Question 15.10: Determine the deflection of the free end of the cantilever b...
Determine the deflection of the free end of the cantilever beam in Fig. 15.20 when subjected to the temperature gradients shown.
The temperature, t, at any section x of the beam is given by
t=\frac{x}{L} t_{0}
Thus, substituting for t in Eq. (15.43),
\Delta_{\mathrm{{Te.B}}}=-\int_{0}^{L}M_{1}\,{\frac{\alpha t}{h}}\,\mathrm{d}x (15.43)
which applies since the variation of temperature through the depth of the beam is identical to that in Fig. 15.19(b), and noting that M_{1}=-1(L-x) we have
\Delta_{\mathrm{Te}, \mathrm{B}}=-\int_{0}^{L}[-1(L-x)] \frac{\alpha}{h} \frac{x}{L} t_{0} \mathrm{~d} x
which simplifies to
\Delta_{\mathrm{Te}, \mathrm{B}}=\frac{\alpha t_{0}}{h L} \int_{0}^{L}\left(L x-x^{2}\right) \mathrm{d} x
whence
\Delta_{\mathrm{Te}, \mathrm{B}}=\frac{\alpha t_{0} L^{2}}{6 h}
