject. A precast panel standing on its edge before it is bolted down is exactly this free body, and the horizontal force acting on it is whatever the wind and the crane between them deliver. Its base is a few hundred millimetres against a height of several metres, so its $\alpha$ is very small — three or four degrees — and it is far inside any plausible friction cone. A panel therefore always tips and never slides, which is why the propping of one is a moment problem and why a temporary condition governs a structure that is perfectly stable when finished.

So the crossing is not a curiosity of freestanding walls. Any body held down only by its own weight, pushed by any force proportional to that weight, tips or slides according to $B/H$ against $\mu$, and the load never enters.

Which of the two failures is worse #

The two are usually given similar factors of safety, and they do not deserve them.

Sliding is bounded and usually recoverable. A body that slides moves until the force falls or something stops it, and the displacement is finite because the wind is finite. Bearings, temporary works and unrestrained plant slide routinely and are put back.

It also matters which of them the rest of the structure is expecting. A body that slides has shed the horizontal force it could not carry, and whatever it was leaning on gets it instead — which is the same transfer a bearing makes when it lets a deck go, and it is a load case somewhere else rather than a failure.

Overturning is not bounded. Past the toe the restoring moment falls with rotation while the overturning moment does not, so the failure accelerates: the shape of the restoring curve in the figure above has a negative slope, which is a mechanism rather than a limit.

That asymmetry argues for treating the two checks differently, and for a reason that is about the shape of the collapse rather than about the reliability of either calculation. It also argues for preferring designs that sit on the sliding side of the crossing when the choice is available — which is why a heavy machine base is made wide rather than merely heavy, and why restraining a body against sliding is often the wrong fix: a dowel that stops it sliding has moved it back across the crossing into the failure that does not stop.

The two factors are not measuring the same thing #

Both checks are quoted as a ratio and both are called a factor of safety, and the two ratios have different contents.

The overturning factor is a ratio of two weights — one of them the body’s own, acting through a geometry that is drawn rather than tested. Nothing in it is a strength, nothing in it has scatter beyond the dimensions, and its uncertainty is the uncertainty in the load. It is in the same family as the check on a substructure against flotation, where the resistance is a weight and the action is a weight of water, and where the modern treatment is to factor the two differently rather than to divide them.

The sliding factor contains one measured material property, and it is the coefficient. That property has a scatter no partial factor was calibrated against, it is a property of a surface state that a specification cannot fully control, and it changes with moisture, temperature and dust in ways nothing on a drawing records. The same numeral in front of the two checks therefore means two different degrees of confidence, which is a general problem with ratios and not a criticism of either number: a characteristic value is a fractile and a factor applied to it is doing something else again.

There is a further asymmetry in how the two respond to being made safer. Doubling the weight doubles both factors and moves neither relative to the other. Widening the base raises the overturning factor in proportion and leaves sliding exactly where it was. Roughening the contact raises sliding and leaves overturning exactly where it was. Only two of the three available moves do anything about the failure that is actually governing, and the crossing height is what says which two.

The base is rigid and so is the ground. Both checks treat the contact as a plane. A flexible base on soft ground redistributes its pressure, and the resultant’s position is then a soil-structure problem rather than a statics one.

The friction and the bearing pressure are treated as independent. They are not: the shear a soil can carry depends on the normal stress on it, and a base that has lifted over part of its width has lost that part’s contribution to sliding as well. Both checks above use the full base for one and the full weight for the other.

The wind is a static pressure at mid-height. A real wind is turbulent, its resultant wanders, and a body near its overturning factor is being pushed by something that varies faster than the body’s own response — which is where a static check stops being the right question.

Nothing here counts a dynamic reserve. A body that begins to tip has to be given energy to go over, and a gust that reaches the tipping moment for half a second does not supply it. The static factor of one is conservative for a transient load by an amount nothing in this calculation reports.

The body is assumed to be standing on the whole of its base. A body on feet, on packers or on a bearing strip has a contact that is a set of patches, and both the friction available and the position of the resultant are then properties of the patches rather than of the outline drawn.

And the coefficient is a single number for a contact that is not uniform. Rain, dust, a membrane, a paint film or a wet screed each change it by more than the difference between a passing and a failing check.

The ladder from here #

Later rungs on this anchor: the same two checks with the weight itself uncertain, where a favourable permanent action is factored down and the arithmetic stops being a ratio. Overturning of a body that is partly submerged, where the effective weight is a buoyant one and both checks move together. The wall on a footing, where the resultant’s eccentricity feeds a bearing calculation rather than a stability one. Restraint against sliding by a shear key or a dowel, and the way it moves a body across the crossing described here. Overturning during construction, where the governing configuration is the one that exists for an afternoon. And the general case of a body on several contacts rather than one, where “the base” is a convex hull and the resultant has to stay inside a polygon rather than a line.

The tilting table is the oldest experiment in this subject. Coulomb’s memoir of 1785 reports friction angles measured exactly that way, and the same apparatus in a modern laboratory measures the same quantity for the same reason — that the crossing between sliding and toppling is visible without instrumentation, and it is the only measurement in structural engineering that a person can make with a plank.