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To find the reaction force at point A for this beam structure, we will use static equilibrium principles. The system consists of a beam supported at A (a pinned support) and B (a roller support). There are two external loads: a 10 kN load acting 3 m from A, and a 50 kN load acting 9 m from A.

We can solve for the reaction forces by using the conditions of static equilibrium, which are:

1. **Sum of vertical forces = 0:**
   \[
   \Sigma F_y = 0
   \]
   This means the sum of all vertical forces acting on the beam must be zero.

2. **Sum of moments about point A = 0:**
   \[
   \Sigma M_A = 0
   \]
   The sum of all moments about point A should be zero to maintain rotational equilibrium.

Let's denote the reaction forces as:
- \( R_A \): the reaction force at point A (vertical).
- \( R_B \): the reaction force at point B (vertical).

### Step 1: Moment Equation About Point A
We will first sum moments about point A to eliminate \( R_A \) and solve for \( R_B \):

\[
\Sigma M_A = 0 \quad \Rightarrow \quad (10 \, \text{kN}) \times (3 \, \text{m}) + (50 \, \text{kN}) \times (9 \, \text{m}) - R_B \times (12 \, \text{m}) = 0
\]

Solving this equation for \( R_B \):

\[
R_B \times 12 = (10 \times 3) + (50 \times 9)
\]
\[
R_B \times 12 = 30 + 450 = 480
\]
\[
R_B = \frac{480}{12} = 40 \, \text{kN}
\]

### Step 2: Vertical Force Equilibrium
Now that we know \( R_B \), we can use the vertical force equilibrium equation to solve for \( R_A \):

\[
\Sigma F_y = 0 \quad \Rightarrow \quad R_A + R_B - 10 - 50 = 0
\]
\[
R_A + 40 = 60
\]
\[
R_A = 60 - 40 = 20 \, \text{kN}
\]

### Final Answer:
The reaction force at point A is \( \mathbf{20 \, \text{kN}} \).

Would you like further details or clarifications? Here are five follow-up questions that could expand on this topic:

1. How would the reaction forces change if the distance between supports A and B increased?
2. What is the significance of a pinned support compared to a roller support?
3. How would you calculate the reaction forces if the beam was inclined?
4. Can you explain how shear force and bending moment diagrams would look for this beam?
5. What would happen if an additional load was applied between the supports?

**Tip:** Always choose the point with the fewest unknown forces when applying the moment equation to simplify your calculations.

Based on the structural system below, the reaction force at A is equal to ____kN.

Learn how to calculate the reaction force at point A for a beam structure with two loads. This problem uses static equilibrium principles, including the sum of vertical forces and moments. Suitable for college-level engineering students.

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