Question 7.14: Use Castigliano’s second theorem to determine the deflection...
Use Castigliano’s second theorem to determine the deflection at point B of the beam shown in Fig. 7.22(a).
Using the x coordinate shown in Fig. 7.22(b), we write the equation for the bending moment in the beam as
M = -Px
The partial derivative of M with respect to P is given by
\frac{∂M}{∂P} = -xThe deflection at B can now be obtained by applying the expression of Castigliano’s second theorem, as given by Eq. (7.60), as follows:
Δ = \int_{0}^{L}{(\frac{∂M}{∂P})(\frac{M}{EI}) dx} (7.60)
Δ_B = \int_{0}^{L}{(\frac{∂M}{∂P})(\frac{M}{EI}) dx}Δ_B = \int_{0}^{L}{(-x)(-\frac{Px}{EI}) dx}
= \frac{P}{EI}\int_{0}^{L}{x^2 dx} = \frac{PL^3}{3EI}
Δ_B =\frac{PL^3}{3EI} ↓
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