A simply supported beam with a span length of 5 m carries a moment of 20 N-m (counterclockwise direction) at the middle of the beam. What will the value of reactions be at both the ends of the beam?

Explanation:

Simply Supported Beam with a Moment at the Center

Problem Statement: A simply supported beam with a span of 5 m carries a moment of 20 N-m (counterclockwise direction) at the middle of the beam. We are tasked with determining the reactions at both ends of the beam.

Solution:

Let us analyze the problem step by step:

1. Understanding the Beam Configuration:
• The beam is simply supported, which means it has a hinge support at one end and a roller support at the other end.
• The span of the beam is 5 m.
• A moment of 20 N-m is applied at the midpoint of the beam in the counterclockwise direction.
2. Reaction Forces in Simply Supported Beams:
• In a simply supported beam, the reactions at the supports are vertical forces. Let these reactions be denoted as R_A (at the left support) and R_B (at the right support).
• Since there is no vertical load applied on the beam (only a moment is applied), the sum of vertical forces must be zero. This implies that the vertical reactions at the supports will be equal in magnitude but opposite in direction.
3. Equilibrium Conditions:

We use the equilibrium conditions for a static system to determine the reactions:

• Sum of Vertical Forces:

The sum of vertical forces acting on the beam must be zero:

ΣF_y = 0

R_A + R_B = 0     (1)

• Sum of Moments:

The sum of moments about any point on the beam must also be zero. Let us take moments about point A (left support):

ΣM_A = 0

Moment due to the reaction at B: (R_B × 5)

External moment applied at the center of the beam: 20 N-m (counterclockwise)

Equating the moments:

(R_B × 5) - 20 = 0

R_B = 20 ÷ 5 = 4 N     (2)

• Substitute R_B into Equation (1):

From Equation (1):

R_A + R_B = 0

R_A = -R_B

R_A = -4 N     (3)

Final Reactions:

• Reaction at the left support (R_A): -4 N
• Reaction at the right support (R_B): 4 N

The negative sign of R_A indicates that the direction of the reaction at the left support is downward, while the reaction at the right support is upward.

Correct Option: Option 1) 4 N, -4 N

Additional Information

To further understand the analysis, let’s evaluate why the other options are incorrect:

Option 2: 8 N, -8 N

This option suggests that the reactions at the supports are 8 N (upward at one support and downward at the other). However, this is incorrect because the applied moment (20 N-m) only results in reactions of 4 N and -4 N. The magnitude of 8 N is incorrect based on the equilibrium equations.

Option 3: 5 N, -5 N

This option assumes that the reactions are 5 N (upward and downward). However, this is inconsistent with the applied moment of 20 N-m. Using the equilibrium equations, the reactions must be 4 N and -4 N, not 5 N and -5 N.

Option 4: N, -2 N

This option is incomplete and does not provide a valid numerical value for the reactions. Additionally, the values do not satisfy the equilibrium equations for the given applied moment.

Conclusion:

The correct reactions at the supports of the simply supported beam are 4 N (upward) at the right support and -4 N (downward) at the left support. This satisfies both the vertical force equilibrium and moment equilibrium conditions. Hence, the correct answer is Option 1: 4 N, -4 N.