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.
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AM_max = w l^2 / 12 at midspan
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BM_max = w l^2 / 8 at midspan
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CM_max = w l / 4 at supports
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DM_max = w l^2 / 16 at quarter points
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EM_max = w l^3 / 24 at midspan
Answer
Correct Answer: M_max = w l^2 / 8 at midspan
Explanation
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:
Reaction at each support: R = w l / 2.Bending moment at a section x from the left: M(x) = R x − w x^2 / 2.Set dM/dx = R − w x = 0 ⇒ x = R / w = (w l / 2)/w = l/2 (midspan).Evaluate M_max: M(l/2) = (w l / 2)(l/2) − w (l/2)^2 / 2 = w l^2 / 8.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