Structural Engineering & Statics
where the load goes
Concept

Slope-deflection — where it appears

The relation giving a member's end moment from the rotations of its two ends, the rotation of its chord and its own fixed-end moment. Every hand method for continuous structures is a way of solving a set of these without writing any of them down, and the modified stiffnesses are what it says when one of the two rotations is already known.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 However heavy the chord, ordinary posts cap its share. The heavy chord's share of the shear in a panel of uniform shear in a Vierendeel girder of 2 m panels, 1.5 m deep, against how many times stiffer it is than the light chord, from the one-panel model (lines) and from a whole girder (dots). Splitting by second moment (dashed) runs from 0.5 towards 1. With posts as stiff as the light chord the share is 0.54 at a ratio of 4 and still only 0.55 at 30; posts 4, 16 and 100 times stiffer reach 0.65, 0.80 and 0.93 at 30. The posts, not the chord, set the ceiling. The dots are a twenty-panel girder at its sixth panel, far from both support and load: with ordinary posts 0.50, 0.53, 0.56, 0.60, 0.65, 0.69 at ratios of 1, 2, 4, 10, 20, 30. They follow the model until the heavy chord's own second moment becomes a noticeable part of the girder's — 31 per cent at a ratio of 30 — and it starts to carry the global moment by bending as a beam.

The heavy chord the posts will not let work

A Vierendeel girder carries its shear as bending in its two chords, and when one chord is four times as stiff as the other the obvious rule gives it four fifths of the shear. With posts as stiff as the light chord it takes 0.56. The heavy chord's end moments have to be carried round each joint by a post, the post bends, and the heavy chord turns with its joints instead of bending between them. No amount of extra steel in that chord takes its share past 0.55 in the middle of a long girder; only the posts can, and they have to be about 150 times as stiff as the light chord to make the rule come true.

structures · Vierendeel

Named alongside it

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

Carry-overDistribution factorFixed-end momentMoment distributionContinuityConvergenceDegrees of freedomIterationRotational stiffnessStiffnessSuperpositionContraflexure

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