The table that cannot be read halfway

Two hand methods for continuous frames were in general use in the middle of the twentieth century. One is moment distribution, published by Hardy Cross in 1930, and it is the one taught in English. The other is Kani’s method, published by Gasper Kani in 1949, and it is the one taught in German, Spanish and Portuguese.

The usual account is that they are rival techniques with different convergence properties. They are not.

That is a stronger statement than it looks, and it is worth saying exactly what has been checked. The two curves are produced by two pieces of code with nothing in common: one balances joints and carries moments over, the other computes rotation contributions and assembles moments at the end. They agree at every sweep to the last digit a double-precision number holds. They are not similar methods. They are one method in two notations.

The same equations, and why that had to be so #

A structure’s joint rotations satisfy a set of simultaneous equations — one per joint, each saying that the moments meeting there sum to nothing. Both methods solve that set by sweeping through the joints, updating each one using the current values of its neighbours, and going round again.

That procedure has a name outside structural engineering. It is Gauss–Seidel, and its convergence is a property of the matrix rather than of the notation used to write the sweep down. The matrix here is the same in both cases, because it is the structure’s stiffness matrix and the structure is the same. The sweep order is the same. So the iterates are the same, and there is nothing left for the two methods to differ in convergence about.

What can differ, and does, is what quantity is written in the boxes.

Rotations instead of moments #

Cross’s table holds moments. Each entry is a share of an out-of-balance moment or a carry-over of one, each is added to a running total, and the total in any column at any time is a bending moment at a member end.

Kani’s table holds rotation contributions. The quantity is written $m'_{ij}$, it is a moment in units but not in meaning, and it becomes a bending moment only at the end through

$$M_{ij} = M^F_{ij} + 2m'_{ij} + m'_{ji}$$

— the fixed-end moment, plus twice the near end’s contribution, plus the far end’s.

The sweep itself is one line. At joint $i$,

$$m'_{ij} = \mu_{ij}\left(M^F_i + \sum_j m'_{ji}\right)$$

where $M^F_i$ is the sum of fixed-end moments at the joint and $\mu_{ij} = -\tfrac12 k_{ij}/\sum k$. The distribution factor is there, halved. The half is the carry-over, absorbed into the factor once at the start rather than written down at every member end of every cycle.

That is the whole of the difference in effort. On this beam Cross writes a share and a carry-over for each of six member ends every cycle; Kani writes one number for each of six member ends every sweep, and the carry-over never appears as an entry at all.

What the factors become #

The factors are computed once, before anything, exactly as they are in the other method.

At a pinned end the factor is $-\tfrac12$ regardless of what the member is, and the reason is worth following because it is the one case where the algebra says something the table does not.

A pin has one member and no moment. Put $\mu = -\tfrac12$ into the sweep and the contribution comes out as $-\tfrac12(M^F + m'_{\text{far}})$. Assemble the end moment from it and every term cancels: the pin carries nothing, exactly, on the first sweep and every sweep afterwards. The released end is handled by the ordinary rule rather than by an exception, which is not true of Cross’s table, where a pinned end is either balanced to zero every cycle or given a modified stiffness to save the trouble.

The first sweep, written out #

The sweep is short enough to follow completely on this beam, and following it once is the fastest way to see that no bending moment is being computed.

The fixed-end moments are $wL^2/12$: 128 kNm on each 8 m span and 200 on the 10 m span. Every rotation contribution starts at zero.

Joint 0, the left pin. One member, $\mu = -\tfrac12$. The joint’s fixed-end sum is $-128$ and its neighbour’s contribution is still zero, so $m' = -\tfrac12(-128) = 64$.

Joint 1, the first interior support. The fixed-end sum is $128 - 200 = -72$. The far-end contributions are the 64 just computed and the zero at the far end of the middle span, so the bracket is $-72 + 64 = -8$. Multiply by the two factors: $-8 \times -0.278 = 2.22$ into the 8 m span and $-8 \times -0.222 = 1.78$ into the 10 m span.

Joint 2 does the mirror image but with the first joint’s numbers already updated, which is what makes this Gauss–Seidel rather than Jacobi, and its contributions come out $-16.40$ and $-20.49$. Joint 3, the right pin, gives $-53.75$.

Six numbers, and the sweep is finished. Assemble the moment at the first interior support: $M = 128 + 2(2.22) + 64 = 196.44$ kNm, against a converged 197.22 — under half a per cent out after one pass over the beam.

That assembly is the point. The 196.44 is not on the sheet. It was constructed from three numbers, two of which live in different boxes, and doing it is a separate piece of arithmetic that the method does not require until the end.

Why the assembly has a two in it #

The factor of two in $M = M^F + 2m' + m'_{\text{far}}$ is not a convention and it is worth a paragraph, because it is where the carry-over went.

Slope-deflection gives the end moment of a member as $M_{ij} = M^F_{ij} + (2EI/L)(2\theta_i + \theta_j)$. Define the rotation contribution as $m'_{ij} = (2EI/L)\theta_i$ — a moment-sized quantity standing in for a rotation — and the expression reads $M_{ij} = M^F_{ij} + 2m'_{ij} + m'_{ji}$ directly.

So the two is the coefficient of the near end’s own rotation in the slope-deflection equation, and the single $m'_{ji}$ is the far end’s. The carry-over of one half in Cross’s method is the ratio of those two coefficients, $1/2$, and Kani’s method never writes it because it never separates them. One method carries a half along a member each cycle; the other multiplies by two at the end, once.

That is the cleanest available statement of what the two notations are doing differently, and it explains the arithmetic saving without any appeal to cleverness. The same quantity is being handled, and one of the two writes it down more often.

A wrong number, and what happens to it #

The two methods differ in one more way and it is the one that decided the argument for a generation of engineers with pencils.

A moment distribution accumulates. Each cycle adds to what the previous one left, so the current value of an entry is a sum of everything written above it in the column. Add a carry-over wrongly and the mistake is in that sum from then on, in every later cycle, and it converges neatly to an answer that is wrong.

Kani’s method recomputes. Each contribution is written afresh from its neighbours’ current values, and the previous value of the box plays no part in the new one.

The second sweep is the interesting one. The run that started wrong is at 6.81 kNm and the clean one at 7.84 — the spoiled table is momentarily the better of the two, which is not a paradox but a consequence of the wrong number having pushed the iterate to the other side of the answer.

What it demonstrates is that the wrong entry has stopped existing. It was overwritten on the first sweep of the joint that owns it, and everything after that is the ordinary iteration from a slightly different starting point — and the starting point of an iteration that converges does not matter.

This is a real property and not a curiosity. A hand computation of a five-storey frame is several thousand pencilled numbers, an arithmetic slip somewhere in it is a certainty rather than a risk, and a method that forgets its mistakes is worth a great deal more than one that is a few per cent faster.

Six numbers that are not yet an answer #

Nothing is free, and the price is in the same property that makes it self-correcting.

A moment distribution stopped halfway hands over a moment diagram. It is wrong by an amount that can be estimated from the out-of-balance still sitting at the joints, and it can be sketched, checked against a free body, and compared against a span’s own $wL^2/8$ to see whether it is plausible. A partial answer is an answer.

Kani’s table stopped halfway hands over six numbers that have to be assembled before any of them can be looked at, and the assembly is itself arithmetic that can go wrong. There is no partial moment diagram, no running check against a free body, and no sanity test available on the table itself.

That distinction is not about accuracy. Both tables at four sweeps contain the same information about the structure, to the same precision, and the assembly is three additions. It is about what a person can see while the work is in progress, and it is the reason a method with a genuine advantage in both arithmetic and error tolerance did not displace the other one.

The irregular beam, where neither is quick #

Both methods slow down on the same beams, and for the same reason, which is another consequence of their being one method.

A more irregular beam is slower to converge and slower to forget, and those are the same statement. Both are governed by how strongly a joint is coupled to its neighbours relative to itself, which is the diagonal dominance that makes the method work at all.

The answer both of them reach #

Three independent computations of one beam is more redundancy than a figure usually gets, and it is here because the claim being made is unusual. These two methods are the same iteration is not a statement that can be supported by two curves lying near each other; it needs both of them checked against something outside the pair, so that the identity is not an identity of two shared bugs.

The largest disagreement between the assembled Kani moments and the moment-distribution ones, over ten sweeps of this beam, is at the last decimal place a double-precision number holds. Against the stiffness solution both are within the convergence they have reached, which after ten sweeps is two ten-thousandths of a kilonewton-metre.

A wrong factor, a sway term, and a pencil nobody holds #

Everything above is a beam. Kani’s method extends to frames free to sway exactly as Cross’s does, by adding a storey term to each sweep — and the storey term is the same missing equation, arriving in the same place, with the same need for a displacement unknown. What Kani’s version does differently is fold that term into the sweep rather than running it as a second pass, which is a genuine simplification and is the strongest practical argument for the method.

The self-correction has a limit. A wrong factor is not corrected, because it is used afresh every sweep. Only wrong entries are overwritten, and the distinction is exactly whether the mistake is in something computed once at the start or in something recomputed each pass. A μ written down wrongly converges perfectly to the wrong answer, in either method.

The comparison of effort is a comparison of pencilled numbers. It counts entries in a table, which was the binding cost in 1949 and is not a cost at all now. Neither method is used to solve a real frame today, and the figures above were produced by machinery that solves the same beam by elimination in less time than it takes to write one entry.

A figure can show the sequence and not the sheet #

None of these plots shows a table, and a table is what both methods are. The layout on the page — where the carry-overs go, how the columns line up, which numbers get a line under them — is most of what a person actually experiences of a hand method, and it is the part that decided which one they used. A figure can show the sequence and cannot show the sheet.

The other absence is the arithmetic slip itself. The spoiled run above starts with a wrong number injected deliberately, which is not the same as a mistake made in the middle of a real calculation. A real slip lands somewhere unpredictable, at an unpredictable time, and its owner does not know it happened; the useful property is that this makes no difference, and that is a claim about all possible slips rather than about the one drawn.

The assumption underneath the identity #

The two methods sweep the joints in the same order here, and the identity of the sequences depends on it. Gauss–Seidel is order-dependent: sweep the joints of this beam from the right instead of the left and the iterates change, though the limit does not.

So the honest form of the finding is that the two methods are the same iteration when they are swept the same way, and both of them are in practice swept whichever way the person doing the arithmetic prefers. The difference that makes is smaller than the difference between beams, which is why nobody ever wrote it down, and it is the reason the comparison above is drawn on one beam with one sweep order rather than presented as a general theorem.

Why one of them travelled and the other did not #

The geography is real and it is not a story about national styles of engineering.

Cross published in 1930 in the Transactions of the American Society of Civil Engineers, in English, at the moment when reinforced concrete frames were becoming the ordinary way to build in North America and Britain. The method was taught from that paper within five years and it arrived with the language.

Kani published in 1949, in German, in a Europe that was rebuilding and whose engineering literature moved along different lines. The method spread through Germany, then Spain and Portugal, then Latin America, and it is still what many engineers educated in those languages were taught first. There is a large Spanish-language literature on it and almost none in English.

Neither method won on its merits, because neither was ever properly compared with the other — which is what the identical error curves above are really evidence of. Two professions used two notations for one iteration for forty years, each convinced the other’s was the inferior technique, and the question was settled by neither. It was settled by the matrix, which arrived in the 1960s and made the comparison irrelevant.

The one place the comparison was genuinely alive is the self-correction, and there the evidence is anecdotal rather than published: engineers who used Kani’s method report that a slip does not spoil a sheet, and engineers who used Cross’s report checking every carry-over. Both reports are correct and the reason is in the arithmetic above.

What survives of a method nobody uses #

It is tempting to file both of these under history and stop, and there are two things in them worth more than that.

The first is the observation that the notation a calculation is written in changes what its user can see. Both tables hold the same information at every stage; one of them can be read at any moment and one of them cannot. That is a fact about the presentation and not about the mathematics, and it is the same fact that decides whether a modern analysis is checkable — a solver reporting a displacement vector and a solver reporting a bending moment diagram are equally correct and are not equally useful, and an answer nobody can sanity-check against a free body is one that has to be trusted rather than believed.

The second is that an iteration which recomputes rather than accumulates is robust to its own mistakes. That is a property worth wanting in anything, and it is the reason numerical libraries prefer fixed-point formulations to incremental ones wherever the choice exists. The property has nothing to do with structures, which is why it was available to be rediscovered decades later under a different name.

There is also an irony in the pair worth naming. The method with the shorter table, the automatic handling of released ends and the tolerance of arithmetic error is the one that lost, and it lost on the single quality a hand method has that a machine does not need: being legible while it is unfinished. As soon as nothing was unfinished by hand, the advantage stopped existing — and by then the argument had been decided by it. The comparison with the force method one rung across is the same shape: a technique’s fate was settled by how it felt to use, at exactly the moment that stopped mattering.

Over-relaxation, and what a hand method is now for #

Later rungs on this anchor: Kani’s sway term, and the version of the whole procedure for a multi-storey frame that folds the storey equations into the sweep. Successive over-relaxation, which is Gauss–Seidel with the update multiplied by a factor greater than one and which would have accelerated either method by a factor of three had anybody in structural engineering been reading numerical analysis at the time. And the question this rung has been circling: what a hand method is for, once nothing is done by hand — which is not a rhetorical question, because choosing what to release is still a judgement a person makes, and the methods that were built around it are where that judgement is written down.