Problem 860 | Deflection by Three-Moment Equation | Strength of Materials Review at MATHalino

Problem 860
Determine the value of EIδ at the end of the overhang and midway between the supports for the beam shown in Fig. P-860.

Solution 860

Click here to expand or collapse this section

$\Sigma M_{R1} = 0$

$aR_3 = w_ob(a + \frac{1}{2}b)$

$R_3 = \dfrac{w_ob}{a}\left( \dfrac{2a + b}{2} \right)$

$R_3 = \dfrac{w_ob}{2a}(2a + b)$

$\Sigma M_{R3} = 0$

$aR_1 = w_ob(\frac{1}{2}b)$

$R_1 = \dfrac{w_ob^2}{2a}$

$M_1 = M_4 = 0$

$M_2 = -R_1(\frac{1}{2}a) = -\dfrac{w_ob^2}{2a}(\frac{1}{2}a) = -\dfrac{w_ob^2}{4}$

$M_3 = -w_ob(\frac{1}{2}b) = -\frac{1}{2}w_ob^2$

For span 1-3-4

$L_1 = a$ → span 1 to 3

$L_2 = b$ → span 3 to 4

$M_1 L_1 + 2M_3(L_1 + L_2) + M_4L_2 + \dfrac{6A_1\bar{a}_1}{L_1} + \dfrac{6A_2\bar{b}_2}{L_2} = 6EI \left( \dfrac{h_1}{L_1} + \dfrac{h_4}{L_2} \right)$

$0 + 2(-\frac{1}{2}w_ob^2)(a + b) + 0 + 0 + \frac{1}{4}w_ob^3 = 6EI \left( 0 + \dfrac{h_4}{b} \right)$

$-w_oab^2 - w_ob^3 + \frac{1}{4}w_ob^3 = \dfrac{6EI}{b}h_4$

$-w_oab^2 - \frac{3}{4}w_ob^3 = \dfrac{6EI}{b}h_4$

$\dfrac{6EI}{b}h_4 = -\frac{1}{4}w_ob^2(4a + 3b)$

$h_4 = -\dfrac{w_ob^3}{6EI}(4a + 3b)$

The negative sign of h₄ indicates that point 4 is below point 3, thus,
$\delta_4 = \dfrac{w_ob^3}{6EI}(4a + 3b) ~ \text{ downward}$ answer

For span 1-2-3

$L_1 = \frac{1}{2}a$ → span 1 to 2

$L_2 = \frac{1}{2}a$ → span 2 to 3

$h_1 = h_3 = h$

$M_1 L_1 + 2M_2(L_1 + L_2) + M_3L_2 + \dfrac{6A_1\bar{a}_1}{L_1} + \dfrac{6A_2\bar{b}_2}{L_2} = 6EI \left( \dfrac{h_1}{L_1} + \dfrac{h_3}{L_2} \right)$

$0 + 2(-\frac{1}{4}w_ob^2)(\frac{1}{2}a) + \frac{1}{2}a) - \frac{1}{2}w_ob^2(\frac{1}{2}a) + 0 + 0 = 6EI \left( \dfrac{h}{\frac{1}{2}a} + \dfrac{h}{\frac{1}{2}a} \right)$

$-\frac{1}{2}w_oab^2 - \frac{1}{4}w_oab^2 = 6EI \left( \dfrac{4h}{a} \right)$

$-\frac{3}{4}w_oab^2 = \dfrac{24EI}{a}h$

$h = -\dfrac{w_oa^2b^2}{32EI}$

The negative sign of h indicates that points 1 and 3 are below point 2, thus,
$\delta_2 = \dfrac{w_oa^2b^2}{32EI} ~ \text{ upward}$ answer

Problem 860 | Deflection by Three-Moment Equation

Navigation

Recent comments