problem 1\ndetermine the resultant couple moment acting on the beam.

Explanation:

Step1: Recall couple - moment formula

The couple - moment of a force $F$ about a point is given by $M = r\times F$, where $r$ is the perpendicular distance from the point of interest to the line of action of the force. For a couple, the moment is independent of the point about which it is calculated.

Step2: Calculate moment due to first couple

The first couple consists of a $10$ kN force. The perpendicular distance between the two forces of the couple is $2$ m. The moment of this couple $M_1$ is $M_1=10\times2 = 20$ kN·m (clock - wise).

Step3: Resolve the inclined force

The inclined $20$ kN force can be resolved into vertical and horizontal components. The vertical component of the $20$ kN force at the right - hand side is $F_{y}=20\sin20^{\circ}$ kN and the horizontal component is $F_{x}=20\cos20^{\circ}$ kN. The vertical component of the $20$ kN force at the left - hand side is also $F_{y}=20\sin20^{\circ}$ kN.

Step4: Calculate moment due to second couple

The perpendicular distance between the vertical components of the $20$ kN inclined forces is $11$ m. The moment of the couple formed by the vertical components of the $20$ kN forces is $M_2 = 20\sin20^{\circ}\times11$. [M_2=20\times0.342\times11=75.24] kN·m (clock - wise)

Step5: Calculate the resultant couple moment

The resultant couple moment $M_R$ is the sum of the moments of the individual couples. [M_R = M_1+M_2] [M_R=20 + 75.24=95.24] kN·m (clock - wise)

Answer:

$95.24$ kN·m (clock - wise)