Hibbeler — Statics and Mechanics of Materials
Deflection of beams — how much sag is too much
A beam can pass the strength check and still fail serviceability — visible sag, ponding, sticking doors. The Chapter 1 roof beam gets checked against that separate standard.
Concepts pulled in
- Deflection (Δ = 5wL⁴/(384EI))
- Serviceability limits (L/240, L/360)
Chapter 1 asked one question: does the roof beam break? This chapter asks a different one: does it sag too much, even if it doesn't break? A beam can pass the strength check and still fail the serviceability check — visible sag, a roof that ponds water instead of shedding it, doors that stick.
The formula, for a simply supported beam under a uniform load: Δ = 5wL⁴/(384EI). Deflection scales with the fourth power of span — double the span and deflection goes up 16x, which is why long spans get deep beams fast, well before they'd fail in pure bending.
The usual limit is a fraction of the span, commonly L/240 for roof live load or L/360 for anything with a finished ceiling or floor below it — not a strength number, a stiffness/appearance number, and it's checked separately from the beam's bending capacity.
Worked case
The Chapter 1 roof beam: 10 ft span, 5 ft of roof depth, 20 psf Charleston snow load. Chapter 1's bending check ruled out a single 2×10 and a single 2×8, so the smallest single board that works is a 2×12 (I ≈ 178 in⁴). The real E for No. 2 Southern Pine is 1,400,000 psi (Southern Forest Products Association, Southern Pine Use Guide, Table 1, 2013, dry service). Snow load on the beam is 20 psf × 5 ft = 100 lb/ft = 8.33 lb/in, and the span is 120 in.
- Sag under the snow load: Δ = 5 × 8.33 × 120⁴ ÷ (384 × 1,400,000 × 178) ≈ 0.09 in, about span ÷ 1,330
- Limit: L/240 = 120 ÷ 240 = 0.5 in. The beam uses about 18% of the allowed sag.
- Under dead plus snow (135 lb/ft) it sags about 0.12 in, still far inside the limit
Answer
Deflection is nowhere near governing on this beam: it uses under a fifth of the allowed sag while bending uses about 85% of the beam's capacity. Even the single 2×10 that failed Chapter 1's bending check would sag only about 0.16 in. That is the pattern to expect on a short span: strength decides the size, and stiffness only takes over on longer spans (sag grows with the fourth power of the span, bending only with the square) or under a stricter limit like L/360.
Open items for a later pass
Creep: wood under a steady load keeps sagging slowly over the years, so the long-term sag is bigger than this number, and the NDS has an adjustment for it that is not applied here. Ponding is not checked either: a flat or low-slope roof that sags can hold water, which adds load and sags more.
Modern Monticello · Technical Library · Worked calculation
Deflection of beams — how much sag is too much
Hibbeler — Statics and Mechanics of Materials · Chapter 15
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Try your own numbers
Starts on the Chapter 1 roof beam as a single 2×12, the smallest single board that passes Chapter 1's bending check. Make the span longer or the board slimmer to see sag grow with the fourth power of the span. Bending strength is shown beside it, so you can see which one governs. Wood also keeps sagging slowly under a steady load over the years (creep), which is not included here.
- Load causing the sag: 100 lb/ft (20 psf snow × 5 ft)
- Stiffness: I = 1 × 1.5 × 11.25³ ÷ 12 = 178 in⁴
- Sag: Δ = 5wL⁴ ÷ (384EI) = 5 × 8.333 lb/in × 120⁴ ÷ (384 × 1,400,000 × 178) = 0.09 in (about span ÷ 1,329)
- Limit: span ÷ 240 = 120 ÷ 240 = 0.5 in, so the beam uses 18% of the allowed sag
- For comparison: snow only sags 0.09 in, dead + snow sags 0.122 in
- Bending strength (Chapter 1 check, table Fb) uses 85% of capacity, so bending governs this beam
- Smallest board that passes the sag limit: single 2×8, doubled 2×6. Passing both sag and bending: single 2×12, doubled 2×8.
The sag limit is a stiffness and appearance number, not a strength number, and it is checked separately from bending. The E value is for dry lumber and does not change with the board size. Not included: creep, shear deflection, point loads, ponding of water on a flat roof, and the beam bearing. Worked example, not a stamped design.
Worked example from the Modern Monticello Technical Library, using published design values and simplified checks. Not a stamped engineering design: your site, materials and loads will differ, and anything that could hurt someone if it fails deserves a licensed engineer's review. Provisional or example values are placeholders, not measurements of your material. Sources for every number: modernmonticellobuild.com/technical-library/foundry-roof-beam-deflection
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