## Question:

The building slab is subjected to four parallel column loadings. Determine ${ F }_{ 1 }$ and ${ F }_{ 2 }$ if the resultant force acts through point (12 m, 10 m).

## Step-by-step

Equivalent Resultant Force. Sum the forces along z axis by referring to Fig. a,

$+\uparrow$  ${ ({ F }_{ R }) }_{ z }$  =  $\sum { F}_{ z }$ ;              –${ F }_{ R }$ = –${ F }_{ 1 }$${ F }_{ 2 }$ – 12 – 6                  ${ F }_{ R }$ = ${ F }_{ 1 }$ + ${ F }_{ 2 }$ + 18

Location of the Resultant Force. Sum the moments about the x and y axes by
referring to Fig. a,

${ ({ M }_{ R }) }_{ x }$$\sum { M}_{ x }$ ;              -(${ F }_{ 1 }$ + ${ F }_{ 2 }$ + 18)(10) = -12(8) – 6(20) – ${ F }_{ 2 }$(20)

10${ F }_{ 1 }$ – 10${ F }_{ 2 }$ = 36

${ ({ M }_{ R }) }_{ y }$$\sum { M}_{ y }$ ;              $({ F }_{ 1 }$ + ${ F }_{ 2 }$ + 18)(12) =  12(6) + 6(26) + ${ F }_{ 1 }$(22)

12${ F }_{ 2 }$ – 10${ F }_{ 1 }$ = 12

Solving Eqs (1) and (2),

${ F }_{ 1 }$ = 27.6 kN ${ F }_{ 2 }$ = 24.0 kN