K3 (medium term load) | K_{3} = **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} × K_{3} × K_{7} × K_{8} = **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} × K_{3} × K_{8} × 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_{eff}²/ 8 = **0.109** kNm |

Bending stress^{ } | σ_{m,a} = (M × 10^{6}) / 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} × K_{3} × K_{8} = **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) | K_{12} = **0.806** |

Permissible compressive stress | σ_{c,adm} = σ_{c,par} × K_{3} × K_{8} × K_{12} = **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} × L_{12} / 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_{eff}^{4}) / (384 × E_{mean} × I) = **0.478** mm |

Shear deflection | δ_{shear} = (12 × F × L_{eff}²) / (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** |