Problem 867 | Deflection by Three-Moment Equation

$M_A = M_C = 0$

$M_B = -4P$
 

$\dfrac{y}{x} = \dfrac{4}{8}$

$y = \frac{1}{2}x$
 

 

$M_A L_{AB} + 2M_B(L_{AB} + L_{BC}) + M_C L_{BC} + \left( \dfrac{6A\bar{a}}{L} \right)_{AB} + \left( \dfrac{6A\bar{b}}{L} \right)_{CB} = 0$

$0 + 2(-4P)(10 + 4) + 0 + \sum \dfrac{Pa}{L}(L^2 - a^2) + 0 = 0$

$\displaystyle -112P + \int_{x_1}^{x_2} \dfrac{(y \, dx)x}{10}(10^2 - x^2) = 0$

$\displaystyle -112P + \dfrac{1}{10} \int_0^8 \left( \dfrac{x}{2} \right)x(100 - x^2) \, dx = 0$

$\displaystyle -112P + \dfrac{1}{20} \int_0^8 x^2 (100 - x^2) \, dx = 0$

$-112P + \dfrac{39\,424}{75} = 0$

$P = 4.693 ~ \text{kN}$       answer