Six equations, and the drawing shows three

Everything in this collection so far has happened on a sheet of paper. A structure is a diagram, forces have two components, moments are numbers with a sign, and equilibrium is three equations: two sums of force and one of moment.

That is not equilibrium. It is a plane section through equilibrium, and the object it describes is a plane section through a structure. The real statement has six equations in it — three forces and three moments, one about each axis — and the three that never appear are doing work that somebody has to do.

The tripod is the three-dimensional twin of the simply supported beam: the smallest arrangement that holds a rigid body and has exactly as many unknowns as there are equations. It is worth starting there because the fourth leg — the one everybody would add without thinking — changes the problem into a different kind of problem entirely.

Where the other three equations went #

In a plane drawing, three of the six equations are satisfied by nothing at all, in the sense that no term in them is non-zero. Every force lies in the plane, so the sum of forces perpendicular to it is 0 = 0. Every moment turns about the axis normal to the page, so the sums about the two in-plane axes are also 0 = 0.

That is not a proof that they are safe; it is a statement that the drawing has assumed there is nothing to satisfy. The moment a real load leans the smallest amount out of plane, or a real member has any width, those three equations acquire terms, and the restraints that answer them have to exist somewhere. In a building they are supplied by the floor slabs, the bracing in the other direction, and the beams framing in at right angles — none of which is on the drawing of the frame being analysed.

A plane frame analysis is therefore a calculation that has been handed three of its answers by an assumption, and the assumption is a real structural requirement being met by real members somewhere else. The habit of naming the free body, which this site insists on everywhere, has a three-dimensional version: name the plane, and name what holds the body in it.

Every reaction in a plane drawing is a member somewhere else #

The habit that makes this tractable is one this collection already has, applied one dimension further out.

The three-dimensional version of that drawing has three more arrows on it, and the discipline is to ask what supplies each. A beam’s plane analysis quietly requires a force perpendicular to the page at each support and a moment about each in-plane axis; in a real building the first is delivered by the floor diaphragm and the second by the beams framing in at right angles. Both are real, both are designed by somebody, and neither is in the calculation that needed them.

Where this goes wrong is not usually that the restraint is missing — buildings have plenty of restraint — but that nobody has costed it. The force that holds a truss upright in its own plane is small and it is not zero, and the member that supplies it is usually a light one whose size was chosen by eye. That member is then the only thing standing between a correct plane analysis and a structure lying on its side, which is a poor position for a member nobody calculated.

A moment is about an axis, not about a point #

The conceptual shift that plane statics conceals is that a moment stops being a number and becomes a vector — and what a structure resists is not the moment but its component along an axis the structure has something to resist with.

The practical consequence appears everywhere once it is noticed. A beam loaded slightly off its shear centre is being asked for a torque as well as a moment, and the torque has to go somewhere. A purlin on a sloping roof is loaded vertically and its section’s axes are not vertical, so the moment resolves into two components and the member deflects out of the roof plane. A bracket bolted to a column face carries a moment whose axis is horizontal and a torque whose axis is not, and the same bolt group resists both by entirely different mechanisms.

None of that is exotic. It is the ordinary condition of a structural member, and the reason it is rarely computed is that the drawing that would show it is a three-dimensional one, which is expensive to draw and hard to read — so the profession has arranged itself around plane drawings and pays for the arrangement in cases like these.

Three legs cannot wobble and four legs always do #

Return to the table. Adding a fourth leg gives four unknowns against the same three equations, and the arithmetic changes character.

The redundancy is exactly one, and it has a shape. The null vector of the equilibrium matrix is $(+1, -1, +1, -1)$: push one diagonal pair down and lift the other by the same amount, and every one of the three equations is still satisfied, with no load applied anywhere. That is a state of self-stress, and it is the same object a redundant plane frame carries — a set of internal forces in equilibrium with nothing.

A three-legged table has no such vector, and that is the whole of why it never rocks. It is not that three legs are better arranged or that the floor is kinder to them; it is that the system of equations has a unique solution, so there is no second state for the table to be in.

The number that matters in that figure is 4 mm. A 12,000 kN/m leg is a stiff one, and four millimetres is a trivial error in a floor, and between them they redistribute half the load. Push the error to 10 mm and two legs carry 50 kN each while the other two carry nothing and the table rocks — which is the everyday experience, arrived at as an eigenvector rather than as an annoyance.

Every indeterminate structure is a table on an uneven floor. What decides its forces is not the load but the difference between the geometry it was built to and the geometry it was built at, and that difference is not on any drawing.

The same arithmetic turns up in a place where somebody has already drawn the conclusion and written it into a rule. A load lifted on four sling legs from one hook is the table turned upside down: four unknown leg forces, and because every leg’s line passes through the hook, the two moment equations contribute nothing at all — so there are three force equations and one redundancy, exactly the table’s count arrived at from the other direction.

Eight point eight millimetres is a chain fabrication tolerance rather than an accident, which is why the rating rule for a four-leg sling is not an approximation but an admission: it is rated as though it had two legs, at 57.7 kN each, the line drawn across the bars. The table’s self-stress vector and the rigger’s derating factor are the same null vector, read once by an engineer and once by a standards committee.

Rank, not the count #

The counting rule that works so well in the plane — members plus reactions against twice the joints — has a three-dimensional analogue with a three in it. It is subject to exactly the same failure, and for the same reason.

The three-dimensional version of the trap is more common than the plane one, because the geometries that produce it are ordinary. Three legs whose plan positions are collinear have a singular matrix even though the count is right: the body can rotate about the line through them. Four columns on a rectangle with pinned bases and no bracing form a mechanism in plan, which no count of the columns will reveal. And a set of supports all of whose reaction lines pass through one axis leaves the body free to spin about it — which is the three-force theorem generalised, and it is the reason that theorem is worth remembering as a picture rather than as a rule about triangles.

The plane taxonomy carries across whole — too few restraints and the body moves, exactly enough and statics answers, too many and the answer needs stiffness — and the only thing that changes is the number to compare against, six rather than three per rigid body. What does not carry across is how easy the geometrical exceptions are to walk into, and the collinear tripod is worth drawing because the failure is not sudden.

Those three numbers sum to 120 and are useless. A structure whose equilibrium matrix is nearly singular does not warn anybody: it returns a solution, to full precision, in which the forces are far larger than the load and are exquisitely sensitive to the geometry — move the middle leg another 40 mm towards the line and they double again. The count sees none of it, and neither does any check that reads the answer rather than the matrix that produced it.

The exact-constraint argument, and why structures ignore it #

There is a discipline in which all of this is treated as a design principle rather than a curiosity. A precision instrument is mounted on exactly six constraints — no more — because a seventh would deform it by whatever the manufacturing error was, exactly as the fourth leg deforms the table. The rule is that a rigid body has six freedoms and each constraint removes one, and the arrangement that removes all six once is the one whose behaviour is predictable.

Structural engineering does the opposite on purpose, and the reason is worth stating because it is not carelessness. A determinate structure has no alternative load path: lose one member and it is a mechanism, which is exactly what a redundancy study measures. Buildings are built with far more constraints than they need because the cost of the extra ones is a redistribution nobody can predict, and the benefit is that the structure survives losing any one of them.

The arithmetic that separates them is one subtraction. Six freedoms, one constraint each, and the seventh constraint is where predictability ends: before it, the forces follow from the load; after it, they follow from the load and from every millimetre by which the parts fail to fit. An instrument builder counts to six and stops. A structural engineer counts to six, keeps going, and accepts that the answer now depends on a set of dimensions that exist only on site.

So the two disciplines make opposite choices from the same arithmetic. An instrument buys predictability and gives up robustness; a building buys robustness and gives up predictability. Neither is wrong, and the reason the trade exists at all is the single fact this essay is about: beyond the sixth constraint, forces stop being determined by loads and start being determined by fit.

The tripod is determinate for one load and a mechanism for the rest #

The figure this essay opened with deserves the treatment the essay recommends, because it fails it.

Three legs, three unknown leg forces, three equations, a unique answer — that is the arithmetic quoted above, and every word of it is true of the vertical load it was given. Count what was actually used: the vertical force sum, and the two moment sums about horizontal axes. Three of the six. The remaining three are the two horizontal force sums and the moment about the vertical axis, and if the legs are what the drawing shows — pin-ended members carrying axial force along their own lines, all three of those lines vertical — then not one of the three has a non-zero term in it.

All three reaction lines are parallel. That is the space version of the parallel-reaction critical form, and the consequences are the ones that form always has: nothing resists a horizontal push in either direction, and nothing resists a twist about the vertical. The tripod is a mechanism with three freedoms, and the equilibrium matrix that returned 56.84, 13.58 and 49.58 kN so decisively has rank three against six rows.

There is a second and smaller way the same figure can be read past its own validity, and it needs no missing equation at all. The arithmetic returns three numbers whatever the load position, including positions where one of them is negative.

A negative reaction is the equations reporting, correctly, that the problem they were given is not the problem in front of them. The support was modelled as a two-way restraint and it is a one-way one, and the whole of overturning is that substitution — which is why the tipping line, rather than the leg force, is what a stability check is written about.

None of which makes the numbers wrong. They are the right answer to the question that was asked, which was how a vertical load divides between three vertical legs, and the load case contains nothing the missing equations would have had to resist. That is the honest description of every plane analysis in this collection, arrived at on the simplest possible object: the answer is exact, and it is exact for a load case that has been chosen to lie inside what the model can see.

What holds a real stool up is then worth naming, because it is not the legs’ axial force. It is friction between the feet and the floor, supplying the two horizontal sums; and the bending stiffness of the legs and the joints between them and the top, supplying the twist. Both are real, both are usually adequate, and neither is in the calculation. A stool with polished feet on a polished floor slides, and a stool whose leg-to-top joints have loosened racks and then folds — which are precisely the two failures the three missing equations predict, occurring in the order the equations are listed.

The design habit that follows is the inventory this essay ends on, and it costs a minute. Write the six equations for the free body. For each, name the term that satisfies it. Any equation whose terms are all zero is a freedom the structure does not restrain, and it is restrained by something outside the drawing or it is not restrained at all. On the tripod that exercise takes three lines and finds three answers that are not in any figure on this page.

Where the third dimension bites in practice #

Three cases account for most of it, and each is a plane analysis whose missing equations were supplied by somebody else.

A beam that is not restrained at its ends against twisting. The plane analysis gives a bending moment; the beam responds by buckling sideways, which is a displacement out of the plane the analysis was drawn in. The restraint that prevents it is a torsional one at the supports, and it appears nowhere in the moment diagram.

A frame braced in one direction and not the other. The plane analysis of the braced direction is complete and correct, and the structure falls over the other way. This sounds too crude to happen and it is the single commonest cause of collapse during construction, when the bracing in one direction has been designed and the sequence has not yet installed it.

A load applied out of the plane of a truss. Wind on the face of a roof truss is resisted by the roof plane and the bracing, which are three-dimensional and were designed by somebody looking at a plan rather than at the truss elevation. The truss’s own analysis has no term for it and cannot be asked whether it is safe.

In all three, the structure is fine if the missing restraint exists and gone if it does not, and the analysis that was performed cannot distinguish the two situations. The check is not a calculation but an inventory: name the six equations, and for each one name the member that satisfies it.

What the picture cannot show #

The legs in every figure above are axial springs and nothing else. A real leg has bending stiffness, a real joint between leg and top transmits moment, and the reactions are then decided by a much larger stiffness problem in which the plan positions are only part of the story.

The load is vertical everywhere. A horizontal load on the same tripod is resisted by whatever restrains it in plan, which the figures do not draw because a plan view of a vertical reaction is a dot. The general case has six unknowns per support and needs a drawing this collection has not yet found a good way to make.

The rank calculations are exact arithmetic on an exact geometry. A frame that is a mechanism by an angle of a tenth of a degree is reported as sound and is, for practical purposes, a very flexible structure rather than a stable one — which is the near-critical form problem, and it is worse in three dimensions because there are more ways to be nearly singular.

Where the ladder goes #

The first rung out of here is the one the wobbling table keeps pointing at: what an indeterminate structure’s forces actually depend on, which is fit and stiffness rather than load, and which this collection has begun to answer in the plane and not in space.

The second is the moment that is left over when a load misses an axis. That component has to be carried by something, and what carries it is torsion — the internal force that has no diagram anywhere else in this collection, for the good reason that a plane drawing cannot contain it.

The third is the practical inventory above, made into a design method. A structure that has been checked equation by equation in three dimensions, with a named member answering each, is a structure whose collapse mechanisms have all been examined — and that is a stronger statement than any amount of member checking, because it asks the question that member checking assumes has already been answered.