Tree Stress — Wind Loading of a Tapered Cantilever

A tree in wind is a vertical cantilever. Wind drag on the crown, F = ½ ρ C_d A v², acts at the crown’s center of pressure and bends the trunk. At the base this produces a bending moment M = F·h_cp, a peak bending stress σ = M/Z and a peak shear stress τ at the neutral axis. A trunk is not solid — the load-bearing sapwood forms a hollow tube (annular section), so Z = π(D⁴−d⁴)/32D and A = π(D²−d²)/4 for outer diameter D and bore d. Because stiffness scales with r⁴, a tube keeps almost all its strength using far less wood. Pick a species, then vary wind speed and tree age — the tree sways, and the base cross-section shows the tension–compression bending gradient and the parabolic shear. Green wood modulus of rupture (MOR) sets the failure threshold and safety factor. Real crowns streamline (reconfigure) in strong wind, so field drag grows more slowly than v² — treat these as conservative upper-bound estimates.