The column given more than its rectangle

The load a beam is given is a decision, and the essay below this one is about what that decision is: where the dividing lines go, what the closing property buys, and why a division that is twenty per cent wrong is nonetheless safe.

Every drawing on that page divided a floor and handed each beam a share. This one is about what happens next, and it is a different failure: the division can be perfectly correct and the column still gets the wrong number, because a floor is continuous over its supports and a continuous beam does not hand each support the load standing above it.

Why the end support is relieved #

The mechanism is the hogging moment over the support, read as a set of vertical forces rather than as a bending moment.

Take the left-hand span as a free body, cut just to the left of the middle support. Its external forces are the end reaction, its own share of the load, the shear on the cut, and — the term a simple span does not have — a moment on the cut, $wL^2/8$ hogging.

That moment has to be balanced by the two vertical forces on either end of the free body. Taking moments about the cut, the end reaction is reduced by $wL^2/8$ divided by the span, which is $wL/8$. So the end reaction is $wL/2 - wL/8 = 3wL/8$, and whatever it gives up has nowhere to go but the middle support, which therefore carries $wL/2 + wL/8$ on each side: $5wL/4$.

Three-quarters and a quarter over. The redistribution is exactly the hogging moment divided by the span, and it is a statement about continuity rather than about stiffness — the same numbers come out for any $EI$, because the spans are equal and a uniform change of stiffness cancels.

That is the one place this differs from most redundant problems. An indeterminate structure usually divides its load by relative stiffness, and its answer therefore depends on a property nobody measures. Here every span has the same stiffness by construction, so the answer is arithmetic and the arithmetic has no material in it.

More spans, less error, and it never goes away #

Adding spans softens the pattern because the second span now has a span beyond it to hog against, which restores some of what the first one took.

Two features of that figure are worth reading carefully.

The first interior support is the worst position at every count, and it converges slowly and not monotonically — 1.25, 1.10, 1.14, 1.13 — because the answer alternates with the parity of the run. There is no number of spans at which it is not the heaviest column on the floor.

The end support never converges at all. It goes from 0.75 to 0.79 and stops, because an end support is always an end support: there is no span beyond it, so nothing ever hogs on its far side to restore what the first interior support took. A perimeter column is systematically over-provided and an internal one next to it systematically under-provided, on every floor plate ever drawn.

Two directions, multiplied #

A floor is not a beam. The slab hands its load to beams, the beams hand theirs to girders, and the girders hand theirs to columns — and each of those handovers is a continuous beam with its own set of factors.

The multiplication is exact rather than approximate, under one stated assumption: that the beams are close enough together for their reactions on a girder to count as a line load rather than as a set of points. Then the beam’s factor scales the load arriving on each girder line, the girder’s factor scales the load arriving at each column, and the two apply in sequence to the same load.

A factor of 2.8 between two columns on one floor plate is not a correction. It is the difference between a 300 mm column and a 500 mm one, and on a small building it is the whole of the column design.

That the error is worst on the smallest building is the part that inverts expectations. A large floor plate has most of its columns in the middle of long runs, where both factors are near 1.00 and the tributary rule is nearly right. A two-bay-by-two-bay structure — a small office, a car park deck, a domestic frame — has no interior columns at all in that sense: every one of its nine columns is at a corner, an edge or the first interior position, and not one of them carries what its rectangle says.

Why a rule wrong about every column is still safe #

Every factor above is wrong and no structure has fallen down because of it, so something is protecting the designer. It is the same thing that protected the tributary division itself, and it is worth stating precisely because the protection is not what it looks like.

The factors, weighted by their own tributary areas, still add to the whole plate. Four bays of load, however they are shared, arrive at the columns and nowhere else. So the tributary allocation is a distribution of load that is in equilibrium with the floor above it, every column is designed to have capacity for the share it was given, and the lower-bound theorem says the frame will not collapse.

That argument has the same three conditions it always has, and one of them bites differently here.

Ductility. The redistribution has to be available. A column that is short of capacity has to shed load to its neighbours, and a column sheds load by shortening — which it does, elastically, by a fraction of a millimetre. That is a very small movement to redistribute a 56 per cent overload with, and it is the reason this argument is weaker for columns than for beams.

The allocation must close, which it does by construction.

And one allocation, used everywhere. This is the condition that gets broken, and the way it gets broken here is specific: the beams are designed from an analysis that is continuous, because a beam’s moments are computed properly, and the columns are designed from tributary areas. The moments are taken from one model and the reactions from another, and the two models disagree about the reactions by 25 per cent.

That is not a conservative simplification. It is two calculations of one structure that are not consistent with each other, and the safety comes from the theorem rather than from either of them being right.

Why the two directions multiply, and when they do not #

The product is worth deriving, because it is the step that turns a 25 per cent discrepancy into a factor of 2.8 and because it has a condition on it.

Follow one bay’s load down. The slab hands a uniform load $q$ to the beams spanning one way. Each beam is continuous over the girder lines, so the load it delivers to girder line $j$ is its tributary share times $f_j$, the factor for that position in the beam direction. Those deliveries land on the girder at the beam positions.

If the beams are close enough together, that set of point loads is a uniform line load of intensity $q L_x f_j$ along girder $j$. The girder is continuous over the columns, so the load it delivers to column $i$ is its tributary share times $g_i$. The column therefore carries $q L_x L_y \, f_j \, g_i$ against a tributary load of $q L_x L_y$, and the factor is the product.

The condition is the smearing, and it is the one place the argument can fail. Where a girder carries three or four beams rather than a dozen, the load on it is genuinely a set of point loads, and a continuous beam under point loads has factors that depend on where the points fall. They are near the smeared ones when the points are spread and can be some way off when a point sits over a support — a beam framing directly into a column delivers its whole reaction there, with no continuity factor at all.

So the product is right for a floor of closely spaced joists on girders, and it is an approximation for a grid of a few heavy beams. Both cases exist, the second is commoner in steelwork, and the direction of the error is not fixed.

There is a second and more important condition, which is that the two directions are genuinely a sequence. A flat slab has no beams and no girders: the load goes from slab to column directly, in two directions at once, and the sharing is a plate problem rather than two beam problems. Its column factors are not a product of anything, and they are larger than these — which is why a flat slab’s punching check is done on a column load taken from an analysis and never from a rectangle.

Why a rule this wrong has lasted #

The tributary rule predates the analysis it disagrees with, and the sequence matters for understanding why it survives.

A floor of timber joists on trimmers, or of precast planks on beams, is a set of simply supported members, and for those the rule is exact rather than approximate: the dividing line is where one member ends and the next begins, and the shear there really is zero. The rule was a description of how the carpentry worked.

Monolithic construction made every floor continuous and did not change the rule, because the rule was already on every drawing and because the thing it computes — a column load — was not what continuity was introduced to improve. Continuity was adopted for the moments, which it reduces substantially, and its effect on the reactions was a side effect nobody had asked about.

The result is the split this essay has been circling: a modern floor’s moments are computed from a continuous model and its column loads from a pre-continuity rule, and the two have coexisted for a century because the theorem covers the gap. That is a legitimate way for a profession to work and it is worth knowing that it is what is happening, because the moment a designer needs an accurate column load — a transfer structure, a foundation on settling ground, an existing frame being assessed — the rule is not merely approximate. It is answering a question about a different building.

Which free body produced the number #

Two free bodies are in play and confusing them is the whole error.

The area is a free body of the floor: a rectangle of slab, cut on four vertical planes, whose weight has to go somewhere. It is in equilibrium with the column below it only if the shears on its four cut faces are zero, and they are zero only if the floor happens to have a point of zero shear exactly on each of those planes.

The reaction is a free body of the column: cut just below the floor, with the true shear on every face of whatever is above it. That free body is the one the column actually carries.

The tributary rule is the assumption that the two coincide — that the dividing lines fall where the shear is zero. In a floor made of separate simply supported members they do, exactly, which is why the rule is a description rather than an idealisation for a joisted timber floor. In a continuous floor the point of zero shear sits at $0.375L$ from an end support rather than at $0.5L$, and the rule has drawn its line an eighth of a span away from it.

So the tributary area is a guess at where the shear is zero, and every factor on this page is the distance between the guess and the answer.

What a designer does about it #

The factor is constant down the height, which is the property that makes it worth acting on. Errors in a load path do not cancel down a column; they compound — and a factor that is the same at every level compounds into the same factor at the base, applied to a number that has grown eightfold. A 56 per cent overload at the base of an eight-storey building is 810 kN, which is a column of its own.

Three habits follow and none is expensive.

Take the reactions from the analysis, not from the plan. Any frame analysis already computes them and prints them; the tributary figure is a hand calculation that exists because it predates the model. Reading the column loads off the output costs nothing and is right.

Where the calculation is by hand, use the factors. They are four numbers — 0.79, 1.14, 0.93, 1.00 — they do not depend on the span or the load, and applying them to a tributary area takes as long as writing them down.

And check the corner. A corner column at 0.56 of its tributary load is the one place the rule is unsafe in the useful direction: it is over-designed, and on a small building the corner columns are a quarter of the columns. That is an economy nobody takes because nobody computes it.

What the picture cannot show #

Every span here is equal and every load uniform. Unequal spans move the factors a great deal — a short end span next to a long one can take an end support below 0.6 or above 1.0 — and the pattern then depends on the ratio of the spans rather than only on their number. Nothing in these figures is a general answer; they are the equal-span case, which is where the rule is least wrong.

The columns are rigid supports. They are not: they shorten under load, and a column that shortens sheds some of its share to its neighbours. A support that moves redistributes, and the redistribution is toward the tributary answer, because a perfectly flexible set of supports gives exactly the tributary shares. So the true factor lies between the figures here and 1.00, nearer the figures for a stiff frame and nearer 1.00 for a slender one.

And load arrangement is absent. Every figure loads every span at once, which is not the arrangement that maximises anything. The pattern that governs a column is usually chequerboard rather than full, and it moves the factors further apart rather than closer.

The assumption that survives all of it #

One thing in this essay is exact and does not depend on spans, stiffness, arrangement or continuity: the reactions add to the load.

That is why the rule’s closing property is the part that is not negotiable, and it is why the failure it produces is a distribution failure rather than a magnitude one. Nothing is lost and nothing is invented; a fixed quantity of load is shared out on a rule that is a little too simple, and the consequence is a set of columns whose sizes are slightly wrong in a pattern that repeats on every floor of every building drawn this way.

A pattern is easier to correct than a scatter, which is the optimistic reading and is also true. Four numbers, applied once, remove the whole of it.

Still open: where the five kilonewtons came from #

Everything on this page and on the essay below it has treated the load per square metre as given, and it is the one input to which every calculation downstream is exactly proportional. Where that number comes from is a question about occupancy, statistics and convention rather than about structures — the reduction a column is allowed for carrying many floors is the one place the collection engages with it, and that essay is about a factor applied to the number rather than about the number.

After that, the descent through a structure that is not a stack: a transfer beam, an outrigger, a column that stops. The tributary rectangle assumes every column runs to the ground, and where one does not the load has to turn a corner — at which point the allocation stops being a sum and becomes a stiffness problem, with all the sensitivity that implies.