Prove Eq. (10–40). Hint: Using Castigliano’s theorem, determine the deflection due to bending of an end hook alone as if the hook were fixed at the end connecting it to the body of the spring....

Prove Eq. (10–40).
Hint:
Using Castigliano’s theorem, determine the deflection due to bending of an end hook alone as if the hook were fixed at the end connecting it to the body of the

spring. Consider the wire diameter
d
small as compared to the mean radius of the hook,

R
=
D/2. Add the deflections of the end hooks to the deflection of the main body to determine

the final spring constant, then equate it to Eq. (10–9).