Step 1: Analyzing the Structure

The given structure is a simply supported beam with various types of loads:

1. Distributed Load: From $A$ to $B$ (left), with a magnitude of 3 kN/m over 3 meters.
2. Point Loads: Located at different intervals:
   • 6 kN at 3 meters from $A$.
   • 10 kN at 5 meters from $A$.
   • 10 kN at 7 meters from $A$.
3. Triangular Load: From 9 to 12 meters with a maximum intensity of 5 kN/m.

The reaction forces at supports are:

• $R_A$ at $A$ (vertical).
• $R_{Bx}$ and $R_B$ at $B$.

Step 2: Calculating the Reaction Forces

1. Calculate the total load of the distributed load: $w = 3 \, \text{kN/m}, \quad \text{Length} = 3 \, \text{m} \quad \Rightarrow \quad F_1 = w \times L = 3 \times 3 = 9 \, \text{kN}$ This acts at the centroid, i.e., 1.5 meters from point $A$.

2. For the triangular load: The triangular load extends from 9 to 12 meters with a maximum value of 5 kN/m. The total equivalent load is: $F_2 = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 3 \, \text{m} \times 5 \, \text{kN/m} = 7.5 \, \text{kN}$ The location of this load is $\frac{1}{3}$ of its base from the larger end, i.e., 1 meter from $B$.

Step 3: Applying Equilibrium Conditions

The structure is in static equilibrium, so the sum of forces and moments should be zero.

1. Sum of vertical forces: $R_A + R_B = 9 + 6 + 10 + 10 + 7.5 = 42.5 \, \text{kN}$

2. Sum of moments about $A$: Taking moments about $A$: $R_B \times 12 = 9 \times 1.5 + 6 \times 3 + 10 \times 5 + 10 \times 7 + 7.5 \times 11$

   $12R_B = 13.5 + 18 + 50 + 70 + 82.5 = 234 \, \text{kN} \cdot \text{m}$ Solving for $R_B$: $R_B = \frac{234}{12} = 19.5 \, \text{kN}$

3. Solving for $R_A$: $R_A = 42.5 - 19.5 = 23 \, \text{kN}$

Step 4: Final Reaction Forces

• Reaction at $A$: $R_A = 23 \, \text{kN}$
• Reaction at $B$: $R_B = 19.5 \, \text{kN}$

Step 5: Checking Equilibrium

The moments and forces are balanced, confirming the structure is in equilibrium.

Do you need more details or have any questions?

Additional Questions:

1. How would the reaction forces change if the triangular load had a steeper slope?
2. What is the effect of changing the location of point loads on the reactions?
3. How can we verify the results using shear and moment diagrams?
4. What happens to the reactions if the beam length is increased?
5. What would be the change in reactions if a moment is applied at the midpoint?

Tip:

When calculating moments, always ensure the correct distance is used relative to the chosen pivot point.