# TRP FEM paper

AbstractTrees are subjected to multiple environmental mechanical loads and adapt the mechanical properties of their stems to an environment changing over time.

## Introduction

Trees, unlike animals have a fixed physical location and throughout their life experience ontogenetic and environmental changes. During development changes in size, shape and wood properties occur for both environmental and ontogenetic reasons. These changes are observed in trends of many physical and chemical wood properties having evolutionary function in mechanical stability and a host of other necessary requirements for survival. The Typical Radial Pattern (TRP) of Micro-Fibril Angle (MFA) and density has been argued to exist to provide superior mechanical stability by some while others have suggested the pattern exists for hydraulic function (see Meinzer et al. (2011) for a review). The aim here is to investigate alternative, unobserved radial patterns of MFA and density with respect to mechanical stability utilizing existing wood property data for green core- and outer-wood (sourced from Davies et al. (2016)) to form mathematical mechanical system models which are sloved using Finite Element Modeling as an analytical tool.

While MFA contributes to the stiffness of the cell wall, basic density measures the amount of cell wall in the tissue. Therefore overall mechanical wood properties rely on both features. Wood properties within ’normal’ stems tend to follow TRP (Meinzer et al., 2011). MFA reduces while density increases from the pith to the periphery of the stem. Note that this is not universal and some species and individuals do not necessarily follow the pattern completely.

Wind is one of the most important mechanical loads (Timell, 1986). Wind loading can cause mechanical failure making the tree worthless in a commercial sense. A substantial amount of research on predicting wind throw and wind damage risk for commercial species has been conducted (Ancelin et al., 2004; Peltola et al., 1999; Mayer et al., 1989; Gardiner et al., 2000; Dunham et al., 2000). These models do not investigate the structural failure within the tree, but attempt to identify how likely failure is to occur in a particular environment. Wind also has less obvious effects. Continued wind loadings from a prevailing direction can cause reaction wood production in order to compensate for this loading (Timell, 1986).

Over the last century or so there have been a number of suggested explanations for why trees grow with the observed TRP. The mechanical hypothesis which is investigated in this study, asserts that the TRP is a result of the tree needing to respond to different mechanical loadings from its environment as it grows. For a seedling, being highly flexible could be important in order to bend out of the path of animals and reduce wind and snow loads. However when the tree grows and a significant size is reached along with a large canopy greater stiffness could be an advantage in outerwood as bending becomes difficult due to the stem diameter. Note that there are other hypotheses, for a good review see Meinzer et al. (2011). The purpose of this paper is to investigate the effect of the TRP of MFA and density on the ability to withstand wind loading.

Structural integrity of both greenwood and corewood have had little attention in literature at the scale of small cellular blocks. Investigating the TRP requires testing at scales small enough to separate corewood and outerwood. Classical mechanics theories have been used, sometimes in conjunction with experimental data from tree pulling and wind tunnel experiments (Rudnicki et al., 2004; Peltola et al., 1999; Spatz et al., 2000). Neither take into account changes in material properties within the stem. Davies et al. (2016) being the only known example where all nine orthotropic elastic material constants have been reported for core- and outerwood.

Elastic deformation of a material occurs when the magnitude of loads applied to a sample are small enough that when released the sample returns to its original state (Hibbeler, 2000). Here we need to define some particular terminology and assumptions. The proportionality limit is the point at which the relationship between stress and strain stops being linear, although not necessarily elastic. The end of the elastic state is characterised by the yield point (elastic limit), after the yield point plastic (irreversible) deformation occurs, although this deformation does not necessarily result in a loss of stiffness (Reiterer et al., 1999).

It was assumed that the proportionality limit and the yield point are equal and the terms yield point, proportionality limit and failure point are used interchangeably to indicate what is strictly the proportionality limit. There is argument for and against the assumption that wood is a linear elastic material in literature (Mackenzie-Helnwein et al., 2005). Within this work models are restricted to the limit of proportionality in order to retain simplicity.