Built to the wrong length

A determinate structure can be built to any lengths at all. Cut every member a little short, a little long, a bit of each: the joints still close, because each member’s length simply decides where the next joint is. The structure ends up in a slightly different place and carries exactly the forces it was designed for.

A redundant one cannot. Its members’ lengths are not independent — choose all but one and geometry decides the last — so a member made a different length has to be pulled or pushed into position, and the force that takes stays in the structure for as long as the structure does.

Nothing has been applied to it. There is no load case, no combination, no partial factor. And the members are carrying real force.

Which free body produced the number #

The released structure, with the redundant force as the unknown — which is the force method with a misfit in place of a load.

Cut the redundant member and the structure becomes determinate. Two things then open up the gap at the cut: the misfit $e$ itself, and the flexibility of the released structure under a unit pair of forces across the cut. Closing the gap gives one equation, and with the self-stress state $\mathbf{s}$ — the set of member forces the structure supports with no load, which the forces that are there with nothing applied computes as the null vector of the equilibrium matrix — it is

$$X = -\frac{\sum s_i e_i}{\sum s_i^2 f_i}, \qquad f_i = \frac{L_i}{EA_i}$$

and every member’s force is $X s_i$. Two sums, and everything about the problem is in one or the other of them.

The numerator is the misfit projected onto the self-stress state. The denominator is the structure’s own flexibility in that state. Each of them says something the design office does not usually have in mind.

The force is proportional to stiffness #

That is the sentence that inverts everything.

Against a load, stiffness is what carries it: double the areas and the stresses halve. Against an imposed deformation, stiffness is what generates the force: double the areas and the force doubles, because the flexibility in the denominator has halved.

So a structure made stiffer is no better off at all under a misfit. Its force doubles and its capacity doubles, and the utilisation is exactly where it was. There is no reward for oversizing, and — worse — a designer who stiffens a frame for a deflection reason has doubled a force that appears in none of the load cases.

The same arithmetic covers a family of effects that are usually filed separately.

A temperature change is a lack of fit that arrives after the building is finished: $e = \alpha\Delta T L$. Thirty degrees on a 6 m member is 2.16 mm, which on this frame produces 43 kN — comparable with the 40 kN a fabrication tolerance produces. The movement nobody applied is the same equation with a different source for $e$.

A settled support is a lack of fit of the ground. The support that moved computes the moment field it produces, which is proportional to EI for the same reason.

A prestress is a lack of fit put in on purpose. The load put on backwards is a deliberate misfit whose sign has been chosen, and the machinery for computing it is this machinery.

And shrinkage, creep and relaxation are lacks of fit that change with time, which is what makes them the only members of the family that go away on their own.

The temperature case is the one most engineers have already met and is the same trade in its most familiar form: a restrained thermal movement produces a force proportional to the restraint’s stiffness, so a temperature effect gets worse as a structure gets better at carrying load. What makes the fabrication case harder to see is only that the misfit arrives before the building is finished rather than after it.

What the misfit has to be parallel to #

The numerator has its own lesson and it is a relieving one.

Only the component of the misfit along the self-stress state produces any force. A misfit orthogonal to it — in this bay, equal and opposite errors in the two diagonals in the right proportion — produces exactly zero, and the computation returns zero to machine precision rather than to a tolerance.

That is why a member can be substantially out of length in a direction the structure does not care about, and it is why some errors on site matter enormously and others do not at all. A frame with one redundancy has one direction in the space of member lengths that it objects to; the rest are free, and a structure with $n$ members and one redundancy is indifferent to $n-1$ dimensions of error.

More redundancies means more directions objected to. A structure with $r$ self-stress states has $r$ compatibility equations, and the misfit vector has to be resolved onto each of them.

So redundancy is bought and this is the price. A determinate structure has $s = 0$: no equation to satisfy, no force to develop, and a short member simply puts the joint somewhere else. One support too many sets out what redundancy buys — alternative load paths, and the robustness that comes with them — and the fit-up force is what it costs. That is a real trade and it is very rarely written down as one.

The tolerances, which are larger than anybody analyses #

The misfits that produce these forces are not hypothetical and are not small.

A steel member is cut to length within a millimetre or two. Five members in a load path accumulate about four and a half millimetres in quadrature. A column is plumbed to about a five-hundredth of a storey height, seven millimetres over 3.5 m. And the erection tolerance on a whole frame — the permitted out-of-plumb of a building — is twenty-five millimetres.

Against the 20 kN per millimetre this bay develops, those are 40, 90, 140 and 500 kN. The last of them exceeds the 400 kN the diagonal was sized for.

The rate at the top of that figure is the whole of it: 20.0 kN per millimetre, a property of the frame rather than of the error. Every row is that rate times a length off a tolerance table, which is why the table can be read straight across and why nothing in it depends on the loading. Halve the rate and the whole column halves with it.

That is the practical form of the proportionality, and it is worth being clear about what it is not. Halving the areas halves the fit-up force and halves the design capacity along with it, so a member sized for load alone gains nothing. What it gains is against the errors, which do not scale with the member: the tolerance table is written in millimetres and stays where it is however light the frame gets, so the ratio of fit-up force to capacity does fall. A light flexible frame absorbs the trade’s ordinary inaccuracy; a heavy stiff one converts it into force.

Why the building is still standing #

The obvious question is why, given the arithmetic above, buildings are not full of members failing under forces nobody applied. It is worth drawing the worst row of that table before answering, because the four mechanisms below are what stands between it and a failure.

Nothing in that figure is a load, and no combination contains it. Four answers, and each is a real mechanism.

Bolt holes are two millimetres oversize. A 22 mm hole for a 20 mm bolt lets a connection take up two millimetres before it starts developing force at all, and a joint with several bolts in slotted or clearance holes accommodates several. That is not sloppiness; it is a designed-in release, and it is why a bearing-type connection is far more forgiving of misfit than a slip-critical one.

The frame is bolted loose, plumbed, and only then tightened. The sequence is the profession’s real answer to this whole essay: while the bolts are loose the frame is effectively determinate, so the members go where they go, and the geometry is corrected before any stiffness is created. The order of operations is a structural decision, and it is made by an erector rather than by a designer.

Steel yields. A member at 25% of its capacity from a misfit and 90% from load reaches yield, sheds the misfit force by a few tenths of a millimetre of plastic strain, and carries on. The amount of plastic strain needed is tiny — the whole misfit is a few millimetres over several metres, which is a few hundred microstrain against a yield strain of about 1,700 — so the shedding costs almost nothing in ductility, and — the moment that was moved on purpose is the same relief seen in bending. An imposed-deformation force is self-limiting in a ductile structure and is not in a brittle one, which is why the effect is far more serious in bolted timber, in cast iron and in unreinforced concrete than in steel.

And everything relaxes. Creep in concrete, relaxation in prestressing steel, and slip at every interface all reduce a force that was imposed as a displacement. Sixty per cent relaxation takes the 100 kN here to 40, over years.

None of those four is an argument for ignoring the effect. They are the reason it is survivable, and the conditions under which it is not — a brittle material, a slip-critical joint, a stiff frame, a fast-acting misfit — are exactly the conditions in which it has caused failures.

Prestress, which is this done deliberately #

The most useful way to hold the whole subject is that a lack of fit and a prestress are the same operation with the sign chosen.

A prestressing tendon is a member made deliberately short and then pulled to length. The force that takes is exactly the $X$ above, and its distribution through the structure is exactly $Xs_i$ — the self-stress state — which is why a prestress applied at one point appears everywhere and why a prestressed indeterminate structure develops secondary moments that nobody applied. The prestress that pushes back is those secondary effects, and they are the self-stress state of the released structure.

Seen that way, three things fall into place.

Those two figures are the same operation and the difference between an accident and a design is which of them somebody meant. Read the second as prestress and the argument is complete: the diagonals are the tendons, the perimeter takes the equal and opposite reaction, and a member deliberately made the wrong length has put a chosen force into every part of a structure with nothing standing on it.

A self-stressed structure is a structure with a chosen misfit. A cable net, a tensegrity, a spoked wheel and a bicycle wheel are all structures whose whole stiffness comes from a lack of fit that somebody put in — and their geometry has to be controlled to the millimetre for exactly the reason this essay is about.

Adjusting a structure to a force is easier than building it to a length. A turnbuckle in a cable is a way of choosing $X$ directly instead of choosing $e$ and accepting whatever $X$ follows. On a cable roof the cables are tensioned to measured forces and their lengths come out wherever they come out, which is the whole problem inverted.

And a misfit can be used to cancel a load effect. Jacking a support, shimming a bearing, or pre-cambering a truss against its dead load are all deliberate misfits chosen so that $Xs_i$ subtracts from the load’s own force distribution. That only works where the two distributions have similar shapes — the self-stress state is fixed by the geometry and cannot be chosen — which is the limitation of every such scheme.

And what the redundancy was bought for is the same vector read the other way. Losing a member in a redundant frame redistributes rather than collapses, and the paths that wake up to carry the loss are the self-stress state — the same one that made the frame object to being built wrong. The benefit and the cost are not two properties in tension with each other; they are one property, and a frame that cannot develop a fit-up force cannot redistribute either.

Which load combination it belongs in #

There is a question this essay has carefully not asked, and it is the one a designer actually faces: where does a fit-up force go in the combinations?

It has none of the properties a load has. It is not variable, so it has no combination factor. It is not permanent in the sense dead load is, because it can relax away. Its sign is unknown, so it cannot be taken as favourable or unfavourable by inspection — it has to be taken in whichever direction is worse for each check, which means it appears twice. And it is present in every combination, because it does not depend on what is standing on the structure.

Practice splits, and the split is instructive because it is drawn along ductility rather than along magnitude.

Where the effect is deliberate and the material is brittle, it is a designed action. A prestress has a partial factor, a set of losses, a short-term and a long-term value, and appears explicitly in every combination. Nobody would omit it, because the whole structure depends on it and concrete will not shed it.

Where the effect is accidental and the material is ductile, it is largely omitted. No load combination in ordinary steel design contains a fabrication misfit. The justification is the four release mechanisms of the previous section, and in particular the third: a steel member sheds an imposed-deformation force by a few hundred microstrain of plastic strain, so at the ultimate limit state the force is simply not there.

That justification is sound and it is worth knowing exactly how far it reaches, because it is a statement about one limit state only.

At ultimate, an imposed-deformation force in a ductile structure is self-cancelling and omitting it is right. At serviceability, nothing has yielded, nothing has shed, and the force is present at its full elastic value — so a deflection, a crack width, a joint slip, or a check on whether a slip-critical connection has slipped is being made on a structure carrying a force the calculation does not contain.

Which is the same conclusion the fatigue and stiffness essays in this field keep reaching from other directions, and it is worth stating as the rule: the argument that lets a misfit be ignored is a plasticity argument, so it is available at exactly the limit state where plasticity is. Anywhere the check is elastic, the force is real and the combinations do not have it.

Where it has gone wrong #

The failures cluster, and the cluster is informative.

Pre-tensioned bolted assemblies in stiff frames, where a slip-critical joint has no clearance to take up and the frame has no ductility to shed with. Long-span trusses assembled on the ground and lifted, where the geometry of the frame on its temporary supports is not the geometry it will have in place, so the closing member is measured in the wrong configuration. Cable-stayed and tensegrity structures, where the whole scheme is a self-stress state and every cable length is a misfit waiting to happen — which is why they are built with turnbuckles and adjusted against measured forces rather than to a length.

And concrete structures cast against existing ones: a new slab poured against an old one, a repair, an underpinning. The new element shrinks against something that has already shrunk, which is a misfit of several hundred microstrain applied to a stiff, brittle, un-ductile connection. That is the standard reason a repair cracks at its interface.

Where the model stops #

The frame was pin-jointed. A rigid-jointed frame has misfits in rotation as well as in length — a beam fabricated with its end plates a degree out of square — and those are just as real and harder to see. The same machinery covers them with a rotational self-stress state.

The misfit was elastic and one-off. A member forced into place beyond yield has been permanently shortened or lengthened, and the locked-in force is then whatever the plastic strain left rather than what the elastic calculation says.

The connection was assumed rigid. It is not, and connection flexibility is the largest of the release mechanisms above — a semi-rigid joint under a misfit rotates, and neither pinned nor rigid is the flexibility that does it.

Only one redundancy was solved. A frame with several self-stress states has a matrix equation rather than a single division, and the states interact — a misfit that is orthogonal to one state is generally not orthogonal to the others. The relieving conclusion above survives only in the sense that the misfit vector has $n - r$ free dimensions rather than $n - 1$.

And the misfit was known. It is not, ever. What is known is a tolerance — a bound on a random quantity whose sign is unknown — so the honest treatment is statistical, and the honest question is how much of the tolerance envelope the structure can absorb rather than what force a particular error produces.

The generalisation #

The idea to take away is that a structure’s members have a geometry as well as a strength, and redundancy makes the geometry a structural quantity.

Every essay in this collection about redundancy has treated it as a benefit — an alternative load path, a warning before collapse, a distribution that can be chosen. It is also a constraint: a redundant structure has fewer independent lengths than it has members, and every difference between the lengths it was given and the lengths it needs is a set of forces.

The practical form of that is a question to ask on any indeterminate structure: what has to fit, and to what? In a braced bay it is one diagonal. In a continuous beam it is the levels of the supports. In a portal frame it is the plumb of the columns and the squareness of the haunches. In a cable roof it is every cable. In a composite section it is the concrete’s shrinkage against the steel’s indifference to it.

And the answer, when the fit cannot be guaranteed, is nearly always the same and is the erector’s rather than the designer’s: make it adjustable, and build it in an order that lets the geometry be right before the stiffness exists. Slotted holes, shims, turnbuckles, packing plates, a joint left ungrouted until the frame is plumbed — every one of them is a small admission that the structure will not be built to the lengths it was drawn to, and each of them is worth more than the material that would otherwise have to carry the difference. The most dangerous day is the same observation about the erection condition generally: the structure that is being assembled is not the structure that was designed, and most of what protects it is decided by people who were not in the design office.