What is Beam Deflection?
Beam deflection is the amount a beam bends downward under load — a critical serviceability check in structural design, since excessive deflection can crack finishes or feel unsafe even if the beam is strong enough not to break. For a simply supported beam with a center point load, maximum deflection is given by δ = PL³/(48EI).
Beam deflection is how much a beam bends under load. For a simply supported beam with a point load at midspan, the maximum deflection is δ = PL³/(48EI), where P is the load, L the span, E the material's stiffness, and I the cross-section's moment of inertia.
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Step-by-step worked examples
A simply supported beam has a 4000 mm span with a 5000 N point load at midspan. E = 200,000 MPa, I = 60,000,000 mm⁴. Find the maximum deflection.
δ = PL³/(48EI) P = 5000 N, L = 4000 mm, E = 200,000 MPa, I = 60,000,000 mm⁴ L³ = 4000³ = 6.4×10¹⁰ mm³ δ = (5000 × 6.4×10¹⁰) / (48 × 200,000 × 60,000,000) = 3.2×10¹⁴ / 5.76×10¹⁴ ≈ 0.56 mm
If the same beam's span is increased to 8000 mm (all else equal), what is the new deflection?
Deflection is proportional to L³, so doubling the span multiplies deflection by 2³ = 8. Original δ (L = 4000 mm) = 0.56 mm New δ (L = 8000 mm) = 0.56 × 8 = 4.48 mm — beams get dramatically more flexible as spans grow.
A beam with a 6000 mm span has a computed deflection of 12 mm. Does it meet a common L/360 serviceability limit?
Allowable deflection limit (typical live-load serviceability) = L/360 L = 6000 mm → limit = 6000/360 = 16.7 mm Actual computed δ = 12 mm 12 mm < 16.7 mm, so the beam satisfies the serviceability deflection limit.
Flashcards
Quick quiz
Q1.What does δ = PL³/(48EI) calculate for a simply supported beam?
Q2.If the span L of a beam doubles (all else equal), the maximum deflection:
Q3.Which change would REDUCE beam deflection?
Q4.A beam has a computed deflection of 20 mm over a 6000 mm span. Does it meet a common L/360 serviceability limit?
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Common mistakes
A beam is safe as long as it doesn't break (strength check only). — Correct: Beams must also pass a serviceability deflection check (like L/360), even if they're strong enough not to fracture.
Deflection scales linearly with span length. — Correct: Deflection is proportional to L³ — a cubic relationship, so span increases have an outsized effect.
A wider beam always deflects the same as a taller one with equal cross-sectional area. — Correct: Moment of inertia I depends heavily on depth (I ∝ depth³ for rectangles), so taller beams resist deflection far better than wider ones of the same area.
Deflection formulas are the same regardless of support and load type. — Correct: The formula changes with support conditions and load pattern — δ = PL³/48EI applies specifically to a simply supported beam with a center point load.
FAQ
What is beam deflection?
Beam deflection is the vertical displacement a beam undergoes when loaded — how much it bends or sags.
What is the beam deflection formula?
For a simply supported beam with a center point load: δ = PL³/(48EI), where P is load, L is span, E is stiffness, and I is moment of inertia.
What are examples of beam deflection calculations?
Examples include floor joists sagging under furniture load, a bridge girder deflecting under traffic, or a shelf bowing under books.
How do you calculate beam deflection?
Identify the support and load type, then apply the matching formula — for a simply supported beam with a center point load, use δ = PL³/(48EI).