Calculating Reactionary Forces on a Beam with Distributed Load

Problem Scenario:

• Beam Length: 6 meters
• Support Points: A and B
• Distributed Load: Starts at 3,000 N/m on the left and increases to 6,000 N/m on the right.

Key Concepts:

• Centroid: Point where the entire distributed load can be considered to act.

Steps to Solve:

1. Finding the Centroid of the Distributed Load

• Assumption: Centroid is more to the right due to increasing force.
• Regions: Divide load into two parts:
  • Rectangular Region
  • Triangular Region

Centroid of Individual Regions:

• Rectangular Region Centroid: Middle of the rectangle.
• Triangular Region Centroid: Located at one-third the base to the peak, i.e., two-thirds from peak to base in this scenario.

2. Calculating Centroid Coordinates

• Formula: [ x_{centroid} = \frac{\sum (x_i \cdot F_i)}{\sum F_i} ]
• Triangular Portion:
  • Base: 6 meters
  • Height Difference: 3,000 N (6,000 N/m - 3,000 N/m)
  • Centroid: 4 meters from the peak
  • Total Force (Area): ( \frac{1}{2} \times 6 \text{ m} \times 3,000 \text{ N/m} = 9,000 \text{ N} )
• Rectangular Portion:
  • Centroid: 3 meters (midpoint)
  • Total Force: ( 6 \text{ m} \times 3,000 \text{ N/m} = 18,000 \text{ N} )_

3. Calculate Combined Centroid

• Total Force: 27,000 N (9,000 N + 18,000 N)
• Centroid Distance (x):
  • Using formula: [ \frac{4 \cdot 9,000 + 3 \cdot 18,000}{27,000} = 3.33 \text{ m} ]

4. Finding Reactionary Forces at A and B

Reaction at B:

• Using Torque Equilibrium:
  • Sum of moments at A = 0
  • Clockwise torque is negative.
  • (-27,000 \text{ N} \times 3.33 \text{ m} + F_B \times 6 \text{ m} = 0)
  • ( F_B = \frac{27,000 \times 3.33}{6} = 15,000 \text{ N} )

Reaction at A:

• Using Vertical Force Equilibrium:
  • Total downward force: 27,000 N
  • ( 27,000 \text{ N} = F_A + 15,000 \text{ N} )
  • ( F_A = 27,000 \text{ N} - 15,000 \text{ N} = 12,000 \text{ N} )

Conclusion:

• Reaction Force at A: 12,000 N
• Reaction Force at B: 15,000 N

Key Takeaway:

• The entire load acts as if it is concentrated at the centroid.
• Reaction forces can be found using torque and force equilibrium principles.