Question 7.18: For the previous problem, calculate the deflection at point ...
For the previous problem, calculate the deflection at point C on the beam.
From the previous problem, we note that the flexure curve is given by
(E I) y=-\frac{R_{ A } x^3}{6}-\frac{M_{ A } x^2}{2}+\frac{w_{ o }}{24}\left\langle x-\frac{L}{2}\right\rangle^4
or (E I) y=-\frac{23 w_{ o } L}{768} x^3+\frac{7 w_{ o } L^2 x^2}{256}+\frac{w_{ o }}{24}\left\langle x-\frac{L}{2}\right\rangle^4 (1)
(by putting the values of R_{ A } \text { and } M_{ A } from the previous problem).
Now substituting x = L/2 in Eq. (1), we obtain
\begin{aligned} \left.(E I) y\right|_{x=L / 2} & =-\frac{23 w_{ o } L}{768}\left\lgroup\frac{L}{2} \right\rgroup^3+\frac{7 w_{ o } L^2}{256}\left\lgroup\frac{L}{2} \right\rgroup^2 \\ & =\frac{7 w_{ o } L^4}{1024}-\frac{23 w_{ o } L^4}{6144}=\frac{19 w_{ o } L^4}{6144} \end{aligned}
Therefore,
\left.y\right|_{x=L / 2}=\delta_{ C }=\frac{19 w_{ o } L^4}{6144 E I}
Thus,
\delta_{ C }=\frac{19 w_{ o } L^4}{6144 E I}(\downarrow)