Castigliano's theorem and unit load method

Strain energy expressions

For a linear elastic member (energy stored equals work done by gradually applied loads):

For slender beams the bending term dominates; the others are usually neglected.

Castigliano's second theorem

Statement: For a linearly elastic structure in equilibrium, the partial derivative of the total strain energy (written in terms of the loads) with respect to a load $P_i$ equals the displacement $\delta_i$ of the point of application of that load, in the direction of the load.

$$\delta_i = \frac{\partial U}{\partial P_i},\qquad \theta_i = \frac{\partial U}{\partial M_i}$$

Proof: Let loads $P_1,\dots,P_n$ act and produce energy $U$. Increase $P_i$ by $dP_i$; the energy becomes $U + \dfrac{\partial U}{\partial P_i}dP_i$.

Now apply the loads in the reverse order: first $dP_i$ alone, then $P_1\ldots P_n$. The energy is

$$\tfrac12\,dP_i\,d\delta_i + U + dP_i\,\delta_i$$

The first term is the work of $dP_i$ while it is applied alone. The last term is the work of $dP_i$ (already acting at its full value) as the other loads are applied and move its point of application through $\delta_i$. The final state, and so the stored energy, does not depend on the order of loading. Equating the two expressions and neglecting the second-order term $dP_i\,d\delta_i$:

$$\frac{\partial U}{\partial P_i}\,dP_i = \delta_i\,dP_i \quad\Rightarrow\quad \delta_i = \frac{\partial U}{\partial P_i}$$

If no load exists at the point of interest, apply a fictitious load $Q$ (or moment $M_0$), differentiate, then set $Q = 0$.

For beams, differentiating under the integral sign:

$$\delta = \int \frac{M}{EI}\frac{\partial M}{\partial P}\,dx$$

Unit (dummy) load method

Apply a unit load (or unit moment for slope) at the point and in the direction of the required deflection, and let $m$ be the bending moment due to it. By the principle of virtual work (external work $1\cdot\delta$ = internal work):

$$1\cdot\delta = \int \frac{M\,m}{EI}\,dx$$

For trusses: $\delta = \sum \dfrac{F\,f\,L}{AE}$, where $F$ is the member force from the real load and $f$ from the unit load. Because the derivation uses only virtual work, the method can also be used for non-linear material by replacing $M/EI$ by the actual curvature.

Use

• Deflections of beams, bent frames, hooks, curved bars and trusses, where double integration is lengthy.
• Deflection at a point where no load acts, or when the loading is discontinuous.
• Solution of statically indeterminate structures: choose the redundant $R$ and impose a compatibility condition, e.g. $\partial U/\partial R = 0$ when the support does not move.