To find the reactions at each end of the beam, we can use the principles of static equilibrium. We'll denote:

• $R_A$ = reaction force at the left end (A)
• $R_B$ = reaction force at the right end (B)
• $W$ = weight of the beam = 1170 pounds
• $P$ = vertical load applied = 13,880 pounds
• $L$ = length of the beam = 19.3 feet
• $d$ = distance from the left end to the applied load = 7.797 feet

Step 1: Sum of Vertical Forces

The sum of the vertical forces must equal zero:

$R_A + R_B - W - P = 0$

Substituting the known values:

$R_A + R_B - 1170 - 13880 = 0 \quad \text{(1)}$

Step 2: Sum of Moments about Point A

Next, we take moments about point A to find $R_B$:

$\sum M_A = 0$

Taking counterclockwise moments as positive, we have:

$P \cdot d + W \cdot \left(\frac{L}{2}\right) - R_B \cdot L = 0$

Substituting the values:

$13880 \cdot 7.797 + 1170 \cdot \left(\frac{19.3}{2}\right) - R_B \cdot 19.3 = 0$

Calculating the moment contributions:

$13880 \cdot 7.797 = 108224.4 \text{ ft-lbs}$ $1170 \cdot 9.65 = 11275.5 \text{ ft-lbs}$

Thus, the moment equation becomes:

$108224.4 + 11275.5 - R_B \cdot 19.3 = 0$ $R_B \cdot 19.3 = 119499.9$ $R_B = \frac{119499.9}{19.3} \approx 6190.5 \text{ pounds} \quad \text{(2)}$

Step 3: Substitute Back to Find $R_A$

Using equation (1):

$R_A + 6190.5 - 1170 - 13880 = 0$

This simplifies to:

$R_A + 6190.5 = 13880 + 1170$ $R_A = 13880 + 1170 - 6190.5$ $R_A \approx 8770.5 \text{ pounds} \quad \text{(3)}$

Final Results

1. Reaction at the left end (A): $R_A \approx 8770.5$ pounds
2. Reaction at the right end (B): $R_B \approx 6190.5$ pounds

If you have any further questions or need additional details, let me know! Here are some related questions you might consider:

1. How would the reactions change if the load were applied at a different point?
2. What would happen if the beam were not uniform?
3. How does the weight of the beam affect the reactions at the supports?
4. What are the safety factors to consider for beam design?
5. How can the reactions be calculated using a different method, such as using the method of joints?

Tip: Always check units and ensure they are consistent throughout calculations.