The addition everything else rests on

Nearly every method in this collection is an addition. An influence line is added up over the axles of a train. The unit-load method sums a product over members. Moment distribution passes corrections round a frame and adds them. A load combination adds actions with factors on them; a prestress is read as a load and added to the applied one; a stiffness matrix multiplies, which is addition in a different notation.

All of it rests on one sentence: the response to a sum of loads is the sum of the responses. That sentence is a theorem, not an axiom, and it has three hypotheses.

Why it is exact rather than close #

The exactness is worth a moment, because it is the difference between a theorem and a rule of thumb.

The governing equation of a beam is $EI\,y'''' = q$. The operator on the left does not contain $y$ or $q$ anywhere except linearly, so if $y_1$ solves it for $q_1$ and $y_2$ for $q_2$, then $y_1 + y_2$ solves it for $q_1 + q_2$ by inspection. The boundary conditions are homogeneous — $y = 0$ at a support, whatever the load — so they add too.

A linear operator distributes over addition by definition, and that is the whole proof. The residual is not small because the numbers are nice; it is zero because the statement is an identity.

That is why superposition is not on the same footing as the approximations elsewhere on this site. Plane sections staying plane is approximately true and the error can be estimated. A shear stress that is uniform across a thin wall is approximately true and gets worse as the wall thickens. Superposition is exactly true, or it is not available at all. There is no regime in which it is a few per cent out.

The three hypotheses #

1. The material is linear elastic. Stiffness must not depend on the load.
2. The geometry does not change enough to matter. Equilibrium must be written on the undeformed shape.
3. The boundary conditions do not depend on the load. What is supporting the structure must be the same in every case.

Each of them can fail, and this site has essays about all three failing — but never with the additivity error measured. Here it is, on one beam.

Cracking, which is the big one #

A reinforced concrete section’s stiffness depends on how far past cracking it is. Branson’s interpolation makes that explicit: the effective $I$ falls from the uncracked value toward the cracked one as $(M_{cr}/M)^3$, so it is a function of the total moment.

Compute the point load alone and the section sees a moment of 120 kNm, giving an effective stiffness of 0.543 of the gross. Compute the uniform load alone: 96 kNm, and 0.726. Compute them together: 216 kNm, and 0.383.

Each case computed alone is riding on a stiffer beam than the pair is, so:

The sum is 62 per cent short of the truth, and it is short in the unsafe direction.

The size of that error is not a property of concrete. It is a property of where the cracking moment sits relative to the three moments being computed, and the same beam with a different $M_{cr}$ gives a different answer — which is worth drawing, because the worst case is not the one an engineer would guess.

That is the worst arrangement available and it is the ordinary one: a section sized so that it does not crack under either action considered alone. The two other rows have not moved, because nothing else about the beam changed.

The error does not keep growing as the section gets weaker, which is the second surprise.

Raise $M_{cr}$ above the 216 kNm the pair produces and the row vanishes altogether: nothing cracks in any case, and the residual is zero like the first row. So the cracking error is a window, largest where the combination crosses a threshold that the individual cases do not, and small on both sides of it. That is the opposite of the intuition that a more heavily cracked structure is a worse-behaved one, and the reason is that superposition is broken by a difference in stiffness between the cases and not by the stiffness being low. It is worth adding that the real member is stiffer than any of this says, because the concrete between the cracks goes on carrying tension — which moves the numbers and not the shape of the argument, since that stiffening is itself a function of how far past cracking the member is.

This is why deflection checks on concrete are done at the service load combination as a whole and never by adding cases, and it is why the quasi-permanent combination is specified rather than left to the engineer: which loads are on at the same time changes the stiffness, and therefore changes the answer to all of them.

The gap, which points the other way #

Put a support under the beam with a small clearance under it, so the beam is a cantilever until it lands and a propped one afterwards. That is hypothesis 3 failing.

For a 4 m cantilever with a 120 mm gap: the uniform load alone deflects 91.4 mm and never touches. The point load alone deflects past the gap, lands, and finishes at 131.5 mm. Both together land earlier and finish at 139.9 mm — against a sum of 223.0 mm.

The sum is 37 per cent too large, and it is too large because the case that was never touching had a much softer structure than the case that was. Adding the two adds a soft answer to a stiff one and produces a structure that is neither.

This one is a window too, and its edges are easier to see because they are a length. Set the same support 400 mm down instead of 120 and nothing reaches it: the point load’s free cantilever deflection is 304.8 mm, the uniform load’s is 91.4, and even together they arrive at 396 mm with the support still untouched.

A support that only exists sometimes appears far more often than the phrase suggests: a bearing that lifts off under wind, a prop struck part way through a pour, a slab resting on a wall it is not connected to, a base plate whose bolts go into tension only past a certain moment. Each of them is a structure with two configurations and a load that decides which, and each has a window of loading in which the addition fails and two ranges either side of it in which it does not.

Second order, which is the small one and the famous one #

This is the failure everybody names and it is the least of the three here.

The usual treatment amplifies the first-order deflection by $1/(1 - N/N_E)$. That factor is a constant for a given axial load, so multiplying by it is a linear operation and a superposition check made with it comes out exact — proving nothing.

The real nonlinearity is subtler. The exact beam-column solutions are

$$\delta_P = \frac{P}{2Nk}\left(\tan u - u\right), \qquad \delta_w = \frac{w}{Nk^2}\left(\sec u - 1\right) - \frac{wL^2}{8N},$$

with $u = kL/2$ and $k = \sqrt{N/EI}$, and they are different functions. Their amplifications at $N/N_E = 0.62$ come out at 2.5925 and 2.6208 — the uniform load is amplified slightly more, because its deflected shape is slightly closer to the buckling half-sine than the point load’s is.

Using one factor for the combination therefore gets it wrong by 0.32 per cent, and the direction is that the single-factor answer is too small. It grows with the axial load, and far more slowly than the other two failures do.

The reason it stays so small is the same one that makes the amplifier work at all: the amplification depends on how close a deflected shape is to the buckling mode, and every ordinary deflected shape is close to it. A point load and a uniform load are two of the closest, which is why the one failure with a clause, a factor and a name in the codes is the one that costs least here.

What is really being assumed when a designer adds #

Superposition’s practical importance is not in deflections. It is in the envelope, and there the assumption being made is a different one and larger.

The chain is worth spelling out because each link is usually invisible:

• a design is checked against a combination of actions;
• the combination’s effects are found by adding the effects of the actions;
• which requires the structure’s response to be linear;
• which requires its stiffness not to depend on what is on it;
• which for a cracked concrete member it does.

Every step of that is standard practice, and the fourth is false for most of the world’s buildings. What saves it is that the forces in a statically determinate member do not depend on stiffness at all, and that in an indeterminate one the redistribution caused by unequal cracking is usually small — but “usually small” is a different kind of claim from “exactly zero”, and the two are routinely conflated.

The one place it is not optional #

There is a use of superposition on which the whole practice of structural design depends and which has no alternative, and it is worth separating from the deflection arithmetic above.

Design is done against combinations, and combinations are enumerated. A building has permanent load, imposed load in several patterns, wind from four directions, snow, thermal actions and an accidental case; the number of combinations runs to dozens and sometimes hundreds. Every one of them is analysed by taking the results of a handful of unit analyses — dead, imposed, wind X, wind Y — and adding them with factors.

If that were not legitimate, each combination would need its own full analysis of the structure, and the number of analyses would be the number of combinations rather than the number of actions. For a large model that is the difference between an afternoon and a fortnight.

Which actions can be on at the same time is a separate question again, and it is answered by convention rather than by mechanics. What superposition supplies is only the arithmetic once the set has been chosen; it has no opinion about whether a full snow load and a full wind can arrive together, and it would produce the same additions just as exactly for a set of combinations nobody would defend.

So the licence is load-bearing in the most literal sense. It is also the one nobody re-derives, because the alternative is unthinkable rather than merely expensive.

The influence line exists only because of this #

The opening list mentions influence lines first and the essay has not returned to them. It should, because an influence line is superposition’s most complete expression and the clearest thing to lose when it fails.

An influence line is the value of one effect — a reaction, a moment at a section, a deflection at a point — as a unit load walks the structure. Compute it once and the response to any loading follows by integration:

$$E = \int q(x)\,\eta(x)\,dx \;+\; \sum_i P_i\,\eta(x_i)$$

That is an enormous compression. A continuum of possible load positions, an unbounded set of load patterns, and every train, every crane position and every arrangement of imposed load — all of them answered by one function computed once. Nothing else in structural analysis buys that much with one calculation, and the entire apparatus of moving-load design rests on it.

And it rests on it completely. The integral above is superposition written as an integral: it says the effect of a distributed load is the sum of the effects of its parts, each acting alone. A structure for which that is false has no influence line at all — not an inaccurate one, none, because the response to an axle at mid-span depends on whether another axle is standing at the quarter point, and a function of one position cannot express a dependence on two.

Which is a sharper loss than the deflection errors above. A cracked concrete bridge at high load has a stiffness that depends on the whole loading, so its influence lines exist only in the range where it is uncracked — and a bridge assessment pushing a structure toward its capacity is working precisely where they do not. The influence lines used in practice are elastic ones, and the honest statement of what they describe is the structure’s behaviour at loads below the one being assessed.

The same failure has a name in the field next door. A transfer function exists only for a linear system, which is why a nonlinear structure does not have a frequency response, does not have a “period” in the ordinary sense, and cannot be characterised by how it answers one input. An influence line is a transfer function in space where a frequency response is one in time, and both are the same theorem’s dividend.

It is worth noticing what this does to the cost of an analysis, because it is the reason influence lines survive in bridge work and are rare in buildings. A structure with an influence line needs one analysis per effect — one for each moment, shear or reaction anybody cares about — and then no analysis at all for any loading. A structure without one needs an analysis per load position, and a moving load has a continuum of them. That is the difference between a table computed once and a sweep run for every vehicle in a code’s fleet, and it is why the loss of superposition is felt in bridges long before it is felt anywhere else.

There is a second casualty worth naming, because it is a search rather than a sum. The worst position of a moving load is found by walking it along and reading where the influence line peaks — which assumes the structure is the same at every position. Where a bearing lifts off, a gap closes, or a member goes slack as the load moves, it is not: the structure has two configurations and the load decides which, so the influence line has a discontinuity in it and the search has to be run separately on each side. That is hypothesis three failing in the one place where its failure is easiest to miss, because the walk produces a smooth-looking curve either way.

Which free body produced the number #

None. Superposition is not a statement about a free body; it is a statement about an operator, and it is proved by inspecting the differential equation rather than by cutting anything.

That makes it unlike every other result in this collection, and it is why the failures are so hard to notice. A free body that is wrong is usually visibly wrong — a force missing, a support in the wrong place. An operator that is not linear looks exactly like one that is, and the only symptom is that two calculations which ought to agree do not.

What breaks it in practice, ranked #

The four rows on the chart are one beam’s illustration. Ranked by how often each one actually bites in a design office, the order is different and worth stating:

1. Cracking, and every other load-dependent stiffness. Reinforced concrete, composite decks with partial interaction, timber connections that slip, bolted joints before they slip. Anything whose stiffness is a function of what is on it.

2. Boundary conditions that switch. A bearing that lifts off, a base plate whose bolts take tension only past a moment, a gap at a movement joint that closes, a slab that is resting rather than fixed. These are common, they are almost never modelled as switches, and the error is large.

3. Plasticity and history. A hinge that has formed does not un-form, and the response depends on the order the loads arrived in. This is why shakedown is a statement about a sequence rather than about a combination, and why any structure that has yielded has left the linear world permanently.

4. Second-order geometry. The one that gets a clause, a factor and a name, and the smallest of the four for anything a designer would build — because a structure at $N/N_E$ high enough for the shape effect to matter is a structure that has failed another check first.

Where the model stops #

The four cases are one beam. The residuals measured are for the specific loads, span and section drawn; they are illustrations of a direction and a magnitude, not coefficients anybody should carry away. Change $M_{cr}$ and the cracking error moves by tens of per cent.

Branson’s interpolation is itself a fitted curve. A more careful cracked-member calculation integrates the curvature along the span with the stiffness varying station by station — and it gives a different number, though the same sign and roughly the same size.

The gap case is idealised as two stiffnesses. A real beam landing on a support does so progressively, through a contact that starts at a point and grows, and the transition is smooth rather than a switch.

Nothing here is about plasticity. A member with a plastic hinge in it has abandoned superposition entirely and permanently, and the sequence in which the loads were applied now matters — which is a fourth failure, of a different kind, since it makes the response depend on the load’s history and not merely on its total.

And the pictures show only deflections. The forces in a determinate structure superpose whatever the material does, because they come from equilibrium alone. Everything on this page is about the quantities that need a stiffness, which is deflections and the internal forces of indeterminate structures — and the second of those is where it matters most and is drawn least.

The habit worth taking away #

There is a single question that catches all four failures and it takes ten seconds to ask.

Does the structure that carries load case B know that load case A is on?

If it does not — if the stiffness, the supports and the geometry it presents to B are the same whether or not A is there — the results add exactly. If it does, they do not, and the size of the error is the size of the difference between the two structures.

That formulation also says where to look for the failure, which a list of hypotheses does not. Cracking makes the beam softer for B when A is on; a gap makes it stiffer; an axial load makes it softer by a little; a hinge makes it a different structure entirely. In every case the question is answered by asking what B’s structure is, and noticing that it has two answers.

The ladder from here #

Later rungs on this anchor: reciprocity, which is superposition’s close relative and says something stronger — that the deflection at A from a load at B equals the deflection at B from a load at A, and which needs the same three hypotheses. Load-history dependence in plastic analysis, and why shakedown is a statement about a sequence rather than about a combination. Nonlinear analysis in practice, where the answer is obtained by loading in increments and superposition survives only within each increment. The cracked-frame problem properly solved, with the stiffness of every member updated to the combination being analysed and the whole envelope recomputed for each. And the modelling question this raises and does not answer: if the answer depends on which loads are on at once, what is the right set of combinations to analyse — a question with no structural content and a great deal of consequence.