1. Castigliano’s First Theorem

Try these first

1. 1.

   Define Castigliano's First Theorem.

   Hint

   Think about how displacement connects to energy.

2. 2.

   What types of energies are associated with deformation in beams?

   Hint

   Recall the acronym ABTS.

3. 3.

   What does Castigliano's First Theorem relate?

   • Stress to strain
   • Energy to displacement
   • Force to load

   Hint

   Think about the connection between energy and motion.

4. 4.

   True or False: The reciprocal relation states that displacement at one point determines the force at another.

   • True
   • False

   Hint

   Focus on the concept of interconnected responses.

5. 5.

   A cantilever beam with a length L is subjected to a point load W at its free end. Derive the expression for the deflection at the free end using Castigliano's theorem.

   Hint

   Start with the basic equations of bending and consider the contribution along the beam.

6. 6.

   Consider a composite beam with varying cross-sections. Use Castigliano’s theorem to find the displacement at a specific segment by first calculating energies in different sections.

   Hint

   Breakdown helps understand how each section contributes to overall behavior.