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 |