The envelope is not a structure

A three-span continuous beam carries a permanent load that is always there and an imposed load that may or may not be. The imposed load is a use of the building — people, storage, a plant room — and it sits on whichever spans happen to be occupied.

That gives eight arrangements: each span loaded or not. Every one of them is a real state of the structure and every one is in equilibrium. The beam has to be designed for all of them at once, and the way that is done is to take the greatest sagging moment and the greatest hogging moment at every station over all eight, and design for those.

The result is called an envelope, it is drawn as a pair of curves, and it looks exactly like a bending-moment diagram. It is not one. Nothing produces it, nothing is in equilibrium with it, and the identity that every one of the eight satisfies exactly, it fails by nearly a quarter.

Which free body produced the number #

Take one span out of the beam with whatever moments the neighbouring spans apply at its ends.

The moment anywhere in it is the free moment of a simply supported span carrying that span’s own load, plus the straight line joining the two end values. At mid-span that gives an identity that holds for any end moments whatever:

$$M_{mid} - \frac{M_A + M_B}{2} = \frac{wL^2}{8}$$

It is statics, it contains no stiffness, and it is true for every arrangement of load on every other span, because everything those spans do arrives here as $M_A$ and $M_B$ and nothing else.

Run it on all eight cases: the residual is machine zero in every one. Run it on the envelope, taking the sagging envelope at mid-span and the hogging envelope at the two supports, and the middle span misses by 23 per cent.

The reason is not subtle once stated. The mid-span ordinate of the envelope comes from the case with that span loaded and its neighbours empty; the support ordinates come from cases with the neighbours loaded. They are ordinates of different beams.

The identity is worth pressing on, because it is the thing the envelope fails and it is much harder to break than it looks. It contains no stiffness. So take the same three spans and change one of them: make the middle span stiffer, or softer, by any factor at all. Every moment in the beam moves — that is what redundancy means, and it is the reason a continuous beam has to be solved rather than read off a table. The support moment moves, the span moment moves, and the two move in opposite directions, because whatever the support gives up the span takes.

The one combination of them does not move. Sweep the middle span’s stiffness over a factor of twenty-five and the mid-span ordinate plus the average of the two end moments sits on the same horizontal line the whole way across, at the free moment for that span and that load. A beam analysis is a redistribution of a fixed quantity, and the fixed quantity is statics.

Which arrangement does what #

The two patterns that matter are easy to state and worth deriving rather than remembering.

To maximise the sagging moment in a span, load that span and leave its neighbours empty — then load the next-but-one, and so on. A loaded neighbour puts hogging moment into the shared support, which pulls the whole diagram of the span under consideration downward. An empty neighbour offers the least restraint it can. The pattern that follows is alternate spans, and on this beam it raises the first span’s sagging peak from 179 to 199 kNm: 11 per cent above loading everything.

To maximise the hogging moment over a support, load both spans adjacent to it and leave the rest. Both contribute hogging at that support; a loaded third span puts sagging curvature into the second and relieves it. On this beam the adjacent pattern gives 240 kNm against 224 for the all-loaded case: 7 per cent above.

Neither maximum is produced by the arrangement carrying the most load. That is the counter-intuitive part, and it is a direct consequence of continuity: a load on one span produces moments of both signs in its neighbours, so adding load somewhere can reduce an effect somewhere else.

Seven of the eight are needed #

A natural question is how many of the arrangements actually matter. On these three spans, seven of the eight appear somewhere on the envelope, and the identity of the one that does not is the essay’s first claim restated as a fact about the picture: the arrangement that never governs anywhere is the one with every span loaded. It is beaten in the spans by the alternate pattern and at the supports by the adjacent pair, and there is no station left for it to own.

The empty case, which looks like the obvious candidate for the useless arrangement, does govern. It owns the hogging envelope through the middle of each span, because the hogging envelope there is the least sagging moment any arrangement produces and the least is what the permanent load alone gives. That ordinate is what decides whether a span needs top steel at mid-span, so it is read rather than ignored.

On four spans it is ten of sixteen; on five spans, thirteen of thirty-two. The count grows and the fraction falls, which is what makes the practical rules worth having, because the alternative is $2^n$ analyses. It is also why moment distribution mattered long after anyone could solve a beam once: Hardy Cross’s method costs a fresh set of cycles per load case and nothing else, so a designer facing eight arrangements paid eight times a small number rather than eight times a large one.

The rules used in practice — load all spans; load alternate spans; load adjacent pairs — are not an approximation to the full set. They are a selection of it, chosen because between them they contain the case that governs each effect. The envelope built from three rules and the envelope built from all $2^n$ arrangements agree at every station where a design number is taken.

Why the envelope is used anyway #

Nothing above is an argument against the envelope. It is the correct design object, and the reason is the lower-bound theorem again.

A section provided for the envelope is a section that can carry every one of the eight cases, because at every station its capacity is at least what any of them demands. That is all a design has to establish. That no single case produces the envelope is irrelevant to the safety of the beam — the beam only ever experiences one case at a time.

What the envelope must not be used for is anything requiring a state. Three examples, all of them mistakes that get made:

Deflections. The deflected shape under the envelope is meaningless; the deflection has to come from whichever arrangement maximises it, which is generally alternate spans.

Reactions. Adding the envelope’s shears at a support gives a reaction no case produces, and column design uses reactions.

Redistribution. Moving moment from a support to a span requires equilibrium to be maintained, and it can only be done to a diagram that has it. Redistributing an envelope is a category error; the redistribution has to be applied case by case and the envelope taken afterwards.

The load that is not all imposed #

Everything so far has treated the permanent load as fixed. It is not quite, and the way it is handled is the second half of the subject.

The permanent load is also uncertain — a screed thickness, a partition allowance, a services load — and its uncertainty acts in both directions. Where it opposes the effect being maximised, its minimum credible value is the dangerous one. So the codes give permanent actions two partial factors, a large one and a small one, and require the small one on spans where the load relieves.

On a continuous beam that produces a further subdivision of every arrangement: the loaded spans get the full factored permanent load and the unloaded ones get the reduced value. It doubles the arithmetic and it moves the answers by a few per cent, and it is the same idea as the empty-jib case for a counterweight — a permanent action that helps is a permanent action whose minimum matters.

There is a prior convention underneath all of it, and it is worth naming before the enumeration is trusted too far. Every arrangement here treats the imposed load as a uniform intensity that is either present over a whole span or absent from it, which is already an idealisation of a load that is spread out in some real and untidy way. Occupancy is neither uniform nor span-shaped. The block pattern is a convention chosen to make the enumeration finite, and its defence is that the two block cases bracket the partial ones closely enough — which is examined below and is not quite true.

Where the arrangement stops being a choice #

Pattern loading applies to loads that can be absent. Several important actions cannot, and the distinction is worth drawing sharply.

A wind load is not patterned span by span; it is a single action with a direction, and its arrangements are directions rather than subsets.

A temperature effect acts on the whole structure at once, and its magnitude rather than its position is the variable.

A settlement is an imposed deformation whose arrangement is which support moved, which is a different enumeration entirely and one where a single support settling is usually worse than all of them settling together.

And a moving load — a train, a crane, a vehicle — is not a subset at all but a continuum of positions, which is the influence line’s own problem and needs a different tool.

The common thread is that “the worst case” is a search over a set, and the shape of the set is decided by the physics of the action rather than by anything about the structure.

The same question exists one level up, and the answer taken there is a different one. Which bays of a floor are occupied is an arrangement problem of exactly this kind, but nobody enumerates it: the load a beam is given comes from a tributary area with no pattern in it at all, and the reduction factors applied to a column carrying many floors are a statistical admission that not every floor is loaded at once. Pattern loading is enumerated on a beam and averaged on a column, and the reason is that the set is small in one case and astronomically large in the other.

Two spans, and the smallest example that shows it #

Three spans is the smallest beam on which the alternate and adjacent patterns are genuinely different. Two spans is smaller still and is worth doing by hand, because everything above appears in it with nothing to hide behind.

Two equal spans, permanent load on both, imposed load available on either. Four arrangements: neither, first, second, both.

• Both loaded gives the largest support moment, $(g+q)L^2/8$, and a span moment of $9(g+q)L^2/128$.
• One loaded gives a smaller support moment — the unloaded span contributes only its permanent share — and a larger span moment in the loaded span, because the support is holding it back less.
• Neither gives the permanent-load diagram, which governs nothing.

So the support is designed from the both-loaded case and each span from its own singly-loaded case, and the empty case supplies the hogging ordinate through the middle of each span, which is why all four of them are needed here. Draw it and the sagging envelope through each span is above the both-loaded curve while its support ordinate is on it — which is the discrepancy of the three-span beam, in miniature and unmistakable.

What the two-span case makes obvious is that the discrepancy is not a rounding or a conservatism introduced by the envelope. It is the envelope’s whole content: if every ordinate came from one case, the envelope would be that case and there would be no reason to compute it.

What the envelope is actually read for, which is where the steel stops #

The envelope’s real working use is not the peak values. Those could be got from three analyses and a comparison. It is read for the whole of its length, because the thing being decided along that length is where each bar can stop.

A reinforced concrete beam has bottom steel where the envelope sags and top steel where it hogs, and the point at which a bar is no longer needed is the point at which the envelope no longer demands it. This is the one design operation that genuinely needs the curve rather than its maxima, and it is the reason the envelope is drawn at all.

Two things then go wrong if it is read naively, and both are worth having.

The envelope’s zero is not any case’s zero. The station where the sagging envelope crosses the axis is the station beyond which no arrangement produces sagging — which is further out than the crossing of any single case. That is the right answer, and it means the point of contraflexure a designer may remember from the all-loaded diagram is in the wrong place: on this beam the all-loaded case changes sign appreciably closer to the support than the envelope does, and steel curtailed at the first point is absent where a different arrangement needs it. The envelope is conservative here, and it is conservative because it is doing its job.

And the tension in a bar is not the moment at the section it sits under. Once a beam is carrying shear, the tension in the bottom steel at any station corresponds to the moment some distance back along the beam — the truss analogy’s inclined compression field pushes the tension force outward, by about the lever arm times the cotangent of the strut angle. So a bar has to extend past the point the envelope says it is needed, by a shift that has nothing to do with load arrangement at all. Codes fold this into a “shift rule” and it looks like an arbitrary detailing margin; it is a statement about equilibrium in the web, and it applies to every case in the envelope identically.

The two effects add rather than cancel, which is the practical summary: a bar stops later than the diagram suggests for two independent reasons, one of them about which load case and one about which free body. And it is a place where the envelope’s not being a state costs nothing, because curtailment is a pointwise question — does this bar need to be here — and pointwise is exactly the kind of question an envelope can answer.

Where the model stops #

The spans are equal and the load intensity is the same on each. Unequal spans change which pattern governs where, and can make a pattern that never governs on a regular beam the critical one — in particular where a short end span abuts a long interior one.

The beam is elastic and the analysis linear. Superposition is what makes eight separate solves legitimate. A beam that has cracked under one arrangement is a different beam under the next, and its stiffness distribution depends on its history.

And the arrangements are of one variable action. With two — an imposed floor load and a moveable partition load, say — the enumeration is over the product of the two sets, and the combination factors that reduce simultaneous variable actions arrive to keep the result from being absurd.

The arrangement that governs the deflection #

Strength is not the only thing the arrangement decides, and the serviceability answer is the one most often taken from the wrong case.

The deflection of a span is largest when that span is loaded and its neighbours are not — the same alternate pattern that maximises its sagging moment, and for the same reason: the neighbours’ hogging restraint is what holds the span up. The difference is not small. On these spans the alternate arrangement deflects the outer spans appreciably more than the all-loaded case, and a deflection computed from the all-loaded arrangement is an underestimate of the number that will be measured.

There is a second reason the arrangement matters more for deflection than for strength, and it is about what the check is against. A strength check compares one number with a capacity that has a factor of safety in front of it. A deflection check compares one number with a span over three-hundred-and-sixty, and there is no factor anywhere — so an error of ten per cent in the arrangement is ten per cent of the answer, not ten per cent of a margin.

What the pictures cannot show #

The eight faint curves are drawn together, which is the one thing the beam never does. Each is a photograph of a different Tuesday.

Nor can the figure show the reason any of this is more than an exercise, which is that the sagging steel and the hogging steel are different bars in different places. An envelope is not really a curve; it is a set of instructions about where reinforcement stops, and the stations at which two arrangements exchange places are the stations at which a bar has to run past.

A third thing the pictures leave out is the column. Every arrangement produces a set of reactions, and the arrangement that maximises a column’s axial load is not the one that maximises the moment the beam delivers into it — a column at the end of a run of spans gets its largest moment when the adjacent span is loaded and the next is not, which is a case in which its axial load is below the maximum. Designing a column from the beam’s worst reaction and the beam’s worst moment simultaneously is designing it for a case that does not exist, and the penalty is real: on an interaction curve the two extremes are far apart.

The assumption the figure rests on #

Each span is either fully loaded or fully empty. That is a convention, and its justification is that partial loading of a span is bracketed by the two cases — which is true for the span’s own moment and not quite true for everything else. A load on half a span produces a slightly different set of end moments than the same total load spread over all of it, and for the support moment the half-span case can be marginally worse. The convention is defended by the margin rather than by the arithmetic, and it survives because the difference is smaller than the uncertainty in the load itself.

The ladder from here #

Later rungs on this anchor: the enumeration for two or more variable actions, and the combination factors that keep it finite. Pattern loading on a two-way slab and on a flat slab, where the “spans” are strips in two directions and the count grows quickly. Redistribution applied case by case, and why the envelope of redistributed diagrams is not the redistribution of the envelope. Reaction envelopes, and why a column cannot be designed from a beam’s moment envelope. And the whole question of what an envelope is for, which reaches past this subject into any design made against a set of scenarios rather than a state.