Skip to main content
Log in

Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees

  • Original Article
  • Published:
Trees Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Alméras T, Gril J, Costes E (2002) Bending of apricot-tree branches under the weight of axillary productions: confrontation of a mechanical model to experimental data. Trees 16:5–15

    Article  Google Scholar 

  • Alméras T, Costes E, Salles JC (2004) Identification of biomechanical factors involved in stem shape variability between apricot tree varieties. Ann Bot 93:1–14

    Article  Google Scholar 

  • Alteyrac J, Fourcaud T, Castera P, Stokes A (1999) Analysis and simulation of stem righting movements in Maritime pine (Pinus pinaster Ait.). In: Connection between silviculture and wood quality through modelling approaches and simulation software, Third Workshop of IUFRO, La Londe-Les-Maures, France, pp 105–112

  • Archer R (1986) Growth stresses and strains in trees. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Bordonné P (1989) Module dynamique et frottement intérieur dans le bois. Mesures sur poutres flottantes en vibrations naturelles. Ph D Thesis, Institut National Polytechnique de Lorraine, Nancy, France (French)

  • Brancheriau L, Baillères H (2002) Natural vibration analysis of clear wooden beams: a theoretical review. Wood Sci Tech 36:347–365

    Article  Google Scholar 

  • Clair B, Fournier M, Prevost MF, Beauchene J, Bardet S (2003) Biomechanics of buttressed trees: bending strains and stresses. Am J Bot 90:1349–1356

    Google Scholar 

  • Fisher J, Stevenson J (1981) Occurrence of reaction wood in branches of dicotyledons and its role in tree architecture. Bot Gaz 142:82–95

    Article  Google Scholar 

  • Fournier M, Chanson B, Guitard D, Thibaut B (1991a) Mechanics of standing trees: modeling a growing structure subjected to continuous and fluctuating loads. 1. Analysis of support stresses (in French). Ann For Sci 48: 513–525

    Google Scholar 

  • Fournier M, Chanson B, Thibaut B, Guitard D (1991b) Mechanics of standing trees: modelling a growing structure subjected to continuous and fluctuating loads. 2. Three-dimensional analysis of maturation stresses in a standard broadleaved tree (in French). Ann For Sci 48: 527–546

    Google Scholar 

  • Fournier M, Chanson B, Thibaut B, Guitard D (1994) Measurement of residual growth strains at the stem surface. Observations of different species. Ann For Sci 51:249–266

    Google Scholar 

  • Jirasek C, Prusinkiewicz P, Moulia B (2000) Integrating biomechanics into developmental models expressed using L-Systems. In: Plant biomechanics, Freiburg-Badenweiler, pp 615–624

  • Mayr A, Cochard H (2003) A new method for vulnerability analysis of small xylem areas reveals that compression wood of Norway spruce has lower hydraulic safety than opposite wood. Plant Cell Environ 26:1365–1371

    Article  Google Scholar 

  • Morgan J, Cannell MJR (1987) Structural analysis of tree trunks and branches: tapered cantilever beams subject to large deflections under complex loading. Tree Physiol 3:365–374

    Google Scholar 

  • Niklas K (1993) The scaling of plant height: A comparison among major plant clades and anatomical grades. Ann Bot 72:165–172

    Article  Google Scholar 

  • Niklas K (1994) The allometry of safety-factors for plant height. Am J Bot 81:345–351

    Google Scholar 

  • Niklas K (1995) Size-dependent allometry of tree height, diameter and trunk-taper. Ann Bot 75:217–227

    Article  Google Scholar 

  • Robards A (1965) Tension wood and eccentric growth in crack willow (Salix fragilis, L.). Ann Bot 29:419–430

    Google Scholar 

  • Robards A (1966) The application of the modified sine rule to tension wood production and eccentric growth on the stem of Crack Willow (Salix fragilis L.). Ann Bot 30:513–523

    Google Scholar 

  • Sinnot E (1952) Reaction wood and the regulation of tree form. Am J Bot 39:69–78

    Google Scholar 

  • Speck T, Spatz H, Vogellehner D (1990) Contributions to the biomechanics of plants. I. Stabilities of plant stems with strengthening elements of different cross-sections against weight and wind forces. Bot Acta 103:111–122

    Google Scholar 

  • Spicer R, Gartner BL (1998) Hydraulic properties of Douglas fir (Pseudotsuga menziesii) branches and branches halves with reference to compression wood. Tree Physiol 18:777–784

    PubMed  Google Scholar 

  • Thibaut B, Gril J, Loup C, Almeras T, Thibaut A, Beauchêne J, Liu S, Badia M, Gachet C, Guitard D (2003) Propriétés physiques et mécaniques des bois de réaction de quelques angiospermes et gymnospermes tempérés et tropicaux. In: Gerardin P (ed) 6èmes Journées Scientifiques de la Forêt et du Bois, Epinal, 3-5 June 2003. GIS Bois Construction Environnement, ARBOLOR , pp 13–16

    Google Scholar 

  • Wilson BF, Gartner BL (1996) Lean in red alder (Alnus rubra): growth stress, tension wood, and righting response. Can J For Res 26:1951–1956

    Google Scholar 

  • Yamamoto H, Kojima Y (2002) Properties of cell wall constituents in relation to longitudinal elasticity of wood. Wood Sci Tech 36:55–74

    Article  Google Scholar 

  • Yoshida M, Okuyama T (2002) Techniques for measuring growth stress. Holzforschung 56:461–467

    Article  Google Scholar 

  • Yoshizawa N, Okamoto Y, Idei T (1986) Righting movement and xylem development in tilted young conifer trees. Wood Fiber Sci 18:579–589

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tancrède Alméras.

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 .

$$\varepsilon \left({r,\theta}\right) = e + C_X r\,\text{sin}\,\theta - C_Y r\,\text{cos}\,\theta $$
(1)

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:

$$\sigma \left({r,\theta}\right) = E\left({r,\theta}\right)\left[{\varepsilon \left({r,\theta}\right) - \alpha \left(\theta \right)}\right]\;\text{for}\;R\left(\theta \right) - \delta R\left(\theta \right) < r < R(\theta ) $$
(2)

Inside mature wood, no maturation strain is induced, so that:

$$\sigma \left({r,\theta}\right) = E(r,\theta )\varepsilon (r,\theta)\;\text{for}\;r < R(\theta) - \delta R(\theta) $$
(3)

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):

$$\begin{array}{*{20}c} {\iint\limits_S {\sigma \left({r,\theta}\right) \cdot r\,dr\,d\theta} = 0}\hfill \\ {\iint\limits_S {r \cdot \text{sin}\,\theta \cdot \sigma \left({r,\theta}\right) \cdot r\,dr\,d\theta} = 0}\hfill \\ {\iint\limits_S {r \cdot \text{cos}\,\theta \cdot \sigma \left({r,\theta}\right) \cdot r\,dr\,d\theta} = 0}\hfill \\ \end{array} $$
(4)

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:

$$\begin{array}{*{20}c} {\iint\limits_S {E\left({r,\theta}\right) \cdot \varepsilon \left({r,\theta}\right) \cdot r\,dr\,d\theta} = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right) \cdot r\,dr\,d\theta}}\hfill \\ {\iint\limits_S {E\left({r,\theta}\right) \cdot \varepsilon \left({r,\theta}\right) \cdot r^2 \cdot \text{sin}\,\theta \,dr\,d\theta} = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right) \cdot r^2 \cdot \text{sin}\,\theta \,dr\,d\theta}}\hfill \\ {\iint\limits_S {E\left({r,\theta}\right) \cdot \varepsilon \left({r,\theta}\right) \cdot r^2 \cdot \cos \,\theta \,dr\,d\theta} = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right) \cdot r^2 \cdot \text{cos}\,\theta \,dr\,d\theta}}\hfill \\ \end{array} $$
(5)

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:

$$\left[{\begin{array}{*{20}c} {K_{00}} &{K_{01}} &{K_{10}}\\ {K_{10}} &{K_{11}} &{K_{20}}\\ {K_{01}} &{K_{02}} &{K_{11}}\\ \end{array}}\right]\left[{\begin{array}{*{20}c} e \\ {C_X}\\ { - C_Y}\\ \end{array}}\right] = \left[{\begin{array}{*{20}c} N \\ {M_X}\\ { - M_Y}\\ \end{array}}\right] $$
(6)

Where:

$$\begin{array}{*{20}c} {N = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right)\,dr\,d\theta}}\hfill \\ {M_X = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right) \cdot r^2 \cdot \text{sin}\,\theta \,dr\,d\theta} - M_Y = \iint\limits_{S'} {E\left({r,\theta}\right) \cdot \alpha \left(\theta \right) \cdot r^2 \cdot \text{cos}\,\theta \,dr\,d\theta}}\hfill \\ {K_{00} = \iint\limits_S {E\left({r,\theta}\right) \cdot r\,dr\,d\theta \quad \quad \quad \quad}\quad \quad \;K_{11} = \iint\limits_S {E(r,\theta) \cdot r^3 \cdot \cos \,\theta \cdot \sin \,\theta \,dr\,d\theta}}\hfill \\ {K_{01} = \iint\limits_S {E\left({r,\theta}\right) \cdot r^2 \cdot \cos \,\theta \,dr\,d\theta}\quad \quad \quad \;K_{10} = \iint\limits_S {E\left({r,\theta}\right) \cdot r^2 \cdot \text{sin}\,\theta \,dr\,d\theta}}\hfill \\ {K_{02} = \iint\limits_S {E\left({r,\theta}\right) \cdot r^3 \cdot \text{cos}^2 \,\theta \,dr\,d\theta}\quad \quad \quad K_{20} = \iint\limits_S {E\left({r,\theta}\right) \cdot r^3 \cdot \sin ^2 \,\theta \,dr\,d\theta}}\hfill \\ \end{array} $$

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).

Fig. 5
figure 5

Discrete representation of eccentric growth in the (X Y) plane of the stem section. For sector i, delimited by angles \(\theta _i^-\) and \(\theta _i^+\), R i is the distance to the pith and δR i is the thickness of the growth ring

The rigidity terms K ij are computed as:

$$\begin{gathered} K_{00} = \sum\limits_i {E_i G_{_{00}}^i \left({R_i}\right)\quad}\;\,K_{11} = \sum\limits_i {E_i G^i _{11}\left({R_i}\right)}\hfill \\ K_{10} = \sum\limits_i {E_i G^i _{10}\left({R_i}\right)}\quad \,K_{01} = \sum\limits_i {E_i G^i _{01}\left({R_i}\right)}\hfill \\ K_{20} = \sum\limits_i {E_i G^i _{20}\left({R_i}\right)}\quad K_{02} = \sum\limits_i {E_i G^i _{02}\left({R_i}\right)}\hfill \\ \end{gathered} $$

with generic geometrical terms \(G_{ab}^i (R)\) defined as:

$$G_{_{ab}}^i \left(R \right) = \int\limits_0^R {\int\limits_{\theta _i^ -}^{\theta _i ^ +} {r^{a + b}\cdot \sin ^a \theta \cdot \text{cos}^b \,\theta \cdot r\,d\theta \,dr}} $$

The normal load N, bending moments M X and M Y are computed as:

$$\begin{array}{*{20}c} {N = \sum\limits_i {\alpha _i E_i \left({G_{00}^i \left({R_i}\right) - G_{_{00}}^i \left({R_i - \delta R_i}\right)}\right)}}\hfill \\ {M_X = \sum\limits_i {\alpha _i E_i \left({G_{01}^i \left({R_i}\right) - G_{01}^i \left({R_i - \delta R_i}\right)}\right)}}\hfill \\ { - M_Y = \sum\limits_i {\alpha _i E_i \left({G_{10}^i \left({R_i}\right) - G_{10}^i \left({R_i - \delta R_i}\right)}\right)}}\hfill \\ \end{array} $$

For a given sector of radius R, with limit angles θ and θ+, the generic geometrical terms integrates as:

$$\begin{gathered} G_{00} = \frac{{R^2}} {2}\left({\theta ^ + - \theta ^ -}\right) \hfill \\ G_{01} = \frac{{R^3}} {3}\left({\text{sin}\,\theta ^ + - \text{sin}\,\theta ^ -}\right) \hfill \\ G_{10} = \frac{{R^3}} {3}\left({\text{cos}\,\theta ^ - - \text{cos}\,\theta ^ +}\right) \hfill \\ G_{11} = \frac{{R^4}} {4}\left({\frac{{\text{cos}\left({2\theta ^ -}\right) - \text{cos}\left({2\theta ^ +}\right)}} {4}}\right) \hfill \\ G_{02} = \frac{{R^4}} {4}\left({\frac{{\theta ^ + - \theta ^ -}} {2} + \frac{{\text{sin}\left({2\theta ^ +}\right) - \text{sin}\left({2\theta ^ -}\right)}} {4}}\right) \hfill \\ G_{20} = \frac{{R^4}} {4}\left({\frac{{\theta ^ + - \theta ^ -}} {2} - \frac{{\text{sin}\left({2\theta ^ +}\right) - \text{sin}\left({2\theta ^ -}\right)}} {4}}\right) \hfill \\ \end{gathered} $$

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 :

$$ D(H) = D_0 - kH\quad \text{where}\,k\,\text{is}\,\text{a}\,\text{taper}\,\text{coefficient} $$

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:

$$\Delta C\left(H \right) \approx \Delta C_0 \times \delta R\left(H \right)/\delta R_0 \times D_0^2 /D^2 \left(H \right) $$

Assuming that the new wood layer has a constant thickness, we have:

$$\Delta C(H) = \Delta C_0 \times D_0^2 /\left({D_0 - kH}\right)^2 $$

The variation of angle ΔΦ at height H is the integral of the variation of curvature along the stem:

$$\Delta \Phi \left(H \right) = \Delta C_0 D_0^2 \int_0^H {\frac{1} {{\left({D_0 - kh}\right)^2}}dh =}\Delta C_0 D_0 H/\left({D_0 - kH}\right) $$

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:

$$\Delta \Phi \left({fH_{\text{tot}}}\right) = \Delta C_0 \times D_0 /\left({k\left({1 - f}\right)}\right) $$

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00468-005-0407-6

Keywords

Navigation