The answer is continuous and the catalogue is not

A beam has to carry 1,200 kNm. The calculation is careful, the load factors are current, the analysis is a frame model rather than a set of coefficients, and the answer comes out as a required plastic modulus of 3,380 cm³.

There is no section with a plastic modulus of 3,380 cm³. There is one at 2,881 and one at 3,994, and the choice is between a member that is 15% short and one that is 18% over.

This is not a rounding error at the end of a calculation. It is a structural feature of the design problem, and it changes what a calculation is for.

Which free body produced the number #

None, and that is the point of this essay. Everything above the choice of section is statics: a free body, a moment, a stress block, a plastic modulus. Everything below it is a set, with no continuity in it and no derivative.

The two halves are usually run together as though they were one calculation, and they behave completely differently. Everything adds to nothing is continuous; a section table is not. The statics is smooth: change the load by 1% and the moment changes by 1%. The catalogue is not: change the moment by 1% and the section either does not change at all or changes by 39%.

So the sensitivity of the design to any assumption has two regimes, and which one is in force depends on where in a tread the requirement happens to land. That is a property of the problem, not of the designer, and it is invisible in a calculation that reports a required modulus to four figures.

The steps, measured #

Across a universal beam series from 203×133×25 to 914×305×201, the ratio between one section’s plastic modulus and the next runs from 23% to 59%, with an average of about 38%.

That is the granularity available. A design requirement landing uniformly across the range is on average about 15% below the capacity of the section it buys, and at worst about 46% below — which happens when the requirement is a kilonewton-metre above a riser and the section below has just been missed.

Fifteen per cent of the steel in the beams of an ordinary building is bought because the list is a list. Not because of load factors, not because of material factors, not because of conservatism anywhere in the analysis: because the requirement is a real number and the catalogue is a set of twelve.

There is a second reading of the same measurement that is more useful in practice. Fifteen per cent of capacity is also fifteen per cent of margin, sitting in every member, and it is margin nobody declared. It is not a safety factor — it is not applied consistently, it varies from member to member depending on where the requirement landed, and it disappears entirely on the members that happened to fall just below a riser. The strength no specimen had is about margins that are stated and computed; this is a margin that is real and arbitrary.

That number is worth comparing with the ones designers argue about. A refinement to the analysis that saves 5% of the moment is a genuine piece of engineering and it is a third of one step. A choice between two load models that differ by 8% is two thirds of a step. Both are frequently smaller than the rounding that follows them.

Why the list is not uniform, and what that buys #

Sections within one series get more efficient as they get deeper, and the rise is large: plastic modulus per kilogram runs from about 10 cm³/kg at the smallest section in the series to 42 at the largest, a factor of four.

The reason is the material far from the middle: a section modulus goes as the depth for a given area of flange, so a deeper section of the same weight has a larger capacity, and there is no penalty for it in bending at all. Depth is the cheapest strength is the general statement, and the same steel in a different shape is the demonstration that the arrangement matters more than the amount.

The consequence for choosing from a list is exact and often left implicit: the cheapest section that carries a moment is always the deepest one available, and the only thing that ever prevents that choice is a constraint that has nothing to do with the moment — a floor-to-floor height, a duct, a soffit level, a headroom.

What a depth limit costs #

Restrict the depth and the answer stops coming from the beam series at all.

A requirement of 900 kNm with no depth limit is a 610×229×101 at 101 kg/m. The same requirement with a 400 mm depth limit has to be met from the column series, and the lightest section is a 356×368×177 at 177 kg/m.

Seventy-five per cent more steel, for a constraint that was not a structural one. The section carries the same moment with the same material at the same stress, and the whole of the difference is that it was not allowed to be deep.

That is a number worth having in a room where a floor-to-floor height is being argued about, and it is not usually available at the time, because the beam sizes are not known until after the geometry is fixed. The order in which the two decisions are taken is the reason the cost is invisible.

The penalty is not only in weight. A shallow section of the same capacity has a much smaller second moment about its minor axis relative to its major one, which changes what governs it: a deep beam is at risk from the beam that fails sideways and a column section used as a beam is very much less so, because the property that makes it heavy is the same property that braces it. Depth and stability pull in opposite directions, and a depth limit quietly moves a member from one regime to the other.

It also changes what the catalogue’s steps mean. A single series is monotonic in efficiency; a real catalogue with beam and column sections in it is not, so the lightest section satisfying a requirement is no longer the first one past it in modulus. The search becomes a genuine search rather than a lookup, which is why every steel design program does it as one.

The sensitivity, which has two regimes #

Here is the practical consequence, and it is one of the few places where the discreteness changes what an engineer should do rather than merely what they should know.

If the requirement sits in the middle of a tread, refining the calculation is worthless. A 10% saving on the moment buys nothing at all: the same section is bought, at the same weight, and the only thing that has changed is the number on the sheet. Time spent there is time spent for no product.

If the requirement sits just above a riser, refining the calculation is worth an entire step. A 2% saving buys 38% of the section’s weight, on every member of that type in the building.

And the designer cannot tell which regime they are in until after the calculation is done. So the sensible working order is: do the calculation quickly, choose the section, then look at where in the tread the requirement landed — and only refine the analysis if it landed just above a riser. That inverts the usual instinct, which is to refine first and choose second.

The same reasoning applies to the load. A designer who is unsure whether a partition allowance should be 1.0 or 1.5 kN/m² is asking a question that matters enormously on some members and not at all on others, and which is which is decided by the catalogue rather than by the partitions.

Where else the set is a set #

Steel sections are the clearest case because the catalogue is published, but the same structure appears wherever the answer has to be bought.

Reinforcement comes in a handful of diameters at a handful of spacings, so the area provided is a lattice rather than a continuum, and the steps near the bottom of the range are large — 10 mm at 200 centres is 393 mm²/m and 12 at 200 is 566, which is a step of 44%. It also interacts with the dimension nobody can measure: choosing the next bar up adds capacity and takes effective depth away, so the step is smaller than the area suggests.

Concrete grades go up in steps of five newtons, and the strength is one of two variables in a capacity that goes as the cube root of the other, so the steps are unevenly valuable.

Timber sections come in a set of sawn sizes that is discrete in both dimensions, and glulam in a set of laminate multiples.

Bolt sizes and grades are a two-dimensional lattice, and a connection design lands on it rather than on a required area — with the further complication that the bolt that carries more than its share makes the required capacity depend on how many bolts there are, so the lattice is not even monotonic.

In every case the design variable a calculation optimises is not the variable that gets built, and the step between the two is usually larger than the precision of the calculation.

What a catalogue is actually for #

It would be possible to roll steel to any modulus. The reason nobody does is not technical, and it is worth stating because it is what makes the waste rational.

A discrete series means a mill can roll long campaigns of one profile, a stockholder can hold it, a fabricator can price it from a table, a contractor can substitute one section for another late in the programme, and a designer can specify it in four characters. The economic saving from all of that is very much larger than the fifteen per cent of steel it costs.

So the staircase is not a defect in the supply chain. It is a deliberate trade of material efficiency for logistical efficiency, and the fifteen per cent is the price. The uncomfortable part is that the price is paid in a quantity — tonnes of steel, and therefore embodied carbon — that has become a design objective in its own right, and the trade was struck when it was not.

That is where the argument goes next, and it goes in an unexpected direction. The response is not a finer catalogue: a series with half the step size would double the number of profiles and destroy the logistics that justified it. The response is to move the requirement, by choosing spans, spacings and depths that put the requirement near the top of a tread rather than just above a riser — which is a geometry decision made in the first week of a project, on the basis of a section table nobody has opened yet.

Where the model stops #

Only plastic modulus was compared. A section can be governed by deflection, by lateral-torsional buckling, by web crippling, by fatigue or by a connection, and each of those orders the catalogue differently. A section list ranked by $I$ rather than by $W_{pl}$ has slightly different steps and a different lightest member.

Serviceability was ignored. A member chosen for strength is frequently governed by deflection instead, and the two orderings of the catalogue are not the same: capacity goes as the section modulus and deflection as the second moment, which rewards depth more steeply again. Stiffness is not strength is the general statement and it has a specific consequence here — a member governed by deflection is chosen from an entirely different column of the same table.

Class was ignored. A section reaches its plastic modulus only if its flanges and web are stocky enough not to buckle locally first, and the deepest sections in a series are the ones nearest that limit — so the efficiency curve flattens at the top for a reason the modulus does not contain.

Cost was taken as mass. It is not. Fabrication, connections, transport, erection and fire protection are all closer to being proportional to area or to piece count than to weight, and a heavier, shallower section can be cheaper than a lighter deep one once the storey height it saves is counted.

And the series quoted is one country’s. The step sizes and the crossover between beam and column sections differ between catalogues, so the fifteen per cent is a property of a particular list rather than of catalogues in general.

The generalisation #

The habit is to ask, at the end of any optimisation, what the answer will be rounded to — and to do it before deciding how much effort the optimisation deserves.

A calculation whose output is a continuous number and whose product is a choice from a set has a resolution, and refining below that resolution produces nothing. It is the same discipline as significant figures and it is applied far less often, because the resolution is not in the calculation: it is in a table at the back of a different book.

There is one more consequence, and it is about how a structure is described rather than how it is designed. A drawing that says 610×229×101 says something exact and something misleading: exact about what will be delivered, misleading about what was required. The requirement was 3,380 cm³ and the drawing records 2,881 — a number chosen from a list — so anyone reading the structure backwards to find out what it was designed for gets the catalogue’s answer rather than the engineer’s. That is one reason a member’s utilisation is worth recording somewhere, and it is why the count that does not see it is a habit worth generalising: a structure’s drawings record what was built and very little of what was known.

The other reading is more cheerful. A discrete answer means most designs have slack in them that nobody put there deliberately — an average of fifteen per cent of capacity, sitting in the structure, available to absorb the surprises this collection is otherwise full of. Which failure arrives first is usually a question about mechanisms. Sometimes the answer is that none of them arrives, because the member that was built is a size larger than the one that was designed.