Question

The unit load method used in structural analysis is

• A. another name for stiffness method
• B. applicable only to statically indeterminate structures
• C. derived from Castigliano's theorem
• D. an extension of Maxwell's reciprocal theorem

Answer 1

The correct option is C derived from Castigliano's theorem

Castigliano's second theorem,
${\delta }_i=\frac{\partial U}{\partial P_i}$

Where,
${\delta }_i$ is the displacement in the direction of $P_i$;
U = strain energy stored in the system due to loading

For a beam,
$U=\int \frac{M^{2}dx}{2EI}$

${\delta }_i$ = $\frac{\partial U}{\partial P_i}$

$=\frac{\partial }{\partial P_i}\int \frac{M^{2}dx}{2EI}$

$=\int \frac{M}{EI}\frac{\partial M}{\partial P_i}dx$

The partial derivative $\frac{\partial M}{\partial P_i}$ represents the value of the bending moment M caused by a unit load of the load $P_i$. Thus this derivative is equal to m, which is the bending moment in the structure due to a unit load corresponding to the desired displacement hence,
${\delta }_i=\int \frac{Mmdx}{EI}$
This is the equation of unit load method when only flexural deformations are considered.