# Problem 840 | Continuous Beams with Fixed Ends

**Problem 840**

For the propped beam shown in Fig. P-840, determine the prop reaction and the maximum positive bending moment.

**Solution 840**

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$M_1L_1 + 2M_2(L_1 + L_2) + M_3L_2 + \dfrac{6A_1\bar{a}_1}{L_1} + \dfrac{6A_2\bar{b}_2}{L_2} = 0$

Where,

$L_1 = 3 ~ \text{m}$
From the overhang

$R_{1R} = 1400(2) = 2800 ~ \text{N}$

$L_1 = 3 ~ \text{m}$

$L_2 = 0$

$M_1 = - 1400(2)(1) = -2800 ~ \text{N}\cdot\text{m}$

$M_3 = 0$

$\dfrac{6A_1\bar{a}_1}{L_1} + \dfrac{6A_2\bar{b}_2}{L_2} = 0$

Thus,

$-2800(3) + 2M_2(3 + 0) + 0 + 0 + 0 = 0$

$M_2 = 1400 ~ \text{N}\cdot\text{m}$ ← maximum positive bending moment *answer*

$M_2 = \Sigma M_{\text{to the left of }R_2}$

$1400 = 3R - 1400(2)(4)$

$3R = 12\,600$

$R = 4200 ~ \text{N}$ *answer*

You can also solve R by the usual span-to-span method

$R_{1R} = 1400(2) = 2800 ~ \text{N}$

From the 3-m span

$3{R_2}' = M_1 + M_2$

$3{R_2}' = 2800 + 1400$

${R_2}' = 1400 ~ \text{N}$

$R = R_{1R} + {R_2}' = 2800 + 1400$

$R = 4200 ~ \text{N}$ *okay*

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