Question

The roof of a building frame is subjected to the wind loading shown. Determine:

a the equivalent force-couple system at D

b the resultant of the loading and its line of action

Answer 1

Free body diagram of the roof:

As all the left hand sides are parallel to each other, and all the right hand side forces are parallel to each other, all the left hand side forces will have same unit vector and all the right hand forces will have the same unit vector.

From the figure, it is clear that,

Unit vector of the left hand side forces, which lie on the line AB,

Unit vector of the right hand side forces, which lie on the line BC,

As the points F and G divide the segment AB into equal parts in the horizontal direction, same must happen with the case of vertical division. Similarly the points H and I divide the segment BC into equal parts in the vertical direction.

Coordinates of the points,

Force acting through A,

Similarly finding the remaining forces, we get

(a)

Resultant force of all the loading,

Moment caused by all of the forces at D,

Hence, the resultant couple force system at the point D is and

b)

To find the distance of the line of action of the resultant, we have to divide the magnitude total resultant moment with the magnitude of the resultant force.

As the moment is in the positive z-direction or in other words anti clock wise, in order to create that moment, line of action of the resultant force must be 6 m below the point D.

Hence, the distance of the line of action from the point D is .