Rigid by one code and not by the other

The floor is a beam lying down showed that a floor plate spanning between walls shares its storey force in proportions that depend on how stiff the plate is compared with the walls, and that the answer moves continuously from one limit to the other. It ended on the question this essay takes up: “the semi-rigid diaphragm as codes define it, and why a binary classification survives when the underlying quantity is continuous.”

The short version is that the two codes most often used for the question do not agree where the boundary is, by two orders of magnitude of stiffness, and that the reason the binary survives is not that it is accurate at its edges. It is that inside each box it lets a designer avoid knowing the one quantity that is hardest to know.

The plate and its three supports #

The plan used throughout is the simplest one that has the problem: a floor 48 m long and 12 m deep, 200 mm of concrete, spanning two 24 m bays between three braced bays that resist the storey force, one at each end and one in the middle. It is a single storey, so a braced bay’s displacement is its storey drift. Each braced bay has a stiffness of 100,000 kN/m, about what a pair of steel cross-braces gives across a 6 m bay, and the plate carries 40 kN/m of storey force along its length.

The plate is a beam lying down, with the braced bays as spring supports. It is a stubby beam — two spans each twice its depth — so its shear deflection matters as much as its bending, and the solve includes both.

Solved, the middle braced bay moves 7.31 mm and the two end ones 5.95. A rigid plate would move all three by 6.40 mm and give each a third of the load. The actual plate bows by a millimetre and a half more at the middle than at the ends, and because the braced bays are springs, a bay that moves more carries more: the middle one takes 38 per cent of the storey force rather than 33.

That is the whole of the physics, and it has nothing binary in it. The rest is how the two codes read it.

Two tests on one floor #

ASCE 7 asks whether the plate is flexible. It measures the plate’s largest deflection from the straight line between the braced bays at the ends of a span — 0.43 mm here — and compares it with the average drift of those two bays, 6.63 mm. If the ratio is above 2 the diaphragm is flexible, and it is analysed as a chain of simple spans that share the storey force by tributary area.

EN 1998-1 asks whether the plate is rigid. It compares the plate’s displacements, modelled with its real in-plane flexibility, against the displacements the rigid plate would have, and calls the plate rigid if nowhere does the first exceed the second by more than 10 per cent. Here the plate at the middle bay moves 7.31 against the rigid 6.40, an excess of 15 per cent.

By ASCE 7 this plate reads a thirtieth of the way to flexible. By EN 1998-1 it is not stiff enough to be rigid. Neither statement is wrong, since they are answers to different questions. What they mean together is that the plate has a stiffness between the two boundaries, and between them each code asks for the plate to be modelled as what it is.

Swept over the plate’s stiffness, the two tests are close to parallel lines, both falling in proportion to the stiffness once the plate is stiff, since both measure a deflection that the plate’s stiffness divides. They cross their limits a long way apart: the plate stops being flexible by ASCE 7 at a fiftieth of its as-built stiffness, and becomes rigid by EN 1998-1 at one and a half times it. A factor of a hundred separates the two boundaries. That factor is the width of the semi-rigid band, and it is not a property of this plan: it follows from the two limits, 2 on a ratio that measures deflection against drift and 0.1 on one that measures it against total displacement.

Why the two tests are so far apart #

The factor of a hundred has a plain cause, visible in the deflected plate. ASCE 7 measures the plate’s deflection from the chord between the braced bays at the ends of a span. The chord moves with the bays, so whatever the bays do between them is subtracted out, and what is left is how much the plate bends between its supports — 0.43 mm. EN 1998-1 measures the plate’s displacement from where a rigid plate would be, and that includes the difference between the bays themselves. The middle bay moving 0.91 mm further than the rigid plate’s 6.40 is most of its 15 per cent.

But the difference between the bays is the load sharing. A braced bay that moves more carries more, in exact proportion, so the part of the deflection that EN 1998-1 counts and ASCE 7 subtracts is precisely the part that redistributes the storey force. ASCE 7’s test is a good measure of whether the plate bends like a chain of simple spans. EN 1998-1’s is a good measure of whether the walls share the load as a rigid plate would make them. They are tests for different idealisations, and each is well chosen for its own.

The numbers can be checked on the back of an envelope. The rigid plate moves $40 \times 48/(3 \times 100{,}000) = 6.40$ mm. One 24 m bay of the plate, simply supported and loaded on its own, bows by $5wL^4/384EI = 0.20$ mm in bending, with $EI = E t d^3/12 = 8.6 \times 10^8\ \mathrm{kN\,m^2}$, and by $wL^2/8GA = 0.12$ mm in shear: a third of a millimetre against six and a half of drift, which is ASCE 7’s ratio of about 0.06 before any analysis. EN 1998-1’s excess cannot be had so cheaply, because it depends on how the springs redistribute the load, which is the thing the classification exists to avoid computing.

One quantity, three names #

The middle bay’s share is one smooth curve. At the soft end the plate behaves as a continuous beam on supports that do not move, and the middle support of two equal spans takes about five eighths of the load — here 61 per cent, a little less than the textbook 62.5 because the plate’s shear flexibility spreads the load towards the ends. At the stiff end the plate is rigid, and each bay takes its stiffness share, a third. The as-built plate sits on the steep part of the curve: halving its stiffness raises the middle share from 38 to 41 per cent, and quartering it to 46.

The codes give the curve three names. Left of ASCE 7’s boundary it is “flexible” and replaced by the tributary line, 50 per cent. Right of EN 1998-1’s boundary it is “rigid” and replaced by the stiffness line, 33 per cent. In between it is “semi-rigid”, and neither code offers a replacement: the plate is modelled.

Notice what the flexible idealisation is. The tributary answer is not the soft limit. It is the answer for a plate that is soft and also discontinuous over every wall, a chain of simple spans. A timber or metal-deck diaphragm with its joints over the walls nearly is; a concrete slab never is. So a concrete plate soft enough for ASCE 7 to call flexible is analysed with a model that gives its middle wall 50 per cent while the plate gives it 59.

Each boundary sits where its own model is wrong #

Set each idealisation against the plate it replaces, and the boundaries acquire a meaning. At ASCE 7’s flexible boundary the tributary share understates the middle bay by 15 per cent and overstates the end bays. At EN 1998-1’s rigid boundary the rigid share understates the middle bay by 9 per cent, and overstates the ends by 5. Neither boundary is placed where its model becomes exact. Each is placed where its model has become wrong by about a tenth, which is roughly the error a designer would accept from any other simplification.

The EN 1998-1 limit makes that explicit: 10 per cent on displacement is very nearly 10 per cent on the force in a spring support, since force is stiffness times displacement. ASCE 7’s limit is on a different quantity — how much the plate bends compared with how much the walls lean — and its error at the boundary is the tributary model’s, which is bounded by how far a continuous plate is from a chain of simple ones. On this plan that is 15 per cent.

Between the two boundaries both models are further wrong. At the plate as built the tributary share overstates the middle bay by 31 per cent and the rigid one understates it by 12. That is why neither code licenses either model there.

The obvious response, when the plate’s stiffness is uncertain, is to envelope: design each wall for the larger of its two idealised shares. That is conservative in the middle of the band, and at the plate as built it overstates the middle bay by 31 per cent — the price of not modelling. It is not conservative at the soft end. A soft continuous plate gives its middle wall 61 per cent against the envelope’s 50, because the flexible model is a discontinuous plate and the continuous one is stiffer over the middle support. That is the second refutation, and it is the case to watch in a concrete building whose slab is heavily cracked or pierced.

The rule that lets a plan be rigid by its proportions #

ASCE 7 also allows some diaphragms to be idealised as rigid without any calculation: a concrete slab with a span no more than three times its depth, in a structure with no horizontal irregularity. The rule reads as a proportion because in-plane stiffness is mostly a matter of proportion — a stubby plate barely bends in its plane — and the proportion is the one the plan fixes.

Held against the plate it describes, the rule is generous. EN 1998-1 stops calling this plate rigid at a span of 1.8 times its depth. At ASCE 7’s limit of three, the plate’s middle moves 34 per cent further than a rigid plate’s, and the middle braced bay carries 44 per cent of the storey force rather than the rigid plate’s 33 — a quarter more than it was designed for. The as-built plan, at a span of twice the depth, is inside ASCE 7’s rule and outside EN 1998-1’s test, which is the first refutation’s situation.

None of this says the rule is careless. It is a rule about concrete slabs in general, calibrated against buildings whose vertical elements are usually concrete walls much stiffer than the steel braced bays here — and a stiffer wall makes the same plate less rigid, since rigidity is a comparison. The rule is a proportion standing in for a comparison, and it is least reliable exactly where the comparison moves: soft plates over stiff walls.

Where the core is in the middle #

Change the walls, and the idealisations change places. With a stiff core in the middle and soft frames at the ends, the rigid plate sends 80 per cent of the storey force to the core, and the tributary model only 50. The plate as built gives the core 76 per cent and each end frame 12. Now the rigid model is the one that overloads the middle and underloads the ends — by 17 per cent for each end frame — while the tributary model overloads the ends by more than double. The stiffest path takes the load, but a floor plate is itself a path, and its flexibility lets the end frames keep some of their share.

This is also the case in which the plan’s torsion matters most, because the rigid plate is what makes torsion a single rotation. A measure of twist that divides by the twist drew the irregularity ratio that codes compute on a rigid plate; on a plate that is not rigid, the ratio is computed on a rotation that is not there, and the edge frames’ forces come from the plate’s own bending instead.

Why the binary survives #

The quantity the classification replaces is the plate’s in-plane stiffness, and that is the one quantity in the problem nobody knows well. A concrete slab cracks under gravity load, under shrinkage and under the earthquake it is being designed for, and its in-plane stiffness can fall to a quarter or a half of the uncracked value; a cracked member can also be stiffer than its cracked section says, because tension stiffening between the cracks gives back some of what the cracks took. Openings for stairs and shafts cut the plate’s web. Metal deck and timber diaphragms get their stiffness from fastener slip, which varies from one fastener to the next.

A binary classification lets a designer avoid putting a number on that. Inside each box the model needs no plate stiffness at all: the tributary shares are geometry, and the rigid shares are the walls’ stiffnesses. The price is an error of about a tenth at the box’s edge. The alternative is a model whose answer depends on a stiffness that is uncertain by a factor of two or more, and on the steep part of the share curve, where the as-built plate here sits, a factor of four in stiffness moves the middle bay’s force by a fifth — more than either idealisation’s error at its own boundary.

So the binary is not a claim that the quantity is binary. It is a claim that, far enough from the middle, the answer does not depend on the unknown. The two codes disagree about how far is far enough by a factor of a hundred in stiffness, and between them the unknown has to be estimated.

Where the model stops #

One storey, three walls, one direction. A multi-storey building has a plate at every level, loaded by storey forces that vary up the building, and the drift that ASCE 7 compares against is an interstorey quantity rather than a displacement; the comparison still has two sides, but both come from the whole structure.

The plate is uniform. Openings, re-entrant corners and changes of slab thickness concentrate the plate’s flexibility, and a plate that is rigid on average can be flexible at a single weak section.

The plate’s stiffness is linear. A cracked slab’s in-plane stiffness falls with the load that cracked it, so the classification can change during the earthquake it was made for.

What the pictures cannot show #

That the two tests are measuring the same curve. The criteria figure draws two lines and the shares figure one, and the three are one physical fact read three ways: the plate bends, the bays that sit where it bends most move more, and a bay that moves more carries more.

Still open: the collector that has to deliver the share #

Everything here is about how much of the storey force each braced bay receives. None of it says how the force gets there. A braced bay is a few metres long in a plate forty-eight metres long, and the plate’s shear has to be gathered along the plate into the bay by a collector, whose force depends on which share the bay takes and is designed with an overstrength factor because it must not fail before the bay does. How the collector force changes across the semi-rigid band, and whether the envelope that protects the bay also protects the member feeding it, is the next question.