Strain energy methods provide an efficient means of calculating specific deflections and rotations in beams and structures. Castigliano's theorem relates displacement to the derivative of the total strain energy and can often avoid the need to derive the complete elastic curve of a member.
This page introduces the application of Castigliano's theorem and includes worked examples for a simply supported beam, a cantilever and a simple frame. For the strain-energy expressions used here, see Strain Energy.
When external forces are applied to an elastic beam or structure, the structure deforms and the external forces perform work. For a conservative linear-elastic system loaded gradually from zero, this work is stored as elastic strain energy in the structure.
Under these assumptions:
where:
When the loads are removed within the elastic range, the deformation is recovered and the stored elastic strain energy is released.
This energy relationship provides an alternative route for determining deflections and rotations. Rather than deriving the complete deflected shape of a beam, an energy method can be used to calculate the displacement at a particular point and in a particular direction.
Castigliano's theorem provides a convenient method for finding a particular displacement or rotation in an elastic structure. Unlike direct integration of the beam differential equation, it does not normally provide an expression for displacement at every point along the member.
For a linearly elastic structure undergoing small displacements, the displacement corresponding to an applied force is obtained by differentiating the total strain energy with respect to that force.
where δi is the displacement at the point of application of force Fi, measured in the direction of that force.
The corresponding relationship for rotation is:
where θi is the rotation corresponding to the applied moment Mi.
Castigliano's theorem is also useful when the required displacement is at a point where no real force or moment acts.
A fictitious force is introduced at the required point and in the direction of the required displacement. The total strain energy is written including this fictitious load, differentiated with respect to it, and the fictitious load is then set equal to zero.
Similarly, a fictitious moment can be introduced when a rotation is required at a point where no real moment acts.
To determine a linear displacement δf in the direction of a real or fictitious force Ff:
To determine a rotation θf in the direction of a real or fictitious moment Mf, use the same procedure, but differentiate the total strain energy with respect to the moment:
If the moment is fictitious, set Mf = 0 after differentiation.
Consider a simply supported rectangular beam carrying a central force F. The vertical deflection at the load point is required.
| Span, l | 2 m |
| Width, b | 0.1 m |
| Depth, h | 0.05 m |
| Force, F | 10,000 N |
| Young's modulus, E | 206 GPa |
| Shear modulus, G | 78.610 GPa |
| Second moment of area, I | 4.17 × 10-6 m4 |
The beam is rectangular and the strain energy due to both bending and transverse shear is included. Because the beam and loading are symmetrical, the total strain energy can be obtained by integrating over half the beam and doubling the result.
For the left-hand half of the beam:
Using the bending and transverse-shear strain-energy relationships developed on the Strain Energy page:
The image is retained so that the original derivation can be checked before it is removed or modified.
The deflection at the point of application of F is:
For the dimensions and elastic properties used in this example, the shear component of deflection is small compared with the bending component. The original calculation gives a transverse-shear contribution of approximately 0.8% of the bending deflection.
Consider the cantilever shown below. A real force F acts at a distance b from the free end. The deflection at the free end is required.
Because no real load acts at the free end in the required direction, a fictitious force Ff is introduced there.
For this example, only bending strain energy is included.
The bending moment must be expressed for the appropriate regions of the cantilever, including both the real and fictitious forces.
Using the bending strain-energy relationship:
the strain energy for the separate regions of the cantilever is:
From Castigliano's theorem:
After differentiation, set:
and integrate over the appropriate beam lengths.
Consider the frame shown below. One leg is free to move horizontally and the other is pinned. A horizontal force F is applied to the free leg, and the horizontal displacement at that point is required.
The original example neglects strain energy due to transverse shear.
For the members identified in the diagram, the original example uses the following internal-force relationships:
The force in member BC is axial, so its axial strain energy must be included in addition to the bending strain energy of the frame members.
The displacement in the direction of the applied force is:
Castigliano's theorem is particularly useful when the displacement or rotation is required at a specific point rather than as a complete equation for the deflected shape. It can be applied to structures in which the total strain energy contains contributions from bending, axial loading, torsion and shear.
The energy terms included in a calculation should reflect the deformation mechanisms that are significant for the structure being analysed. For many slender beams bending dominates, while shear, axial or torsional deformation may become important in other geometries.
The theorem in the form used on this page assumes elastic behaviour and the conditions appropriate to the strain-energy formulation. For nonlinear material behaviour, large deformation or other non-linear effects, the applicable energy relationship must be considered separately.