Beam Mechanics – Purlin under Uniformly Distributed Load For a purlin of span L subjected to a uniformly distributed load ω (per unit length), what is the maximum bending moment assumed for a simply supported condition?
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Aω L^2 / 8
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Bω L^2 / 12
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Cω L^2 / 10
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Dω L^2 / 16
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E—
Answer
Correct Answer: ω L^2 / 8
Explanation
Introduction / Context:Purlins span between roof trusses or rafters and carry roofing loads idealized as uniform line loads. Determining the maximum bending moment for a given support condition is a fundamental step in sizing the purlin section and its connections.
Given Data / Assumptions:
- Span: L (simply supported).
- Loading: uniform line load ω (force per unit length along the span).
- Objective: compute the maximum bending moment magnitude for design.
Concept / Approach:
For a simply supported beam under uniform load, reactions are equal at supports and the bending moment diagram is a parabola with a peak at midspan. The standard closed-form result for the midspan moment is ω L^2 / 8, widely used in preliminary and detailed design.
Step-by-Step Solution:
1) Write support reactions: R_A = R_B = ω L / 2.2) Bending moment at distance x from the left: M(x) = R_A x − ω x^2 / 2.3) Set dM/dx = 0 ⇒ R_A − ω x = 0 ⇒ x = L / 2 (midspan).4) Evaluate M_max = M(L/2) = (ω L / 2) * (L/2) − ω (L/2)^2 / 2 = ω L^2 / 8.Verification / Alternative check:
Area-under-shear-diagram or energy methods yield the same value. Standard tables of beam formulas list ω L^2 / 8 for this case, confirming the result.
Why Other Options Are Wrong:
- ω L^2 / 12 and ω L^2 / 16: Too low; these belong to other support/load cases.
- ω L^2 / 10: Not a standard exact result for classical boundary conditions.
Common Pitfalls:
Confusing simply supported with continuous purlins (which reduce peak moment due to continuity), or neglecting load components from roof slope and cladding self-weight.
Final Answer:
ω L^2 / 8