Critical wind speed at which trees break

Data from storms suggest that the critical wind speed at which trees break is constant ($\simeq 42\mathrm{m}/\mathrm{s}$), regardless of tree characteristics. We question the physical origin of this observation both experimentally and theoretically. By combining Hooke's law, Griffith's criterion, and tree allometry, we show that the critical wind speed indeed hardly depends on the height, diameter, and elastic properties of trees.

Figure 1

Early studies on the resistance of wood. (a) Sketch in Leonardo's notes (in mirror writing); “ab” and “ef” are, respectively, three times thicker and nine times smaller than “cd”. We can read “27”, “3”, and “27” for the dimensionless resistance, which suggests a power law (adapted from Ref. [13]). (b) Wooden beam illustrating the Galileo's Discorsi (adapted from Ref. [15]). (c) Engraving from Buffon's Natural History where the man at the bottom loads an oak beam (adapted from Ref. [17], p. 1).

Figure 2

Storm Klaus (South-West of France, January $24\text{th}$, 2009). (a) Maximal wind speed recorded during the storm Klaus. (Data: Météo France [19]; calculated from an average over 1 s). (Inset) A trunk breakage in a pine forest [in the area A of Fig. 2], attributed to storm Klaus. (Photograph: Saint Julien en Born City Hall). (b) Percentage of broken trees attributed to the storm Klaus. (Data: Inventaire Forestier National [26]). Area A is a forest of pines, whereas area B is mainly a forest of oaks. The highest wind speeds were recorded in these areas, leading to extreme damage regardless of tree species. (Inset) Correlation between wind speed and tree damage. There is no statement of damage below 20 m/s, whereas a majority of trees gets broken for wind speeds exceeding 42 m/s.

Figure 3

Deformation of a rod induced by a point mass $m$ suspended at the free end. (a)–(d) Wood rod. (e)–(h) Pencil lead. While the length scales are roughly divided by ten, the mass scale is lowered by a factor one thousand.

Figure 4

Two regimes for the curvature of the rod. (a) Small deformation ($R\gg L$). (b) Large deformation ($R\ll L$). (c) Normalized radius of curvature $R/L$ as a function of the normalized mass $m/m_{el}$, where $m_{el}$ is $EI/g{L}^2$: (- - -) small and large deformation limits, () Eq. (2), and ($•$) numerical resolution of Eq. (1).

Figure 5

Critical radius of curvature $R_c$ at breakage. (a) The rod diameter is kept constant ($D=5$ mm) and its length $L$ is varied. The error bars are standard deviations observed for ten successive measurements. (b) The rod length is kept constant ($L=950$ mm) and its diameter $D$ is varied. The error bars are standard deviations observed for ten successive measurements. (c) Critical radius of curvature measured as a function of Eq. (4). All the data collapse on a straight line of slope 1 in logarithmic scale, with an adjusted numerical parameter $\delta$ in the prefactor (see Table 1).

Figure 6

Curvature induced by wind loading. (a) Small deformation ($R\gg L$). (b) Large deformation ($R\ll L$). (c) The curvature of a straw placed in a wind tunnel increases with the wind speed. (d) Radius of curvature observed for wood rods, normalized by the rod length $L=960$ mm. The solid line is given by Eq. (5). Error bars correspond to estimations of error due to the method of measurement.

Figure 7

Breakage induced by torsion. (a) A tree broken after a trunk torsion. (Photograph: Keraunos, Fontainebleau forest, France, June 2010). (b)–(e) Snapshots from experiments. Wood rod of length $L=900$ mm and diameter $D=4$ mm. Snapshots are stretched vertically to emphasize the deformation out of its axis (note the two scale bars).

Figure 8

Critical angle of rotation. (a) The rod diameter is kept constant ($D=5$ mm) and its length $L$ is varied. (b) The rod length is kept constant ($L=900$ mm) and its diameter $D$ is varied. Error bars indicate the standard deviation observed on five successive experiments. (c) Critical angle of rotation as a function of the predicted one, Eq. (A1). All the data collapse on a straight line of slope 1 in logarithmic scales.