Counting the unknowns, and finding out whether statics can answer

Before a structure is analysed it is worth asking whether it can be. The answer comes from counting, it takes ten seconds, and it decides which body of theory the problem belongs to.

Count the unknowns: one axial force per member, one component per restrained direction at each support. Count the equations: two per joint in a plane. Subtract. The sign of what is left says everything.

The three cases #

Fewer unknowns than equations. The equations cannot all be satisfied, and the structure moves. It is a mechanism: a four-bar square with pinned corners folds into a rhombus without stretching anything, and no amount of member strength prevents it. This is the failure mode that kills people during construction, when a frame is complete but the bracing has not gone in.

Exactly as many. The system has a unique solution and statics finds it. The structure is statically determinate, and everything on this site that is solved by equilibrium alone lives here.

More unknowns than equations. The system has infinitely many solutions satisfying equilibrium, and equilibrium alone cannot choose between them. The structure is statically indeterminate, or redundant, and finishing the job needs to know how stiff each member is.

The counting rule for a plane pin-jointed truss is

$$m + r - 2j,$$

with $m$ members, $r$ restraint components and $j$ joints. Negative is a mechanism, zero is determinate, positive is the degree of redundancy.

The second and third frames in that figure differ by one diagonal. The extra member adds an unknown without adding an equation, so the count goes from zero to one and the structure passes out of reach of statics.

Why an extra member is not free #

The intuition that more members means a stronger structure is correct and incomplete. The extra diagonal in the second figure does make the frame stronger. It also makes it a different kind of problem.

With two diagonals rather than one, the load can travel by two routes, and equilibrium does not say how it splits. Any split that adds up is in equilibrium — all the load down one diagonal, all down the other, or any mixture. Choosing among them requires a further principle, and the principle is compatibility: the two diagonals share their end joints, so whatever the load does it must leave both members fitting the same deformed shape.

That converts the problem from a force problem into a force-and-displacement problem. The stiffer route takes more load, in proportion to stiffness, and stiffness depends on the material, the area and the length — none of which appeared anywhere in statics.

The practical consequence is that a redundant structure is sensitive to things a determinate one is not. A support that settles a few millimetres redistributes load in a redundant frame and does nothing at all in a determinate one. Temperature change does the same. A member fabricated slightly short is stressed before any load arrives.

What redundancy buys #

Given the cost, redundancy is nonetheless normal, and for one reason: it survives losing something.

A determinate structure has exactly enough members, and every one of them can be found by statics. Remove one and the count goes negative — it becomes a mechanism, and it comes down. There is no alternative route because the count says there is none.

A redundant structure has a spare. Remove a member and it is still standing, with the load redistributed to the remaining routes, which is the load path being re-chosen under duress. That property is called robustness, and after the progressive collapse at Ronan Point in 1968 it became a design requirement rather than a bonus.

So the trade is explicit. Determinacy buys analysability, insensitivity to settlement and temperature, and clean force paths. Redundancy buys survival. Most real structures choose redundancy and pay the analysis cost, which since about 1960 has been paid by a computer.

The two members carrying nothing are worth a moment, because they look like the spare capacity this section is about and are not. A zero-force member in a determinate truss is zero for this load case only; it is holding a joint’s geometry, it appears in the count like any other member, and taking it out still turns the structure into a mechanism.

The other thing redundancy buys is efficiency. Building in both ends of a beam drops the peak moment to two-thirds of the simply supported value, because the ends now take a share. The material saving is real, and it is unavailable to a determinate structure.

Counting a truss #

The rule generalises with care. For a pin-jointed truss the formula above is right. For a rigid-jointed frame each member carries three unknowns rather than one, and the count changes accordingly.

Change the web arrangement and the count changes with it, because a different pattern of diagonals is a different number of members and, usually, a different number of joints. The sum still has to come out at zero for the same reason.

The arithmetic can also come out identical for two trusses that behave quite differently, which is the clearest demonstration that the count is about solvability and about nothing else.

Both are determinate, and the solver behind these figures proves it by returning a unique answer. When a frame is redundant the same solver has to fall back on a least-squares solution, which is a warning sign rather than an answer.

There is a subtlety the count cannot see. A structure can have the right number of members and still be a mechanism if they are arranged badly — three collinear members restraining a joint, or a panel with no diagonal while another has two. The count is necessary and not sufficient, and the honest test is whether the equations are independent, which is what the solver actually checks.

The count is a statement about a matrix #

The subtlety at the end of the last section — that a frame can have the right count and still be a mechanism — is not a wart on the rule. It is the rule being a shadow of something exact, and the exact thing is worth having, because it explains every exception at once.

Assemble the equilibrium equations for the whole frame rather than one joint at a time. Each of the $2j$ equations is a row; each of the $m + r$ unknowns is a column. The result is an equilibrium matrix $A$, and the statement $A\mathbf{f} = \mathbf{p}$ says that the member forces $\mathbf{f}$ balance the applied loads $\mathbf{p}$. That is the matrix the generators on this site build, and the elimination they run is nothing more than its solution.

Now the count $m + r - 2j$ is just the number of columns minus the number of rows, and shape is not rank. Two quantities matter that the shape cannot see:

The null space of $A$ — force systems that satisfy equilibrium with no load applied at all. Each independent one is a state of self-stress: a set of member forces in balance with nothing. Call the number of them $s$.

The null space of $A^{\mathsf{T}}$ — joint displacement patterns that stretch no member. Each independent one is an inextensional mechanism, a way the frame can move without any bar changing length. Call the number of them $k$.

Linear algebra then supplies the exact statement, established by Calladine in 1978:

$$m + r - 2j = s - k.$$

Maxwell’s rule is the special case where both $s$ and $k$ are zero. Everything that looked like an exception falls out of the general form. A frame with the right count and a badly arranged member has one self-stress state and one mechanism, which cancel in the count and do not cancel in reality. Three collinear bars restraining a joint give a mechanism perpendicular to them and a self-stress along them, and the arithmetic reports zero while the structure moves.

The most spectacular case is the one where the count is negative and the structure stands. A tensegrity — cables and struts with the cables in tension throughout — counts as a mechanism, sometimes as several. It has, however, at least one state of self-stress, and prestressing the cables into that state stiffens the mechanisms: the structure resists load not because the geometry forbids motion but because moving would slacken something already tight. A cable net roof is the same argument at building scale. Counting says these cannot work; rank says exactly how they do.

There is a practical residue for anyone reading a figure on this site. When the count is not zero, the solver here falls back on a least-squares solution, which quietly returns one answer from a family of them — a perfectly reasonable numerical act and a completely unreasonable structural claim. A least-squares result on a redundant frame is the answer for one particular assumed stiffness distribution, and on a mechanism it is the answer to a question that has none.

A structure with a self-stress can be stressed by nothing at all #

The self-stress states counted by $s$ above are usually introduced as a bookkeeping curiosity — force systems in balance with no load. They are not a curiosity. They are the reason a redundant structure can be fully stressed on a still day with nothing on it.

A determinate frame has $s = 0$, and the consequence is worth stating plainly: there is no set of member forces in equilibrium with zero load other than zero. So if a member is fabricated ten millimetres short, the frame simply assembles into a slightly different shape and no force appears anywhere. The same is true of a support that settles, and of a temperature rise: the structure changes shape freely, and shape change with no force is exactly what having no self-stress state means.

Add one redundant member and $s$ becomes one. Now there is a non-zero force system in balance with nothing, and any imposed distortion that the structure cannot accommodate by changing shape has somewhere to go — into that state, scaled by however much distortion was imposed. The short member is stretched into place on assembly and pulls on everything it connects to, permanently, before any load is applied.

The numbers are not small. A steel member 10 m long fabricated 5 mm short is being asked for a strain of $5\times10^{-4}$, and at $E = 205$ kN/mm² that is 102 N/mm² — over a quarter of the yield stress of ordinary structural steel, from a fabrication error well inside what a tape measure on a windy site will detect. A uniform temperature change of 30 °C in a fully restrained member gives $\alpha \Delta T E = 12\times10^{-6} \times 30 \times 205\,000 = 74$ N/mm², again with nothing standing on the structure.

Three consequences follow, and they are the practical content of the whole section on redundancy above. Redundant structures need their fabrication tolerances specified and their erection sequence controlled, because both decide what self-stress is locked in. They need movement joints, or a deliberate accounting of temperature as a load case, because a restrained structure has no free way to expand. And the same mechanism run deliberately is prestress: choosing the self-stress state rather than inheriting it, and using it to put the structure into a helpful state before the load arrives — which is the whole of prestressed concrete and of every cable structure that is tightened rather than merely connected.

The count applies to a moment in time #

A structure is determinate or redundant in a particular configuration, with a particular set of members present and a particular set of supports acting. Every one of those is a fact about a snapshot, and the finished building is not the dangerous snapshot.

A steel frame is erected column by column and beam by beam. The bracing that makes it stable typically arrives after the members it braces, sometimes days after, because the bays have to be plumbed and the connections adjusted before the diagonals can be fitted. Between those two moments the frame is a mechanism by count, held up by the friction in its bolted connections and by temporary guys, and it is carrying the weight of everything erected so far plus whatever plant is standing on it.

The same reversal occurs in concrete. A continuous slab is redundant when it has cured and is propped as a series of simple spans while it has not, so the moment diagram it is designed for is not the moment diagram it first experiences. Striking the props transfers the load in a sequence, and a slab struck too early sees a load case nobody drew. In bridge construction the effect is deliberate and enormous: a balanced cantilever is a cantilever throughout its construction and a continuous beam only at the moment the closure pour sets, so the permanent structure carries the memory of the forces locked in on the way to existing.

The general statement is that determinacy is a property of the structure and its history, and analysis of the finished article does not contain the history. Which is why temporary works are designed by specialists, why the erection sequence is a drawing rather than an assumption, and why so large a share of structural fatalities happen to structures that would have been perfectly safe an hour later.

What the count does not tell #

Determinacy is a statement about solvability, not about strength, stiffness, or whether the structure is any good.

A determinate truss can be hopelessly under-designed. A redundant frame can be a mechanism in disguise. A structure with a count of zero and one member ten times too small collapses at a tenth of the intended load, and the count is entirely silent about it.

The count also says nothing about which arrangement is better. The Pratt and the Howe have identical counts and put their diagonals in tension and compression respectively — a difference that decides which one to build in steel and which in timber, and one the count cannot express.

The other thing the count decides #

Redundancy is not only an analysis problem. It changes what the structure does under load.

A built-in beam peaks at two-thirds of the simply supported moment and deflects a fifth as far, which for a member sized by how far it moves is the difference between two sections. That gain is unavailable to a determinate structure, and it is bought with the analysis cost the count predicts.

A deflected shape is obtained by integrating the moment diagram twice, and for a redundant structure the moment diagram itself depends on that curve.

The circularity in that last sentence is the whole difficulty. In a determinate structure the forces come first and the deflections follow; in a redundant one each depends on the other, and the solution has to satisfy both at once.

Where the model stops #

Pin joints. The formula above assumes joints that transmit no moment, which is the truss idealisation. Real connections are bolted or welded and do transmit moment, which adds unknowns and usually adds redundancy. A truss analysed as pin-jointed and built with welded joints has secondary bending stresses that the analysis does not contain.

Plane structures. In three dimensions there are three equations per joint and the formula becomes $m + r - 3j$. Space frames are counted differently and fail differently.

Small deflections. The count is a statement about the equations of the undeformed structure. A cable net has a count that says mechanism and stands up anyway, because its geometry changes under load until it can carry it — the analysis is genuinely nonlinear and the linear count does not apply.

No internal releases. A hinge inserted in a member adds an equation, and a three-pinned arch is determinate for exactly that reason. Any count that ignores releases will call it redundant.

The figures have a limitation of their own: they show the count as an arithmetic caption beside a drawing, which makes it look like a property that can be seen. It cannot. Two frames that look nearly identical can differ in count by one member, and the difference in what can then be said about them is total.

The ladder from here #

Later rungs: internal releases and how they change the count. The three-pinned arch, determinate by design. Rigid-jointed frames and their different formula. Space frames. Kinematic indeterminacy, which counts degrees of freedom rather than forces. The force method and the compatibility equations that resolve redundancy. The displacement method, which is what every structural program actually does. Robustness and progressive collapse. And prestress in a mechanism, which is how a cable net or a tensegrity stands up while counting as a mechanism throughout.

Maxwell wrote the counting rule down in 1864 in a paper about reciprocal figures, as a side remark. It carries his name and is the least of what that paper contains.

He also, in the same remark, noted the exception — that a frame satisfying the count might still be movable if its members were arranged so that the equations were not independent — and then left it there, since the paper was about something else. It took until 1978 for Calladine to put the correction in the form given above and to point out that Maxwell’s aside was not a caveat but the general case. A rule quoted in every textbook for a century, with its own exception written down by its own author in its own founding paper, and the exception overlooked because the rule was so much easier to apply than to justify. That sequence is not unusual, and it is worth remembering whenever a piece of structural arithmetic seems too cheap for what it claims to deliver.