Problem 870 | Beam Deflection by Three-Moment Equation | Strength of Materials Review at MATHalino

Problem 870
Compute the value of EIδ at the overhanging end of the beam in Figure P-870 if it is known that the wall moment is +1.1 kN·m.

Solution 870

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$\dfrac{y}{3} = \dfrac{2}{5}$

$y = 1.2 ~ \text{kN}$

$M_B = -\frac{1}{2}(2)(2)(\frac{2}{3} \cdot 2) - \frac{1}{2}(2)(1.2)(\frac{1}{3} \cdot 2)$

$M_B = -\frac{52}{15} ~ \text{kN}\cdot\text{m}$

$\begin{align}
\left( \dfrac{6A\bar{a}}{L} \right)_{AB} & = \dfrac{7w_oL^3}{60} + \dfrac{8w_oL^3}{60} \\
& = \dfrac{7(2)(2^3)}{60} + \dfrac{8(1.2)(2^3)}{60} \\
& = \dfrac{236}{75} ~ \text{m}^2
\end{align}$

$\begin{align}
\left( \dfrac{6A\bar{b}}{L} \right)_{CB} & = \dfrac{8w_oL^3}{60} \\
& = \dfrac{8(1.2)(3^3)}{60} \\
& = \dfrac{108}{25} ~ \text{m}^2
\end{align}$

Another Way to Solve the Three Moment Equation Factors (by Integration)

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$\begin{align}
\displaystyle \left( \dfrac{6A\bar{a}}{L} \right)_{AB} & = \sum \dfrac{Pa}{L}(L^2 - a^2) \\
& = \int_{x_1}^{x_2} \dfrac{(y \, dx) \, x}{2}(2^2 - x^2) \\
& = \frac{1}{2}\int_{x_1}^{x_2} xy(2^2 - x^2) \, dx \\
& = \frac{1}{2}\int_0^2 x(-0.4x + 2)(2^2 - x^2) \, dx \\
& = \frac{236}{75} ~ \text{m}^2
\end{align}$

$\begin{align}
\displaystyle \left( \dfrac{6A\bar{b}}{L} \right)_{CB} & = \sum \dfrac{Pb}{L}(L^2 - b^2) \\
& = \int_{x_1}^{x_2} \dfrac{(y \, dx)(3 - x)}{3} \left[ 3^2 - (3 - x)^2 \right] \\
& = \frac{1}{3}\int_{x_1}^{x_2} (3 - x)y \left[ 3^2 - (3 - x)^2 \right] \, dx \\
& = \frac{1}{3}\int_0^3 (3 - x)(-0.4x + 1.2) \left[ 3^2 - (3 - x)^2 \right] \, dx \\
& = \frac{108}{25} ~ \text{m}^2
\end{align}$

Apply Three-moment Equation to Span A-B-C
$M_A L_{AB} + 2M_B(L_{AB} + L_{BC}) + M_CL_{BC} + \left( \dfrac{6A\bar{a}}{L} \right)_{AB} + \left( \dfrac{6A\bar{b}}{L} \right)_{CB}$
$= 6EI \left( \dfrac{h_{BA}}{L_{AB}} + \dfrac{h_{BC}}{L_{BC}} \right)$

$0 + 2(-\frac{52}{15})(2 + 3) + 1.1(3) + \frac{236}{75} + \frac{108}{25} = 6EI\left(\dfrac{\delta_A}{2} + 0 \right)$

$3EI \, \delta_A = -\frac{239}{10}$

$EI \, \delta_A = -\frac{239}{30} ~ \text{kN}\cdot\text{m}^3$ answer

Problem 870 | Beam Deflection by Three-Moment Equation

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