PLAIN AND CIVIL: EXAMPLE 4.3.11. BEAM SUPPORT REACTIONS FOR UNIFORMLY DISTRIBUTED LOADS AND CONCENTRATED LOADS

Determine the support reactions of the beam with the applied loads as shown.

There are two ways in which we can convert the distributed loads into concentrated loads.

METHOD A: Converting the distributed loads into three separate concentrated loads.

$P_{1}=P_{3}=\omega _{2}L=4\frac{kN}{m}\left ( 3m \right )=12kN$

$P_{2}=\omega _{1}L=10\frac{kN}{m}\left ( 8m \right )=80kN$

$\sum F_{X}=0]\rightarrow$

${\color{Red} \mathbf{R_{DX}=0}}$

$\sum M_{D}=0$

$12kN\left ( 5m \right )+R_{A}\left ( 14m \right )-P_{1}\left ( 12.5m \right )-P_{2}\left ( 7m \right )-P_{3}\left ( 1.5m \right )=0$

$R_{A}\left ( 14m \right )=12kN\left ( 12.5m \right )+80kN\left ( 7m \right )+12kN\left ( 1.5m \right )-12kN\left ( 5m \right )$

${\color{Red} \mathbf{R_{A}=47.714kN}}$

$\sum M_{A}=0$

$P_{1}\left ( 1.5m \right )+P_{2}\left ( 7m \right )+P_{3}\left ( 12.5m \right )+12kN\left ( 19m \right )-R_{DY}\left ( 14m \right )=0$

$R_{DY}\left ( 14m \right )=12kN\left ( 1.5m \right )+80kN\left ( 7m \right )+12kN\left ( 12.5m \right )+12kN\left ( 19m \right )$

${\color{Red} \mathbf{R_{DY}=68.286kN}}$

METHOD B. Converting distributed loads by layer.

$P_{1}=\left ( \omega _{1}-\omega _{2} \right )L=\left ( 10-4\frac{kN}{m} \right )\left ( 8m \right )=48kN$

$P_{2}=\omega _{2}L=4\frac{kN}{m}\left ( 3+8+3m \right )=56kN$

$\sum F_{X}=0]\rightarrow$

${\color{Red} \mathbf{R_{DX}=0}}$

$\sum M_{D}=0$

$R_{A}\left ( 14m \right )+12kN\left ( 5m \right )-\left ( P_{1}+P_{2} \right )\left ( 3+4 \right )=0$

$R_{A}\left ( 14m \right )=\left ( 48+56kN \right )\left ( 7 \right )-12kN\left ( 5m \right )$

${\color{Red} \mathbf{R_{A}=47.714kN}}$

$\sum M_{A}=0$

$\left ( P_{1}+P_{2} \right )\left ( 3+4m \right )+12kN\left ( 19m \right )-R_{DY}\left ( 14m \right )=0$

$R_{DY}\left ( 14m \right )=\left ( 48+56kN \right )\left ( 7m \right )+12kN\left ( 19m \right )$

${\color{Red} \mathbf{R_{DY}=68.286kN}}$

RELATED TOPIC:

4.0. SHEAR AND MOMENT IN BEAMS

PLAIN AND CIVIL

Pages

Thursday, 28 March 2019

EXAMPLE 4.3.11. BEAM SUPPORT REACTIONS FOR UNIFORMLY DISTRIBUTED LOADS AND CONCENTRATED LOADS

No comments:

Post a Comment