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

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.

Introduction

Trees must maintain the vertical orientation of their stem against mechanical perturbations due to physical or biological events. Long-term deviation of the stem from vertical can be due to anchorage failure (caused by soil instability or storm), wood damage (caused by wind forces or snow weight), or to permanent loads (self-weight in relation to growth, external loads such as contact with other plants, weight of epiphytes, or dead trees) (Speck et al. 1990). The effect of such perturbations can be reduced by mechanical optimization of the stem, i.e. adaptation of its dimensions and mechanical properties of the wood (Niklas 1993, 1994, 1995). However, it cannot be completely compensated. Because of the instability of the vertical orientation, a small deviation from this orientation would always be further accentuated by self-weight increments (Morgan and Cannel 1987; Alméras et al. 2002). Therefore, a complementary mechanism is necessary in order to correct minor changes in orientation and thus ensure long-term stability (Archer 1986; Fournier et al. 1991a, 1991b).

This is achieved through circumferential variations in cambial activity. Indeed, when a stem is bent or tilted, it produces reaction wood on one side of the stem, in order to restore an optimal shape (Sinnot 1952; Fisher and Stevenson 1981; Wilson and Gartner 1996). Reaction wood differs from normal wood by its mechanical state of pre-stressing, achieved at the end of cell maturation. Maturation strains can be observed through releasing internal stresses in the peripheral wood layers of a living stem (Fournier et al. 1994; Yoshida and Okuyama 2002). In angiosperms, reaction wood is usually located in the upper part of the stem, characterized by high values of tension and called tension wood. In gymnosperms, it is located in the lower part of the stem, characterized by high values of compression and called compression wood. The production of reaction wood is often associated with eccentric cambial growth (Robards 1965, 1966; Yoshizawa et al. 1986) and circumferential variations of wood mechanical properties.

The circumferential heterogeneity of maturation stress induces a change in the curvature of the stem, and, consequently, its re-orientation (Archer 1986). The difference in maturation strain between normal and reaction wood is identified as the main cause of the circumferential heterogeneity of maturation stress. However, two other factors are recognized to play a role. The modulus of elasticity of reaction wood is expected to differ from that of normal wood. As the maturation stress is a function of the induced strain and the modulus of elasticity, these differences also influence the efficiency of re-orientations. In addition, eccentric growth associated with the production of reaction wood generates circumferential variations in the width of the new wood layers. This creates an asymmetry in the area on which maturation stress is integrated, and also influences the efficiency of re-orientations.

The relationship between the amount and quality of reaction wood and the efficiency of re-orientations can be simulated by biomechanical models (Fournier et al. 1991a, 1991b; Alteyrac et al. 1999; Jirasek et al. 2000; Alméras et al. 2004). For convenience, the developed models often assumed axisymmetry of cambial growth or material properties, thus neglecting the effect of eccentricity and variations of the modulus of elasticity. The present work aims to test the effect of these variations. A mechanical formulation accounting for these variations is proposed. This model is used with data measured on various tree species, in order to quantify the efficiency of their re-orientations and to analyze the effect the three factors of asymmetry promoting re-orientation.

Materials and methods

Principles of the biomechanical model

The biomechanical model can be briefly introduced as follows. During their maturation, wood fibers are submitted to induced longitudinal strains, i.e. a spontaneous tendency to shrink or swell in the longitudinal direction. Because the new wood layer sticks to the rigid core of already mature wood, this deformation is almost entirely blocked. This results in a state of stress inside the new layer: a compression stress if the induced strain was a swelling (case of compression wood), a tension stress if the induced strain was a shrinkage (case of normal or tension wood). The magnitude of maturation stress is the product of the induced strain and the modulus of elasticity of the wood.

The deformations of the stem due to the maturation can be computed in the context of beam theory. It depends on the distribution of maturation stress within the section. The sum of all stresses results in a macroscopic force N normal to the section. Its asymmetric distribution results in bending moments M _X and M _Y around the two axes of the plane of the section (Fig. 1). These actions induce macroscopic deformations of the stem: a small extension or shortening in the longitudinal direction, and variation of curvature around both axes. The relationship between macroscopic efforts and deformations depends on the rigidity of the whole stem section. A complete mathematical formulation of the mechanical problem and its resolution is given in Appendix 1.

Fig. 1

Normal resultant (N) and bending moments (M _X , M _Y ) of a stem section submitted to a non-axisymmetric field of maturation stress σ(r,θ)

Based on this model, three kinds of data are needed to compute the variation of curvature of the stem due to maturation: the distribution of induced strain inside the new layer, the section geometry (radius and thickness of the new layer) and the distribution of the modulus of elasticity. These data must be given at various positions around the circumference.

Material

Trees of various species coming from different geographical locations were selected for the study. A large sample of severely tilted trees was initially chosen, so that they were presumably in a stage of active re-orientation. Growth stress indicators were measured, and only trees exhibiting contrasting values were kept. Their diameter at breast height ranged between 20 cm and 40 cm.

A first sample of 11 angiosperm trees from different botanical families was collected in French Guyana during September 2001. This sample was extended with three different hybrids of poplar, collected in Spain and China, and finally completed by collecting data from three gymnosperm trees growing in the south of France. A complete list of species and their botanical families is given in Table 1.

Table 1 Main characteristics of the measured trees: diameter at breast height (DBH in cm), dry density (10³ kg m⁻³), modulus of elasticity of green wood (MOE in GPa), growth strain indicator (GSI in μm), for normal wood (NW) and reaction wood (RW); ring width ratio (reaction wood side/normal wood side)

Most species are represented by a single tree. This study does not pretend to characterize the reaction of each species, but was designed to represent the diversity of reactions among re-orienting adult trees.

Field observations and measurements

For each standing tree, the diameter at breast height was recorded, and the circumference was divided into eight equal sectors. At the middle of each sector, a growth stress indicator (GSI) was measured at the stem periphery using the single-hole method (Fournier et al. 1994). This method measures the relative displacement of two pins inserted in the wood after releasing longitudinal growth stress in the outermost wood layer. A positive displacement, resulting from the shortening of fibers, corresponds to the release of a tensile stress.

The trees were then felled. A disc of wood was sawn at the height where GSI measurements were performed to observe and quantify eccentricity. The distance (R _i ) between the pith and the cambium was recorded at each angular position i. The thickness of the last annual ring (δR _i ) was measured at the same eight positions for temperate trees. In the case of tropical trees where annual rings could not be observed, “false rings” were used instead. These continuous anatomical features were produced at the same time, thus allowing the eccentricity to be quantified.

Laboratory measurements

The part of the stem just below measurement points was taken to the laboratory for mechanical testing. A green rod (dimensions: 500 mm in the longitudinal direction of wood fibers, 25 mm in the radial and tangential directions) was sawn below each GSI measurement point. The longitudinal modulus of elasticity of green wood (E _i ) was measured using a vibration test (Brancheriau and Baillères 2002). A prismatic beam of wood lying on elastic supports is hit with a hammer, while resulting acoustic vibrations are recorded with a microphone. The modes of natural vibration are related with elastic properties, green density and geometry of the beam. The longitudinal MOE can be deduced using a formula that takes into account the effect of shear (Bordonné 1989). The dry density was estimated by dividing the mass by the volumes in air-dry state (about 12% moisture content). Other technological properties of wood, not used in this paper, were measured on the same sample and are detailed in a separate paper (Thibaut et al. 2003).

Computing total variation of curvature

To simulate and compare the re-orientation of each tree, the biomechanical model detailed in Appendix 1 was applied for an elementary growth ring. The macroscopic deformation of the stem due to the maturation of the growth ring was computed using a discrete formulation in eight equal angular sectors. The following data were used to characterize each of the eight sectors: the radius (R _i ); the thickness of an elementary growth layer (δR _i ); the modulus of elasticity (E _i ), assumed homogeneous on each sector and set at the value measured on the periphery; the maturation strain in the growth layer (α_i), set proportional to the measured GSI using an approximate conversion factor α=−12.10⁻⁶×GSI (Fournier et al. 1994). The thickness of the elementary growth layer (δR _i ) was standardized by setting it proportional to the measured ring thickness so that the mean δR _i was an arbitrary 5% of the mean radius. This standardization aims at canceling any specific effect of the ring width, so that results can be compared between trees, for a given growth increment.

The reference system was chosen so that the center was the pith and the Y-axis was the direction of the peak of reaction wood (corresponding to the direction of lean). The efficiency of re-orientation was quantified by the variation of curvature around the X-axis, further denoted as ΔC. The curvature of a stem portion is the difference between the angle at its two ends, divided by its length.

Computing isolated effects

Three other sets of simulations were performed to quantify the isolated effect of the circumferential heterogeneity of each factor (radial growth, modulus of elasticity and maturation strains). In each case, only one factor was kept heterogeneous in the growth ring. The effect of other factors was cancelled by replacing δR _i and/or E _i by their mean values within the ring, and/or by replacing α _i by the mean value of maturation strains of normal wood. Normal wood is defined as the wood with maturation strain lower than 1/3 of the maximal value obtained around the circumference.

The variation of curvature subsequent to the isolated effect of the heterogeneity of radial growth, modulus of elasticity and maturation strains are denoted respectively ΔC _R , ΔC _E ,ΔC _α .

Computing residual effects

Isolated effects quantify the action of each factor in a situation where the action of other factors is ignored. However, the effects of factors are not independent of each other, so that the total re-orientation is not equal to the sum of all isolated effects. The “residual effect” was defined to take into account the interaction between factors.

First, three additional sets of simulation were performed to quantify the effect of each pair of factors. They were quantified by canceling the heterogeneity of one factor, while keeping the two other factors heterogeneous. The subsequent variations of curvature are denoted ΔC _RE , ΔC _Eα ,ΔC _αR . Then, the residual effect of a given factor was computed as the difference between the total variation of curvature and that obtained after canceling this factor:

$$\begin{array}{@{}c@{}} \Delta C_{\sim R} = \Delta C - \Delta C_{E\alpha} \\ DC_{\sim E} = \Delta C - \Delta C_{\alpha R} \\ \Delta C_{\sim \alpha} = \Delta C - \Delta C_{ER}\\ \end{array} $$

Results

Circumferential variations

A large amount of reaction wood was found in the external wood layers of all trees. This was expected because the trees were severely tilted and at a stage of active re-orientation. In gymnosperm trees, reaction wood was located on the downward side of the stem section. In angiosperm trees, it was located on the upper side. The presence of reaction wood was generally correlated with variations of the other measured properties. Figure 2 shows the circumferential variations of the distance to the pith (R), growth strain indicator (GSI) and modulus of elasticity (MOE) for some representative trees.

Fig. 2

Circumferential variations in the distance to the pith (R), growth strain indicator (GSI) and modulus of elasticity (MOE) for some representative trees; variables were standardized between −1 and 1 by dividing by the maximal absolute value, in order to facilitate comparison between trees. The angular positions are represented in the abscissa. Position 0° is the position of the peak of reaction wood. Position 180° (or −180°) is the opposite side

The three gymnosperm trees had the same typical pattern (e.g. Pinus pinaster and Picea abies, Fig. 2). Strongly negative values of GSI (corresponding to compression stress) were observed on the side with compression wood, and weakly positive values (corresponding to tension stress) were found on the opposite side. Eccentric growth was clearly observed in the direction of compression wood. The MOE was always lower on the side with compression wood than on the opposite side.

All angiosperm trees showed large positive values of GSI (i.e. shrinkage) on the side of tension wood, and weaker positive values on the opposite side. Eccentric growth was observed on the side of tension wood for most trees (e.g. S. amara and V. surinamensis), and unclear for some of them (e.g. O. guyanensis and Populus hybrid I-214). Different patterns were observed for the modulus of elasticity. For many trees (e.g. O. guyanensis), it was found to be higher in tension wood than in normal wood. This was expected from the high cellulose content and low microfibril angle usually observed in tension wood. For some trees, the pattern of variation was not so clear (ex. Populus hybrid I2). For two trees (e.g. V. surinamenesis), the pattern was reversed with lower MOE in tension wood than in normal wood.

Considering the heterogeneity of maturation strains, each sector was classified into “reaction wood” (|GSI/GSI_max|>2/3) or “normal wood” (|GSI/GSI_max|<1/3). Mean values of properties for reaction and normal wood are indicated for all trees in Table 1. Figure 3 shows a representation of the properties of reaction and normal wood of each tree. This representation allows groups of trees with a similar pattern of wood properties to be defined:

Fig. 3

Growth strain indicator (GSI) and modulus of elasticity (MOE) of all trees, for normal wood (circles), compression wood (triangles) and tension wood (squares). Lines join the normal and reaction wood of the same tree

• Gymnosperm trees are characterized by the presence of compression wood. They also had low values of MOE and GSI for normal wood, when compared to angiosperm trees.

• Angiosperm trees with highest MOE for normal wood (E. decolorens, M. fragilis) had higher MOE for tension wood.

• Angiosperm trees with low MOE (Populus hybrids, J. copaia, S. amara) had similar MOE values for normal and tension wood (with the exception of Populus hybrid I-69).

• Angiosperm trees with intermediate MOE for normal wood had the highest values of GSI in tension wood. Some of them (L. procera, E. falcata, O. guyanensis, C. procera, Q. rosa) had MOE higher for tension wood than for normal wood. The two other trees (C. sciadophylla, V. surinamensis) had lower MOE in tension wood.

Simulated re-orientations

Table 2 shows the simulated variation of curvature due to the maturation of an elementary growth ring. The variation of curvature differs between trees, ranging between 0.65×10⁻³ rad/m and 3.76×10⁻³ rad/m. This large variability is a consequence of the combined variability of the three biomechanical factors. The isolated and residual effects are indicated in Table 2 as a percentage of the total variation of curvature.

Table 2 Simulated variation of curvature (ΔC); isolated and residual effect of circumferential variations of radial growth (R), modulus of elasticity (E) and maturation strains (α) expressed as a percentage of ΔC

Isolated effect of each factor

For angiosperms, the mean isolated effects show that there is a clear hierarchy between factors: asymmetry of maturation strain is pre-eminent on eccentricity, which is pre-eminent on the asymmetry of MOE. Asymmetry of maturation strain alone allows 38–80% of the total re-orientation (57% in average). Eccentricity allows on average 9% of total re-orientation. It is slightly negative for some trees whose direction of eccentricity is not clear. This effect is subject to important variations: in one case (C. sciadophylla), it allows 25% of the total effect. Asymmetry of MOE allows on average 2% of the total re-orientation, and does not exceed 7%. Weak negative values were obtained for the trees that have normal wood stiffer than tension wood.

For gymnosperms, most of the re-orientation is due to the effect of asymmetry of maturation strain. Once isolated, this factor allows on average 90% of the total re-orientation. In one case, it exceeds 100%, meaning that this factor is more efficient when it is isolated than when it is combined with other factors. The effect of MOE asymmetry does not exceed 2%. The isolated effect of eccentricity is negative, ranging between −16% and −34%. These negative values are related to the change in sign of maturation strain between normal and compression wood. If normal wood was produced on all sides, the observed direction of eccentricity would induce negative curvature, i.e. accentuated bending.

Non-additivity and residual effects

Results of isolated effects clearly show the non-additivity of the factors. Indeed, the sum of all isolated effects is always below 100%, meaning that factors are more efficient when combined to each other than when isolated.

Residual effects provide complementary information about the effect of a factor, integrating the interactions with other factors. According to this criterion, the effect of maturation strains remains largely pre-eminent for both angiosperms and gymnosperms. However, the effect of eccentricity is larger than when considering its isolated effect. Canceling this factor decreases mean total re-orientations by 26% in the case of gymnosperms and by 31% in the case of angiosperms (up to more than 45% for some trees). The residual effect of MOE asymmetry for angiosperms also appears to be more important than when considering isolated effects. Its suppression decreases re-orientations by 13% on average, and in some cases by more than 30%. For gymnosperms, this factor has a non-negligible negative effect (−24% in average). Indeed, the MOE of compression wood is lower than that of normal wood, so that suppressing the variation of this factor would lead to increased re-orientation.

Discussion and conclusion

Re-orientation induced by circumferential heterogeneity of wood properties

The variations of curvature were quantified for an arbitrary 5% increment of radius. The computed values can be considered as an indicator of the biomechanical reactivity of the trees, i.e. their efficiency to recover verticality once tilted. The possible dependence on the magnitude of the stimulus or on a size effect was limited by the use of trees with similar size and degree of tilting. The differences in computed re-orientations can be reasonably attributed to variations in genetic aptitude to active re-orientation.

These results are computed as a local variation of curvature for a given section of the tree. They were extrapolated to provide a practical quantification of the efficiency of this mechanism. The predicted change in orientation of the whole tree was computed using a simple model whose equations are detailed in Appendix 2. This model is based on the following main assumptions: stem is uniformly tapered (i.e. conical) with taper coefficient k, the thickness of the new wood layer is uniform along the stem, circumferential variations of induced strain and MOE are uniform along the stem.

Then, the variation of the stem angle at a fraction f of the total height is: ΔΦ_f=ΔC₀×D₀/[k(1−f)]. This value was computed for all trees at 90% of the total height (f=0.9) assuming k=0.015 m/m (Table 2). It ranges from 2.5° to 21.1°, with a mean value 10.6° for angiosperms and 6.2° for gymnosperms. Although this estimation is based on very simplified assumptions, it gives a reasonable order of magnitude of change in orientation due to a 5% radial growth with active production of reaction wood. Mean values show that the mechanism of active re-orientation allows substantial correction in stem orientation.

However, the amount of radial growth needed to correct a given perturbation of stem orientation is highly variable. If an initial perturbation is not corrected quickly enough, it is necessarily accentuated by later growth, thus altering long-term stability of the tree and diminishing its chances to survive and reproduce. The ability to efficiently produce reaction wood then appears as an important biological trait of the tree species.

Data concerning dry density of wood (Table 1) show that reaction wood is often denser than normal wood, especially for gymnosperms. This shows that active re-orientation of the tree implies a supplementary investment in matter. In addition, reaction wood has lower hydraulic efficiency (Spicer and Gartner 1998) and safety (Mayr and Cochard 2003), so that it has a non-negligible physiological cost. This investment is optimized through efficient repartition (eccentricity) and quality of the produced wood, i.e. adaptation of its physical properties (asymmetric induced strain and MOE).

Complementary effects of eccentricity and heterogeneity of MOE

The analysis of individual effects confirmed the generally accepted idea that heterogeneity of maturation strains is the main factor of active re-orientation. The effect of other factors (eccentricity and MOE variations) is lower if not combined with this one. However, the calculation of residual effects showed that these secondary factors have an important complementary role through their interaction with maturation strains. The magnitude of this effect is in some cases surprisingly high.

For a Populus tree (hybrid I-69), the residual effect of variations in MOE accounted for 36% of the re-orientation. The important effect of this factor is found for all trees having contrasted values of MOE between normal and reaction wood (33% for C. procera, 27% for Q. rosa, 26% for O. guyanensis, 17% for E. falcata). For these trees, heterogeneity of MOE has a non-negligible complementary effect. There is a general trend for trees differentiating stiff tension wood to have more efficient re-orientations. Indeed, the mean total variation of curvature for the seven trees with the highest MOE in tension wood (E. decolorens, M. fragilis, E. falcata, C. procera, O. guyanensis, L. procera and Q. rosa) is 2.38×10⁻³ rad/m and the mean for the seven trees with lower MOE in tension wood is 1.83×10^-3 rad/m.

The complementary effect of eccentricity is more pronounced. For 7 of the 14 studied angiosperm trees, its residual effect was more than 30% of total re-orientation. For 3 trees (S. amara, C. sciadophylla, E. decolorens) it was more than 45%. The magnitude of these effects shows that the assumption of axisymmetry in models of stem re-orientation can lead to substantial errors. For example, in the case of Populus hybrid I-69, neglecting both eccentricity and variations in MOE reduced re-orientation to 38% of the total (then, an axisymmetric model would result in underestimating re-orientations by 62%).

The efficiency of re-orientation is strongly increased by concomitant eccentricity. Figure 4 shows the relation between eccentricity (quantified by the ratio of ring width on the tension and opposite side) and total variation of curvature for angiosperms. The two parameters are clearly correlated (r=0.66, P<0.05). Besides its biomechanical efficiency, eccentric growth has the advantage of being realized without any supplementary investment in material. It is simply based on a better location of a given amount of wood, and is then an “economic” solution. Indeed, recent works on buttresses (Clair et al. 2003) showed that they are an extension of such an economic re-orientation mechanism; their mechanical role is not only to sustain the tree, but also to actively re-orient it by the production of reaction wood. This is achieved by an optimal location of reaction wood, rather than by an increase in the amount of reaction wood. For angiosperm trees with strong eccentricity, the isolated effect of eccentricity can reach up to 25% of the total re-orientation. In this case, variation in properties of the wood is not strictly necessary. An important part of re-orientations can be achieved only by quantitative variations of cambial activity.

Fig. 4

Relation between total variation of curvature and ring width ratio (width on the tension wood side/width on the opposite wood side) among angiosperms trees. Coefficient of correlation is r=0.66

In the case of gymnosperms, the MOE variations do not act in synergy with other factors, because compression wood is less stiff than normal wood. If compression wood were stiffer than normal wood (as for angiosperms), the mechanism of re-orientation would be clearly more efficient. One can wonder why the evolution of gymnosperms did not select such a mechanism. Various arguments can be given. The first is related to a physical limitation of wood properties. Indeed, the properties of wood are related to its ultra-structure (Yamamoto and Kojima 2002). Several works showed that, in gymnosperms, the induced strain and MOE of wood mainly depend on the orientation of microfibrils inside the fiber wall. Compression strains are induced if the microfibrillar angle is large, whereas longitudinal stiffness is achieved if it is small. Then, it seems impossible to make fibers with both high stiffness and induced compression. Furthermore, active re-orientation is not the only mechanical role of wood. Indeed, its primary role is to sustain the tree without breaking. Because of its specific ultrastructure, compression wood might be stronger and tougher relative to compression failure. This advantage may be a counterpart to the loss of optimality in the re-orientation mechanism. Finally, a physiological argument can be given in relation to the chemical composition of wood fiber. In compression wood, the lignin content is higher and the cellulose content is lower than in tension wood. A difference in the physiological cost of these elements (lignin being supposed to be cheaper than cellulose) could explain why compression wood is a good trade-off between physiological cost and biomechanical efficiency.

For gymnosperms, eccentricity appears to have as important a complementary effect as for angiosperms. However, as eccentric growth is realized on the downward side, its isolated effect (assuming normal wood on all sides) would be counter-efficient. In such conditions, the production of a wood with specific properties (compression wood) is necessary to achieve efficient re-orientations.

Conclusions

This study aimed at quantifying the respective role of three morphogenetic factors implicated in the control of tree orientation (maturation strains, eccentric growth and asymmetry of the MOE). Maturation strains are clearly the main factor fulfilling this role. Eccentricity was shown to have an important complementary effect. For many angiosperm trees, asymmetry of MOE also has a non-negligible effect, mainly through its interaction with maturation strains. In the case of gymnosperms, the effect of this factor is paradoxical, because MOE is lower in compression wood than in normal wood. These results open up some interesting questions concerning the evolution of the mechanisms of re-orientation, their ecological role and their optimality in relation to other tasks the wood has to fulfill.

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

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 H_tot is: H_tot=D₀/k.

Then, at a fraction f of the total height, the variation of angle is:

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