Module 13 · Course notes

Moment distribution: balancing joints by hand

Moment distribution is the stiffness method carried out by hand, one joint at a time. It was the workhorse of design offices for decades, and it is still the quickest way to feel how a frame shares moment. You lock the joints, look at the out-of-balance, release a joint and share the correction by stiffness, carry a little over to the far ends, and repeat until it settles.

The method starts by telling a comfortable lie and then correcting it.

1. Lock every joint so no rotation can occur. Each member is fixed-ended, carrying its fixed-end moments. 2. At a real joint the member moments will not sum to zero — that out-of-balance is held by the imaginary clamp. 3. Release the joint. The out-of-balance is shared among the members in proportion to their stiffness — the stiffer member takes more. 4. Carry over: when one end of a member rotates, a moment appears at its far end — half of the distributed moment, for a fixed far end. 5. Repeat. The carried-over moments unbalance the neighbours, so go round again. The corrections shrink fast; two or three cycles usually suffice.

Lock every joint and each span is fixed-ended, carrying wL²/12 hogging at each end. These are the starting moments; releasing the joints and balancing them corrects the diagram to the real one.

Two quantities run the arithmetic. The distribution factor at a joint is a member's share of the total stiffness there, DF = K / ΣK, and the factors at any joint must sum to 1 — the out-of-balance has nowhere else to go. The carry-over factor is ½ towards a fixed far end and 0 towards a pinned one (there is nothing at a pin to carry to).

A member's stiffness is 4EI/L with a fixed far end, or 3EI/L with a pinned one — so a member with a pinned far end is three-quarters as stiff, and attracts a smaller share.

Distributing and carrying over until the joints balance yields the real continuous-beam moment: hogging over the central support, sagging in the spans — the same answer the stiffness method gives, reached by hand.

Worked example

Worked example — distribute at one joint

Two members meet at a rigid joint, both with fixed far ends: one 6 m long, one 4 m, the same EI. Share an out-of-balance moment there.

Step 1 — distribution factors
Stiffness is 4EI/L, so K₆ = 0.667 EI and K₄ = 1.000 EI, total 1.667 EI. The distribution factors are 0.40 and 0.60 — they sum to 1.0, and the shorter, stiffer member takes the larger share.
Step 2 — distribute and carry over
A 50 kNm out-of-balance splits as 20 kNm to the 6 m member and 30 kNm to the 4 m member. Half of each carries over to the fixed far end — 10 kNm and 15 kNm. The stiffer member did more work, exactly as its distribution factor said.

In the exercises, compute distribution factors, apply carry-over, and connect the hand method to the stiffness method it mimics.

How to read these problems

The three-step method

1Points of certainty. The deflected curve must pass through every support and deflect downward under the load. Mark what each support prevents before drawing anything.
2Deflected shape and reaction directions. Sketch the compatible deflected shape. To find a reaction's direction, imagine removing that support: the direction that pushes the structure back to its place is the reaction's sense (it may be a hold-down).
3Bending moment and contraflexure. Draw the bending-moment diagram on the tension side and check it against the shape: hogging where the curve is convex-up, sagging where convex-down, zero at pins and at every contraflexure.

Rules that must always hold

1.The bending moment is zero at a simple support and at an internal pin or hinge.
2.A bending-moment diagram crosses the baseline exactly at a point of contraflexure.
3.Under a distributed load the bending-moment diagram is curved; under point loads alone it is straight lines.
4.At a fully fixed support the deflected shape leaves the support with zero rotation (tangent along the member).
5.If a part of the structure stays straight after loading, it carries no bending moment there.
6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Moment distribution exercises

5 exercises on distribution factors, carry-over, the starting fixed-end moments and the link to the stiffness method. Predict, then reveal an explanation.

Start the exercises →

This lesson is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions.