The gap nobody computed

A partition runs up to the underside of a floor slab and stops. The gap between the two is filled with a compressible seal and covered with a bead, and the drawing says 25 mm.

Where did 25 come from?

The honest answer, on most projects, is that it came from a standard detail. And the reason that is not indefensible is that computing it properly is a job nobody’s discipline owns: the structural engineer computes one of the five terms, the concrete specialist knows two more, the contractor owns the fourth, and the fifth is a statistical question about all of them together.

Which free body produced the number #

There is no free body here, and saying so is worth doing because it explains why the calculation is orphaned.

Every other quantity in this collection is a force or a stress, arrived at by cutting something and insisting the sums cancel. A movement budget is none of those. It is an accumulation of displacements, each computed by a different method, most of them not forces at all, and the answer is a length rather than a stress.

That is why the calculation has no natural home in a set of calculations organised by member. It belongs to the joint, and a joint is a place where two members are not, which is precisely the region every member-based method leaves out.

The five terms, and their ordering #

For a concrete frame with a 6 m storey and a 30 m plan dimension, the movements at a wall head are roughly these.

Thermal — the slab above expands and contracts relative to the wall below, at $\alpha\Delta T L$ over whatever length is restrained. Nine or ten millimetres, and it reverses, which is what makes it different from the others.

Shrinkage — a concrete slab shrinks by 300 to 500 microstrain over its life, most of it in the first year. Six millimetres over the same length, one way only, and it never comes back. Where it is restrained it becomes a curvature rather than a movement, which is the curvature nobody applied and adds to the deflection term as well.

Creep — the slab shortens under sustained compression by two or three times its elastic shortening. Four or five millimetres, again one way.

Deflection — the slab above sags under its own weight and its live load, and the sag at the wall’s position is what closes the gap. Three millimetres, and the only one anybody calculates properly. It is also the only one with a code limit attached to it, which is part of why it receives the attention: span over 360 is a number a check can be written against, and there is no equivalent clause for any of the other four.

Tolerance — the slab is cast within a construction tolerance of its intended level, and the wall is built to within another. Ten millimetres, comfortably the largest term, and it is not a structural quantity at all.

The smallest term is the computed one and the largest is a builder’s tolerance. Every hour of analysis spent refining the deflection prediction is being spent on a tenth of the answer, and the term that dominates it is decided by a spirit level.

Two combination rules, and the answer between them #

Add the extremes and the answer is 33 mm. That assumes the hottest day, the fullest floor, the whole of the shrinkage, the whole of the creep, and the worst-placed wall all arrive at once.

Treat them as independent and take the root-sum-square, and the answer is 16 mm — less than half.

Neither is right and the reason is structural rather than statistical. Some of the terms are not random at all: shrinkage and creep are one-way, they arrive with certainty, and there is no sense in which they might not happen. Some are random and reversible: thermal and live-load deflection are both at their extremes for a small fraction of the time. And one — the tolerance — is a fixed offset once the building exists, unknown before it is built and constant afterwards.

The rule that fits that mixture is the one already used for actions: one term at its full value and the rest at their coincidence factors, taken as the maximum over which term leads. On the budget drawn that gives 25 mm — which happens to be the number on the standard detail, arrived at by a route nobody took.

The envelope is not a structure is the same argument for load arrangements: the envelope of a set of cases is a curve no single case produces, and designing to it produces a member that no state of the structure ever asks for. A sum of movement extremes is that envelope, applied to displacements.

Why the conservative answer is not free #

The instinct is to add the extremes and be safe, and it is worth being explicit about why that is not obviously right.

A joint that is too large is not a joint with a margin. It is a detail that has to work at every opening between zero and its full size, and the wider it is the harder that is: the seal has to compress and recover through a bigger range, the cover bead has to hide a bigger gap, the fire and acoustic performance of the joint has to hold at its widest, and the aesthetic of a 40 mm dark line at every wall head is a thing an architect will notice.

More practically, an over-sized joint on a partition is an under-sized joint somewhere else. The movement is a fixed quantity: it goes into the joints provided or into the elements that were not supposed to move. A design that provides generously at one joint and forgets another has moved the problem rather than solved it.

That last point is the movement nobody applied in its most useful form: the stress from restraining a thermal movement is $E\alpha\Delta T$ and contains no length whatever, so a small unprovided-for movement produces the same stress as a large one. There is no such thing as a movement small enough to ignore; there are only movements small enough to be absorbed by cracking something that does not matter.

That last clause is the one to be careful with, because “something that does not matter” is a choice somebody has to make rather than a property of the structure. A concrete wall cast against a base that has already hardened is the standard case, and the base does not have to hold it very firmly to break it.

Note what the restraint fraction does and does not depend on. It is a ratio of two stiffnesses, so it carries the wall’s proportions and not its size: the same figure at twice the length and twice the height gives the same fractions. The movement a joint would have accommodated does not disappear when the joint is left out — it is redistributed into cracks whose spacing and width are the only remaining design variables.

There is a third failure mode of the conservative answer that is worth naming because it is the one that actually happens. A joint sized by adding extremes is a joint that spends its entire life partly open, because the coincidence it was sized for never occurs. A 40 mm gap that is at 12 mm on an ordinary day is a 28 mm void behind a bead, and voids behind beads collect dust, admit sound, defeat fire seals and get filled by somebody with a mastic gun who has not been told why the gap is there. Built to the wrong shape is the same idea for a beam: a member cambered for a movement that does not arrive is a member that is visibly wrong for the whole of its life, and being wrong in the safe direction is still being wrong.

A joint has two budgets, not one #

The five terms have been added as magnitudes, and they do not all have the same sign. Sorting them by direction turns one number into two, and the two are not equal.

Some terms close the gap: the element above sagging under load and creep, a support beneath settling relative to the one above, the element below expanding as it warms. Others open it: the element below shrinking, contracting as it cools, or losing a camber it was built with.

So the joint has a required closure capacity and a required opening capacity, computed from two different subsets of the same five terms, and quoting a single dimension conflates them. On a typical wall head the closing sum is dominated by the slab’s long-term deflection and the opening sum by the wall’s own shrinkage, and the two are rarely within a factor of two of each other.

That distinction is what selects the component rather than the dimension. A gap that mostly closes needs a compressible filler with enough travel and enough recovery to be squeezed repeatedly. A gap that mostly opens needs a seal that stays sealed as it is stretched, which is a different material property and a different failure mode — a compressible filler pulled apart simply parts.

And the tolerance term sits in both, with an unknown sign. It is a fixed offset once the building exists, so it does not consume travel in either direction repeatedly — it just decides where in the joint’s range the building starts, which is the reason a joint sized correctly can still be at the end of its travel on the day it is built.

The differential is what does the damage #

One further distinction turns the budget from an accounting exercise into a design problem.

Nothing is damaged by moving. Things are damaged by moving relative to something else, and the relative movement is a difference of two of the quantities above rather than either of them.

A wall head gap closes by the difference between the slab’s downward movement and the wall’s upward growth. A cladding panel is distressed by the difference between the frame’s shortening and its own. A column and a core standing side by side shorten by different amounts because one is more heavily stressed than the other, which is the columns are shorter than the core and is a differential of two large numbers.

It follows that a joint is not the only remedy and often not the best one. Making the two elements move together removes the differential without removing either movement — which is why a partition fixed rigidly to the slab above and free at the bottom can be a better detail than one free at the top, and why a cladding system hung from one floor and restrained laterally at the next has no differential to accommodate in the vertical direction at all. The design variable is the connection, and the budget is what says which connection to make free. Stiffness is not strength has a corollary here: what matters at a joint is neither, it is whether the two things are allowed to move apart.

Differences of large numbers are the hardest quantities to predict, because the errors do not cancel — a 20% error in each of two 15 mm movements is a 6 mm error in a 3 mm difference. So the term of the budget that matters most is the one computed least reliably, and the response has to be a detail that tolerates the answer being wrong rather than a calculation that gets it right.

The same budget everywhere else #

The wall head is the smallest instance and the easiest to draw. The same accounting, with the same ordering and the same neglect, decides several much larger things.

A cladding support. A curtain wall bracket has to accommodate the frame’s floor-to-floor movement, the panel’s thermal movement, the frame’s shortening differential and the erection tolerance of two components made in different factories. The tolerance term dominates again, and the failure — a bracket at the end of its adjustment — is discovered on site with the panel in the air.

A settlement joint. Two parts of a building founded differently settle by different amounts, and the settlement that matters is the difference — which is another one-way term with a large uncertainty and no code limit.

A bridge bearing. Its movement rating is the sum of thermal, shrinkage, creep, live-load rotation effects and the setting error at installation, and where the structure is allowed to move shows that the articulation scheme decides how much of the total any one bearing sees.

A lift shaft. The guide rails are fixed to a shaft wall that shortens under creep while the rails do not, so the brackets have to slide — and the accumulated differential over forty storeys is tens of millimetres.

And a movement joint through a whole building. Here the budget decides a spacing rather than a dimension: how far apart the joints have to be for a joint of the available size to cope, which converts the whole argument into a plan decision made at scheme stage.

The bridge version of that last one is the clearest, because somebody has actually built the alternative. An integral bridge has no joint at all: the deck is cast into the abutments and the movement goes into the soil behind them.

That is the budget argument with the joint deleted and the terms unchanged: the same thermal movement, the same shrinkage, the same creep, now accommodated by something that answers back. A jointless bridge therefore has a length, and the length is set by the budget rather than by the deck.

In every case the pattern is the same. The structural term is computed, the others are estimated or inherited, the combination rule is chosen by habit, and the result is a dimension that decides whether a component works.

Where the model stops #

The terms were taken as independent when they are not. Creep and shrinkage share a moisture mechanism and both depend on the member’s thickness and the ambient humidity, so a dry environment increases both together. The root-sum-square is optimistic to the extent that they are correlated.

Nothing here is about time. Shrinkage and creep are mostly over within a year or two, and a wall built at six months has missed some of both — so the movement a joint has to accommodate depends on when the wall was built, which is a programme decision.

The tolerance was taken as a single number. It is a distribution, with its own mean and scatter, and it is the term with the least published data.

And a term can grow after the building is finished, which is the assumption most invisible in a budget written once. The integral abutment above is the case where it has been measured.

Nothing in the five terms of the wall-head budget ratchets like that, and the reason is worth stating: none of them is accommodated by a material that changes when it is worked. A soil is, a sealant is, and a compressible filler that has been squeezed ten thousand times is not the filler that was specified. The budget is written against the joint’s first year and the joint has to survive its fiftieth.

The reversible and irreversible terms were added as though they were the same kind of thing. They are not, and the distinction decides the detail rather than the dimension: a joint accommodating a reversible movement has to work at both ends of its range for a hundred years, while one accommodating a one-way movement passes through its range once and stays there. The first needs a seal that recovers; the second needs a seal that can be pointed once the movement has happened, which is a much easier component and a much harder programme.

And the joint has to be built. A 25 mm gap detailed and a 25 mm gap constructed are different things: the seal has a compression range, the bead has a coverage, and the two together define a working window narrower than the nominal dimension.

The generalisation #

The habit worth taking away is to write out every term of a quantity before refining any of them.

The instinct in an engineering calculation is to compute the term that has a method behind it, carefully, and to allow for the rest. That is the right instinct when the computed term dominates and it inverts the effort exactly when it does not — and a movement budget is the clearest case in this collection of a quantity whose largest term is outside the calculating discipline altogether.

The second habit is about combination. Any quantity that is a sum of several uncertain things needs a rule for how they combine, and there are only three candidates: add the extremes, add them in quadrature, or take one at its extreme and the rest at a factor. The profession has a well-developed version of the third for loads, has thought about it for decades, and applies it to actions and to nothing else. Movements, tolerances, temperature effects and construction stages are combined by whichever of the first two the author reached for, and the difference between them is a factor of two.

There is a diagnostic that follows from the ordering and takes no calculation at all. On any budget of this kind, write the terms in order of size and then mark which discipline owns each. If the largest terms belong to disciplines that are not in the room, the number on the drawing was not designed — it was inherited, and it is right or wrong by whatever margin the standard detail happened to have. That is a defensible position on a repeat building and an indefensible one on a long-span, a tall building, or anything with an unusual grid, which is exactly where the standard detail’s assumptions have been left behind.

Scale changes everything has a version here that is easy to miss. Every term in the budget except the tolerance is proportional to a length, and the tolerance is not. So on a short element the tolerance dominates completely and on a long one it hardly matters — and a standard detail that works on a 6 m grid is the wrong detail on a 20 m one, for a reason that has nothing to do with the structure.

Read that against the budget at the top of the page and the two ends of the argument meet. On a short element the movement terms shrink toward nothing while the tolerance stays at ten millimetres, so the joint is sized by the builder; on a long one the movement terms dominate the budget and, if the joint is not provided, the restraint is more nearly complete rather than less. Both ends of the length scale are worse than the middle, for opposite reasons, and the standard detail was drawn for the middle.