Module 15 · Lesson 15.5
Castigliano's theorems
Differentiate the strain energy, and the deflection falls out.
Why this matters
The unit-load method needs you to invent a second, virtual system. Castigliano's theorem does not: it works with the real structure alone, and turns the deflection into a derivative. For a structure whose strain energy you can write down — trusses especially — it is often the quickest route there is, and it comes with a trick for finding a deflection at a point where no load acts.
By the end of this lesson you should be able to
- State Castigliano's first and second theorems
- Use ∂U/∂P to find a deflection
- Apply the dummy-load technique where no load acts at the point of interest
- Explain how Castigliano relates to the unit-load method
What you should already know
- Strain energy in axial members and bending (this module)
- The unit-load method (this module)
- Standard deflection formulae (Module 13)
Castigliano's second theorem — the one that gets used — says:
For a linearly elastic structure, the deflection at the point of application of a load, in the direction of that load, is the partial derivative of the total strain energy with respect to that load.
δ = ∂U/∂P
The argument behind it is short. Load the structure gradually to P, storing energy U(P). Now add a small extra δP. The extra energy is (∂U/∂P)δP. But you could equally have applied δP first and then P — the final state, and therefore the final energy, must be identical because the structure is elastic. Working that second order through, the extra energy is δP times the deflection at that point. Equating the two gives the theorem.
Castigliano's first theorem is the mirror image: P = ∂U/∂δ, giving the force required to produce a displacement when the energy is expressed in terms of displacements. It underpins the stiffness method, but it is the second theorem you will actually use by hand.
What it calculates: deflection or rotation from the strain energy
- U
- total strain energy of the structure (N·mm)
- P
- load at the point and in the direction wanted (N)
- M
- moment applied at the point where a rotation is wanted (N·mm)
This assumes
- Linearly elastic behaviour
- Small deflections, so the geometry does not change
- The load P must actually act at the point and in the direction of the deflection sought
In plain terms: Differentiate with respect to a force and you get a displacement. Differentiate with respect to a moment and you get a rotation. The pairing is always force with displacement, moment with rotation — the quantities whose product is work.
The obvious limitation is in the last assumption: P has to be there. What if you want the deflection at a point carrying no load, or in a direction nothing pushes?
The dummy load. Apply an imaginary load Q at exactly that point, in exactly that direction. Carry it symbolically through the whole analysis. Differentiate with respect to Q. Then set Q = 0 at the very end.
The result is the deflection under the real loading alone. Q never physically existed; it was a handle to differentiate with respect to.
It is worth noticing what this really is. Differentiating with respect to Q and then setting Q to zero extracts exactly the rate at which the internal forces change per unit of Q — which is the n of the unit-load method. The two techniques are the same calculation, arrived at from opposite directions. Castigliano is more mechanical; the unit-load method makes the physical content clearer. Use whichever suits the problem.
Predict first
You want the mid-span deflection of a beam that carries only a UDL — no point load at mid-span. What does Castigliano's theorem require?
Worked example
Cantilever tip deflection by Castigliano
Given
- Cantilever of span L = 3.00 m carrying a point load P = 15.0 kN at its free end
- E = 205 000 N/mm², I = 1.20 × 10⁸ mm⁴
- Bending strain energy only
Find
The tip deflection, from the strain energy.
Practice
A cantilever of span 3.00 m carries a point load P at its free end. Its bending strain energy is U = P²L³/6EI. What is the coefficient in the expression δ = ∂U/∂P = PL³/(n·EI)? Enter n.
Practice
For that cantilever with P = 15.0 kN, L = 3.00 m, E = 205 000 N/mm² and I = 1.20 × 10⁸ mm⁴, what is the tip deflection, in mm?
Practice
A beam carries only a UDL, and you want the deflection at a specific point. After applying a dummy load Q there and differentiating, what value must Q be given?
Summary
- Castigliano's second theorem: δ = ∂U/∂P, the deflection at and along a load
- Differentiate with respect to a moment instead and you get a rotation
- The load must act at the point of interest — otherwise use a dummy load
- Carry the dummy load through symbolically, differentiate, then set it to zero
- Castigliano and the unit-load method are the same calculation from opposite directions
- The first theorem, P = ∂U/∂δ, is what underlies the stiffness method
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint