The table statics cannot settle

A rigid object resting on point supports, with a weight on it, is the first problem in every statics course. Draw the free body, resolve the forces, take moments, solve. The method is complete and it works.

It works for three supports. For four it returns a line of answers, and the line is not an approximation or a numerical artefact: every point on it satisfies all three equations exactly. This is not a hard problem. It is a problem with no answer, and the reason is worth following, because it recurs everywhere from a table leg to a bridge to a heap of sand.

The three equations, and where they come from #

A rigid body in the plane of a floor, loaded vertically, has three ways it can fail to be in equilibrium: it can accelerate downwards, and it can turn about either of two horizontal axes. Equilibrium is therefore three equations.

Three unknowns and three equations close. The answer, for a triangular arrangement, has a pleasant geometric form — each reaction is the area of the sub-triangle opposite it divided by the whole, which is why the three sum to the load and why moving the load towards a leg loads that leg more.

Nothing about the material enters. A top of steel on legs of steel and a top of cheese on legs of cheese give the same three numbers, provided both are stiff enough for the geometry to hold. That independence is exactly what makes the free-body method so powerful and is the thing that is about to be lost.

It is worth being precise about why there are three equations and not six. A rigid body in three dimensions has six ways to accelerate — three translations and three rotations — so equilibrium is six equations in general. Here the loads are all vertical and the supports all push vertically, so three of those six are satisfied identically: nothing pushes sideways, and nothing turns the top about the vertical. What is left is the vertical force balance and the two moments about horizontal axes. Adding a sideways load, or a leg that can grip the floor, brings the missing three back and changes every count in this essay.

The same count also decides when the top tips over. Push the load outside the triangle of support and one of the three reactions comes out negative, which is a leg pulling downwards — something a leg standing on a floor cannot do. That is the tipping condition, read off the arithmetic rather than argued about.

Four unknowns, three equations #

Add a fourth support and the count breaks. There are now four reactions and still three equations, and the solutions form a one-parameter family.

The direction along that family is easy to see once it is named. Press two diagonally opposite corners harder and the other two less, by the same amount: the total force is unchanged, and so is the moment about both axes, because the increases and decreases are placed symmetrically. That pattern — plus, minus, plus, minus around the square — adds nothing to any of the three equations, so any multiple of it can be added to a solution and the result is still a solution.

What closes it is elasticity. If each leg gives a little under load, the rigid top settles into some plane, and a plane is specified by three numbers rather than four. The reactions then follow from how far each leg has been compressed, the count comes back into balance, and there is one answer.

That answer lies exactly on the family the rigid calculation produced, at a definite point along it. Elasticity has not overruled statics; it has supplied the one piece of information statics did not have. Which is a fair summary of what the rigid-body idealisation costs: it throws away the deformations, and the deformations were carrying the missing equation.

What the extra information actually is #

Saying “solve it with springs” makes the problem sound closed. It is worth looking at what has to be known.

The stiffness of a table leg depends on its material, its cross-section, its length, how it is joined to the top, and how the joint has aged. None of those is known to better than a factor of two in practice. Worse, the four legs do not stand on four independent springs: they stand on a floor, which couples them, and on a carpet, which couples them more.

There is a second difficulty hiding in the word “spring”. A leg compresses, but it also bends, and it is joined to a top that twists; each of those is a different stiffness with a different dependence on the geometry, and which of them dominates depends on the proportions. A short thick leg is governed by compression and a long thin one by bending, and the two differ by orders of magnitude for the same piece of wood. Choosing one number to stand for all of it is a modelling decision, and the answer inherits whatever that decision got wrong.

So the honest position is that the four-legged problem has an answer in principle and, in most real cases, an answer nobody knows. This is not an unusual situation in engineering; it is the normal one for any structure with more members than the equilibrium equations can determine, which is nearly every structure built since iron became cheap.

The leg that is one part in a thousand short #

The most striking consequence is that the load sharing is dominated by manufacturing errors rather than by loads.

A leg made short by a small amount does not simply fail to touch. It redistributes the load along the very direction the rigid problem left free, by an amount proportional to the shortfall and to the stiffness, and independent of the load entirely — computed at two quite different load positions in that figure and agreeing to a part in ten thousand billion.

The scale of the effect is the part that surprises. Take a table leg a metre long carrying a quarter of a hundred-kilogram load: it compresses by a few micrometres. A leg short by a tenth of a millimetre is therefore short by tens of times its own working compression, and the redistribution it forces is correspondingly enormous — enough, as the figure shows, to take a reaction through zero. A tolerance that would be invisible on any drawing decides the load sharing completely.

That independence is what makes a wobble a property of the table rather than of what is on it. It is also why a stiffer table wobbles worse: the same tolerance produces a redistribution proportional to the stiffness, so a heavy rigid table on a stone floor rocks where a light one on a carpet does not.

Push the shortfall a little further and one reaction goes negative. Since a leg cannot pull, what happens instead is that the top lifts off and rocks between two three-legged states. A three-legged stool cannot do this, and the reason is exactly the count: with three supports there is no free direction for a manufacturing error to excite.

The same count, one level down #

The identical arithmetic decides whether a heap of grains is a fluid or a solid, and it is worth seeing because the granular case makes the count visible.

Each grain in the plane has two coordinates and each contact removes one relative motion, so a pack of $N$ grains with mean coordination $Z$ has $2N$ freedoms against $NZ/2$ constraints. The freedoms run out at $Z = 4$, which is twice the dimension, and that is the isostatic point: below it the pack has genuine mechanisms and flows, above it the pack is rigid.

The same grains twice, with and without enough contacts, make the point without any arithmetic: nothing about the grains has changed — not their size, their friction or their density — and yet one pack will flow under any load and the other will not. What changed is a count.

Above the isostatic point the extra contacts are redundant in exactly the sense a fourth table leg is: their forces are not determined by any balance of forces or moments.

This is why the force network in a granular pack is not reproducible. Fill two identical silos identically and the chains of heavily loaded grains run in different places, because the packing history selects a point on the same kind of family the table’s fourth leg left open. The silo that does not weigh what it holds is the same indeterminacy seen from the outside, and the angle a heap settles at is what it looks like when the pack gives way.

Why anything is built this way #

If redundancy makes the load sharing unknowable, the obvious question is why structures are not all built to be exactly determinate.

Some are. A tripod stands on any floor, a three-point mount on a telescope holds its mirror without twisting it, and a truss designed as a mechanism-free but redundancy-free assembly has forces that can be read off from the loads alone. Machine tools and optical mounts are built this way on purpose, because the point of them is that nothing is unpredictable.

A moment about a point is one of the three equations available, and the count is the whole difficulty. Two force balances and one moment balance, in a plane; six in three dimensions. There are three such statements and no more, whatever the structure, because the count comes from the body’s freedoms rather than from its complexity. A table with four legs has four unknowns and three equations, and nothing about being more careful adds a fourth.

The reason most things are not is failure tolerance. A determinate structure has no spare members: remove one and the count goes the other way, the structure acquires a mechanism, and it collapses. A redundant one loses a member and redistributes, which is the whole argument for redundancy and is a genuinely good argument.

The two properties are in tension and cannot both be had. Predictable load sharing requires exactly as many members as equations; surviving a broken member requires more. Every real structure is a choice about where on that trade to sit, and the choice is usually made in favour of not falling down.

What indeterminacy does to a calculation #

There is a practical consequence for anybody doing the arithmetic, and it is the reason this distinction is taught early.

A determinate problem looks quite different, and it is worth having one in view. Two pistons connected by a fluid: one equation, one answer, and the answer depends on no material property whatever — not on what the fluid is, not on what the pistons are made of, not on how stiff anything is. Everything in this essay is about what happens when a problem stops being of that kind and the material starts to appear in the answer.

In a determinate problem the forces can be found before anything is known about the material, and the material properties are then used only to check that nothing breaks. In an indeterminate one the two calculations are entangled: the forces depend on the stiffnesses, the stiffnesses depend on the sections, and the sections are chosen to carry the forces. That circularity is why indeterminate structures are solved iteratively, and it is a good part of what structural analysis software is for.

There is a related trap in measurement. Strain gauges on a redundant structure report the forces that structure actually has, which are not the forces the drawing implies, and the discrepancy is not an instrument error. A bolted joint tightened in a different order, a foundation that settled a millimetre, a beam installed on a warm day — each moves the answer along the free direction without moving the loads at all. The chatter a stiffer holder removes is a cousin of the same effect in a machine: a quantity that ought to be decided by the forces turns out to be decided by the stiffnesses instead.

It also changes what a safety factor means. In a determinate structure, doubling a member’s strength doubles its margin. In an indeterminate one, doubling a member’s stiffness draws more load into it, so a stronger member can end up no safer than before — and, occasionally, less so.

The table that can always be levelled #

There is a piece of folk advice about wobbly tables — turn it a little and it will settle — and it happens to be a theorem.

Take a table whose four feet are of equal length and form a square, standing on a floor that is uneven but continuous and not too steep. Rest it on three feet and measure the gap between the fourth foot and the floor, calling that gap $g$. Now rotate the table about its centre. After a quarter turn the four feet occupy the same four positions on the floor, but the roles have been permuted: the foot that was hanging is now one of the three touching, and one of the others is hanging instead. That relabelling reverses the sign of $g$.

A continuous function that is positive at one angle and negative a quarter of a turn later must be zero somewhere between. So there is an orientation at which all four feet touch the floor at once, and it can be found by rotating through less than ninety degrees. The argument is due to Fenn and has been made rigorous with an explicit bound on the floor’s slope; for anything one would call a floor, the bound is not close to being violated.

Two caveats keep it honest, and both are the subject of this essay.

The theorem requires the legs to be equal. A table whose legs differ — which is to say the case the preload figure computes, and the case almost every real wobbly table is — has a fourth reaction whose zero crossing may not exist at any orientation, and no amount of turning helps. The advice works when the floor is the problem and fails when the table is, which is a diagnostic worth having: if turning it helps, the floor was uneven; if nothing helps, a leg is short.

And the resulting position is stable rather than level. All four feet touch, so the table does not rock; the top may be tilted by whatever the floor demands, and a wine glass on it will still lean. The theorem solves the indeterminacy and not the geometry.

Designing so the question does not arise #

The essay has said that determinate structures are built on purpose where predictability matters. It is worth saying how, because the technique is a discipline with a name and it is the direct engineering answer to everything above.

A rigid body in space has six freedoms. Constrain it with exactly six point contacts, each removing one freedom, and the body is fully located with nothing left over — no redundancy, no indeterminate forces, and no dependence on stiffness. That is exact-constraint or kinematic design.

The canonical realisation is the Kelvin clamp: three balls on the moving part, one landing in a trihedral cone that removes three freedoms, one in a V-groove that removes two, and one on a flat that removes one. Six contacts, six constraints. The alternative three-groove mount uses three V-grooves radiating at a hundred and twenty degrees, two constraints each, and is preferred where temperature changes matter because it expands symmetrically about its centre.

What such a mount buys is repeatability. Lift the part off and put it back, and it returns to the same place — to tens of nanometres in a good design, because there is exactly one geometric solution and the contacts have nowhere else to go. That is why optical mounts, metrology fixtures, wafer stages and the supports under large telescope mirrors are built this way, and why a three-point mount does not deform the mirror it carries.

The contrast is a bolted flange with six bolts on a machined face, which is what most things are. That joint is enormously over-constrained: the contact pressure distribution is indeterminate, it depends on the tightening sequence, it changes when the temperature does, and the part distorts by an amount nobody computed. It is stiffer and stronger than a kinematic mount by a wide margin, and it does not go back to the same place twice.

The trade is the essay’s own trade, stated in hardware. Exact constraint gives determinate forces and perfect repeatability and has no spare load path — every contact is critical, and all the load goes through six points, so the contact stresses are high and a single damaged ball loses the location. Over-constraint gives stiffness, load capacity and tolerance of a failed member, at the cost of not knowing what any member carries. Most machines want the second; the parts that have to be put back where they were want the first.

What the pictures cannot show #

Everything here is linear and small. The springs are Hookean, the top does not bend, and the deflections are far too small to change the geometry. A real table top sags, and a sagging top is a fifth freedom that changes the count again.

Another determinate case, for contrast: three shapes released on a slope arrive in an order set by how the mass is arranged and not at all by how the bodies deform. Rigid-body mechanics is exactly the set of problems in which that separation holds. Every problem in this essay is one where it fails — where the answer depends on stiffness, and the stiffness has to be known before the forces can be.

Friction is left out entirely. A real leg on a real floor also resists sideways motion, which adds constraints and can make an apparently indeterminate arrangement more indeterminate still. Whether a contact grips at all is its own difficulty.

The granular count ignores friction too, and friction changes the isostatic number: a frictionless pack needs $Z = 4$ in the plane, and a perfectly rough one needs only 3, because each contact then blocks a rotation as well. Real packs sit between.

And the rigidity matrix is drawn for one configuration. A pack moving along one of its free motions breaks contacts and makes new ones, so the count is a statement about the arrangement in front of it rather than about a trajectory.

The ladder from here #

Later rungs on this anchor: the force method and the displacement method, which are the two systematic ways of adding the missing equations; thermal loads in a redundant structure, which produce forces with no external load at all; prestressing, which is redundancy used deliberately to put a member into a state it could not otherwise reach; and the plastic hinge, where a redundant structure redistributes as members yield and the collapse load is determinate even though the working load sharing is not.

The neighbouring ladders are slide or topple, which is the determinate version of the same free-body question, the silo that does not weigh what it holds, where indeterminacy is what the scale reads, and the chatter a stiffer holder removes, where stiffness decides an outcome the force balance does not.