Determine the horizontal and vertical components of force at pins A and D.

Steps to Solve

1. Free Body Diagram (FBD) of the Entire Structure

Draw a free body diagram of the entire structure. This involves representing the structure with all external forces acting on it. These forces include the 12 kN load acting downwards, and the reaction forces at pins A ($A_x$, $A_y$) and D ($D_x$, $D_y$).

2. Equilibrium Equations for the Entire Structure

Apply the equations of static equilibrium to the entire structure. These equations are:

• Sum of horizontal forces = 0: $\sum F_x = 0$
• Sum of vertical forces = 0: $\sum F_y = 0$
• Sum of moments about a point = 0: $\sum M = 0$

3. Applying $\sum F_x = 0$

$A_x + D_x = 0$ (Equation 1)

4. Applying $\sum F_y = 0$

$A_y + D_y - 12 = 0$ (Equation 2)

5. Applying $\sum M_A = 0$ (Moments about point A)

Taking moments about point A eliminates $A_x$ and $A_y$ from the equation. We have: $D_y \cdot 0 - D_x \cdot 2 - 12 \cdot (1.5+1.5+0.3) = 0$ $-2D_x - 12(3.3) = 0$ $-2D_x - 39.6 = 0$ $D_x = -19.8 \text{ kN}$

6. Solve for $A_x$ using Equation 1

Substitute $D_x$ into Equation 1: $A_x + (-19.8) = 0$ $A_x = 19.8 \text{ kN}$

7. Free Body Diagram of Member DB

Draw a FBD of member DB. This member is under two forces, $D_x$ and $D_y$ at $D$, and a force at $B$ which would be equal and opposite to the force that the rod $ABC$ applies to the member $DB$.

8. Sum of the moments about point B should be zero

Take moments about point B. $D_x * 2 - D_y * 3 = 0$ $-19.8 * 2 - D_y * 3 = 0$ $-39.6 - 3D_y = 0$ $D_y = -13.2 \text{ kN}$

9. Solve for $A_y$ using Equation 2

Substitute $D_y$ into Equation 2: $A_y + (-13.2) - 12 = 0$ $A_y = 25.2 \text{ kN}$