# Problem 727 | Fully restrained beam with uniform load over the entire span

**Problem 727**

Repeat Problem 726 assuming that the concentrated load is replaced by a uniformly distributed load of intensity w_{o} over the entire length.

**Solution**

$\frac{1}{2}(\frac{1}{2}L)(\frac{1}{4}w_oL^2) - \frac{1}{2}L(M_{wall}) - \frac{1}{3}(\frac{1}{2}L)(\frac{1}{8}w_oL^2) = 0$

$\frac{1}{16}w_oL^3 - \frac{1}{2}M_{wall}L - \frac{1}{48}w_oL^3 = 0$

$\frac{1}{2}M_{wall}L = \frac{1}{24}w_oL^3$

$M_{wall} = \frac{1}{12}w_oL^2$ *answer*

Note that the actual bending moment is a negative moment (bending the beam downward) as shown in the figure. The answer above is positive which indicates that our assumption of downward arrows is correct. Had we assumed positive M_{wall} (upward arrows), the answer would be negative pointing out that the actual moment is negative (or downward arrows).

$\delta_{max} = t_{A/B}$

$EI ~ \delta_{max} = EI ~ t_{A/B}$

$EI ~ \delta_{max} = (Area_{AB}) \cdot \bar X_A$

$EI ~ \delta_{max} = \frac{1}{2}(\frac{1}{2}L)(\frac{1}{4}w_oL^2)[ \, \frac{2}{3}(\frac{1}{2}L) \, ] - \frac{1}{2}L(M_{wall})[ \, \frac{1}{2}(\frac{1}{2}L) \, ]$

$~ ~ ~ ~ ~ ~ ~~~~ ~~~ ~ ~ ~ - \frac{1}{3}(\frac{1}{2}L)(\frac{1}{8}w_oL^2)[ \, \frac{3}{4}(\frac{1}{2}L) \, ]$

$EI ~ \delta_{max} = \frac{1}{16}w_oL^3(\frac{1}{3}L) - \frac{1}{2}M_{wall}L(\frac{1}{4}L) - \frac{1}{48}w_oL^3(\frac{3}{8}L)$

$EI ~ \delta_{max} = \frac{1}{48}w_oL^4 - \frac{1}{8}M_{wall}L^2 - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{48}w_oL^4 - \frac{1}{8}(\frac{1}{12}w_oL^2)L^2 - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{48}w_oL^4 - \frac{1}{96}w_oL^4 - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{384}w_oL^4$ *answer*

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