The Unit Load Method
Two separate analyses of the same structure, multiplied member by member. Handles temperature and fabrication error just as easily.
Skip to the animationAnalyse the structure twice — once under the real loads and once under a single unit load at the point you care about — then multiply the two member force sets together and sum, and the result is the deflection at that point.
The procedure
- 1Analyse the real structure under the real loads. Call the member forces
P. - 2Remove every real load and apply a single unit load at the point, in the direction of the deflection you want. Call these forces
k. - 3Multiply, member by member, and sum:
δ = Σ PkL/AEfor a truss, or∫ Mm dx / EIfor a beam. - 4A positive result means the point moves in the direction of the unit load.
Both analyses are ordinary determinate ones — nothing new is needed. A truss problem becomes a five-column table with one row per member, which is why this is the method actually used by hand.
Why it works
The unit load system is in equilibrium and the real deformation is compatible, which is all virtual work requires. Equate external and internal virtual work:
- External: the 1 kN unit load moving through the real deflection — that is
1 × δ. - Internal: the
kforces moving through the real member extensionsPL/AE— that isΣ k·PL/AE.
The unit load makes the left-hand side simply δ, so it is isolated immediately. The two systems need not be related in any way — that independence is what makes the method so flexible.
What it handles that other methods do not
The real system supplies deformations and the virtual system supplies forces, and nothing requires them to share a cause. So any source of deformation substitutes straight into the real-extension term:
| Cause | Real extension term |
|---|---|
| Applied load | PL/AE |
| Temperature change | α·ΔT·L |
| Fabrication error | the actual misfit, e.g. 3 mm short |
| Support settlement | Enters as external virtual work directly |
No other hand method absorbs a member fabricated to the wrong length this cleanly, and lack-of-fit is a real construction problem rather than a textbook curiosity.
Practical points
- Keep tension positive in both analyses; two compressive forces multiply to a positive contribution, which is correct.
- For a rotation rather than a deflection, apply a unit *moment* instead of a unit force.
- The
kvalues are dimensionless, since the load is 1 — they are influence coefficients. - For beams,
Mandmmust be written for the same origin and sign convention, which is where most errors occur.
What it leads to
Indeterminate analysis needs a compatibility condition — typically "the deflection here is zero". You can only impose that once you can *compute* a deflection, which is exactly what this method provides. That is why the deflection module precedes the indeterminate one.
The numbers you will be asked for
- Truss deflection
δ = Σ (P·k·L) / (A·E)
One row per member.
- Beam or frame deflection
δ = ∫ (M·m dx) / (E·I)
M from the real load, m from the unit load.
- Rotation
θ = ∫ (M·m_θ dx) / (E·I)
Apply a unit moment instead of a unit force.
- Temperature effect
δ = Σ k·α·ΔT·L
The real extension is thermal rather than elastic.
- Virtual work
1 × δ = Σ k·(real extension)
External equals internal.
Advantages and disadvantages
Advantages
- Two straightforward analyses instead of one awkward one.
- Tabulates cleanly, so errors are easy to spot.
- Handles temperature, lack of fit and settlement with no extra theory.
- Gives rotations as readily as deflections.
Disadvantages
- One deflection at a time, at one point, in one direction.
- Requires a full analysis of the structure twice over.
- Assumes linear elastic behaviour throughout.
- For beams, sign and origin conventions must match between M and m, and often do not.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.