1. Lecture 2 : Castigliano’s Energy Theorems
Hans Welleman

2. Work and Energy methods
Castigliano ua ux Fa Fx Fb ub Fc uc
Ir J.W. Welleman
Work and Energy methods

3. Differentiate the Work to a specific force, e.g. Fx:
Use Maxwell’s notation :
Ir J.W. Welleman
Work and Energy methods

4. Work and Energy methods
Elaborate …
Ir J.W. Welleman
Work and Energy methods

5. Castigliano’s 2nd theorem
Result:
Maxwell :
Castigliano’s 2nd theorem
Ir J.W. Welleman
Work and Energy methods

6. Work and Energy methods
Castigliano
Work done by external loads Aext is stored in the strain energy Ev
Differentiate the strain energy to a force at location x to find the displacement u at x
Ir J.W. Welleman
Work and Energy methods

7. Work and Energy methods
Example
0,5 l F
wmax EI
Ir J.W. Welleman
Work and Energy methods

8. Strain Energy in terms of the load
Ir J.W. Welleman
Work and Energy methods

9. Steps of the solution strategy
Castigliano : differentiate EV to a force.
EV is integral of the square of the moment distribution so
…. differentiate after integration …
IS THIS SMART TO DO ??
Ir J.W. Welleman
Work and Energy methods

10. Work and Energy methods
Smart help
Ir J.W. Welleman
Work and Energy methods

11. Work and Energy methods
Castigliano
Second theorem
First theorem
Ir J.W. Welleman
Work and Energy methods