A simple tree swaying model for forest motion in windstorm conditions

Abstract

A simple tree swaying model, valid for windstorm conditions, has been developed for the purpose of simulating the effect of strong wind on the vulnerability of heterogeneous forest canopies. In this model the tree is represented as a flexible cantilever beam whose motion, induced by turbulent winds, is solved through a modal analysis. The geometric nonlinearities related to the tree curvature are accounted for through the formulation of the wind drag force. Furthermore, a breakage condition is considered at very large deflections. A variety of case studies is used to evaluate the present model. As compared to field data collected on three different tree species, and to the outputs of mechanistic models of wind damage, it appears to be able to predict accurately large tree deflections as well as tree breakage, using wind velocity at tree top as a forcing function. The instantaneous response of the modelled tree to a turbulent wind load shows very good agreement with a more complex tree model. The simplicity of the present model and its low computational time make it well adapted to future use in large-eddy simulation airflow models, aimed at simulating the complete interaction between turbulent wind fields and tree motion in fragmented forests.

Appendix

A mode shape and frequency

The mode shapes $\varphi_{j}$ and frequencies f _j of a cylindric cantilever beam of height h, with a clamped origin and a free end, can be deduced from the linearised Euler–Bernoulli equation, assuming that d _i (s, t) <  < h (Voltera and Zachmanoglou 1965).

The mode shapes are deduced from:

$$\varphi_{j}(s)=\left(\sinh\left(\alpha_{j}s\right)-\sin\left( \alpha_js\right) - \frac{\sin\left(\alpha_jh\right)+\sinh\left(\alpha_jh\right)}{ \cos\left(\alpha_{j}h\right)+\cosh\left(\alpha_jh\right)}\right) \left(\cosh\left(\alpha_{j}s\right)-\cos\left(\alpha_{j}s\right)\right),$$

(6)

where the variable α _j is deduced from $\cosh(\alpha_{j}h)\cos(\alpha_{j}h)+1=0$ by using a Newton–Raphson method.

These modes form an orthogonal modal basis where the scalar product of two modes is defined as:

$$\left\langle \varphi_{i},\varphi_{j}\right\rangle=\int\limits_{0}^{h} \varphi_{i}\left(s\right)\varphi_{j}\left(s\right){\rm d}s= \left\lbrace \begin{array}{lll} 1 & \hbox{if} & \varphi_i=\varphi_j\\ 0 & \hbox{if} & \varphi_i\ne\varphi_j \end{array}\right.$$

(7)

The frequency of each mode j can be obtained from:

$$f_j= \frac{\alpha_j^2}{2\pi}\sqrt{\frac{EI}{\rho_{\rm w}S_{\rm w}}},$$

(8)

where E, ρ _w, I (=π D ⁴/64), and S _w (=π D ²/4) are the Young modulus, the wood density, the second moment of inertia and the cross-sectional area of the beam, respectively. As in this approach all ingredients of f _j in Eq. (8) are considered constant, whereas in reality they vary along the tree, the frequencies of the various modes calculated this way may differ from the measured frequencies. This parametrisation turns out to be well adapted for the first frequency as it is closely related to the trunk vibration. For example, using Eq. (8) for the Maritime pine and taking a diameter value at mid-canopy height, a frequency value of 0.16 Hz is obtained, close to the measured value of 0.18 Hz (Sellier et al. 2008).