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Castigliano’s second theorem obtains displacement in the direction of a force by differentiating elastic strain energy with respect to that force.
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Formula
δ in m; U in J; F in N; ∂U/∂F in J/N = m
| Symbol | Meaning | Unit |
|---|---|---|
| Displacement at the force location in the force direction | m | |
| Elastic strain energy of the system | J | |
| Generalized force used for differentiation | N |
\delta_i=\frac{\partial U}{\partial F_i}The theorem applies when strain energy can be expressed as a differentiable function of generalized forces; standard introductory use assumes linear elasticity and small deformation. Forces and moments must be treated consistently as generalized loads. To find a displacement where no load acts, introduce a dummy force, differentiate, then set it to zero. Plasticity, large deformation and dissipative energy require another model.
Differentiable elastic energy relation; standard use assumes small deformation and linear elasticity.
Differentiating energy with respect to generalized load gives the conjugate displacement.
Cantilever tip deflection
Displacement and force must use the same point and consistent direction.
Check the cantilever with P=100 N, L=0.50 m, E=200 GPa, I=10−6 m4.
δ=PL3/(3EI)≈0.0208 mm; the energy derivative matches the standard result.
Differentiating by a load not conjugate to the target displacement.
Use the conjugate force or moment and check units J/N=m.
Formula cluster
The energy theorem complements equilibrium and direct deflection integration.
A cantilever with end load P has U=P2L3/(6EI), so δ=∂U/∂P=PL3/(3EI). With P=100 N, L=0.50 m, E=200 GPa and I=10−6 m4, δ≈2.08×10−5 m=0.0208 mm, matching the standard cantilever result.
Find displacements and rotations of elastic structures using energy methods.
Here is how to work through a typical Castigliano’s second theorem (δi = ∂U/∂Fi) task step by step:
Task
Check the cantilever with P=100 N, L=0.50 m, E=200 GPa, I=10−6 m4.
Solution path
δ=PL3/(3EI)≈0.0208 mm; the energy derivative matches the standard result.
The partial derivative of stored strain energy with respect to a generalized force gives the associated displacement in that force direction. Differentiation with respect to a moment gives a rotation. The result depends on the chosen load coordinate, with signs set by the model’s virtual-work convention. The energy method requires a suitable elastic model. Displacement and generalized force must refer to the same point and consistent direction. Application: Find displacements and rotations of elastic structures using energy methods. The numerical example A cantilever with end load P has U=P2L3/(6EI), so δ=∂U/∂P=PL3/(3EI). With P=100 N, L=0.50 m, E=200 GPa and I=10−6 m4, δ≈2.08×10−5 m=0.0208 mm, matching the standard cantilever result anchors the idea in a concrete measurement case.
For an end-loaded cantilever, U=P2L3/(6EI). Differentiation by P gives PL3/(3EI). Substitution gives 100·0.53/(3·200·109·10−6)=2.08×10−5 m, or about 0.0208 mm. E·I has units N·m2, so displacement is in metres. The energy method requires a suitable elastic model. Displacement and generalized force must refer to the same point and consistent direction. The corresponding units are δ in m; U in J; F in N; ∂U/∂F in J/N = m. A cantilever with end load P has U=P2L3/(6EI), so δ=∂U/∂P=PL3/(3EI). With P=100 N, L=0.50 m, E=200 GPa and I=10−6 m4, δ≈2.08×10−5 m=0.0208 mm, matching the standard cantilever result. This checks that the reference quantities are compatible.
To find displacement at a point with no actual load, add a symbolic dummy force Q there in the desired direction. Express U in terms of Q and the real loads, differentiate with respect to Q, then set Q=0. This captures the displacement without permanently changing the original loading problem. The energy method requires a suitable elastic model. Displacement and generalized force must refer to the same point and consistent direction. Scope: The theorem applies when strain energy can be expressed as a differentiable function of generalized forces; standard introductory use assumes linear elasticity and small deformation. Forces and moments must be treated consistently as generalized loads. To find a displacement where no load acts, introduce a dummy force, differentiate, then set it to zero. Plasticity, large deformation and dissipative energy require another model. A practical application is Find displacements and rotations of elastic structures using energy methods.
Introduce a dummy generalized force at the target location in the desired direction. Express total elastic energy in terms of that force and the real loads, take the partial derivative, then set the dummy force to zero. The sign follows the chosen direction. This yields an otherwise unloaded displacement without applying a physical load there. The energy method requires a suitable elastic model. Displacement and generalized force must refer to the same point and consistent direction. Check the calculation with this case: A cantilever with end load P has U=P2L3/(6EI), so δ=∂U/∂P=PL3/(3EI). With P=100 N, L=0.50 m, E=200 GPa and I=10−6 m4, δ≈2.08×10−5 m=0.0208 mm, matching the standard cantilever result. Keep the model limits and units in view.
U must include stored elastic deformation energy from all relevant modes. Bending, axial force, torsion and shear may contribute; neglecting shear can matter in short deep beams. Use consistent generalized forces and moments. Dissipation, plastic work and large deformation are not automatically covered by the standard linear form. The energy method requires a suitable elastic model. Displacement and generalized force must refer to the same point and consistent direction. The relation describes Castigliano’s second theorem obtains displacement in the direction of a force by differentiating elastic strain energy with respect to that force. This helps distinguish its target quantity from neighbouring quantities.
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Curated exam set for "Castigliano’s second theorem":
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Which formula describes Castigliano’s second theorem?
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How do you rearrange Castigliano’s second theorem for Cantilever tip deflection?
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Which common mistake happens with Castigliano’s second theorem?
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