Question 9.7: Use Castigliano’s second theorem to find the reaction force ......
Use Castigliano’s second theorem to find the reaction force in the support at C due to the load P at B. The beam has a constant E I.
Given: Statically indeterminate cantilever beam with a point load.
Find: Reaction force at redundant support.
Assume: Hooke’s law applies; long slender beam.
The work we have done in Example 9.6 allows us to quickly solve this problem. There is now a real force at C, so we can consider the force Q to be −R_C (noting Q was pointing down and R_C is pointing up). Our intermediate result from Example 9.6 for ΔC can be set to equal the real, physical deflection at C, which due to the support is zero:
\Delta_C=\frac{\partial U}{\partial Q}=\frac{L^3}{48 E I}(5 P+16 Q)=0 .
From this compatibility equation, we find that R_C = −Q = 5P/16.
The problem does not ask for the deflection at B, but we could find it with other equations from Example 9.6 now that R_C is known.