The most dangerous day is before it is finished

Every analysis in this collection is of a finished structure. Its bracing is in, its slabs are cast, its joints are made, its bases are grouted, and every restraint that the effective lengths assume is present. Loads are applied to that object and the answers follow.

A real structure spends weeks or months in states that are not that object, carrying real loads, in weather. Nobody draws those states, no analysis model contains them, and the arithmetic that governs them is the arithmetic of this collection with terms removed.

The mistake worth dismantling is the intuition that a partly built structure is safe because it is lightly loaded. The load falls, and the capacity falls faster.

Restraint is worth more than load #

A restraint does not add strength; it changes what the member is. Removing it does not reduce a capacity by a percentage, it moves the member to a different point on a curve — and the curves in stability are steep.

That is a factor of 8.8 between the beam as designed and the same beam on the day it was landed, and nothing about the beam has changed. The deck that will restrain its compression flange at 3 m centres has not arrived yet; the beam is carrying its own weight and two people, and its capacity is a sixth of what every calculation about it assumed. It is held and not held in the most literal available sense.

The load side of the comparison moves by nothing like as much. A composite floor beam in service carries its own weight plus the slab plus the imposed load — the self-weight alone is perhaps a fifth of the total. So the load has fallen by a factor of five and the capacity by a factor of nearly nine.

A ratio that improves by five and worsens by nine is a ratio that has got worse. That single comparison is the whole subject.

Which free body produced the number #

Take the beam as it is on the day: simply supported at its ends, unrestrained between them, carrying a uniform load equal to its own weight and whatever is on it.

The check is the ordinary lateral-torsional one, with the unbraced length equal to the whole span rather than the deck spacing. $M_{cr}$ contains $\sqrt{EI_z GJ}$ and a warping term, and both are divided by the unbraced length — so the answer scales roughly as $1/L$ for a long member and the loss is not a small correction.

The same argument applies to a column and to a frame, and it applies with the same steepness because every stability capacity goes as an inverse square of a length.

A factor of four, from a member whose job is to be stiff rather than strong, and which is not present until somebody installs it.

Joints that are not yet joints #

The second thing missing from a partly erected structure is the fixity every analysis assumed.

A steel frame is erected with bolts in place and not fully tightened, so the connections behave as something much closer to pins than to the semi-rigid joints they will be. A base plate is set on levelling nuts and not grouted, so the base is a pin — or worse, a point support on a plate that can rock.

For a beam this is usually benign, since a pinned end is the conservative assumption in bending. For a frame it is not, because a frame’s stability against sway depends on its joints. A moment frame with its joints not yet tightened has no lateral system at all — which is the redistribution nobody chose with the whole structure as its subject.

Second-order effects arrive early #

A structure at a fraction of its buckling load has its deflections amplified, and the amplification is not linear.

Put the two reductions together. If the erection load is 25% of the design load and the erection capacity is 34% of the design capacity — the sway factor of 2.90 above — then a frame designed to sit at 20% of its buckling load in service sits at

$$0.20 \times \frac{0.25}{0.34} = 0.15$$

during erection, which is better. That is the honest arithmetic for gravity load and it is the reason most steelwork is erected without incident.

The trouble is that the other load does not scale. Wind does not know the building is unfinished. The lateral load on a partly erected frame is very nearly what it will be on the finished one — the frame is the same height, the cladding may or may not be on, and the reference wind speed is reduced only by the shorter exposure period. Meanwhile the lateral system is the part that is missing.

So the ratio that matters during erection is not gravity load over gravity capacity. It is full wind over a third of the lateral resistance, and that ratio is several times worse than anything the finished structure ever sees.

The frame that is stable only when it is complete #

There is a class of structures for which the finished load path does not exist until the last member is in, and for those the temporary condition is not a reduced version of the permanent one. It is a different structure.

An arch is the standard example: incomplete, it is two cantilevers, and its whole mechanism depends on a closure that happens last. A cable-stayed bridge is another, built out from its towers as a balanced cantilever with the deck carrying moments it will never see again. A space frame assembled on the ground and lifted is a third — the lifting configuration has different supports from the final one, and the member forces during the lift can exceed the in-service ones.

What redundancy is worth here #

The relationship between redundancy and erection is not the one intuition supplies. A highly redundant finished structure is robust; the sequence that builds it passes through configurations that are less redundant than either end, and the least redundant configuration is not necessarily the earliest one. A frame with half its bracing installed can be less stable than the same frame with none, if the half that is in has attracted load to a path that is not yet complete.

Which is the argument for the erection sequence being designed rather than left to be worked out on site: the sequence is a series of structures, and each of them needs to be a structure.

Four states nobody has a drawing for #

It is worth listing the specific configurations, because each is a real check that a permanent-works model cannot produce.

The beam landed and not yet decked. Its compression flange is unrestrained over its whole span. Every composite floor beam in a steel building passes through this state, and the check is a lateral-torsional one at the unbraced length of the span.

The frame erected and not yet braced. The bay containing the vertical bracing is often erected last, because the bracing gets in the way of the crane. Until then the frame’s stability comes from temporary guys or from the crane itself, and its effective lengths are the sway ones rather than the braced ones.

The floor cast and not yet cured. A concrete slab is a diaphragm when it has strength and a load when it does not — so for some days it is the heaviest thing on the frame and contributes nothing to holding the frame together. The diaphragm’s whole distribution argument waits on a cube test.

And the column erected and not yet loaded. A column is stabilised in part by the beams framing into it, and a column standing with two of its four beams connected is restrained about one axis and not the other. Its slenderness in that state is about the axis nobody computed.

What the four have in common is that each is a reduction of restraint, not of load. The load lines on a construction programme go up monotonically; the restraint lines do not, and the gap between them is where the risk is.

Where the model stops #

Nothing here computes a temporary works design. Guys, plumbing struts, kentledge, tower cranes tied into the frame and the frame’s own use as a crane base are all real loads and real restraints, and they are outside every figure on this page.

The wind during erection is not the design wind. Codes allow a reduced return period for a short exposure, which reduces the pressure by perhaps 20% for a period of months. That is a real reduction and it is nothing like the factor of three the lateral system has lost.

The lifting condition has been ignored entirely. A member picked at two points is a beam with two overhangs; a member picked at one is a member in bending it was never designed for; and a long slender member picked flat is a lateral-torsional problem in its own right.

And the numbers here are one beam and one frame. The factor of 8.8 is a property of a 457 mm universal beam over 12 m; a shallower section restrained more closely loses less, and a deep plate girder loses more.

The one number to carry away #

Stability capacities go as the inverse square of a length. That single fact is why removing a restraint is so much more expensive than removing a load, and it can be turned into a rule of thumb that survives without any of the arithmetic above.

Doubling an unbraced length quarters a capacity. Removing every second restraint from a beam does exactly that. Removing every restraint but the ends, on a beam braced at quarter points, divides the capacity by sixteen — before the warping term, which makes it worse still for a rolled section.

Against that, loads during construction fall by factors of two to five. There is no arrangement of those two numbers in which the temporary state is comfortably better than the permanent one, and there are many in which it is much worse.

The corollary is the useful part. The cheapest erection stability measure is always a restraint, never a strengthening. A temporary tie at mid-span of an unrestrained beam costs almost nothing, needs to be stiff rather than strong, and buys a factor of four. Making the beam heavier to survive being unrestrained buys a factor equal to the weight added. The whole of temporary works design is that comparison, made repeatedly.

What the pictures cannot show #

Every figure on this page draws a finished structure with something removed. What actually happens on a site is a structure being added to, with each addition changing the whole, and no static figure has a way to show a load path in the process of being connected.

Nor can they show the thing that makes this subject difficult in practice: the erection condition is decided by a sequence that is chosen after the design is complete, by people who were not part of it, and it is checked — when it is checked — against a model built for a different purpose.

And the ratios quoted are for one moment in the sequence. A partly erected frame passes through hundreds of configurations, and the governing one is not usually the one anybody would have chosen to draw.

The ladder from here #

Later rungs on this anchor: temporary bracing design, and how much of the permanent bracing’s job it has to do. The lifting analysis of a long member, including the two-point pick and the effect of sling angle on the compression it induces. Balanced cantilever construction, where the temporary condition governs the whole design of the permanent structure. Formwork and falsework as structures in their own right, which they legally are. Wind on a partially clad frame, where the pressure coefficients are not the finished building’s. And the historical case: the great majority of structural collapses happen during construction rather than in service, and the ones that are remembered — Quebec, Tacoma’s erection, the box girder bridges of 1970 — are remembered because each of them found a state the design had not considered.