Half as far between the legs

A stand on three legs carries a 60 kN tank. Its legs sit on a circle 1.5 m from the centre, the wind on the tank is 16 kN, and the wind acts 3 m above the ground. Checked against overturning the way a freestanding body is checked — restoring moment over overturning moment — it passes comfortably when the wind blows toward one of the legs. When the wind swings round by sixty degrees and blows toward the gap between two legs, the stand goes over.

Nothing about the stand, the tank or the wind changed except the direction. The stand does not have a factor of safety against overturning. It has one for every direction, and they differ by a factor of two.

The base of a body on legs is the polygon its feet enclose #

A body on a continuous base — the hoarding of the tipping-or-sliding comparison — tips when the resultant of its weight and the lateral force reaches the edge of the base. That rule survives the move to legs intact. What changes is where the edge is.

A foot on the ground can push and cannot pull, so the ground can hold the body up anywhere a combination of pushes can put a resultant. The set of those places is the convex hull of the feet: the smallest convex polygon that contains them all, the shape a rubber band would take if it were stretched around them. Inside it the three feet can share the weight in positive amounts. On its boundary one foot has nothing left to give. Outside it some foot would have to pull, and the equations will report that pull as a negative reaction with perfect confidence while the body lifts off that foot and rotates about the line joining the other two.

So the tipping lines of a body on supports are the edges of the hull, and the question a stability check has to answer is how far the resultant has to travel from under the weight to reach one.

The wind moves the resultant by the overturning moment over the weight:

$$e = \frac{F\,h}{W} = \frac{16 \times 3}{60} = 0.80\ \text{m}$$

in the direction the wind blows. The reach is the distance from under the weight to the first edge of the hull along that direction. The factor of safety is the reach over $e$.

Toward a leg the reach is the full 1.5 m to that leg’s foot, because the two edges that meet there both have to be crossed at once. Toward the gap it is the distance from the centre to the middle of the opposite edge, which for an equilateral triangle is half the distance to a corner: 0.75 m. The ratio of the two factors is exactly two.

Every edge governs over a band of directions #

The two figures are two samples of one function, and it is worth drawing the whole of it, because the shape of that function is the result.

Take moments about each edge of the triangle in turn. The weight restores about an edge by the weight times its perpendicular distance from the edge. The wind overturns about the same edge by its moment times the cosine of the angle between the wind and the edge’s outward normal. Each edge therefore has a factor of safety that depends on the wind’s direction, finite only over the directions that push toward that edge, and smallest when the wind blows straight at it.

The envelope has three low points and three cusps. Each low point faces the middle of an edge, where that edge is squarely in the way. Each cusp faces a leg, where the two edges meeting at that leg are equally in the way and the resultant has further to go before it clears both.

That is the reverse of the intuition the word leg invites. A leg looks like the thing a body leans on, and pushing a body toward a leg looks like the dangerous direction. It is the safest direction there is. The weak direction points at nothing — at the empty middle of an edge, between two supports, where there is no foot to lean on and the tipping line runs closest to the weight.

The same pair of curves can be found without taking a single moment. From under the weight, draw a ray in the direction of the wind and see which edge it meets first and how far away. The distance is the reach, and the reach over $e$ is the factor. The two procedures share no arithmetic — one is a set of moment equations, the other a line intersecting a polygon — and they give the same number at every direction, which is the reason for trusting either of them.

Toward an edge the reach is cos(π/n) of the reach toward a support #

A tripod is the extreme case, and the extremity has a formula.

For supports equally spaced on a circle, the reach toward a support is the circle’s radius and the reach toward the middle of an edge is the distance from the centre to that edge, the radius times $\cos(\pi/n)$ for $n$ supports. The ratio between a body’s weakest and strongest directions is therefore a number that depends only on how many supports it stands on.

Three legs: one half. Four: 0.707. Six: 0.866. Eight: 0.924. The curve never flattens, because a finite number of supports always leaves a straight edge between two of them, and a straight edge is always closer to the centre in its middle than at its ends.

That makes the second claim a body’s designer is tempted to rely on — that more supports mean more stability — true in a precise and limited sense. On the same circle a fourth leg raises the weakest reach from 0.75 m to 1.06 m, and a sixth to 1.30 m. None of them raises the strongest reach at all: it is 1.50 m on every one of those bases, because it is the circle. More supports make a base more uniform, not larger, and the thing they buy is the removal of a weak direction, which is exactly what a body whose load can come from any direction needs.

Four corners, and the checks that happen to be right #

The ordinary structural case is not a tripod. It is a rectangle on its four corners — a scaffold tower on base plates, a plant skid on four pads, a frame on four feet — and for that case the habit of checking two directions turns out to be correct, for a reason that is worth knowing so that it is not carried into cases where it fails.

A rectangle’s edges are parallel to its own axes, so its weak directions are the two axis directions and its strong ones are the diagonals. Checking the two principal directions of a rectangular footprint therefore checks exactly the directions that govern, and misses nothing: the diagonal, which looks like the long way across and the place a designer might worry about, is the direction with the most reach.

The check is right because the edges are where the axes are, and that is a coincidence of the rectangle. A triangular base, a hexagonal one, a base with a leg out of line, a mast guyed from three points — none of them has its edges on the drawing’s axes, and for all of them the two-direction habit checks the wrong directions. The direction that has to be checked is the normal to each edge, however the plan happens to be oriented on the sheet.

An offset weight makes one side weak #

The reach is measured from under the weight, not from the middle of the base, and moving the weight moves the whole curve.

An offset of a tenth of a metre on a base 1.2 m deep removed a sixth of the reach on one side and added it to the other. Most bodies that tip do so on the side their weight is already nearest, and most of the ways a weight gets offset are ordinary: a cantilevered platform on a mobile tower, a counterweight not yet fitted, a tank half-drained on one side of a baffle, a load slung from a jib whose radius grows as the load swings out.

There is a machine built around this figure, and it is the most common cause of fatal overturning in industry. A counterbalance forklift truck has a steer axle that pivots at its middle, so the truck does not stand on four wheels at all. It stands on the two front wheels and the pivot — a triangle whose apex is at the back — and its centre of gravity has to stay inside that triangle. Raising a load moves the centre of gravity up and forward; turning puts a sideways inertia force on it; and the sideways direction faces one of the triangle’s two long edges, from a front wheel back to the pivot, which are close to the centre of gravity along their whole length. The truck tips over sideways in a turn with the load raised because that is the weak direction of the triangle it stands on.

One support lost #

The last reading is the one with the most consequence for anything temporary, and it takes one support away.

A four-legged stand that loses a leg does not keep three-quarters of its stability. It keeps most of it in most directions and none of it in one: the remaining corners enclose a right triangle whose hypotenuse passes exactly under the weight, so any horizontal force at all toward the missing corner rotates the body about that diagonal. It does not fall over at once, because a weight exactly on a tipping line is in equilibrium; it falls over the first time anything pushes it that way.

That is the overturning version of the argument that a determinate structure has no robustness at all: a body on three supports is a statically determinate one, and it has no redundant support to redistribute to. The structure that survives losing a member does so because what remains can still find a load path. A body that loses a foot needs the remaining feet to enclose its weight with room to spare, and a square does not. A base on four supports whose weight is at the centre is one support away from zero, and the way to make it tolerant of losing one is to keep the weight well inside the triangle that any three of them enclose — a fifth support, a wider base in the direction that matters, or a weight lower and further from the corners.

The cases where a support is lost without anybody deciding to remove it are exactly the ones this matters for: an outrigger pad punching into soft ground, a base plate on a sole board that splits, a leg that buckles, a foot that was never quite in contact. The most dangerous day in the life of most temporary structures is one on which a support is less than it was drawn as.

The diagonal is where a leg lifts first #

The rectangle’s diagonal came out as its strongest direction against tipping, and it is worth saying immediately that it is not its strongest direction against everything. Tipping is the end of the story. The beginning is a leg losing its load, and for that the diagonal is the worst direction there is.

Take the four-legged stand with a rigid top and four legs of equal stiffness — the assumption a four-legged table needs before statics can share its load at all. With the legs at the corners of a square of half-side $a$, the reactions vary linearly across the plan, and the leg furthest from the resultant carries

$$R = \frac{W}{4}\left(1 - \frac{e_x + e_y}{a}\right)$$

when the resultant is at $(e_x, e_y)$ from the centre. That leg lifts when $e_x + e_y = a$. Along an axis that is at $e = a$, which is the edge of the base: the leg lifts exactly as the stand tips, and nothing is lost. Along a diagonal it is at $e = a/\sqrt 2$, while the tipping reach that way is $a\sqrt 2$. A leg lifts at half the tipping reach.

The region inside which all four legs keep some load is therefore a square turned through 45 degrees, touching the middle of each edge of the base — the same kind of object as the middle third of a continuous base, and smaller than the base in exactly the directions where the base looks largest.

For the stand on the 1.5 m circle, $a$ is 1.06 m, so a leg lifts at 0.75 m along a diagonal. The 16 kN wind blowing toward a corner moves the resultant 0.80 m. The stand does not tip — its factor that way is 1.88, the best it has in any direction — and one of its legs is already off the ground. If that leg’s foot was relied on for anything else, for friction against sliding or for a holding-down bolt that was never meant to carry uplift, the corner that looks safest is where it failed.

Three routes to the same number #

The free body is the whole stand, cut at the ground. Across the cut pass the three vertical reactions at the feet and whatever horizontal friction the feet develop, and across nothing else. The factor of safety found above can be reached three ways from that one cut, and it is worth seeing that they are three.

By the ray: from under the weight, along the direction of the push, to the first edge of the hull; the distance over $e$.

By moments about each edge: the weight’s restoring moment over the push’s overturning moment about that edge, minimised over the edges facing the push.

By the reactions: for three supports the reactions are determinate, and as the push grows the reaction at the support opposite the governing edge falls in proportion, reaching zero exactly when the resultant reaches that edge. On the stand pushed toward the gap, that support would have to pull 1.3 kN, which a foot on the ground cannot do.

The first is geometry, the second is statics and the third is the solution of three simultaneous equations, and they agree. For four or more supports the third route is no longer available from statics alone — the fourth leg makes the reactions indeterminate, and they depend on the stiffness of the legs and the flatness of the floor — while the first two still work, because tipping is decided by the hull and the hull does not care how the reactions were shared out before it was reached.

What the hull leaves out #

The body and the ground are rigid. A rigid body tips at the hull and not before. Flexible legs, a top that is not rigid and pads on soft ground all change where the first leg lifts, and the rotated square found above is the answer for one particular idealisation of them rather than a property of four legs.

The push has one direction and one size. A real wind on a real stand changes both, and the projected area it acts on changes with its direction, so the demand curve and the reach curve both vary around the compass and the factor is the smallest ratio of the two rather than a ratio at one angle.

The body does not slide. Every figure above assumes the feet grip. Feet on a smooth floor may slide first, and a body on several supports that slides can do so in a direction that is neither the push nor any single foot’s, because each foot’s friction is a force of limited length and free direction.

The push is static. A gust that reaches the tipping moment for a moment does not tip a body that needs to be given energy to rotate over, and a block rocking on its corners survives accelerations a static check says it cannot.

The weight stays where it is. A counterweight that slews, a load that swings and a liquid that sloshes all move the point the reach is measured from, and a crane’s empty jib is the case that leans the other way.

Still open: how much of the hull a flexible base lets the body use #

The hull is the answer for a rigid body on rigid ground, and nothing real is either. A tower on four pile caps, a tank on four legs on a slab that deflects, and a mobile crane on outrigger pads over made ground all start to lose a support long before their resultant reaches the hull, and what happens in between depends on how stiff each support is against the others. Where along the way from the middle of the base to its edge a real body stops being a body on four supports and becomes a body on three — and what its reach is from then on — is decided by the stiffness of the supports, not by where they are.