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2Université Bordeaux 1, Mixed Research Unit: Unité Sciences du Bois et des Biopolymères (UMR US2B), Talence, F-33405 France; 3Institut pour le Développement Forestier (IDF), Paris, F-75007 France; 4Centre de Coopération International en Recherche Agronomique pour le Développement (CIRAD), Mixed Research Unit: Botanique et Bioinformatique de l'Architecture des Plantes (UMR AMAP), TA A-51/PS2, Boulevard de la Lironde, 34398 Montpellier Cedex 5, France; 5Institute National de la Recherche Agronomique (INRA), Mixed Research Unit: Botanique et Bioinformatique de l'Architecture des Plantes (UMR AMAP), TA A-51/PS2, Boulevard de la Lironde, 34398 Montpellier Cedex 5, France
Received for publication January 5, 2007. Accepted for publication July 10, 2007.
ABSTRACT
Understanding the mechanism of tree anchorage in a forest is a priority because of the increase in wind storms in recent years and their projected recurrence as a consequence of global warming. To characterize anchorage mechanisms during tree uprooting, we developed a generic finite element model where real three-dimensional (3D) root system architectures were represented in a 3D soil. The model was used to simulate tree overturning during wind loading, and results compared with real data from two poplar species (Populus trichocarpa and P. deltoides). These trees were winched sideways until failure, and uprooting force and root architecture measured. The uprooting force was higher for P. deltoides than P. trichocarpa, probably due to its higher root volume and thicker lateral roots. Results from the model showed that soil type influences failure modes. In frictional soils, e.g., sandy soils, plastic failure of the soil occurred mainly on the windward side of the tree. In cohesive soils, e.g., clay soils, a more symmetrical slip surface was formed. Root systems were more resistant to uprooting in cohesive soil than in frictional soil. Applications of this generic model include virtual uprooting experiments, where each component of anchorage can be tested individually.
Key Words: biomechanics • poplar • Populus • root anchorage • tree stability • windthrow
In recent years, attention has focused on the consequences of climate change on the composition and sustainability of forest ecosystems around the world. One likely consequence is an increase in the number and intensity of wind storms (Bloomfield, 2000
), which would cause major economic damage and disturbance to temperate forests. Although much experimental research has been carried out on tree anchorage and biomechanics, particularly since the storms Lothar and Martin hit Europe in 1999 (Cucchi et al., 2004
; Achim et al., 2005
; Nicoll et al., 2005
; Stokes et al., 2005
; Brüchert and Gardiner, 2006
; Peltola, 2006
), mechanistic numerical models of such phenomena are lacking (Blackwell et al., 1990
; Dupuy et al., 2005a
, b
; Fourcaud et al., 2007
). The use of such models is three-fold: they can be used to investigate the contribution of different plant or soil parameters to tree anchorage; virtual experiments can be carried out, thus reducing destructive and expensive fieldwork; and finally, the models can be incorporated into decision support systems for forest managers (Gardiner et al., 2000
; Cucchi et al., 2005
). Although the initial development of such models is time consuming, once calibrated, they could be adapted fairly easily to different species in a variety of situations. To our knowledge, mechanistic tree anchorage models based on a description of root architecture have not yet been linked to field data, but would be a major step toward developing a powerful tool to predict tree uprooting during storms.
Vulnerability to uprooting has been traditionally investigated at the population level, by correlating forest damage to certain stand characteristics, i.e., tree species, silviculture, soil type, and wind speed (Putz et al., 1983
; Ruel, 2000
; Cucchi and Bert, 2003
; Mayer et al., 2005
). However, this empirical analysis of windthrow is incomplete because root system characteristics and other variables influencing tree stability (Khuder et al., 2007
) are not always taken into account. Experimental studies whereby trees are winched until failure and the force necessary to uproot or cause stem breakage is measured are useful in that they provide detailed data at the tree level (Fraser, 1962
; Stokes, 1999
; Cucchi et al., 2004
; Nicoll et al., 2005
; Stokes et al., 2005
; Peltola, 2006
). Nevertheless, even if root system architecture is measured, data are often incomplete because roots are usually damaged during the tests (Stokes et al., 2007
). The role of root system architecture in tree anchorage has been alluded to in studies of root system morphology, but without the appropriate experimental data, it is difficult to quantify the influence of each parameter during the uprooting process (Nicoll and Ray, 1996
; Mickovski and Ennos, 2003
; Danjon et al., 2005
; Di Iorio et al., 2005
; Soethe et al., 2006
; Khuder et al., 2007
).
While root architecture strongly affects tree anchorage, soil physical and mechanical properties are major determinants of uprooting. Because the root–soil plate of a tree is a composite structure, soil type and its interaction with the root system will contribute to determining the overall anchorage capacity of a tree (Moore, 2000
). When soil is waterlogged, both shear strength and the root–soil bond are reduced, and therefore, the entire system fails more easily during wind loading (Coutts, 1986
). In dry soils or soils with a high shear strength, trees tend to be better anchored but the stem breaks during loading, whereas trees in waterlogged soils or soils with a lower shear strength are usually uprooted (Coutts, 1986
; Moore, 2000
). In simulations of uprooting experiments of simplified model trees, soil internal friction angle modified the position of the root system rotation axis during overturning (Dupuy et al., 2005b
). Such an adjustment to the mechanics of the composite root–soil system in turn strongly influences the force required to cause failure of this system. However, although the models by Dupuy et al. (2005a
, b
) and others (Ennos, 1990
; Niklas et al., 2002
; Fourcaud et al., 2007
) explicitly consider soil and root system properties, the models do not use real root system architectures in uprooting simulations. Hence, testing models that can accurately predict overturning behavior of trees in different types of soils is a priority.
The development of automatic and generic numerical methods that use accurate descriptions of plant morphology for biomechanical analyses can be extremely useful, and such models have already been applied to both real and simulated architectures subjected to mechanical loading in different environments (Fourcaud et al., 2003a
, b
; Sellier et al., 2006
). We focused on the use of finite element modelling (FEM) to simulate the uprooting mechanism of realistic root architecture in different soil types.
The validation of FEM models with real data is, however, particularly difficult as mechanical testing often greatly damages the root system. Therefore, trees selected for complete uprooting were different from trees used for root system measurements. We also chose two poplar cultivars (Populus trichocarpa Torr. Gray and P. deltoides Batr. Marsh) to illustrate interspecific variations in root morphological properties and anchorage (e.g., branching density, rooting depth) (Harrington and DeBell, 1996
).
Winching tests were carried out in the field, and the force required to cause failure was recorded. Root system architecture was then measured, and these data were fed into the model, which calculated the force necessary to uproot the trees in two types of soil. Results were compared with experimental data to better understand the major parameters influencing tree uprooting during a wind storm.
MATERIALS AND METHODS
Tree material and site
To provide data from real trees, which could be used to test the 3D model developed later, we performed static bending tests and measured root architecture of forest trees. We chose poplars (Populus sp.) because poplar plantations suffered severe losses during the 1999 storms in Europe (Bergonzini and Laroussinie, 2000
). The poplar trees selected for this study were planted as live pole cuttings, each pole being several meters long and pushed into the ground at a depth of 1 m. Adventitious lateral roots then grow from this pole, which can be considered analogous to a central tap root. Therefore, root systems of plantation-grown poplars are unusual but simpler to describe than seed-grown, tap-rooted conifers such as maritime pine (Pinus pinaster Ait., Danjon et al., 2005
). Because these data were to be fed into the 3D root and soil model in a first attempt to calibrate the model with real data, we preferred to use simple systems, thus reducing computation time.
Six 7-yr-old trees were used, three each of two North American poplar cultivars: ‘Beaupré' (Populus trichocarpa Torr. and Gray) and ‘Raspalje' (Populus deltoides Batr. ex Marsh). Stem diameter at breast height (DBH) was 0.24 ± 0.01 m (mean ± SE) for Beaupré and 0.28 ± 0.01 m for Raspalje. Trees were planted at a density of 200 stems·ha–1 in two neighboring stands located in northwest France (Moulin de Bariteau, Chinon, 0°14' E, 47°10' N). The study area was flat with an elevation of 37 m. Mean annual precipitation is 820 mm·yr–1, and the prevailing wind direction is northwest. Mean wind speed ranges from 4.5–5.5 m·s–1 (Météo-France). The soil consisted of a clayey sand (69% sand, 21% clay, and 10% loam) and lacked a hard pan in the top 1.0 m (F. Charnet, Institut pour le Développement Forestier, personal communication). During the investigation, the ground water table was 1.2 m deep. Soil was sampled at a depth of 0.30 m from the root–soil plate of trees winched sideways. The wet soil mass was measured immediately, and samples were then oven-dried for 5 d at 105°C to calculate soil moisture content (g water/100 g soil). In both plantations, soil moisture was 21.7 ± 1.4% (Raspalje) and 21.7 ± 0.9% (Beaupré) and thus was not thought to influence anchorage between trees.
Static bending tests
The method used for uprooting trees was similar to that used by Coutts (1983)
, Moore (2000)
, Mickovski and Ennos (2003)
, Cucchi et al. (2004)
, and Nicoll et al. (2005)
. Two trees of each cultivar were winched sideways until anchorage failure, which occurred when the tree was deflected 
10° from the vertical, forming an 80° angle with the soil. One other tree for each species was winched sideways almost to the point of failure, i.e., the tree was deflected 5° at the stem base, then allowed to return to the vertical. Trees that were bent 5° were not damaged and were used for measuring root system architecture, which required intact root systems. A motorised winch (type Hit-Trac 16B, Habegger, Switzerland), with a 16 kN maximal strength capacity, was used to winch trees sideways. The winch was attached to the base of an anchoring tree at the longest possible distance to the winched tree to obtain a small deviation from pure horizontal forces (Fig. 1). The cable was attached to the winched tree at only 6.0 m so as to induce root failure instead of stem breakage. The tension applied to the winched tree was measured by a load cell (type K25H-20kN Scaime S.A., France), attached between the winch and the anchoring tree, and was recorded every second using a data logger (type Almemo 2290–8, Ahlborn, Germany). An inclinometer was attached to the tree stem at a height of 6.0 m to measure the stem deflection angle during the winching test. Deflection angles were recorded every second with a second data logger, which was synchronized and simultaneously activated with the load cell data logger. Trees were winched along the direction of the westerly prevailing wind.
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) was then used to remove the soil and clean the root system. Root system architecture was measured manually in the field, and during these measurements, each root system was discretized into small sections. Following a recursive pathway along the central tap root and each lateral root, we measured the distance (link) between each root base and daughter root, along with the diameter at the root section base and halfway along the link. The depth, branching angle, and azimuth of each link were also noted, as well as any abrupt changes in azimuthal direction. Data were recorded and encoded simultaneously in the software ARCHIROOT (Dupuy, 2003
, http://www.archiroot.org.uk). ARCHIROOT stores data in the format of a directed Tree Graph (TG) using the AMAP MTG format as described by Danjon et al. (1999)
. The data were used to compute the total root biomass and the distribution of diameter size and to produce 3D images of the root system (Fig. 1).
Finite element modelling
Soil and roots material definition
Modelling was carried out using the FEM software Abaqus (ABAQUS, Inc., http://www.abaqus.com/). Soil was modelled as an elastic perfectly plastic material using a linear Mohr Coulomb model available in the Abaqus software (see Abaqus user's manual for details). For frictional material, the model was used with nonassociated flow, with a dilation angle 
= 0 (see Whitlow [1995
] for an introduction to soil mechanics). Root material was considered to be elastic linear with a plastic threshold modelled by a von Mises yield criterion (Kachanov [2004
]).
Results from the 3D mechanical model were then to be compared with the experimental data from the static bending tests. However, information concerning the mechanical properties of soil at the field site was not available. Therefore, we used theoretical values (Whitlow, 1995
), which were considered representative of the site conditions. Because the soil to be modelled comprised 70% sand particles, we chose the following values: Young's modulus E = 20 MPa, Poisson's coefficient 
= 0.3, density 
= 20k N/m3, and friction angle 
= 25° (Table 1). A low cohesion value c = 1.5 kPa was also used to account for the presence of about 20% clay in the soil (Table 1).
|
= 0.49, and = 0°. The remaining mechanical parameters were kept the same as in the preliminary soil model (Table 2).
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r) associated with the von Mises criterion equalled 15 MPa.
Geometry and meshing
The root architecture data, which were recorded in the TG files, provided the topology and geometry of the whole root structure. These data are the list of consecutive nodes, the type of connection (link) these nodes had with the antecedent node of the same branching order, and the associated geometrical attributes.
In this FEM model, root systems were discretized using 2-node linear beam elements B21 available in the structural element library of ABAQUS. The FEM mesh of the root system was automatically derived from the architectural data using a computer program specifically dedicated to this task (see flowchart in Fig. 2). This program gave a generic character to the meshing process, so that the program is applicable to any branching structure encoded with an AMAP MTG format.
|
The soil domain consisted of a parallelepipedic block discretized with 20 nodded brick elements, i.e., the 3D continuum element C3D20 in the ABAQUS elements library.
Root–soil interaction
The following stage consisted of creating the physical links between the soil and roots. Different techniques exist to model explicit surface to surface interaction through the use of constraints implemented with Lagrange multipliers or interface elements. Such methods are computationally extensive and may sometimes lead to convergence difficulties. However, the hypothesis of a rigid root–soil interaction was supported by field observations that most of the tree roots remained embedded in the mass of soil after uprooting, suggesting that most of the soil lifted during uprooting was trapped in the network of roots and that little relative displacement between soil and roots occurred.
Therefore, a node to node interaction between soil and roots was implemented using kinematic coupling (see the ABAQUS theory manual). Let Xr denote the position of a root node (master node) and Xs be the coupled soil position (slave) in reference configuration. The initial reference configuration is expressed as the translation vector N:
|
| (1) |
|
| (2) |
Soil–root node pairs were connected using the ABAQUS command *TIE.
Boundary conditions and bending test simulation
The reconstructed root system model was placed at the center of a parallelepipedic domain of soil, 4.4 x 4.4 m wide and 2.0 m deep. Boundary conditions were defined to reproduce the action of the surrounding infinite earth: translation on lateral faces was fixed horizontally, and the underside face was built in (Fig. 3). A rigid beam 6 m long was fixed to the top of the root system, representing the tree trunk beneath the crown. To simulate uprooting, lateral displacements of 1.0 m were applied incrementally on the rigid trunk up to a height of 6.0 m. The reaction force was recorded at this point, and the force–displacement curve then determined. The maximum force at the end of the lateral displacement was used to compare the soil–root anchorage efficiency. The field of equivalent plastic strain was monitored in the vertical plane in the direction of pulling. The plastic area around the roots was also visualized in the complete root system, and the motion of the stump was monitored for each increment. These variables were used to understand the different mechanisms occurring during uprooting.
|
Sensitivity analysis of material properties
Because we did not possess any quantitative data on soil and root mechanical properties at the field site for the static bending tests on poplar trees, a sensitivity analysis was conducted to study the influence of small variations in frictional soil properties (c, , ) and root mechanical parameters. Each parameter of the initial configuration was varied successively by 15% (Table 2). The sensitivity S of a parameter 
was then quantified as the percentage increase in anchorage resistance for an increase in 1% of the specific parameter:
|
| (3) |
RESULTS
Static bending tests
Of the four trees winched sideways until failure, the stem of two of each cultivar broke at heights of 2 m; the remaining two trees, which were deflected to 5 rather than 10°, failed through uprooting. The mean force that caused failure was 8.75 kN in Beaupré poplars and 13.5 kN in Raspalje (Table 3). Because we were simulating uprooting behavior in these trees, we could not compare results from the model with data from trees that had failed in the stem; therefore, only two trees could be used to compare observed and predicted uprooting results. For the two trees, which were winched sideways but not to failure, only the initial part of the force displacement curve could be compared with results from the model.
|
The root system of the Raspalje cultivar had fewer but larger roots than the Beaupré. One massive central vertical root with a length of 0.95 m and a basal diameter of 0.34 m bore secondary laterals with an average density of 24 laterals/m and a mean diameter of 0.55 ± 0.04 m. The root system had a total volume of 0.21 m3 and had 82 structural roots (branching order <5) with a total root length of 82.0 m (Fig. 1b).
Mesh size sensitivity analysis
The study of mesh size influence on the numerical results demonstrated that the calculated reaction force opposed to a given stem displacement decreased with the size of the soil elements (Fig. 4). The calculated reaction force vs. mesh size did not stabilize to a constant value with the range of element size used in this sensitivity analysis (Fig. 4); however, the absolute difference between the mechanical responses of the two root systems remained constant. On the other hand, tthe calculation time (CPU) increased exponentially with the number of soil elements; therefore, a mesh with 1372 elements was finally used for the comparative analysis. This choice provided a good compromise between the accuracy of the numerical results and the calculation time required for the simulations.
|
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, , and MOR, showed that the cohesion c was the most influential parameter with regard to uprooting, with a sensitivity of 0.5 (Fig. 8). The properties and MOR had a similar effect on uprooting resistance, with a sensitivity of less than 0.3 (Fig. 8), while had the least influence on uprooting resistance (Fig. 8).
|
Understanding how wind forces are transferred to the soil via the root system and what determines the anchorage resistance of trees is crucial for predicting windthrow in forests. To date, our knowledge and ability to predict tree uprooting has been limited by our ability to understand the nature of the root–soil interaction in the context of real root system architectures. Root morphologies are usually very complex and their variability poorly understood. Also, by current experimental approaches we are unable to measure the essential mechanical variables (e.g., strains, stresses, and displacement fields) in both roots and soils during in situ tree pulling tests. Finite element and other numerical methods used in physics and engineering are potentially very powerful tools for obtaining quantitative in-depth information on such mechanisms, and the model presented here was a first attempt in using such a method to understand the mechanical properties of root–soil composites.
In this paper we primarily described the development and testing of the finite element method in tree anchorage studies. The developed model allowed us to simulate the uprooting of two different poplar cultivars while accounting for an accurate geometrical and topological description of their two distinct root system architectures. To compare the model's predictions with the anchorage of real trees, destructive winching tests were conducted on four additional trees. The use of cultivars from the poplar plantation at Moulin de Bariteau aimed at reducing the morphological variability among species (Harrington and DeBell, 1996
), while the Beaupré and Raspalje cultivars were chosen for their morphological differences. However, significant intraspecific variability occurred during the pulling tests, and the model predictions can only provide indications about the differences in anchorage between cultivars. More trees need to be tested to validate our model, and more elaborated poro-viscoplastic models could be developed to incorporate the effect of pore water in the mechanisms of uprooting.
The properties of the field soil (clayey sand) used are between those of sand and clay, and yet the model simulations predicted much higher resistance than was measured in the field. Results from the simulations seem, therefore, to have overestimated the actual uprooting resistance. Apart from the fact that the trees that were winched to failure in the field were not those for measured for root architecture, the difference in uprooting resistance between simulated and real anchorage tests could be explained in several ways. The rigid node-to-node interaction used in the model does not enable consideration of sliding or opening of the root–soil surface interaction during failure. Studies on the influence of interface properties have also shown that roughness of the interface on foundation piles can account for 15–50% of the variation in resistance, depending on the type of loading and pile configuration (Chen and Martin, 2002
; Jeong et al., 2004
). Therefore, the rigid root–soil adhesion used in our model can be expected to artificially increase the uprooting resistance. Finally, the material model used for roots and soils is probably not suitable for large deformations. Highly anisotropic and fragile models, which can involve failure of individual root elements, will more realistically represent root breakage.
Although the generic 3D uprooting model developed in this study does not yet include all the elements necessary to accurately predict uprooting resistance, it can nevertheless provide useful information on how root system structures interact within a soil. In the simulations of uprooting in clay soil, failure in the soil was symmetrical on both the windward and leeward sides of the root system. In analogous studies on laterally loaded piles on cohesive clay soils (Yang and Jeremic, 2002
), deformations were also symmetrical. However, in the same study on a frictional soil, Yang and Jeremic (2002)
found that the plastic deformation not only occurred close to the soil surface but also expanded laterally further away from the pile. In our model of uprooting in a frictional soil, plastic deformation was greater on the windward than on the leeward side of the root system. Although this phenomenon occurs in real situations of tree overturning (Coutts, 1983
, 1986
; Crook and Ennos, 1996
; Cucchi and Bert, 2003
), it has not been explained comprehensively. We show that soil mobilized on the windward side of the tree was subjected to low or negative isostatic pressure when lifted up by roots. Therefore, failure occurred and uprooting was induced.
The geometrical structure of the root systems also affected soil behavior during overturning in our simulations. The two root systems, which were from two different cultivars, were distinctly different from each other with regard to the number and diameter of first order lateral roots borne on the central tap root. The Raspalje cultivar possessed few but large lateral roots, whereas the Beaupré had twice as many smaller lateral roots and also more total roots (115 structural roots in Beaupré, 85 structural root in the Raspalje root system). Because the stiffness of a root is a function of its diameter to the fourth power, numerous small roots will decrease anchorage rigidity but increase resistance in tension (Stokes et al., 1995
). The "group effect" of large lateral roots near the tree trunk will cause plastic failure in the soil to occur further from the tree axis, thus augmenting resistance to failure (Coutts, 1983
; Dupuy et al., 2005a
). The simulated failure zone was larger with the Raspalje than with the Beaupré cultivar. This difference in the size of plastic failure in the soil is probably due to the effect of the large lateral roots in the Raspalje tree.
Clonal, morphological differences affect the anchorage efficiency of poplar trees in a plantation (Harrington and DeBell, 1996
). Uprooting resistance was higher in the Raspalje cultivar than the Beaupré in both the numerical and real uprooting tests. This increase in anchorage efficiency can be explained by the high root volume and number of large lateral roots in the Raspalje tree. The resistance of lateral roots held in tension during uprooting is a major component of anchorage (Coutts, 1983
, 1986
; Stokes et al., 1995
; Crook and Ennos, 1996
) because of the high tensile resistance of wood (Genet et al., 2005
). Numerous thin lateral roots, even if they are not large, will augment uprooting resistance. However, our model does not yet include the anisotropic material properties of root wood, and including such properties is a priority for future modelling studies.
Although root architecture has long been considered a major component of root anchorage (Coutts, 1983
, 1986
; Danjon et al., 2005
; Dupuy et al., 2005a
, b; Khuder et al., 2007
), we showed that soil cohesion c was also an important factor affecting uprooting resistance. The weight of the system alone was not sufficient to produce enough isostatic pressure to prevent soil failure according to the Mohr–Coulomb theory. Additionally, the soil that is lifted on the windward side of the tree during uprooting is subject to low or negative isostatic pressure, thus increasing sensitivity to variations in soil shear strength, which was triggered by the soil cohesion c. Soil friction angle also influenced uprooting resistance, but dilation angle less so. Chen and Martin (2002)
showed that these two parameters were more important than soil cohesion when considering the efficiency of landslide stabilizing piles, but pile failure is concerned with soil cohesion in compression in the direction of pulling. In our model, soil failure was predominant on the windward side of the tree, where roots and soil were held in tension during overturning.
Changing root strength had little effect on overturning resistance in the model. To our knowledge, no quantitative studies of the influence of root strength on anchorage efficiency of trees has been done, although indicators of its importance, particularly in shallowly rooted forest trees, are available (Stokes and Mattheck, 1996
).
With regard to the technical aspects of the generic 3D model described here, CPU time was high, and the problem of convergence does not yet enable models to be compared readily to real data. Nevertheless, such a numerical tool is very promising and can be applied to any root architecture from which topology and geometry have been accurately described. Recording the structure with an AMAP MTG format also allows the topological data of the root system to be read automatically and incorporated in the FEM anchorage model.
The numerical approach developed in our study can also be used to explore in detail the role of each of the structural elements involved in tree uprooting. In particular, how any given root interacts within the root–soil matrix can be determined by isolating and removing that root from the model, or by adding new roots (Dupuy et al., 2005a
; Fourcaud et al., 2007
). Such virtual experiments are easier to perform than complex field experiments, whereby root systems are physically manipulated (Khuder et al., 2007
).
LIST OF SYMBOLS
: Addition of a new line in the NODE or BEAM list : Soil friction angle : Soil dilation angle : Poisson coefficient : Volumetric weight
FOOTNOTES
1 The authors thank M. Maitrot and C. Drenou (IDF) for help in the measurements of root architecture and F. Charnet (IDF) for data on soil properties at Chinon. Funding was provided by a French governmental CIFRE bursary, the poplar ECOFOR project (no 2002.06), and the E.U. project Eco-Slopes (QLK5-2001-00289). Authors are also grateful to the two anonymous reviewers for very constructive advice that helped improve this manuscript. 
6 Author for correspondence (e-mail: lxd20{at}cam.ac.uk
), present address: Department of Plant Sciences, Downing Street, Cambridge CB2 3EA, UK, phone: +44 01223-7-66545
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indicates the insertion of a new line <i> or <i, j> in the NODE or BEAM list.
) and crosses (x) indicate the maximum reaction force for Beaupré and Raspalje, respectively, for a given mesh size. The solid line (—) indicates the calculation time (CPU) required for a given mesh size.
