Simply supported beam of span L under a uniformly distributed load (total load notation) A simply supported beam of clear span L carries a uniformly distributed load whose total magnitude over the span is W. What is the maximum bending moment M (sagging) in the beam?

Introduction / Context:The maximum bending moment for a simply supported beam under a uniformly distributed load (UDL) depends on whether the symbol denotes load per unit length or total load. Many texts use w for intensity (force per length) and W for total load. This question explicitly states W is the total load acting over the span L.

Given Data / Assumptions:

• Simply supported beam, span L.
• Total uniformly distributed load W across the span.
• Static equilibrium and small deflection theory.

Concept / Approach:

For a UDL of intensity w (force per unit length), the maximum bending moment is M_max = wL^2/8 at mid-span. If W is the total load, then W = wL, so M_max = (W/L)L^2/8 = WL/8.

Step-by-Step Solution:

Verification / Alternative check:

Shear at mid-span is zero; bending moment diagram is parabolic, peaking at mid-span with the stated value, consistent with classical solutions.

Why Other Options Are Wrong:

• W L / 4 doubles the correct value.
• W L^2 / 8 corresponds to using W as w (confusing total with intensity).
• W^2 L / 8 and W/(8L) have incorrect dimensions.

Common Pitfalls:

• Mixing symbols W and w; remember W = w*L for a uniform distribution.

Final Answer:

M = W L / 8.