Question:

The tube supports the four parallel forces. Determine the magnitudes of forces ${ F }_{ C }$ and ${ F }_{ D }$ acting at C and D so that the equivalent resultant force of the force system acts through the midpoint O of the tube.

Step-by-step

Since the resultant force passes through point O, the resultant moment components
about x and y axes are both zero.

$\sum { M}_{ x }$ = 0 ;    ${ F }_{ D }$(0.4) + 600(0.4) – ${ F }_{ C }$(0.4) – 500(0.4) = 0

${ F }_{ C }$${ F }_{ D }$ = 100

$\sum { M}_{ y }$ = 0 ;    500(0.2) + 600(0.2) – ${ F }_{ C }$(0.2) – ${ F }_{ D }$(0.2) = 0

${ F }_{ C }$ + ${ F }_{ D }$ = 1100

Solving Eqs. (1) and (2) yields:

${ F }_{ C }$ = 600 N              ${ F }_{ D }$ = 500 N