A uniform horizontal beam is 19.3 feet long and weighs 1170 pounds.  It is supported at either end.  A vertical load of 13,880 pounds is applied to the beam 7.797 feet from the left end.  Find the reaction at each end of the beam.

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.

A uniform horizontal beam is 19.3 feet long and weighs 1170 pounds. It is supported at either end. A vertical load of 13,880 pounds is applied to the beam 7.797 feet from the left end. Find the reaction at each end of the beam.

Solve for the reaction forces at each end of a 19.3-foot beam with a 13,880 lb load applied 7.797 feet from the left end. This problem involves concepts from statics, equilibrium, and moments, making it ideal for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation