Module 10 · Course notes
Virtual work: a spotlight on movement
Virtual work turns a question about deformation into a question about overlap. To find how much a chosen point moves, you shine a unit action on it and see which parts of the structure respond. It is the most reliable way to reason about deflections qualitatively — and the sign and location of a movement fall out before any integration.
To find a movement at a chosen point, the unit-load method uses two systems on the same structure. First the real loads, giving the real internal actions (call the bending moment M). Then a unit action applied exactly at the movement you want — a unit force for a translation, a unit moment for a rotation — giving the virtual actions (m).
The movement is the overlap of the two, weighted by flexibility: for bending, it is the integral of M·m ÷ EI along the structure. Where the real and virtual actions are both large, and the member is flexible, that region contributes most. Where either is zero, that region contributes nothing.
Two things trip people up. First, the unit action goes where you want the answer, not beside the real load: it represents the movement you are asking about. Second, the biggest real moment does not automatically dominate — the virtual diagram and the stiffness weight the contribution, so a region with a large M but a near-zero m contributes little.
Used this way, virtual work is a qualitative spotlight: it tells you which regions matter, and whether the movement is positive or negative, before you compute anything.
Worked example
Worked example — which region moves the beam most?
To find the mid-span deflection of a uniformly loaded beam, pair its real moment with the virtual moment of a unit load at mid-span.
Step 1 — the two diagrams
The real moment M is a sagging parabola. The virtual moment m, from a unit load at mid-span, is a triangle peaking at mid-span. Both are largest in the middle third of the span.
Step 2 — where the overlap is largest
The product M·m is largest where both diagrams are large — the middle of the span. So the mid-span deflection is built mainly from the central region; the ends, where m is nearly zero, contribute almost nothing. The overlap is positive, so the deflection is downward, in the direction of the unit load.
In the exercises, decide where the unit action belongs, which regions contribute, and what the sign of the movement is — all before any integral.
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
Virtual-work exercises
5 exercises on where the unit action goes, the real-virtual overlap, which regions contribute and the sign of the movement. 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.