Castigliano's Theorem, Maxwell Betti's Reciprocal Theorem

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If you are a Civil engineering student then these two theorems are important to understand. These both theorems are based on the energy principles.

 You can browse a numbers of results after you search them in the Google or you can find them in any books available on structural analysis.

Castigliano's first theorem and second theorem are used to determine the deflection and force respectively at the given points on a given structural system . 

Maxwell's reciprocal theorem is quite important because it allows you to skip a number of calculations when the relevant conditions arise. Let me explain a little more:

• Castigliano's 1st Theorem: According to this theorem, partial derivative of the strain energy of a structural system with respect to a point load at the given coordinate gives us the amount of deflection along that coordinate.
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• Castigliano's second Theorem: According to his second theorem, partial derivative of strain energy with respect to deflection at the given coordinate gives the corresponding force.                                                                                     

To approach at the solution i.e. either force or deflection, first you have to write down the equation of strain energy of the system in terms of the force or deflection(to determine deflection and force respectively) and then you have to apply either first theorem or second theorem, according to the desired results. 

example:  application of Castigliano's theorem to solve for the Deflection and force in a Beam: 

• Maxwell Betti's Reciprocal Theorem: According to this law the amount of work done by first load system due to displacements due to second load system, is equal to the amount of work by the second load system due to displacements due to first load system at their respective co-ordinates. 

In other words, if a load W is applied at A resulting a deflection "d" at B, then if same load W is applied at B, it will give the same deflection "d" at A.

This theorem can make you skip the calculations because if you know the deflection at B due to load at A, and you are asked about deflection at A due to load at B, you already know it, so why to do the calculations?

These theorems are really important to understand. I told same thing to my student in the class.

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