Report - Rafter Calculator to BS 5268-2:2002 and BS 5268-7.5:1990

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	L_cl,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	E_mean = 8,800 N/mm²
Minimum modulus of elasticity	E_min = 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)	ρ_mean = 370 kg/m³
Self weight per linear metre (g = 9.81 m/s²)	F_rafter = b × h × ρ_mean × g = 0.0131 kN/m

Consider long term loading (1 kN/m² dead UDL. K3 = 1)

K3 (long term load)	K₃ = 1
Total load	F = F_dead × cos(α) × (s / 1000) + F_rafter × cos(α) = 0.502 kN/m
Grade bending stress for C16 (BS5268-2:2002 Table 8)	σ_m,par = 5.3 N/mm²
Permissible bending stress	σ_adm = σ_m,par × K₃ × K₇ × K₈ = 6.62 N/mm²
Compression perpendicular to grain for C16 (BS5268-2:2002 Table 8)	σ_c,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 = (L_cl,slope × F / 2) / (σ_c,per × K₃ × K₈ × b - (F / 2)) = 3.55 mm
Effective span on slope	L_eff = L_cl,slope + a = 1 m
Check bending stress
Bending moment	M = F × L_eff²/ 8 = 0.0632 kNm
Bending stress	σ_m,a = (M × 10⁶) / Z = 1.11 N/mm²
	σ_m,a <= σ_adm ( 1.106 N/mm² <= 6.616 N/mm² ) therefore OK
Check shear stress
Grade shear stress for C16 (BS5268-2:2002 Table 8)	τ_par = 0.67 N/mm²
Permissible shear stress	τ_adm = τ_par × K₃ × K₈ = 0.737 N/mm²
Shear stress	τ = (3 × F × L_eff / 2 × 10³) / (2 × b × h) = 0.105 N/mm²
	τ <= τ_adm ( 0.105 N/mm² <= 0.737 N/mm² ) therefore OK
Check compressive stress parallel to grain
Compression stress parallel to grain	σ_c,par = 6.8 N/mm²
Minimum modulus of elasticity	E_min = 5,800 N/mm²
Slenderness ratio	λ = L_eff / i = 36.6
Compression member factor (calculated using equation in BS5268-7.5 clause 5.3.1)	K₁₂ = 0.815
Permissible compressive stress	σ_c,adm = σ_c,par × K₃ × K₈ × K₁₂ = 6.1 N/mm²
Applied compressive stress	σ_c,a = 0.246 N/mm²
	σ_c,a <= σ_c,adm ( 0.246 N/mm² <= 6.096 N/mm² ) therefore OK
Check combined bending and compressive stress
Euler critical stress	σ_e = π² × E_min / λ² = 42.7 N/mm²
Euler coefficient	K_eu = 1 - (1.5 × σ_c × L₁₂ / σ_e) = 0.993
Combined axial compression and bending check	= σ_m,a / (σ_m,adm × K_eu) + σ_c / σ_c,adm = 0.209
	0.209 <= 1 therefore OK
Check deflection
Permissible deflection	δ_adm = 0.003 × L_eff = 3.01 mm
Bending deflection	δ_bending = (5 × F × L_eff⁴) / (384 × E_mean × I) = 0.278 mm
Shear deflection	δ_shear = (12 × F × L_eff²) / (5 × E_mean × b × h) = 0.0382 mm
Total deflection	δ_total = δ_bending + δ_shear = 0.316 mm
	δ_total <= δ_adm ( 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)	F_imposed,udl = 1 × ((75 - α) / 45) = 0.889 kN/m²
Total load	F = ( F_imposed,udl × cos(α)² + F_dead × cos(α)) × (s / 1000) + F_rafter × cos(α) = 0.86 kN/m
Grade bending stress for C16 (BS5268-2:2002 Table 8)	σ_m,par = 5.3 N/mm²
Permissible bending stress	σ_adm = σ_m,par × K₃ × K₇ × K₈ = 8.27 N/mm²
Compression perpendicular to grain for C16 (BS5268-2:2002 Table 8)	σ_c,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 = (L_cl,slope × F / 2) / (σ_c,per × K₃ × K₈ × b - (F / 2)) = 4.87 mm
Effective span on slope	L_eff = L_cl,slope + a = 1 m
Check bending stress
Bending moment	M = F × L_eff²/ 8 = 0.109 kNm
Bending stress	σ_m,a = (M × 10⁶) / Z = 1.9 N/mm²
	σ_m,a <= σ_adm ( 1.899 N/mm² <= 8.27 N/mm² ) therefore OK
Check shear stress
Grade shear stress for C16 (BS5268-2:2002 Table 8)	τ_par = 0.67 N/mm²
Permissible shear stress	τ_adm = τ_par × K₃ × K₈ = 0.921 N/mm²
Shear stress	τ = (3 × F × L_eff / 2 × 10³) / (2 × b × h) = 0.18 N/mm²
	τ <= τ_adm ( 0.18 N/mm² <= 0.921 N/mm² ) therefore OK
Check compressive stress parallel to grain
Compression stress parallel to grain	σ_c,par = 6.8 N/mm²
Minimum modulus of elasticity	E_min = 5,800 N/mm²
Slenderness ratio	λ = L_eff / i = 36.6
Compression member factor (calculated using equation in BS5268-7.5 clause 5.3.1)	K₁₂ = 0.806
Permissible compressive stress	σ_c,adm = σ_c,par × K₃ × K₈ × K₁₂ = 7.53 N/mm²
Applied compressive stress	σ_c,a = 0.422 N/mm²
	σ_c,a <= σ_c,adm ( 0.422 N/mm² <= 7.532 N/mm² ) therefore OK
Check combined bending and compressive stress
Euler critical stress	σ_e = π² × E_min / λ² = 42.6 N/mm²
Euler coefficient	K_eu = 1 - (1.5 × σ_c × L₁₂ / σ_e) = 0.988
Combined axial compression and bending check	= σ_m,a / (σ_m,adm × K_eu) + σ_c / σ_c,adm = 0.289
	0.289 <= 1 therefore OK
Check deflection
Permissible deflection	δ_adm = 0.003 × L_eff = 3.01 mm
Bending deflection	δ_bending = (5 × F × L_eff⁴) / (384 × E_mean × I) = 0.478 mm
Shear deflection	δ_shear = (12 × F × L_eff²) / (5 × E_mean × b × h) = 0.0656 mm
Total deflection	δ_total = δ_bending + δ_shear = 0.544 mm
	δ_total <= δ_adm ( 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

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