PLAIN AND CIVIL: EXAMPLE 4.4.4. EQUATION METHOD: BEAM WITH TRIANGULAR LOADS

Determine the shear and moment equations for the beam shown. Draw the shear and moment diagrams.

Example 4.3.14. shows the solution for the reactions of the beam.

There are two major load changes in the beam thus there are two cutting planes.

Segment AB [cutting plane a-a'] 0<x<5.50m:

$\sum F_{Y}=0]\uparrow$

$4.757kN+V_{AB}=0$

${\color{Red} \mathbf{V_{AB}=-4.757kN}}$

$\sum M_{cut}=0]$

$4.757\left ( x \right )-M_{AB}=0$

${\color{Red} \mathbf{M_{AB}=4.757x}}$

Segment BC [cutting plane b-b'] 5.5<x<10.50m:

For y: Use ratio and proportion:

$\frac{y}{x}=\frac{6\frac{kN}{m}}{5m}$ $y=\frac{6}{5}x$

$P_{1}=\frac{1}{2}\left ( x-5.5 \right )\left ( 6-y \right )$

$P_{1}=\frac{1}{2}\left ( x-5.5 \right )\left ( 6-1.2x \right )$

$P_{1}=6.3x-0.6x^{2}-16.5$

$P_{2}=y\left ( x-5.5 \right )$

$P_{2}=1.2x\left ( x-5.5 \right )$

$P_{2}=1.2x^{2}-6.6x$

$\sum F_{Y}=0]\uparrow$

$4.757-P_{1}-P_{2}+V_{BC}=0$

$V_{BC}=\left ( 6.3x-0.6x^{2}-16.5 \right )+1.2x^{2}-6.6x-4.757$

${\color{Red} \mathbf{V_{BC}=0.6x^{2}-0.3x-21.257}}$

$\sum M_{cut}=0]$

$4.757x-P_{1}\left [ \frac{2}{3}\left ( x-5.5 \right ) \right ]-P_{2}\left ( \frac{x-5.5}{2} \right )-M_{BC}=0$

$M_{BC}=4.757x-\frac{1}{2}\left ( x-5.5 \right )\left ( 6-1.2x \right )\left [ \frac{2}{3}\left ( x-5.5 \right ) \right ]-1.2x\left ( x-5.5 \right )\left ( \frac{x-5.5}{2} \right )$

$M_{BC}=4.757x-\frac{1}{3}\left ( x-5.5 \right )^{2}\left ( 6-1.2x \right )-0.6x\left ( x-5.5 \right )^{2}$

$M_{BC}=4.757x-\left ( x-5.5 \right )^{2}\left ( 2-0.4x \right )-0.6x\left ( x-5.5 \right )^{2}$

$M_{BC}=4.757x-\left ( 2-0.4x \right )\left ( x^{2}-11x+30.25 \right )-0.6x\left ( x^{2}-11x+30.25 \right )$

$M_{BC}=4.757x+\left ( 0.4x^{3}-6.4x^{2}+34.10x-60.50 \right )-0.60x^{3}+6.6x^{2}-18.15x$

${\color{Red} \mathbf{M_{BC}=-0.20x^{3}+0.20x^{2}+20.707x-60.50}}$

PLAIN AND CIVIL

Pages

Monday, 1 April 2019

EXAMPLE 4.4.4. EQUATION METHOD: BEAM WITH TRIANGULAR LOADS

Segment AB [cutting plane a-a'] 0<x<5.50m:

Segment BC [cutting plane b-b'] 5.5<x<10.50m:

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