Problem 730 | Uniform loads at each end of fully restrained beam

 

 

$EI ~ \theta_{AC} = 0$

$\frac{1}{3}(\frac{1}{2}L - a)[ \, \frac{1}{2}w_o(\frac{1}{2}L - a)^2 \, ] + \frac{1}{2}(\frac{1}{2}L)(\frac{1}{2}w_oLa) - \frac{1}{2}LM - \frac{1}{3}(\frac{1}{2}L)(\frac{1}{8}w_oL^2) = 0$

$\frac{1}{6}w_o(\frac{1}{2}L - a)^3 + \frac{1}{8}w_oL^2a - \frac{1}{2}LM - \frac{1}{48}w_oL^3 = 0$

$\frac{1}{2}LM = \frac{1}{6}w_o(\frac{1}{2}L - a)^3 - \frac{1}{48}w_oL^3 + \frac{1}{8}w_oL^2a$

$\frac{1}{2}LM = \frac{1}{48}w_o(L - 2a)^3 - \frac{1}{48}w_oL^2(L - 6a)$

$\frac{1}{2}LM = \frac{1}{48}w_o [ \, (L - 2a)^3 - L^2(L - 6a) \, ]$

$M = \dfrac{w_o}{24L}[ \, L^3 - 3L^2(2a) + 3L(2a)^2 - (2a)^3 - L^3 + 6L^2 a \, ]$

$M = \dfrac{w_o}{24L}[ \, L^3 - 6L^2a + 12La^2 - 8a^3 - L^3 + 6L^2 a \, ]$

$M = \dfrac{w_o}{24L}[ \, 12La^2 - 8a^3 \, ]$

$M = \dfrac{w_o}{24L}[ \, 4a^2(3L - 2a) \, ]$

$M = \dfrac{w_oa^2}{6L}(3L - 2a)$           answer
 

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

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

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

$EI ~ \delta_{max} = \frac{1}{3}(\frac{1}{2}L - a)[ \, \frac{1}{2}w_o(\frac{1}{2}L - a)^2 \, ][ \, a + \frac{3}{4}(\frac{1}{2}L - a) \, ]$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + \frac{1}{2}(\frac{1}{2}L)(\frac{1}{2}w_oLa)[ \, \frac{2}{3}(\frac{1}{2}L) \, ] - \frac{1}{2}LM[ \, \frac{1}{2}(\frac{1}{2}L) \, ]$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \frac{1}{3}(\frac{1}{2}L)(\frac{1}{8}w_oL^2)[ \, \frac{3}{4}(\frac{1}{2}L) \, ]$

$EI ~ \delta_{max} = \frac{1}{6}w_o(\frac{1}{2}L - a)^3(\frac{1}{4}a + \frac{3}{8}L) + \frac{1}{8}w_oL^2a(\frac{1}{3}L)$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \frac{1}{2}LM(\frac{1}{4}L) - \frac{1}{48}w_oL^3(\frac{3}{8}L)$

$EI ~ \delta_{max} = \frac{1}{384}w_o(L - 2a)^3(2a + 3L) + \frac{1}{24}w_oL^3a - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{384}w_o[ \, L^3 - 3L^2(2a) + 3L(2a)^2 - (2a)^3 \, ](2a + 3L)$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + \frac{1}{24}w_oL^3a - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{384}w_o(L^3 - 6L^2a + 12La^2 - 8a^3)(2a + 3L)$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + \frac{1}{24}w_oL^3a - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{384}w_o(2L^3a - 12L^2a^2 + 24La^3 - 16a^4 + 3L^4 - 18L^3a + 36L^2a^2$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 24La^3) + \frac{1}{24}w_oL^3a - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{384}w_o(24L^2a^2 - 16a^4 + 3L^4 - 16L^3a) + \frac{1}{24}w_oL^3a - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = (\frac{1}{16}w_oL^2a^2 - \frac{1}{24}w_oa^4 + \frac{1}{128}w_oL^4 - \frac{1}{24}w_oL^3a) + \frac{1}{24}w_oL^3a$
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \frac{1}{8}L^2M - \frac{1}{128}w_oL^4$

$EI ~ \delta_{max} = \frac{1}{16}w_oL^2a^2 - \frac{1}{24}w_oa^4 - \frac{1}{8}L^2M$

$EI ~ \delta_{max} = \frac{1}{16}w_oL^2a^2 - \frac{1}{24}w_oa^4 - \frac{1}{8}L^2\left[ \dfrac{w_oa^2}{6L}(3L - 2a) \right]$

$EI ~ \delta_{max} = \frac{1}{16}w_oL^2a^2 - \frac{1}{24}w_oa^4 - \frac{1}{48}w_oLa^2(3L - 2a)$

$EI ~ \delta_{max} = \frac{1}{16}w_oL^2a^2 - \frac{1}{24}w_oa^4 - \frac{1}{16}w_oL^2a^2 + \frac{1}{24}w_oa^3$

$EI ~ \delta_{max} = -\frac{1}{24}w_oa^4 + \frac{1}{24}w_oLa^3$

$EI ~ \delta_{max} = \frac{1}{24}w_oa^3(L - a)$           answer