Abstract
Active mechanisms of re-orientation are necessary to maintain the verticality of tree stems. They are achieved through the production of reaction wood, associated with circumferential variations of three factors related to cambial activity: maturation strain, longitudinal modulus of elasticity (MOE) and eccentric growth. These factors were measured on 17 mature trees from different botanical families and geographical locations. Various patterns of circumferential variation of these factors were identified. A biomechanical analysis based on beam theory was performed to quantify the individual impact of each factor. The main factor of re-orientation is the circumferential variation of maturation strains. However, this factor alone explains only 57% of the re-orientations. Other factors also have an effect through their interaction with maturation strains. Eccentric growth is generally associated with heterogeneity of maturation strains, and has an important complementary role, by increasing the width of wood with high maturation strain. Without this factor, the efficiency of re-orientations would be reduced by 31% for angiosperms and 26% for gymnosperms. In the case of angiosperms, MOE is often larger in tension wood than in normal wood. Without these variations, the efficiency of re-orientations would be reduced by 13%. In the case of gymnosperm trees, MOE of compression wood is lower than that of normal wood, so that re-orientation efficiency would be increased by 24% without this factor of variations.
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Acknowledgements
This work was supported by the French Ministry of Agriculture and ADEME Agency, though project 61.45.47/00 on physical and mechanical properties of reaction wood.
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Appendices
Appendix 1: Biomechanical model of stem re-orientation in a non-axisymmetric case
General formulation
Let us consider the transverse section of a growing stem. The section geometry is given by its radius R(θ) in an arbitrary reference system (O, X, Y). Let E(r,θ) denote the longitudinal MOE of the wood at any position in the section. Let δR(θ) denote the thickness of the newly formed wood layer. Because of the process of maturation, a tendency to strain α(θ) is induced in the new wood layer. This strain is partly restrained by the geometric compatibility of the whole section, and results in a strain field ɛ(r, θ) and a stress field σ(r, θ). In the context of beam theory, we assume that all sections remain plane, so that the strain field is described by the strain at the center of the coordinate system e, and the variation of curvature around the two axes, C X and C Y .
Our objective is to compute C X , as a function of the data R, E, δR and α.
Inside the layer of maturing wood, the stress and strains are related by:
Inside mature wood, no maturation strain is induced, so that:
In order to concentrate on the effect of wood maturation only, we will assume that the variation of weight of the upper part of the stem is negligible. Then, the section is in static equilibrium if the resultant normal force and bending moments are null. These are calculated by integrating the stress and its first-order moments (relative to the X and Y axes) over the section (denoted S):
Introducing Eqs. 2 and 3 into 4, we can rewrite this equation so that the terms relative to the actual strain ɛ are gathered on one side and the term relative to the induced strain α in the new wood layer (denoted S′) are gathered on the other side:
Introducing Eq. 1 into Eq. 5, it is seen that the macroscopic deformations (e, C X, C Y ) leading to an equilibrium solution are solutions of the following linear system:
Where:
The parameters macroscopic load N, bending moments M X and M Y , and rigidities K ij are functions of the data. Values of e, C X and C Y are deduced by inversion of the linear system.
Discrete formulation
In a practical case, parameters of Eq. 6 (N, M X , M Y and the K ij ) can be numerically computed using a discrete formulation. Let us assume that the section is divided into n angular sectors. Any sector i is characterized by its radius R i , its MOE E i , the thickness of the new wood layer δR i , maturation strain α i , and its limit angles \(\theta _i^+\) and \(\theta _\text{i}^-\) (Fig. 5).
The rigidity terms K ij are computed as:
with generic geometrical terms \(G_{ab}^i (R)\) defined as:
The normal load N, bending moments M X and M Y are computed as:
For a given sector of radius R, with limit angles θ− and θ+, the generic geometrical terms integrates as:
Appendix 2: Longitudinal extrapolation
The previous model can be applied to a stem section located at any height in a tree. Making some simplifying assumptions, it is possible to extrapolate the results obtained for a particular section, and to estimate the total re-orientation of the tree.
Let us assume that the stem has a conical shape. Then, its diameter D at height H can be given as a linear function of its diameter at the base D 0 :
Let us assume that the circumferential distribution of induced strains and MOE is uniform along the tree stem. It can be shown from Eq. 6 that the variation of curvature ΔC at height h is roughly proportional to that at the base ΔC0:
Assuming that the new wood layer has a constant thickness, we have:
The variation of angle ΔΦ at height H is the integral of the variation of curvature along the stem:
The total height of the tree Htot is: Htot=D0/k.
Then, at a fraction f of the total height, the variation of angle is:
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Alméras, T., Thibaut, A. & Gril, J. Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees. Trees 19, 457–467 (2005). https://doi.org/10.1007/s00468-005-0407-6
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DOI: https://doi.org/10.1007/s00468-005-0407-6