Structural Engineering & Statics
where the load goes
Concept

Carry-over — where it appears

The moment that appears at a member's far end when its near end is rotated, equal to half the applied moment for a prismatic member fixed at the far end. It is what makes moment distribution an iteration rather than a single pass, since every balanced joint disturbs its neighbours.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The answer arrives in instalments. The hogging moment at support 1 of a three-span beam, cycle by cycle. It starts at the fixed-end moment of 53.3 kNm — the value with every joint clamped — and settles at 64.0 kNm against an exact 64.0. The error falls by about a factor of four per cycle: 14.00, 3.50, 1.95, 0.59 kNm after one, two, three and four. Two cycles is an engineering answer and nobody had to invert anything.

Solved by passing it around

An indeterminate structure needs simultaneous equations, and for thirty years engineers solved them without writing any down. Clamp every joint, release one, share out what is left over, pass half of it along, and repeat — and the answer walks in, three figures correct after four cycles.

deflection · Moment distribution The answer arrives in instalments. The hogging moment at support 1 of a three-span beam, cycle by cycle. It starts at the fixed-end moment of 98.0 kNm — the value with every joint clamped — and settles at 156.9 kNm against an exact 156.9. The error falls by about a factor of four per cycle: 21.03, 5.92, 1.54, 0.60 kNm after one, two, three and four. Two cycles is an engineering answer and nobody had to invert anything.

Why it converges, and how fast

Moment distribution is an iteration, and iterations do not always converge. This one always does, at a rate the beam's own proportions fix — about a factor of four per cycle on a regular beam and considerably worse on an irregular one, which is where the method's reputation for two cycles being enough comes from and where it stops being true.

deflection · Moment distribution A portal on a stepped base — the two passes added. Bending moments on a single-bay portal with columns of 5 m and 3.5 m under one horizontal beam of 9 m, carrying 10 kN/m down and 60 kN across, with fixed bases. This frame is the two passes added. The corner moments are 5.8 and 99.1 kNm, and the short column's top carries 17.01 times what the tall one does. The diagram is drawn on the tension side of each member.

Every joint balanced, and the frame still leaning

Moment distribution enforces one equation per joint, and a frame free to translate has one more equation than it has joints. So a table that balances perfectly can describe a structure held up by a prop nobody built — and finding the prop, then removing it, is a second pass whose unknown is a distance rather than a rotation.

deflection · Moment distribution Four things a far end can be doing, and what each is worth. The rotational stiffness of a member at one end, for four conditions at the other, each drawn as the shape the member takes when the near end is rotated through one unit. They are the same expression evaluated four times — M = (2EI/L)(2θ_a + θ_b) — and the only thing that changes is what the far end is known to be doing. A held far end gives 4EI/L and carries over a half; a free one gives 3EI/L and carries over nothing; a far end rotating equally and oppositely gives 2EI/L and carries over minus one, which is what a symmetric structure does to a member crossing its axis; and a far end rotating equally and in the same sense gives 6EI/L and carries over one. None of the four is an approximation. Each removes a freedom that was going to be discovered by iteration.

Told what the far end is doing

Moment distribution discovers, cycle by cycle, that the pinned end of a beam carries no moment — a fact known before any arithmetic started. Telling it instead changes one stiffness from 4EI/L to 3EI/L and the work from thirty numbers to eight, for the identical answer. Cutting the beam on its own axis of symmetry gets it in two.

deflection · Moment distribution Every number in the table is a rotation, and none of them is a moment. The rotation contributions of a three-span beam of 8, 10, 8 m under 24 kN/m, sweep by sweep. There are six of them, one per member end, and not one is a bending moment: the moment is assembled at the end from M = FEM + 2m′ + m′ of the far end, and until that is done the table holds quantities that mean nothing on their own. That is the trade. A moment distribution stopped after two cycles hands over moments that are wrong by a known amount and can be used; this table stopped after two sweeps hands over nothing that can be read at all — and it gets there in four sweeps against Cross's own count on the same beam, writing six numbers a sweep rather than one per distributed member end plus a carry-over.

The table that cannot be read halfway

Kani's method converges at exactly the rate moment distribution does, sweep for sweep and digit for digit, because it is the same iteration. What it changes is what is written in the boxes — rotations rather than moments — and that buys a shorter table that repairs its own mistakes and cannot be stopped early.

deflection · Moment distribution Stressed bay by bay, the beam keeps half its secondary moment. The secondary moment along a 4-span beam of 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, straight between supports: stressed in one operation, stressed one bay at a time, and bay by bay after creep with a coefficient of 2.0. At the first interior support it is 350 kN·m stressed at once, 164 bay by bay and 307 after creep; at the next, 234, 160 and 216.

The beam stressed one bay at a time

A long continuous beam is rarely stressed in one pull. It is cast and stressed a bay at a time, each new length of tendon coupled to the last, and at every stage the structure is a different beam. The secondary moments the finished beam carries are the sum of what each stage did to the beam that existed then — about half of what the one-stage analysis gives — and creep spends the following years taking most of the other half back.

internal-forces · Secondary prestress The order of building decides which supports keep the secondary moment. The secondary moment at each interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built and stressed from one end, from both ends towards a closure bay, from the middle outwards, and all at once. From one end: 161, 175, 160, 219 kN·m; from both ends: 161, 172, 172, 161 kN·m; from the middle out: 215, 120, 120, 215 kN·m; all at once: 344, 258, 258, 344 kN·m.

The beam built towards its own middle

A continuous prestressed beam built one bay at a time from one end keeps about half the secondary moment a beam stressed at once would carry, and most of it at the end where it was finished. Built from both ends towards a closure bay, it is symmetric, and the closure's supports keep only a little more than the rest: 172 kN·m against 161. Built from the middle outwards, the first interior supports — the ones the load hogs hardest — keep 215, 63 per cent of the at-once value, for any length of beam. The order of construction is a design variable. Creep then takes back most of the choice.

internal-forces · Secondary prestress

Named alongside it

The objects these essays reach for when they reach for this one.

ContinuityDistribution factorFixed-end momentMoment distributionIterationConvergenceIndeterminacySlope-deflectionSuperpositionConstruction sequenceCreepDegrees of freedom

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