Module 4 · Course notes

Stiffness and compatibility: how a structure shares its load

When a structure has more restraints than statics needs, equilibrium alone cannot say how the load is shared — the answer depends on stiffness. This module builds the second half of the picture: parallel paths share load in proportion to their stiffness, movement must fit together (compatibility), and forcing a shape onto an over-constrained structure generates force even with no applied load.

For a statically indeterminate structure — one with redundant restraints — equilibrium gives fewer equations than there are unknowns. The missing information is stiffness. Where two paths share a load, the one that has to move less for a given force — the stiffer one — attracts more of it.

This is easy to get backwards. It is tempting to think the flexible member "gives", so it takes the load. In fact, for paths that share the same movement, the stiffer path needs a larger force to reach that movement, so it carries more.

A rigid beam forces both column tops to move sideways together. The stiffer column develops the larger moment and carries most of the load; the flexible one carries little. Stiffness attracts force.

The two ideas behind this are compatibility and limiting cases.

Compatibility is the statement that the movement must fit together: members meeting at a rigid joint share one rotation; two paths that share a point share its displacement; a settling support imposes a prescribed movement. These geometric conditions supply the equations equilibrium is missing.

Limiting cases are the fastest way to sanity-check sharing: take one stiffness to zero and to infinity. As one column's stiffness tends to zero it carries almost nothing and the other takes it all; as it tends to infinity it takes almost everything. The real answer sits between.

There is no load on this continuous beam, yet forcing the middle support down bends it — continuity resists the imposed movement. In a determinate structure the same settlement would cause movement without force. Compatibility, not equilibrium, produces this.

Worked example

Worked example — which column works harder?

A sideways load is shared between two columns of different stiffness, tied together by a rigid beam.

Step 1 — the compatibility condition
The rigid beam forces both column tops to move sideways by the same amount. That shared displacement is the compatibility condition: whatever the forces, the tops move together.
Step 2 — stiffness sets the share
To reach the same sideways movement, the stiffer column needs a larger force. So it carries the larger share of the load, and develops the larger bending moment. Take its stiffness to infinity and it would carry nearly all the load; take it to zero and the other column would.

In the exercises, decide how load is shared, what movement generates force, and — importantly — when there simply is not enough information to say.

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

Stiffness & compatibility exercises

5 exercises on load sharing by stiffness, limiting cases, settlement and when there is not enough information. 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.