To determine the maximum bending moment for a simply supported beam with a uniformly distributed load (UDL), let's analyze the problem using the fundamental concepts of strength of materials.

Conceptual Explanation:

A simply supported beam subjected to a uniform load per unit length \( w \) creates a bending moment across the span of the beam. For a simply supported beam with UDL, the maximum bending moment occurs at the mid-span of the beam.

The formula to calculate the maximum bending moment \( M \) for a simply supported beam of span \( L \) under a uniformly distributed load \( w \) is given by:

M = \frac{wL^2}{8}

Derivation of Formula:

1. **Starting with the Total Load**: For the entire span of the beam, the total load \( W \) caused by the uniformly distributed load is given by:

W = w \times L

2. **Reaction Forces**: Because the beam is simply supported, the reactions at both supports (\( R_A \) and \( R_B \)) are equal and can be calculated as:

R_A = R_B = \frac{W}{2} = \frac{wL}{2}

3. **Calculate Bending Moment at Mid-Span**: The maximum bending moment occurs at the middle of the beam (at x = \frac{L}{2}).

The bending moment at a distance \( x \) from support \( A \) is:

M(x) = R_A \cdot x - \frac{w \cdot x^2}{2}

Substituting \( x = \frac{L}{2} \), we find the maximum bending moment:

M_{max} = \frac{wL}{2} \cdot \frac{L}{2} - \frac{w \cdot (\frac{L}{2})^2}{2}

M_{max} = \frac{wL^2}{4} - \frac{wL^2}{8}

M_{max} = \frac{wL^2}{8}

Thus, the maximum bending moment for the beam is \frac{wL^2}{8}.

Conclusion:

The correct answer is indeed \frac{wL^2}{8}, which confirms that the maximum bending moment for a simply supported beam under a uniformly distributed load is correctly computed.