Maximum bending moment for a simply supported beam carrying a uniformly distributed load w per unit length over the entire span l: select the correct expression.

Introduction / Context:
Beams under uniformly distributed load (UDL) are fundamental in structural analysis. The maximum bending moment for a simply supported beam under full-span UDL occurs at midspan, and its value is a standard result used for sizing and checking stresses/deflections.

Given Data / Assumptions:

• Simply supported beam, span l.
• Uniformly distributed load w per unit length over entire span.
• Linear elastic behavior.

Concept / Approach:

The support reactions are each R = w l / 2. The bending moment diagram is parabolic, peaking at midspan. Using statics or standard formulas yields the maximum value M_max at the center.

Step-by-Step Solution:

Verification / Alternative check:

Area under shear diagram (triangle) from support to midspan equals w l^2 / 8, matching the computed maximum moment.

Why Other Options Are Wrong:

Other expressions give incorrect numerical constants, locations, or dimensions (e.g., w l^3 / 24 is dimensionally inconsistent for bending moment).

Common Pitfalls:

Forgetting that maximum bending moment occurs where shear is zero; misplacing the location at quarter points (that is for point loads or other cases).

Final Answer:

M_max = w l^2 / 8 at midspan