Question

If a simply supported beam AB of span 6 m is subjected to a concentrated load of W acting at a distance of 2 m from the left support A, then

• A. maximum bending moment occurs at the support A.
• B. maximum shear force occurs at support B.
• C. maximum bending moment occurs at the centre.
• D. maximum bending moment occurs under the load.

Hint:

Remember this critical rule: Maximum bending moment occurs where shear force is zero.
For a simply supported beam with only point loads, the SFD is a series of horizontal lines.
The shear force will change sign at one of the load points, and that's where the BMD will have its peak.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the location of the maximum bending moment in a simply supported beam subjected to a single, off-center point load.

Step 2: Key Formula or Approach:
A fundamental principle of beam theory states that the maximum bending moment occurs at a point where the shear force is zero or changes sign.

Step 3: Detailed Explanation:
Let the beam have length $L=6$ m. The load $W$ is at $a=2$ m from A and $b=4$ m from B.

1. Calculate the Reactions:
Take moments about A: $R_B \times 6 - W \times 2 = 0 \implies R_B = W/3$.
Take moments about B: $R_A \times 6 - W \times 4 = 0 \implies R_A = 2W/3$.

2. Analyze the Shear Force Diagram (SFD):
- From A to the load (0 $\lt x \lt $ 2 m): The shear force is constant and equal to $V = R_A = +2W/3$.
- At the point of the load ($x=2$ m): The shear force drops by the magnitude of the load, from $+2W/3$ to $+2W/3 - W = -W/3$.
- From the load to B (2 $\lt x \lt $ 6 m): The shear force is constant and equal to $V = -W/3 = -R_B$.

3. Locate Maximum Bending Moment:
The shear force diagram changes sign (from positive to negative) exactly at the point where the concentrated load $W$ is applied. Therefore, the bending moment is maximum at this location.
The value of the maximum bending moment is $M_{max} = R_A \times a = (2W/3) \times 2 = 4W/3$.

Step 4: Final Answer:
For a simply supported beam with a single concentrated load, the maximum bending moment always occurs at the point of application of the load.