To solve for the reactions $R_1$ and $R_2$ in the beam problem, we must perform static equilibrium analysis. Here's the approach:

Given Data:

• Triangular distributed load with a maximum of $2 \, \text{kip/ft}$ over a length of $10'$.
• Concentrated load of $10 \, \text{kip}$ applied at $3'4''$ (or $3.33'$) from the right support.
• The beam has two supports: a pin support at $R_1$ and a roller support at $R_2$.

Step 1: Calculate the Total Load of the Triangular Distributed Load

The area of the triangular load gives the total force: $F_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10' \times 2 \, \text{kip/ft} = 10 \, \text{kip}$ This force acts at the centroid of the triangle, which is located $\frac{1}{3}$ of the distance from the left (pin support).

Thus, the location of this force from $R_1$ is: $\text{Location of centroid} = \frac{1}{3} \times 10' = 3.33' \text{ from the left}$

Step 2: Write the Equilibrium Equations

We will use the three static equilibrium conditions:

1. Sum of vertical forces: $R_1 + R_2 = F_{\text{triangle}} + 10 \, \text{kip} = 10 + 10 = 20 \, \text{kip}$

2. Sum of moments about $R_1$ (to eliminate $R_1$ and solve for $R_2$): Taking counterclockwise moments as positive: $\text{Moment at } R_1: 10 \, \text{kip} \times 3.33' + 10 \, \text{kip} \times (10' + 3.33') - R_2 \times (10' + 3.33') = 0$ Simplifying: $33.3 + 133.3 - R_2 \times 13.33 = 0$ $R_2 = \frac{33.3 + 133.3}{13.33} = 12.5 \, \text{kip}$

3. Sum of forces (to find $R_1$): From the first equilibrium equation: $R_1 + 12.5 = 20 \quad \Rightarrow \quad R_1 = 7.5 \, \text{kip}$

Final Reactions:

• $R_1 = 7.5 \, \text{kip}$
• $R_2 = 12.5 \, \text{kip}$

Would you like any additional details or further clarification on the solution?

Here are 5 related questions for deeper understanding:

1. How would the reactions change if the concentrated load were moved closer to the pin support?
2. What is the significance of the location of the centroid of the triangular load?
3. How do different types of loads (uniform vs triangular) affect the reaction forces?
4. How does the pin and roller support system differ from a fixed support system in terms of reaction forces?
5. What would be the effect of increasing the magnitude of the distributed load?

Tip: Always verify that the sum of moments and forces balances to ensure your calculations are correct.