Module 12 · Course notes

The stiffness method: displacements as the unknowns

The stiffness method flips the flexibility method on its head. Instead of choosing forces to release, it takes the joint displacements as the unknowns, works out how stiff each member is against them, and solves one matrix equation. It needs no judgement about which redundant to release — which is exactly why it, and not the force method, is what runs inside every analysis program.

The displacement (stiffness) method asks a different question from the force method. Rather than which forces are redundant, it asks how far does each joint move. The joint displacements and rotations are the unknowns.

The recipe is mechanical:

1. Lock every joint so nothing can move. Each member is then fixed-ended, and its end actions are its fixed-end moments. 2. Work out each member's stiffness against the joint movements. 3. Impose equilibrium at every joint — the member end-moments there must balance. This gives one equation per unknown displacement. 4. Solve for the displacements, then substitute back to recover every member force.

With both ends locked, the beam is fixed-ended and carries its fixed-end moments (wL²/12 hogging at each end, wL²/24 sagging at mid). These locked-in actions are the starting point; releasing the joints and letting them rotate corrects them.

In matrix form, each member contributes an element stiffness matrix; these are assembled into one global matrix K, the loads into a vector f, and the response is the solution of Kd = f. Two facts matter for judgement:

Stiffness attracts action. A member's rotational stiffness is 4EI/L (or 3EI/L with a pinned far end). At a joint, the out-of-balance moment is shared in proportion to stiffness, so the stiffer member takes more.

K is symmetric. The entry coupling degree of freedom i to j equals the one coupling j to i — which is Maxwell's reciprocal theorem written as a matrix. Analysis programs rely on it to halve their work.

Two columns of unequal stiffness share a sideways load through a rigid beam. The stiffer column develops the larger moment — stiffness attracts action, exactly as the distribution factors in the stiffness method predict.

Worked example

Worked example — lock, then release

Read a two-span continuous beam through the stiffness method's two moves.

Step 1 — lock every joint
Clamp the joint over the central support so it cannot rotate. Each span is then fixed-ended and carries its fixed-end moments. But the real joint is free to rotate, so this locked state is not yet right — there is an out-of-balance moment at the clamp.
Step 2 — release and solve
Release the joint and let it rotate until the member moments there balance. That rotation is the unknown the method solves for. Substituting it back gives the real continuous-beam response: hogging over the support, sagging in the spans.

In the exercises, identify the unknowns, use stiffness to share action, and see why this method — not the force method — is the one built into software.

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

The stiffness method exercises

5 exercises on the unknowns, locking joints for fixed-end moments, stiffness sharing, matrix symmetry and why software uses it. 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.