Question

The maximum bending moment of a simply supported beam of span \( l \) and carrying a point load \( W \) at the centre of the beam is

• A. \( \dfrac{Wl}{2} \)
• B. \( \dfrac{Wl}{4} \)
• C. \( Wl \)
• D. \( 2Wl \)

Hint:

For a simply supported beam with a central point load, maximum bending moment always occurs at mid-span.

Answer 1

The Correct Option is B

Step 1: Understanding the loading condition.
A simply supported beam carries a concentrated load \( W \) at its mid-span. Due to symmetry, the reactions at both supports are equal.
Step 2: Calculating support reactions.
Each support reaction is: \[ R = \frac{W}{2} \]
Step 3: Finding maximum bending moment.
Maximum bending moment occurs at the centre of the beam: \[ M_{\max} = R \times \frac{l}{2} = \frac{W}{2} \times \frac{l}{2} \] \[ M_{\max} = \frac{Wl}{4} \]
Step 4: Conclusion.
The maximum bending moment is \( \dfrac{Wl}{4} \).

Answer 2

For a simply supported beam of span \( l \) with a single point load \( W \) at mid-span, the bending moment diagram is triangular, rising linearly from zero at each support to a peak value at the centre. Let's check which of the listed values is consistent with this peak.

1. \( \dfrac{Wl}{2} \): This is twice the true peak moment. Using this value would imply a moment larger than what the support reactions and geometry actually produce at mid-span, over-estimating the true bending effect.
2. \( \dfrac{Wl}{4} \): Each support reaction is \( \dfrac{W}{2} \), and the moment at the centre is this reaction multiplied by the distance to the centre, \( \dfrac{l}{2} \), giving \( \dfrac{W}{2}\times\dfrac{l}{2} = \dfrac{Wl}{4} \). This matches the standard triangular moment diagram for a central point load and is consistent with the support reactions found from equilibrium.
3. \( Wl \): This value is four times too large; it would only arise if the entire load \( W \) acted at a distance \( l \) from a single support without any load sharing between two supports, which does not describe a simply supported beam.
4. \( 2Wl \): This is eight times the correct peak value and does not correspond to any consistent equilibrium condition for a simply supported beam carrying a single central point load.

Only \( \dfrac{Wl}{4} \) is consistent with the reactions found from statics and the standard triangular bending moment diagram for this loading case.

Therefore, the correct answer is \( \dfrac{Wl}{4} \).