# Problem 728 | Isosceles triangular load over the entire span of fully restrained beam

**Problem 728**

Determine the end moment and maximum deflection of a perfectly restrained beam loaded as shown in Fig. P-728.

**Solution**

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$\frac{1}{2}(\frac{1}{2}L)(\frac{1}{8}w_oL^2) - \frac{1}{2}L(M_{wall}) - \frac{1}{4}(\frac{1}{2}L)(\frac{1}{24}w_oL^2) = 0$

$\frac{1}{32}w_oL^3 - \frac{1}{2}M_{wall}L - \frac{1}{192}w_oL^3 = 0$

$\frac{1}{2}M_{wall}L = \frac{5}{192}w_oL^3$

$M_{wall} = \frac{5}{96}w_oL^2$

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

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

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

$EI ~ \delta_{max} = \frac{1}{2}(\frac{1}{2}L)(\frac{1}{8}w_oL^2)[ \, \frac{2}{3}(\frac{1}{2}L) \, ] - \frac{1}{2}L(M_{wall})[ \, \frac{1}{2}(\frac{1}{2}L) \, ]$

$~ ~ ~ ~ ~ ~ ~~~~ ~~~ ~ ~ ~ - \frac{1}{4}(\frac{1}{2}L)(\frac{1}{24}w_oL^2)[ \, \frac{4}{5}(\frac{1}{2}L) \, ]$

$EI ~ \delta_{max} = \frac{1}{32}w_oL^3(\frac{1}{3}L) - \frac{1}{2}M_{wall}L(\frac{1}{4}L) - \frac{1}{192}w_oL^3(\frac{2}{5}L)$

$EI ~ \delta_{max} = \frac{1}{96}w_oL^4 - \frac{1}{8}M_{wall}L^2 - \frac{1}{480}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{96}w_oL^4 - \frac{1}{8}(\frac{5}{96}w_oL^2)L^2 - \frac{1}{480}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{96}w_oL^4 - \frac{5}{768}w_oL^4 - \frac{1}{480}w_oL^4$

$EI ~ \delta_{max} = \frac{7}{3840}w_oL^4$ *answer*

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