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## Rafter details

 Timber strength class C16 Rafter width b = 38 mm Rafter depth h = 95 mm Rafter spacing s = 600 mm Rafter slope α = 35 degrees Clear span on slope Lcl,slope = 1 m Brittle finishes (e.g. plasterboard) on the underside of the rafter? Yes Diagrams not to scale

## Modification factors

 Depth factor K7 = (300 / h)0.11 = 1.13 Load sharing modification factor (BS5268-2 clause 2.10.11) K8 = 1.10

## Section properties

 Mean modulus of elasticity Emean = 8,800 N/mm² Minimum modulus of elasticity Emin = 5,800 N/mm² Cross section area A = b × h = 3,610 mm2 Second moment of area I = b × h³ / 12 = 2,720,000 mm4 Section modulus Z = b × h² / 6 = 57,200 mm³ Radius of gyration i = √(I / A) = 27.4 mm Average density for C16 timber (BS 5268-2:2002 Table 8) ρmean = 370 kg/m³ Self weight per linear metre (g = 9.81 m/s²) Frafter = b × h × ρmean × g = 0.0131 kN/m

 K3 (long term load) K3 = 1 Total load F = Fdead × cos(α) × (s / 1000) + Frafter × cos(α) = 0.502 kN/m Grade bending stress for C16 (BS5268-2:2002 Table 8) σm,par = 5.3 N/mm² Permissible bending stress σadm = σm,par × K3 × K7 × K8 = 6.62 N/mm² Compression perpendicular to grain for C16 (BS5268-2:2002 Table 8) σc,per = 1.7 N/mm² Notional bearing length(Note from BS 5268-7.5 Clause 4.2: 'The bearing length required at each end of the rafter, calculated in accordance with 5.6, may not be sufficient for practical construction purposes.') a = (Lcl,slope × F / 2) / (σc,per × K3 × K8 × b - (F / 2)) = 3.55 mm Effective span on slope Leff = Lcl,slope + a = 1 m Check bending stress Bending moment M = F × Leff²/ 8 = 0.0632 kNm Bending stress σm,a = (M × 106) / Z = 1.11 N/mm² σm,a <= σadm ( 1.106 N/mm² <= 6.616 N/mm² ) therefore OK Check shear stress Grade shear stress for C16 (BS5268-2:2002 Table 8) τpar = 0.67 N/mm² Permissible shear stress τadm = τpar × K3 × K8 = 0.737 N/mm² Shear stress τ = (3 × F × Leff / 2 × 10³) / (2 × b × h) = 0.105 N/mm² τ <= τadm ( 0.105 N/mm² <= 0.737 N/mm² ) therefore OK Check compressive stress parallel to grain Compression stress parallel to grain σc,par = 6.8 N/mm² Minimum modulus of elasticity Emin = 5,800 N/mm² Slenderness ratio λ = Leff / i = 36.6 Compression member factor (calculated using equation in BS5268-7.5 clause 5.3.1) K12 = 0.815 Permissible compressive stress σc,adm = σc,par × K3 × K8 × K12 = 6.1 N/mm² Applied compressive stress σc,a = 0.246 N/mm² σc,a <= σc,adm ( 0.246 N/mm² <= 6.096 N/mm² ) therefore OK Check combined bending and compressive stress Euler critical stress σe = π² × Emin / λ² = 42.7 N/mm² Euler coefficient Keu = 1 - (1.5 × σc × L12 / σe) = 0.993 Combined axial compression and bending check = σm,a / (σm,adm × Keu) + σc / σc,adm = 0.209 0.209 <= 1 therefore OK Check deflection Permissible deflection δadm = 0.003 × Leff = 3.01 mm Bending deflection δbending = (5 × F × Leff4) / (384 × Emean × I) = 0.278 mm Shear deflection δshear = (12 × F × Leff²) / (5 × Emean × b × h) = 0.0382 mm Total deflection δtotal = δbending + δshear = 0.316 mm δtotal <= δadm ( 0.316 mm <= 3.011 mm ) therefore OK

## Consider medium term loading (1 kN/m² dead UDL + 1 kN/m² imposed UDL. K3 = 1.25)

 K3 (medium term load) K3 = 1.25 BS 5268-7.5 Clause 4.3: For a roof slope greater than 30° and not exceeding 75°: an imposed load obtained by linear interpolation between the values at 30° roof slope, e.g. 0.75 kN/m2, and zero for a 75° roof slope. Imposed load (UDL) Fimposed,udl = 1 × ((75 - α) / 45) = 0.889 kN/m² Total load F = ( Fimposed,udl × cos(α)² + Fdead × cos(α)) × (s / 1000) + Frafter × cos(α) = 0.86 kN/m Grade bending stress for C16 (BS5268-2:2002 Table 8) σm,par = 5.3 N/mm² Permissible bending stress σadm = σm,par × K3 × K7 × K8 = 8.27 N/mm² Compression perpendicular to grain for C16 (BS5268-2:2002 Table 8) σc,per = 1.7 N/mm² Notional bearing length(Note from BS 5268-7.5 Clause 4.2: 'The bearing length required at each end of the rafter, calculated in accordance with 5.6, may not be sufficient for practical construction purposes.') a = (Lcl,slope × F / 2) / (σc,per × K3 × K8 × b - (F / 2)) = 4.87 mm Effective span on slope Leff = Lcl,slope + a = 1 m Check bending stress Bending moment M = F × Leff²/ 8 = 0.109 kNm Bending stress σm,a = (M × 106) / Z = 1.9 N/mm² σm,a <= σadm ( 1.899 N/mm² <= 8.27 N/mm² ) therefore OK Check shear stress Grade shear stress for C16 (BS5268-2:2002 Table 8) τpar = 0.67 N/mm² Permissible shear stress τadm = τpar × K3 × K8 = 0.921 N/mm² Shear stress τ = (3 × F × Leff / 2 × 10³) / (2 × b × h) = 0.18 N/mm² τ <= τadm ( 0.18 N/mm² <= 0.921 N/mm² ) therefore OK Check compressive stress parallel to grain Compression stress parallel to grain σc,par = 6.8 N/mm² Minimum modulus of elasticity Emin = 5,800 N/mm² Slenderness ratio λ = Leff / i = 36.6 Compression member factor (calculated using equation in BS5268-7.5 clause 5.3.1) K12 = 0.806 Permissible compressive stress σc,adm = σc,par × K3 × K8 × K12 = 7.53 N/mm² Applied compressive stress σc,a = 0.422 N/mm² σc,a <= σc,adm ( 0.422 N/mm² <= 7.532 N/mm² ) therefore OK Check combined bending and compressive stress Euler critical stress σe = π² × Emin / λ² = 42.6 N/mm² Euler coefficient Keu = 1 - (1.5 × σc × L12 / σe) = 0.988 Combined axial compression and bending check = σm,a / (σm,adm × Keu) + σc / σc,adm = 0.289 0.289 <= 1 therefore OK Check deflection Permissible deflection δadm = 0.003 × Leff = 3.01 mm Bending deflection δbending = (5 × F × Leff4) / (384 × Emean × I) = 0.478 mm Shear deflection δshear = (12 × F × Leff²) / (5 × Emean × b × h) = 0.0656 mm Total deflection δtotal = δbending + δshear = 0.544 mm δtotal <= δadm ( 0.544 mm <= 3.015 mm ) therefore OK

## Design summary

 Permissible Applied/Actual Utilisation Result Long term load shear stress (N/mm²) 0.74 0.1 14.2 % OK Long term load bending stress (N/mm²) 6.62 1.11 16.7 % OK Long term load deflection (mm) 3.01 0.32 10.5 % OK Long term compressive stress parallel to grain (N/mm²) 6.1 0.25 4 % OK Long term combined bending and compressive stress (N/mm²) 1 0.21 20.9 % OK Medium term load shear stress (N/mm²) 0.92 0.18 19.5 % OK Medium term load bending stress (N/mm²) 8.27 1.9 23 % OK Medium term load deflection (mm) 3.01 0.54 18 % OK Medium term compressive stress parallel to grain (N/mm²) 7.53 0.42 5.6 % OK Medium term combined bending and compressive stress (N/mm²) 1 0.29 28.9 % OK

## Notes

This design is in accordance with BS 5268-2:2002 Structural use of timber - Part 2: Code of practice for permissible stress design, materials and workmanship and BS 5268-7.5:1990 Structural use of timber - Section 7.5 Domestic rafters.

These calculations apply to systems of at least four rafters, and having tiling battens adequate to provide lateral distribution and lateral support.

Timber to be covered, this calculation is not to be used for timber which is fully exposed to the elements.

For roof slopes greater than 30 degrees, the concentrated (point) 0.9 kN load can be ignored in accordance with BS 5268-7.5 Clause 4.3.

These calculations are only applicable for roofs consisting of four or more rafters.

Wane as allowed in BS 4978:2007 + A2:2017 is permitted.

DEMO