Answer
Minimum Required Width of Timber Beam Under Load
This page provides a complete calculation for determining the minimum width of a timber beam based on given bending and shear stress limits.
🔹 Bending Stress Analysis
Formula:
σ = (M × c) / I ⇒ σ = (6M) / (b × h²) ⇒ b = (6M) / (σallow × h²)
Given:
- Load, P = 4800 N
- Span length, L = 2400 mm
- Beam height, h = 120 mm
- σallow = 85 MPa
Calculation:
- Maximum Bending Moment: M = (P × L) / 4 = (4800 × 2400) / 4 = 2,880,000 N⋅mm
- Width from bending stress:
b = (6 × 2,880,000) / (85 × 120²) = 17,280,000 / 1,224,000 = 100 mm
🔹 Shear Stress Analysis
Formula:
τ = (3V) / (2bh) ⇒ b = (3V) / (2τallow × h)
Given:
- V = P / 2 = 4800 / 2 = 2400 N
- τallow = 12 MPa
Calculation:
b = (3 × 2400) / (2 × 12 × 120) = 7200 / 2880 = 36.36 mm
✅ Final Answer
The beam must satisfy both bending and shear criteria. Therefore, the minimum required width is:
bmin = max(100 mm, 36.36 mm) = 100 mm
📘 Explanation of Variables
| Symbol | Meaning | Unit |
|---|---|---|
| P | Applied load | N |
| L | Span length of the beam | mm |
| M | Maximum bending moment | N⋅mm |
| V | Shear force at support | N |
| b | Beam width | mm |
| h | Beam height | mm |
| I | Moment of inertia | mm⁴ |
| c | Distance from neutral axis | mm |
| σ | Bending stress | MPa |
| τ | Shear stress | MPa |
| A | Cross-sectional area | mm² |