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Tropical Biology |
2UMR, Ecologie des Forêts de Guyane, CIRAD ENGREF INRA, Campus Agronomique, BP 709, 97310 Kourou, Guyane Française, France; 3UMR, Botanique et Bioinformatique de l'architecture des plantes AMAP, IRD Centre de Cayenne, BP 165, 97323 Cayenne, Guyane Française, France; 4Laboratoire de Mécanique et de Génie Civil, UMR 5508 CNRS Université de Montpellier II, cc 081, Bat 13, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France
Received for publication November 12, 2002. Accepted for publication April 4, 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Key Words: biomechanics • buttress • Eleaocarpaceae • French Guiana • Sloanea spp • tropical trees • wood
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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). A number of different biological theories have tried to explain the function and the formation of buttresses (Kaufman, 1988
; Richards, 1995
). Some authors assumed that physiological stresses force aerial development of shallow root systems (Halle et al., 1978
). Others assumed that buttresses prevent climbing by woody vines or reduce competition with other trees by occupying available space (Black and Harper, 1979
). Lastly, as the term "buttress" suggests, there is a general consensus that these structures are of mechanical importance (Kaufman, 1988
; Mattheck, 1991
; Ennos, 1993
, 1995
; Young and Perkocha, 1994
; Chapman et al., 1998
). However the actual function has not been empirically demonstrated, and such theories assume that buttresses are of adaptive value in resisting wind or gravity forces or are formed via mechanistic constraints based on hypotheses of stimulated growth in mechanically stressed zones of the trunk base.
There is only limited experimental and theoretical work on how buttresses act mechanically. Mattheck's group (Mattheck, 1990
, 1991
; Mattheck and Prinz, 1991
) assumed that cambial growth is activated by stress and simulated stresses induced by a bending moment at the stem base via a finite element model. By an incremental process that was supposed to simulate peripheral secondary growth in the tree (Mattheck, 1990
), material was added step by step at the wood surface, where mechanical stresses are greater. Mechanical stresses were analyzed at each step, until they became constant all around the trunk and buttress surfaces, under different testing boundary conditions that simulated anchorage between buttresses and underground soil or roots. The forms predicted by Mattheck's model were finally compared to existing ones in nature. Mattheck's model can explain many patterns of buttress form and development observed by ecologists (Ennos, 1993
). Although some of the assumptions from such approaches are questionable, such types of mechanical hypotheses need to be validated by experimental measurements of geometries, material properties, and strains.
Ennos (1995)
and Crook et al. (1997)
discussed Mattheck's theory with simulated windthrow tests and morphological observations of roots after working on Malaysian trees with and without buttresses. They observed root movements and trunk displacement, strains along buttresses, trunk and anchorage strength, and crack modes on both the leeward and windward sides.
In this paper, we present experimental results based on a typical buttressed genus in the lowland tropical rainforest of French Guiana. The aim was to localize the risks of failure by directly measuring strains and to discuss Mattheck's constant stress hypothesis in the light of these results. We also tested the ability of a simple beam model to describe strain patterns under external bending forces.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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, p. 395). This genus is represented by 17 species in French Guiana (Hollowel et al., 2001
), 60–70 species in tropical America, and 120 species in the world, all of which are tropical in distribution (Gentry, 1993
; Mabberley, 1997
). The two trees used in the this experiment were of the same size (20 cm DBH) and located in rainforest in the "Piste St. Elie" field station (53°0' W–5°20' N). Both trees had three well-developed and thin buttresses. However, in the first tree, buttresses are straight, approximately triangular "plates." In the second one, the buttress flanges have lateral curvatures. Specimens have been donated to the international herbarium of Cayenne. The first tree has been identified as Sloanea cf. tuerckheimii J. D. Smith; the second one is a different species that has not yet been identified.
Field experiments
Local strains were measured at the periphery of standing trees. The strain sensors were attached on peripheral wood, after removing the thin bark, in order to measure wood behavior because wood is the main support tissue. In buttresses, strains were measured with strain gauge sensors (DD1 type; Hottinger Baldwin Messtechnik, Darmstadt, Germany) connected in full bridge mode to a battery-powered strain bridge (Alco system, Captels, S. Matthieu de Treviers, France). The sensors were fitted with steel pins, spaced 13 mm apart. The accuracy of this system of measurement is estimated at 30 microstrains (i.e., 30 µm/m).
The bending experiment is illustrated in Fig. 1. The trunk was pulled with a manual winch, using a steel cable anchored to the base of another tree nearby. The applied force was measured with a field load cell (Captels). Forces were applied at a height of 5 m in five stages. The main aim during testing was to remain within the intact structural and linearly elastic material properties; strains were measured at different locations along the ridges and on the sides of the three buttresses by moving the sensors between successive experiments. This approach assumed and checked that the behavior was linear and elastic, so that experiments were repeatable. A mechanical extensometer (Fournier et al., 1994
) was fixed on the trunk just above the buttresses and was not moved between successive tests. This ensured repeatability while defining the maximal applied loads. A strain control was chosen with a maximum of –1000 microstrains in the compressed wood of the trunk above the buttresses. After checking the linearity of each test, the slopes 
L/F, T/F, 45/F (L represents strain along the grain or longitudinal [L] strain, T represents strain perpendicular to the grain or tangential [T] strain, 45 represents strain parallel to the diagonal between L and T direction, and F represents applied force) were computed from all five levels of applied force and analyzed for each measurement. According to the definition of the strain tensor and mathematical formulas for axes changes (Berthelot, 1999
), the shear strain LT/F was calculated as
 | (1) |
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On the second tree, the force direction was also parallel to one buttress, put in compression. We measured longitudinal strains along the ridge of this buttress and shear strains at two opposite points of the lateral sides. In a final test, we increased the load until the buttress broke and observed how and where the break was initiated.
Model of bending strains
A simple model for longitudinal bending strains has been designed at the cross section level based on beam theory (strength of materials) that is often used to compute stresses and strains in slender plant axes (Niklas, 1992
). The model (see Appendix) uses the following data: the two components (vertical parallel to the trunk and horizontal) of the applied force, the lever arm (vertical distance from the cross section to the applied force), the geometrical characteristics of the cross section (the cross sectional area and the second moments of area that are the relevant geometrical variables to predict flexural stiffness at the cross section level [Niklas, 1992
]), and wood stiffness (i.e., Young's modulus of elasticity of green wood as measured parallel to the grain).
As the model predicts that the longitudinal strain L is proportional to the applied force F and to the modulus of elasticity EL, we will compare theoretical and experimental results by analyzing linear regressions between experimental slopes L/F and theoretical values of L/ELF. The slopes of these regressions are estimations of the modulus of elasticity EL.
Additional data
At the end of the field experiment, the tree was felled above the buttresses. To characterize cross section geometries, slices 10 cm thick with parallel and horizontal surfaces were carefully sawn from the buttresses. In the laboratory, each surface was photographed and analyzed with the software Optimas 6.5 (Media Cybernetics, Silver Spring, Maryland, USA) to define the outlines and compute (standard functions) for each height x, the center of mass position, the area S(x), and the second moments of area Iy, Iz(x) and Iyz(x).
Wood basic density (ratio of oven-dried mass to saturated volume) was measured within buttress A (seven heights, three radial positions for each height, two samples for each position), as dried wood density is a good predictor of elastic properties (Kollman and Cote, 1968
; Guitard, 1987
; Niklas, 1992
). Moreover, we adjusted the measured values of basic density to estimate air-dried (i.e., wood moisture content of 12%) density: drying shrinkage is supposed to account for 0.6% of volume per percentage of moisture content loss (CIRAD Forêt database on tropical woods, properties of Sloanea spp., J. Gérard, CIRAD Forêt, Montpellier, France, personal communication) and therefore air-dried density is 1.25 times basic density. We then calculated the modulus of elasticity ELad of air-dried wood, from Guitard's regressions (Guitard, 1987
). The green wood assessment 0.73 ELad (Kollman and Cote, 1968
) were compared to the mentioned slopes of regression. Lastly, Young's moduli have been measured by tensile tests on air-dried wood samples by the method still used by Clair et al. (2003)
and compared to other estimations.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Strain and force magnitudes
During all the bending experiments, measured strains in the different parts of buttresses remained between –3000 microstrains and 2000 microstrains along the grain, –5000 microstrains and 1200 microstrains in the tangential direction, below green wood elastic limit, but often much higher than the controlled maximal strain in the trunk of –1000 microstrains. The maximum bending moments about the base of the tree were 23 500 Nm in the first set and 20 000 Nm in the second. Maximal forces were about 5000 N.
As expected, longitudinal strains, which are the most intuitively predictable, were positive (stretching) on buttresses under tension, compressive (shrinking) in the opposite ones. When the force direction was reversed (second set of tests), strains were obviously of opposite sign, but of the same magnitude, along the ridges (Fig. 2) and on the buttress sides.
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L/FL cos 
and LT/FL cos , had the same scale of magnitude, 0.022 microstrains/Nm and 0.017 microstrains/Nm, respectively. Tangential strains were slightly lower (mean value 0.013 microstrains/Nm). Longitudinal and tangential strains were opposite and had a significant negative correlation (r2 = 0.30), with a slope of –0.64 (Fig. 3). When the data were split, the correlation was greater (r2 = 0.53) when keeping only the points closest to the ridge (<10 cm), but it was not significant when these points were removed. Although an exhaustive study of shear strains was not performed in every part of the three buttresses, smaller values for all strain components have been measured in buttresses B and C not parallel to the applied force. Therefore, the following discussion is mainly focused on observations of buttress A.
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T/L was –1.5 on the concave side, 1 on the opposite one.
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L/F) and theoretical ones L/ELF, computed from the model and independently measured geometrical data, were calculated for different samples (by splitting the whole set of data into different zones) (Table 1). In a general sense, the model and experimental values fit well, with an estimation of the mean modulus of elasticity of 16 400 MPa, with no significant difference between tension and compression. An example of the evolution of both modeled and experimental strains around the circumference at a given height is given in Fig. 8. However, the correlation coefficient and the modulus of elasticity fit for each cross section increase with height (Table 1). Without great changes of modulus of elasticity from the buttress top to the base, the model does not properly describe developments of strains with height, as indicated by the observed bias on regressions split by ridge or trunk positions. On the same longitudinal line, theoretical strains decrease continuously from the top to the base and measured ones are almost constant.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Modeled and experimental values of longitudinal strains
In a general sense, the model and experimental values fit well. The estimations of the mean green wood Young's modulus (16 400 MPa), when all data were pooled, seemed very realistic, when compared to independent values estimated from tensile tests on air-dried wood specimens or from wood density, although we neglected bark that was very thin compared to stem diameter. The model predicted well the evolution of strains around a circumference (cross section level) and could be used to analyze, for instance, the effect of different force azimuths. However, without great changes of modulus of elasticity from the buttress top to the base, the model did not describe properly the evolution of strains with height. As such variations were not observed from tensile tests or density measurements on wood specimen, these great and regular changes of EL were not realistic, neither on the whole cross section regardless of the radial position, nor on the ridges that would contribute most to flexural stiffness. The odds are that some hypotheses of the model are not robust enough, especially for low and continuous taper and vertical fibers. Such results emphasize the capacities and limits of beam models that are often used in plant biomechanical theoretical studies (Niklas, 1992
) and scarcely validated by direct assessment of strains and stresses.
Stresses and strains with respect to failure risks
Strains and stresses are three-dimensional. In the principal axes of wood (L, fiber direction; T, tangential direction), at the trunk surface, assuming linear wood behavior, stress components are related to strain components:
LT and TL are Poisson's ratios, and is the GLT shear modulus in LT plane (Guitard, 1987
; Niklas, 1992
). Because we measured T and L strains of the same order of magnitude, and as TL << 1 (TL 
–0.03 [Niklas, 1992
]), T has no significant effect on 
L in Eq. a, which then becomes
 | (2) |
), is verified and wood follows approximately a simple one-dimension Hooke's law in the longitudinal direction.
However, values of tangential stress T, often neglected, could be of great importance as wood is very weak when submitted to tangential tensile stresses. When tangential stresses are zero, the ratio T/L is –LT, and LT 0.67 (Guitard, 1987
; Niklas, 1992
). Statistical correlation between T and L was significant, with a ratio T/L of –0.64. Thus, mean tangential stresses T are proved to be statistically near zero. They are tensile for points above the regression line T/L = –LT and compressive for points below. For example, sides of buttress A are on average put into tangential tension during the first set of tests. Tangential stresses are more likely to be low near the ridge where the relation T/L = –0.64 is the most significant. They can be high on buttress sides where no relation between T/L has been found. For example, in the second tree, by substituting in Eq. b values of the measured ratio T/L on the buttress side, we prove that T is safe in compression on the convex side, but dangerous in tension on the concave one. Indeed, this tangential tension, superposed on the high shear, provoked the crack observed during breakage of the tree (Fig. 8).
Mattheck's theory
To analyze our data with respect to the constant stress theory, we need to define stress, because the stress field is defined by three components at the stem surface. Then we must design a stress index, which is calculated from these three components and able to predict locally the risks of failure. The choice of such a failure criterion is still a vexing problem in highly anisotropic materials (Berthelot, 1999
): Standards are maximal strain or stress criterions (each strain or stress component is compared to a limit experimentally determined by straight forward uni-axial tests); however, additional interactive criteria able to describe the material behavior under complex triaxial loadings can be designed (Berthelot, 1999
). Although Mattheck (Mattheck, 1990
; Mattheck and Prinz, 1991
) used the interactive Von Mises' criterion, we prefer to use a simple maximal strain or stress criterion that was implicitly used by other authors (Ennos, 1995
; Crook et al., 1997
). Indeed, Von Mises' criterion is not adapted to anisotropic materials (Berthelot, 1999
). For instance, it predicts the same risk of failure for a tension of 10 MPa in the longitudinal direction than in the tangential one, although it is well known that the former is safe and the latter not. Therefore, in bending experiments in which longitudinal normal stresses L (tension and compression along wood grain) are obviously greater, Von Mises' criterion focuses on these stresses and neglects other components. Actually, along and near the ridges, where L would be the only significant stress, Von Mises' and maximal stress criteria are similar. In this part of the buttress, we found two very different patterns: the first tree follows the constant stress law, but the second one is much more stressed at the buttress top. Both behaviors have been reported (Ennos, 1995
; Crook et al., 1997
). According to Mattheck's theory, the first tree buttress would have reached its morphologically optimized steady state. The second tree is still optimizing its buttress form. However, such assertions should be demonstrated and, as pointed out by Ennos (1995)
, Mattheck's theory could not be verified by static data without chronological observations of dynamic growth in natural or experimental conditions. The mechanism of strain perception that controls growth locally is still a puzzling question. Nevertheless, regardless of adaptive theories, the different buttress shapes (plate or shell) of the two trees can explain the different observed patterns of strains, because in the second tree, the buttress base is almost perpendicular to the bending plane.
Moreover, neglecting stress components other than longitudinal ones is not relevant, because breakage modes are obviously not simple compression cracks on the buttress ridges where longitudinal stresses are the highest. In the studied trees, the greatest risks of failure are related to shear and tangential tensions, even though values of these stress components are smaller than the longitudinal ones (wood is less stiff, but also less strong in these strain modes). Although some authors discussed the mechanisms of transverse or shear strengthening in trees (Mattheck, 1991
), potential adaptations of tree growth in respect to these strain or stress modes in buttresses have not yet been studied. If, as often supposed, buttresses are material-saving devices, selected for more anchorage stiffness and strength with less volume and less physiological expenditure, in an environment with intense biotic competition for light and space, they have to deal with a mechanical trade-off between minimizing uprooting on one hand and risks of shear and tangential failure in the plates on the other hand. The ecological question is then to estimate the importance of the second constraint: do buttresses fail at the plates? Obviously, the answer will depend on site conditions such as soil types, on buttress shapes, and ontogenetic stages of species.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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L(x,y,z) within the cross section (y and z are coordinates in a horizontal plane, y in the same vertical plane of the applied force F) are a function of only three unknown parameters: one elongation 0(x) and two curvatures k1(x) and k2(x) as
 | (1) |
 | (2) |
Henceforth, the modulus of elasticity EL will be assumed to be constant within the cross section, but this assumption is easily modified if significant variations of EL with y or z are observed.
Following this, forces that act on each cross section (height x) during the tests are calculated as
is the angle between F and the horizontal. N is the normal force (compression due to the vertical component of the force F), Mz and My are, respectively, the bending moments around z and y (i.e., the products of the horizontal components of the force F by the lever arm).
Using the strength of materials (Niklas, 1992
; Berthelot, 1999
), 0(x), k1(x) and k2(x) can be calculated from Eqs. 1, 2, and 3 as
L(x,y,z) can be calculated everywhere from Eqs. 4 and 1.
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FOOTNOTES |
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5 Author for reprint requests (fournier{at}engref.fr)
.
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Black H. L. K. T. Harper 1979 The adaptive value of buttresses to tropical trees: additional hypotheses. Biotropica 11: 240.[CrossRef][ISI]
Chapman C. A. L. Kaufman L. J. Chapman 1998 Buttress formation and directional stress experienced during critical phases of tree development. Journal of Tropical Ecology 14: 341-349[CrossRef][ISI]
Clair B. J. Ruelle B. Thibaut 2003 Relationship between growth stresses, mechano-physical properties and proportion of fibres with gelatinous layer in chestnut (Castanea sativa Mill). Holzforschung 57: 189-195[CrossRef][ISI]
Crook M. J. A. R. Ennos J. R. Banks 1997 The function of buttress roots: a comparative study of the anchorage systems of buttressed (Aglaia and Nephelium ramboutan species) and not-buttressed (Mallotus wrayi) tropical trees. Journal of Experimental Botany 48: 1703-1716
Ennos A. R. 1993 The function and formation of buttresses. Tree 8: 350-351
Ennos A. R. 1995 Development of buttresses in rainforest trees: the influence of mechanical stress. In M. Coutts and J. Grace [eds.], Wind and trees, 293–301. Cambridge University Press, Cambridge, UK
Fournier M. B. Chanson B. Thibaut D. Guitard 1994 Mesures des déformations résiduelles de croissance à la surface des arbres en relation avec leur morphologie. Observations sur différentes espèces. Annales des Sciences Forestières 51: 249-266[CrossRef][ISI]
Gentry A. H. 1993 A field guide for the families and genera of woody plants of North West South America (Colombia, Ecuador, Peru) with supplementary notes on herbaceous taxa. Conservation International, Washington, D.C., USA
Guitard D. 1987 Mécanique du bois et composites, Cepadues Editions, Toulouse, France
Halle F. R. A. A. Oldeman P. B. Tomlinson 1978 Tropical trees and forests. An architectural analysis. Springer Verlag, New York, New York, USA
Hollowel T. P. Berry V. Funk C. Kellof 2001 Preliminary check list of the plants of the Guiana Shield. Smithsonian Institution, Washington, D.C., USA
Kaufman L. 1988 The role of developmental crises in the formation of buttresses: a unified hypothesis. Evolutionary Trends in Plants 2: 39-51
Kollman F. F. P. W. A. Cote 1968 Principles of wood science and technology. I. Solid wood. Springer-Verlag, New York, New York, USA
Mabberley D. J. 1997 The plant book. A portable dictionary of the vascular plants. Cambridge University Press, Cambridge, UK
Mattheck C. 1990 Design and growth rules for biological structures and their application to engineering. Fatigue and Fracture of Engineering Materials Structures 13: 535-550
Mattheck C. 1991 Trees: the mechanical design. Springer Verlag, Heidelberg, Germany
Mattheck C. M. Prinz 1991 Buttress root: why they grow and what they are good for. Internal Report. Karlsruhe Nuclear Research Centre, Karlsruhe, Germany
Niklas K. J. 1992 Plant biomechanics: an engineering approach to plant form and function. University of Chicago Press, Chicago, Illinois, USA
Richards P. W. 1995 The tropical rainforest. Cambridge University Press, Cambridge, UK
Young T. P. V. Perkocha 1994 Treefalls, crown asymmetry, and buttresses. Journal of Ecology 82: 319-324[CrossRef]
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