Branch Weight & Load Estimator
Estimates the weight of a tree branch and the bending moment (load) it exerts at its base attachment point, using a tapered cylinder model and wood density.
Formulas Used
Volume (Frustum / Truncated Cone):
V = (π · L / 3) · (rb² + rb·rt + rt²)
Branch Weight:
Wwood = V · ρ (ρ = wood density in lb/ft³)
Snow/Ice Weight:
Wsnow = qsnow · L · (db + dt) / 2 (projected surface area)
Center of Mass of Frustum from Base:
xcm = (L/4) · (rb² + 2rbrt + 3rt²) / (rb² + rbrt + rt²)
Bending Moment at Base:
M = Wtotal · xcm · sin(θ) (θ = angle from vertical)
Bending Stress at Base (solid circular section):
σ = M / S where S = π·rb³ / 4
Assumptions & References
- Branch modeled as a solid truncated cone (frustum) — no hollow core or decay.
- Wood density values are for green (freshly cut) wood; dry wood is ~15–20% lighter.
- Snow load is applied to the projected horizontal surface area (trapezoidal approximation).
- Bending moment assumes the branch weight acts at its center of mass with a horizontal moment arm = xcm · sin(θ).
- Bending stress uses the elastic flexure formula (σ = M/S) for a solid circular cross-section at the base.
- No dynamic (wind) loads, foliage weight, or secondary branches are included.
- Wood density references: USDA Forest Products Laboratory, Wood Handbook (2021), Table 5-3.
- Frustum centroid formula: standard solid geometry derivation.
- For professional tree risk assessment, consult a Certified Arborist (ISA) or structural engineer.