## Abstract

Failure of a tree can be caused by a stem breakage, tree uprooting, or branch failure. While the pulling test is used for assessing the first two cases, there is no device-supported method to assess branch failure. A combination of the optical technique, pulling test, and deflection curve analysis could provide a device-supported tool for this kind of assessment. The aim of the work was to perform a structural analysis of branch response to static mechanical loading. The analyses were carried out by finite element simulations in ANSYS using beam tapered elements of elliptical cross-sections. The numerical analyses were verified by the pulling test combined with a sophisticated optical assessment of deflection evaluation. The Probabilistic Design System was used to find the parameters that influence branch mechanical response to loading considering the use of cantilever beam deflection for stability analysis. The difference in the branch’s deflection between the simulation and the experiment is 0.5% to 26%. The high variability may be explained by the variable modulus of the elasticity of branches. The finite element (FE) sensitivity analysis showed a higher significance of geometry parameters (diameter, length, tapering, elliptical cross-section) than material properties (elastic moduli). The anchorage rotation was found to be significant, implying that this parameter may affect the outcome in mechanical analysis of branch behavior. The branch anchorage can influence the deflection of the whole branch, which should be considered in stability assessment.

## INTRODUCTION

The probability of tree or tree section failure is a part of tree risk assessment (Ellison 2005). There are a number of visual methods available for tree stability assessment corresponding to level 1 and 2 by arboricultural standards (ANSI 2011). Even though these methods are commonly used, they are significantly influenced by the assessor’s background (Norris 2007). The Visual Tree Assessment (VTA) method has a specific position: it is based on a presumption of constant stress distribution and the ability of a tree to self-optimize (Mattheck and Breoler 1994; Mattheck 2004). The calculation method for tree stability can be used if visual assessment confirmation is necessary (Mickovski et al. 2005; Praus et al. 2006; van Wassenaer and Richardson 2009). Another option for tree stability assessment is to use a device-supported method, corresponding to level 3 by arboricultural standards (ANSI 2011). Device-supported methods are used to evaluate and quantify possible tree breakage or uprooting, but there is no such approach for branch stability assessment.

One of the device-supported methods is the Static Integrated Method (SIM), a pulling test, which assesses a tree’s resistance against breakage and uprooting (Brudi and Wassenaer 2001). SIM is based on a static approach that employs the standard equilibrium state of forces. The method employs the relationships between applied force (wind load), geometry (shape of trunk/tree), and material (greenwood). The linear relationship between the stem deformation and stress is considered in the case of an assessment of resistance to breakage. In the case of a tipping resistance analysis, the non-linear relation between the applied bending moment and root-plate inclination is investigated (Brudi and Wassenaer 2001; Detter 2012; Buza and Divos 2016). Both the device-supported methods and the computational methods simplify the stem mechanical response due to the assumption of the fixed cantilever beam condition in the stem breakage assessment. Neild and Wood (1999) used the tree flexibility instead of traditional stress-strength evaluation and created a mathematical model for the deflection curve calculation, including the interaction of stem and root-plate. In the case of branches, there is the assumption of fixed anchorage for mechanical analysis as well (King and Loucks 1978; Shahbazi et al. 2015). Although a significant influence of root-plate inclination to the branch is not expected, the material properties at the branch base shows high flexibility of branch anchorage (Jungnikl et al. 2009). On the contrary, Dahle and Grabosky (2010) found that the modulus of elasticity (*MOE*) decreases along the branch. They measured modulus of elasticity on the beams which were fully fixed, and the nearest point was 5 cm (1.9 in) from the branch base.

The stress distribution along the primary branches seems to be similar to the distribution at the tapered cantilever beam, linearly increasing from the tip. The value of the maximum bending stress within a variety of species is relatively constant (Evans et al. 2008). The detailed branch stress analysis, built on the assumption of the tapered cantilever beam, showed that stress increased with branch length, non-tapered shape, curved shape, and with a higher change in modulus of elasticity in diameter. There was a negligible difference between a model with elliptical cross-section and a circular one (Shahbazi et al. 2015). Except the modulus of elasticity change along the diameter, the influence of change along the branch was not investigated (Dahle and Grabosky 2010). Kane (2007) experimentally found that the branch taper, location of failure, and specific gravity, modulus of rupture, and modulus of elasticity showed low significance for calculating breaking stress.

The importance of aspect ratio (Gilman 2003; Kane 2007) and the effect of tree pruning on the strain distribution along branches have been discussed too (Gilman et al. 2015). The right pruning technique helps to keep the aspect ratio between branch and trunk small and reduces the risk caused by branch failure (Gilman 2015). According to Gilman (2003), the strength of branch attachment is correlated to the aspect ratio, while Kane and Farrell (2008) highlighted the fact that there is still a lack of strong correlation coefficients for the prediction of branch failure.

To understand the mechanical response of branches to loading, the traditional calculation of stress distribution could be supported by the deflection curve (King and Loucks 1978; Morgan and Cannell 1987). The advantage of a deflection curve is that it can be measured directly in contrast to the stress analysis. The branch deflection can be measured directly by the optical technique, which seems to be a prospective tool for the measurement of branch/tree reaction to loading due to the obtaining of full-field data (Lundström et al. 2007; Sebera et al. 2014; Sebera et al. 2016). Additionally, the results from the optical technique are well comparable with the results of finite element (FE) analyses too. The FE analyses are widely used nowadays for the description of branch growth and understanding the influence of maturation strain (Fourcaud et al. 2003; Coutand et al. 2011; Guillon et al. 2012).

The aim of this study is to find the significant parameters that influence the branch deflection curve within the frame of relationship amongst the load, the material, and the branch geometry. The parameters studied are branch diameter and length, shape of cross-section, curvature, angle of attachment, anchorage rotation, and modulus of elasticity. The emphasis is put on finding a device supported tool for branch stability assessment by a combination of pulling test, optical measurement, and beam deflection theory. Moreover, numerical simulation using FE analyses will be employed to further investigate the mechanical behavior of branches.

## MATERIALS AND METHODS

### Trees and Site

Four branches from four lime trees (*Tilia cordata* L.) were measured in April 2016. Trees were chosen from two locations in Brno–Cerna Pole, Czech Republic. Lime no. 1 (GPS: 49°12′28.6″N 16°37′18.2″E) was located in a park with compacted soil condition. Limes no. 2, 3, 4 (GPS: 49°12′36.2″N 16°37′01.5″E) were found in a street line with one site-limited root system. The purpose of the selection was to choose branches with different geometrical parameters to verify FE simulations. The branches differ in diameter, shape of cross-section (from circle to elliptical), tapering, and angle of attachment (Figure 1).

The diameter of branches was measured in two perpendicular directions at each tracked point. The angle of attachment was measured by hypsometer Nikon Forestry Pro (sensitivity 0.01°), where the horizontal level with the ground was considered to be 0°. The branches of limes no. 1 and 4 have more elliptical cross-sections, while branch no. 1 has a base diameter of 36 cm (14 in) and is significantly tapered at the base. Branches no. 2 and 3 are similar regarding the size and more circular cross-section, while branch no. 2 is more curved. Branches no. 3 and 4 are attached to the stem in a high angle to the stem (45°, 33°, respectively) (Table 1).

### Measurement

The branches were painted white to obtain markers at the neutral axis (20 to 30 cm [7.8 to 11.8 in] distance from each other) from the branch base to the point of force application. The point of force application was determined according to the size and shape of the branch. The aim was to measure in the elastic range with sufficient deflection of the branch. The elastic range of measurement was controlled by strain monitoring at the branch base (10% of maximum allowed strain for lime wood is the limit of elasticity 0.24 according Wessolly and Erb 1998). The motion of the markers was recorded during loading using a Canon EOS 700D camera with continuous shooting at 0.2 fps (sensitivity ±1.6 mm). Loading force was applied on the branches by manual winch. The force was applied by rope in the vertical direction to induce branch bending and redirected through fixed pulley to winch placed in horizontal direction. The pulley was placed directly under the anchorage point to ensure vertical loading. Force was recorded by the load cell (Hottinger Baldwin U2b type). At the base of each branch, between markers no. 1 and 2, inclinometer Sitall STS 110 (sensitivity 0.001°) was placed to check the motion of branch anchorage. The loading, inclination, and displacement measurement were time-synchronized. The displacements in vertical direction (deflections) were computed using 2D digital image correlation method implemented in Mercury software (Sobriety Ltd.) (Figure 2).

The origin of the coordinated system was located at the stem below the branch base, where X-axis direction was horizontal and Y-axis direction was vertical. For computations, the subset size was 50 × 50 pixels, and the correlation was searched in an intermediate neighborhood of 100 × 100 pixels. This setup helped reduce computational time. The full affine transformation was used for correlation of parameters. The matching criterion used in DIC computation was a normalized cross-correlation coefficient (Sutton et al. 2009).

Bending of branches provided data for the calculation of the modulus of elasticity (*MOE*) similarly to the approach applied by Cannell and Morgan (1987) and Dahle and Grabosky (2010). The equation of beam deflection was used for *MOE* calculation, where the boundary conditions for branch anchorage were subtracted from the measurement of stem deflection. The *MOE* was calculated from the deflection of the markers (*y*) as follows:
1

where *F* is the pulling force [N], *α* is the branch attachment angle [°], *L* is the length of the branch to the loading point [m], *x _{1,2...n}* is the marker position [m],

*l*is the moment of inertia at the marker position [m

_{1,2...n}^{4}],

*y*is the marker displacement (branch deflection) along the branch, and

_{1,2...n}*y*is the branch base deflection representing stem deflection [m]. The

_{0}*MOE*for each point along the branch was calculated (variable

*MOE*). Then the mean value of

_{var}*MOE*from variable values was calculated for each branch (mean

*MOE*).

_{x}### Finite Element Modelling

Parametrical numerical simulations of branch mechanical response to loading were carried out using finite element software ANSYS (Mechanical APDL v. 18.0). The static structural analysis with assumption of large deformation was chosen as appropriate for this study due to geometrical nonlinearities. The shape of branches was described by several segments being placed along circle or spline sectors. The segments are built from tapered beam elements (Beam type 189) with elliptical cross-sections (Figure 3). The maximum element size of beam elements was set to 0.01 m.

As the object of study, the branch is tested below the occurrence of permanent deformations, and linear-elastic material behavior was chosen. The orthotropic material model defined by cylindrical coordinate system was set up by elastic moduli (*E*), shear modulus, Poisson’s ratios, and density. The *E* was calculated by eq. 1 and then, based on that value, two simulation scenarios were used: 1) with a mean value of *E _{x}*; and 2) with a change of

*E*along the branch. The value of bending modulus of elasticity (

_{var}*MOE*) was used for longitudinal elastic moduli (

*E*), because there is the presumption of similar properties for greenwood. The tangential, radial value of

*E*and shear modulus were derived from

*E*by the following ratios:

_{L}*E*= 0.066,

_{R}/E_{L}*E*= 0.27,

_{T}/E_{L}*G*= 0.046,

_{LT}/E_{L}*G*= 0.02,

_{TR}/E_{L}*G*= 0.056 (Kretschmann 2010). The ratios and Poisson’s numbers were chosen for lime dry wood, since there is no evidence of values for greenwood. The Poisson’s numbers are μ

_{LR}/E_{L}*= 0.022, μ*

_{TL}*= 0.346,*

_{TR}*μ*= 0.034 (Kretschmann 2010). The density value was set up for lime greenwood to ρ

_{RL}*= 700 kg/m*

_{gw}^{3}(Wessolly and Erb 1998).

Boundary conditions were defined according to physical conditions of the measurement. The branch was fixed at the base with a small displacement in the horizontal and vertical directions (*y _{0}, x_{0}*) and rotation (θ

*). Force was applied at the branch top to simulate branch bending.*

_{0}The verification of FE simulation was carried out based on the comparison of branch deflection (*y*) along the branch length. The deflections obtained from markers were compared with those extracted from nodes at the same positions in the FE model and compared by relative error.

### Probabilistic FE Analysis

The sensitivity analysis in probabilistic design system (PDS) was carried out to find parameters which influence branch mechanical response significantly. The PDS analysis randomly generates values of selected input parameters with the principle of the Monte Carlo method. The significance for input to output parameters was evaluated by Spearman correlation coefficients.

The selection of input parameters was based on the input of equation for beam deflection and studies with three basic parts: geometry, material, and boundary conditions. The selected input parameters were: a) geometrical: attachment angle (*α*), branch length (*L*), branch diameter (*d _{base_}*), tapering (

*d*), ellipse (

_{top}*e*), curvature (

*R*); b) material: elastic moduli (

*E*); and c) for boundary conditions: branch anchorage rotation (

*ϴ*

_{0}).

Two sensitivity analyses were performed: 1) analysis with a wide variety of input parameters; and 2) detailed analysis of dendrometry and material. The first analysis filtered the most significant parameters, the second analysis aimed to find which parameters can influence the result of measurement significantly (Table 2).

## RESULTS AND DISCUSSION

The numerical model that was developed within this study was compared and verified by the experimental assessment of branch bending. The experiments provided modulus of elasticity, deflections, rotations, and forces. The maximum values of applied forces were *F _{lime no. 1}* = 2024 N,

*F*= 351 N,

_{lime no. 2}*F*= 231 N,

_{lime no. 3}*F*= 1193 N (Table 3). The maximum deflection ranged from 12.6 mm to 33.7 mm depending on the branch. The branch rotations near the base (between points 1 and 2) ranged from 0.057° to 0.35° (Table 3). The results show a high variability among the branches, which is attributed to their variation in geometry and the material variability.

_{lime no. 4}The deflection along the branches and their curvatures (difference between the minimum and the maximum deflection along the branch) increases with load (Figure 4). Branch no. 3 experiences a little drop between load step at 43% and 48% of maximum load, which may be caused by measurement inaccuracies near the branch base.

The average *MOE _{x}* is 6.01 GPa ± 2.97 GPa SE. The results show the high

*MOE*variability along the branches and amongst them (Figure 5).

The results correspond with variability which is presented in the literature except for branch no. 1. The values of elastic moduli (*E,MOE*) for lime wood in a green state range from 5.5 GPa to 9.2 GPa (Lavers 1983; Wessolly 1998; Maly 2012; Kretschmann 2010; Niklas and Spatz 2010). Lavers (1983) presented a standard deviation for *MOE* of lime greenwood to be −0.7, +0.024 GPa, and Kretschmann (2010) showed the influence and variability of moisture content of wood on the elastic modulus. Cannell and Morgan (1987) expected lower values in branch wood due to the lower specific gravity and higher moisture content. The reported/literature values of elastic moduli are provided only for trunk wood, but the correlation between the density of branch and trunk wood is known (Swenson and Enquist 2008; Sarmiento et al. 2011), which should be reflected in the correlation between the *MOE* of branch and trunk wood as well (Niklas and Spatz 2010). The wood from stem is 9% denser than branch wood, but there is variation within the site (5.4%) and family (16.1%) (Sarmiento et al. 2011). The wood variability itself, moister content, and differences between stem and branch wood could contribute to the *MOE* variability in the case of the observed branches. Branch no. 1 exhibits a significantly different mean value of *MOE* (2.98 GPa) than the other branches. Moreover, such a low value may be influenced by the unusual convex tapered shape of the branch (Figure 1a).

The *MOE* along the branch shows a high variability as well, but a slowly increasing trend towards the tip of the branch may be observed (Figure 5). Dahle and Grabosky (2010) reported the decreasing *MOE* in maple branches ranging from 9.7 GPa at a 5 cm (1.9 in) distance from the branch base to 7.8 GPa at the branch midpoint. In that case, the branch base was not influenced by the branch anchorage since the samples were measured with full anchorage. The anchorage is considered to be flexible and deformable due to the high microfibril angle and the low density of wood around the branch junction (Jungnikl et al. 2009). Due to the variability of MOE along the branches, both configurations—the *E _{x}* mean value of each branch and the variable

*E*in the sections—were considered in numerical models. The variable results for different material properties suggest that a detailed study of the branch greenwood should be done in future work.

_{var}### Finite Element Simulation

The mean relative error (RE) of branch deflection obtained from simulation is 0.5% to 26% compared to the measurement, depending on the branch and material properties (Table 4). In the literature, there are no results about comparing FEM simulations and measurements of branch deflection, but in the case of tree anchorage simulation, the difference between simulation and experiment varied from 20% to 70% based on the soil composition (Dupuy et al. 2007). In one study of tree stability, the numerical simulations overestimate the maximum overturning forces from 15% to 25% (Rahardjo et al. 2014). The overestimation of forces corresponds with the fact that the branch simulations in this study shows smaller deflections than experiments. The RE of branch no. 1 is significantly (−26% RE) higher than would be expected for simulation. This may be caused by the unusual geometry or specific kind of branch anchorage (Figure 1a).

The shape of the experimental deflection curve (Figure 6) better corresponds with a numerical model with non-variable *E* along the branch length except branch no. 4. The rotation at the branch base could influence the calculation of *E*. The RE’s near the branch base (points 2 and 3) of branches no. 2 and 3 are significantly higher (47% to 140%) than the other deflections along the branches. The deflection at this part is up to 1.5 mm, with a small absolute difference (0.27 mm). This caused a high relative error and could be influenced by the measurement inaccuracies. The measurement at the branch base showed high variability, which implies certain inaccuracy in the measurement. In future, the inaccuracy should be eliminated using a camera of higher resolution.

### Probabilistic Analysis

The first probability analysis showed a close correlation (*R* = 0.8 to 0.9) between the branch diameter and all output parameters (Figure 7), which may have been predicted from the beam deflection theory. Even though Gilman (2003) found experimentally that the aspect ratio is more significant to the strength of branch attachment than the diameter itself. The branch length together with the loading point had lower significance (*R* = 0.29).

Because of the wide range of input parameters and a high number of calculations, even the low coefficients show significant influence on the branch deflection. The branch curvature was expected to be a significant factor because of their influence to the stress distribution (Shahbazi et al. 2015) and due to the fact that material properties have a significant impact on the form of the branch curvature during growth (Coutand et al. 2011). The analysis did not show a significant correlation of the branch curvature to the branch deflection or rotation. The correlation of growth stress and curvature is one of the driving factors for young branches in their shape development (Coutand et al. 2011; Guillon et al. 2012). According to the results, this influence can be eliminated by measuring mature branches where curvature is not a determining factor for branch response to loading. Ennos and van Casteren (2010) presented the significance of curvature for transversal stress distribution, which is the phenomenon that needs more attention. As Gilman (2003) found, there was no significant influence of the attachment angle to branch deflection in the study, with the exception of a small correlation to the middle branch rotation (*R* = 0.05). In comparison to other parameters, the anchorage rotation showed a significant influence (*R* = 0.26) on the branch maximum deflection. The influence of anchorage rotation is more evident in the second analysis (*R* = 0.75). The anchorage is the most significant at the base of branch, but it affects the behavior of the whole branch as well (Figure 8).

The significance of branch anchorage rotation should be considered since there is no full-fixed anchorage at the tree branch–stem junction and as is commonly assumed in published mechanical analyses (Evans et al. 2008; Ennos and van Casteren 2010; Shahbazi et al. 2015). The significance of small rotations at the branch base corresponds with the aberration in verification results. This could be improved by using a high-resolution camera for the branch base experiencing small rotations. A high influence of the elliptic shape to the branch deflection was found, especially in the middle part of the branch (*R* = −0.58). This corresponds with the conclusion that simplification of cross-section to a circular shape is inappropriate if there is no close similarity in shape (Kane 2007). In the stress analysis made by Shahbazi et al. (2015) the difference between the maximum stress of elliptical and circular cross-section was not proved. The maximum stress can be usually found near to the branch base but the results shows that significance of elliptical shape to deflection is increasing in the middle part of branches (Figures 7 and 8). There was a high negative correlation within the tapering and the branch top response (*R* = −0.67). Kane and Farrell (2008) observed the taper to be insignificant to the breaking stress calculation with the comparison of aspect ratio significance, which could correspond to our results for the anchorage rotation. The results show a higher influence of the anchorage rotation at the branch base (*R* = 0.75) and a higher influence of tapering at the branch top (*R* = −0.67). The significance of material properties (*E*) is lower than that of geometrical parameters (R_{max} = −0.29). The *E* at the branch base is more significant for the deflection than the other segments. The results showed that the deflection of points is more influenced by a change of *E* in the previous sections.

## CONCLUSION

The shape of the deflection curve could be predicted by the numerical computations when precise geometry is known. The significant deviation of the measured deflection curve at a certain point could be consequently considered as the indicator of defect, which can be used in branch stability assessment. Sensitivity analysis showed that the parameter’s diameter, elliptical shape, and tapering have a high impact on a branch deflection; simultaneously, the curvature and angle of attachment can be neglected within the analysis. If the precise branch geometry is known, the deflection curve may reflect the changes of material properties along the branch length. The branch mechanical response is very sensitive to the branch anchorage properties, which requires high accuracy of its detection during the measurement. The anchorage of a branch which is not full-fixed should be considered in future examinations. The results showed that the branch anchorage properties influenced the overall branch deflection, which requires more attention in the case of the branch stability assessment, and this corresponds with the conclusion of Kane (2008). The use of more sophisticated tools for the assessment of branch anchorage should be considered where optical techniques seem to be a promising new tool.

## ACKNOWLEDGMENTS

This article is supported by the Internal Grant Agency of Mendel University (project no. 2016036).

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