The analysis that assumes the answer

A four-storey, three-bay rigid frame has forty-eight bending moments to find and twelve equations of statics to find them with. It is indeterminate to thirty-six degrees, which is a polite way of saying that no amount of adding forces will finish it, and until a machine would do the arithmetic nobody could finish it any other way either.

They finished it anyway, and the trick is worth understanding long after the necessity has gone. Assume where the answer is zero. A bending moment that is zero is a hinge as far as statics is concerned — a point that transmits shear and axial force and no moment at all — and putting enough hinges into a frame makes it determinate. Then the whole of it comes apart into free bodies, and every force in it can be found by adding.

Which free body produced the number #

Cut the frame horizontally through the assumed hinges in one storey’s columns, and take everything above the cut as the free body.

What crosses that cut is a shear and an axial force in each column, and no moment, because the cut was made where the moment is zero. What is applied above it is the lateral load of every floor above. Three equations of statics act on that free body: two force sums and one moment sum.

The horizontal force sum gives the total shear the storey’s columns carry — the storey shear, which is the sum of the loads above and which no assumption is needed for. The vertical force sum and the moment sum are two equations for the column axial forces, and there are four columns, so one more statement is needed. That statement is the whole difference between the two hand methods, and everything else about them is identical.

The portal method supplies it as a shear rule. Each interior column stands between two bays and does twice the work of an exterior column standing beside one, so the storey shear divides as 1 : 2 : 2 : 1. The column moments follow as the shear times half the storey height; the girder moments follow from joint equilibrium along each floor; the girder shears follow from the girder moments and the assumed hinge at mid-span; and the column axial forces follow from accumulating those girder shears down the building. Nothing after the first rule is an assumption.

The cantilever method supplies it as an axial rule. The frame is a vertical cantilever, so the axial force in each column varies linearly across the plan about the centroid of the column areas, exactly as the fibre stresses in a section vary about its neutral axis — and the overturning moment of the loads above fixes the constant. From there everything runs the other way: axial forces give girder shears, girder shears give girder moments, joint equilibrium gives column moments, and the column moments give the shears.

Two methods, and which one is for which building #

The portal method’s rule is about shear and the cantilever method’s is about bending, and that is exactly the distinction between a squat frame and a tall one. A low, wide frame racks: each storey shears sideways, the columns bend in double curvature, and the axial forces in them are an afterthought. A tall, narrow frame behaves like a beam standing on end: the axial forces in the outer columns are the whole of its lateral resistance, and the storey shear is the afterthought.

So the two methods are not rivals. They are the two limits of the same structure, and the received rule — the portal method up to about thirty-five metres, the cantilever method above it — is a statement about which of the two mechanisms dominates rather than about which arithmetic is better.

Where the hinge actually is #

It is not at mid-height, and the two places it is furthest from mid-height are the two places a hand check is most likely to be made.

In the bottom storey the zero sits at 0.61 of the height. A fixed base is stiffer than the joint above it — the base has no beam framing into it and no column above it to share the joint’s rotation — so it takes more of the column’s moment and pushes the zero upward. In the top storey the zero sits at 0.36, for the mirror-image reason: there is no column above to share the top joint, so the beam takes all of it and the moment there is large, which pulls the zero down.

Between those two the frame settles down. The middle storeys of this one sit at 0.48 and 0.49, and the average over every column in the frame is 0.475 — which is why the assumption survives. It is right on average and wrong everywhere, and averaging is precisely what a designer proportioning a whole building does.

The distribution error follows the same shape. The portal method gives the outer columns a sixth of the storey shear where the exact solution gives them 0.222 in the top storey — a quarter low — and 0.183 in the bottom one, which is within a tenth. The method is best where the frame is most nearly a uniform stack and worst where it stops being one, which is at both ends.

Why the error is bounded, and not merely small #

Here is the part that makes an approximate method respectable rather than merely convenient.

The set of forces the portal method returns satisfies equilibrium exactly. The column shears add to the storey shear to machine precision; every joint balances; every free body closes. What it does not satisfy is compatibility — the members, given those forces, would not fit together at the joints without kinks in them.

That is the condition of the lower-bound theorem, and two ways of being wrong sets out what it buys: a set of internal forces in equilibrium with the applied load, nowhere exceeding the capacity of the member carrying it, is a safe set. A frame proportioned for the portal method’s forces will carry the load, whatever the real elastic distribution turns out to be — provided every member has enough ductility to redistribute into that distribution, which for a steel frame is exactly what the moment that was moved on purpose is about.

So the error is not an unknown quantity. It is a known kind of quantity: a distribution error, bounded below by the theorem and bounded above by whatever ductility the sections have. That is a completely different situation from an arithmetic mistake, and it is the reason the hand methods were trusted to design buildings that are still standing.

The identity that fences it in #

There is a second bound, and it belongs to the girders rather than the columns.

Take any single span of any girder in the frame, cut it free, and draw its free body. The load on it, its two end shears and its two end moments must balance, and for a span with a uniform load that balance gives

$$M_{mid} + \tfrac{1}{2}(M_{left} + M_{right}) = \frac{wL^2}{8}$$

which is two of these move and the third cannot in one line. The right-hand side contains the load and the span and nothing else — no stiffness, no assumption, no method. So however badly an approximate analysis divides the moment between the middle of the girder and its ends, it cannot get the total wrong, and the total is what the girder has to be able to carry somewhere.

What the assumption costs, and where #

Two costs are worth separating, because only one of them is about the frame.

The first is the drift. An approximate analysis of forces says nothing whatever about displacement, and sway is what decides a tall frame long before strength does — two motions with one name is the accounting for it. A hand method can be used to size members and then has to be handed over to something that solves the compatibility it ignored.

The second is second-order effect. The frame that has swayed carries its gravity load out of plumb, and the moment that produces is not in the analysis: the load that makes itself worse is the whole of it, and it is proportional to the sway the hand method did not compute. On a slender frame this term is not a correction; it is a substantial fraction of the answer.

Both of those are compatibility problems, and they are the half the method threw away. That is not a coincidence — it is the same fact stated twice.

The history, which is about who could not afford a computer #

The methods were not devised to be quick. They were devised because there was no alternative, and the dates say so plainly: the portal method is attributed to Albert Smith in 1915 and the cantilever method to A. C. Wilson in 1908, both of them published in the decade when the steel frame stopped being an experiment and became the way tall buildings were built. Every skyscraper of the 1920s and 1930s was designed with one of them, checked by hand, on a frame nobody could solve.

What is striking, looking back, is how little the sizes changed when the frames were finally re-analysed. The buildings were not wrong — they were slightly wrong in the distribution and quite right in the total, which is the exact signature of a method that satisfies equilibrium and not compatibility. The stiffness method arrived in the 1950s and settled the distribution; it did not overturn the buildings.

Where the model stops #

The bases were assumed fixed. A pinned base has no moment at all, so the column has no point of contraflexure in it: the zero is at the base, not at 0.61 of the height, and the portal method’s hinge assumption is wrong by half a storey in exactly one place. The remedy is the obvious one — put the hinge at the base — and the fact that it has to be remembered is the method’s weakest point.

The bays were assumed equal. A frame with one long bay and two short ones does not divide its storey shear 1 : 2 : 2 : 1, because the interior columns are not serving equal halves. The honest generalisation is to divide in proportion to the bay widths a column serves, which reduces to 1 : 2 : 2 : 1 when they are equal and does something sensible when they are not.

The columns were assumed similar. A frame with a heavy corner column and light interior ones attracts shear to the heavy one in proportion to its stiffness, which is the stiffest path takes the load applied across a storey. Neither hand method knows this, and both are wrong in the direction that matters — they give the stiff member less than it will actually take.

And a transfer level breaks it completely. A frame with a storey missing, or with columns that stop, has no repeating unit for the assumption to be about — the column that stops is a discontinuity that neither the shear rule nor the axial rule survives.

The check that comes for free #

A hand method that assumes its way to determinacy has one property no solver has: every number in it can be recovered by a second free body.

Take the column axial forces at the base of the frame. The portal method gets them by accumulating girder shears down the building, floor by floor. But the whole frame is also one free body, and taking moments about the base of the windward column gives the same set directly from the applied loads and the plan dimensions — one line of arithmetic against thirty-two. If the two disagree, something upstream is wrong, and the disagreement points at which storey.

That is the same discipline as the check that cannot see the error, and it is available here precisely because the structure has been made determinate: a determinate structure’s forces can be found by more than one route, and a redundant one’s cannot. The assumption bought a check as well as an answer, which is a return nobody lists among the method’s advantages.

The generalisation #

The idea worth carrying is that an approximate method is a different structure, analysed exactly, rather than the same structure analysed carelessly.

That distinction decides what can be said about the answer. A carelessly analysed structure has an error of unknown sign and unknown size. A different structure analysed exactly has an error whose sign is available from a theorem, because the assumed structure is in equilibrium with the real load — and the difference between the two is the difference between an engineering guess and an engineering bound.

It also decides how to improve one. The way to make the portal method better is not to add a correction factor to its answers; it is to make a better assumption about where the moment is zero, and then to redo the exact statics that follows. Assume the base hinge low in a pinned-base frame; assume it high in the bottom storey of a fixed-base one; assume it at 0.4 in the top storey. Each of those is a better structure, and the arithmetic after it is unchanged.

And there is a third thing, which is why any of this survives into an age with a solver on every desk. An analysis nobody can check is an analysis nobody can be responsible for. A stiffness solution of a forty-storey frame arrives as a table of numbers with no way in; the same frame by the portal method arrives as a drawing whose every number can be recovered from the one before it by adding forces on a free body. The free body is a choice is the skill, and the hand method is where it is practised — which is the argument for keeping it, on a site that would otherwise have very little use for a method whose whole purpose was to avoid arithmetic that is now free.