The resultant of the line loads acting on the beam $AB$ is a couple $CR$. Determine the distance a and the couple $CR$.

Question

I know how to convert the distributed loads to concentrated. I am able also to get the moment of each of the concentrated loads in order to get the moment summation. However, I am confused about determining the couple $CR$. How is couple $CR$ related to forces resultant or moment summation of forces. Any help or guide to solving the problem will be much appreciated.

Answer 1

In order to have a couple both resultant forces (of triangle and rectangle) have to be equal in magnitude:

$$F_{\Delta}=0.5 \cdot (8-a)\cdot 100lb/ft$$ $$F_{rect}=a\cdot 75lb/ft.$$

Solve this system to obtain $a$. If you have calculated $a$ you can determine the distance of the couple by the distance between both forces $d$. Then $C^R=d\cdot F_{\Delta}$.