142 Additionally, knowing the center of gravity for a branch is important in determining how the load from a lateral branch acts upon the parent branch (see Formula 2), and thus is an im- portant component of a branch’s structural evaluation. Since the actual center of gravity cannot be known without destruc- tive sampling, this point would need to be estimated. Research is sparse in non-destructive methods to predict the center of gravity. Researchers have utilized two-dimensional images to estimate center of pressure during dynamic loading (Kane and Smiley 2006; Smiley and Kane 2006; Kane et al. 2008b), but this is not directly related to center of gravity. This paper examines if it is possible to develop a predictive relationship to estimate the location of the center of gravity of branches. Finally, ap- plying the load at the center of gravity may be important when considering dynamic loading over the whole tree or branch, such as during wind interception (Smiley and Kane 2006). METHODS Data was collected from August 24–26, 2010, at the Davey Research Farm, Portage County, Ohio, U.S., approximately 41°14’19”N and 81°10’21”W. Soil was Ravenna silt loam. Three exterior growing first-order branches were selected from five Tilia cordata Mill. (littleleaf linden) trees, for a total of fifteen branches. Trees were between 38- and 46-years-old. The mean diameter at breast height (1.4 m above ground) for the five trees was 13.9 cm ± 1.4 SD. Branches were selected to represent typi- cal first-order branches in the lower half of the crown that likely acted as structural branches. Branches were removed from the tree and lowered to the ground to avoid loss of lateral branch- es. Overall branch length (using a string to follow the contour), angle of branch departure (A), diameter, and branch mass (M) were measured. Center of gravity (CG) was defined as the lin- ear distance from the proximal base of the branch to the point at which the branch balanced evenly in one hand. All second order branches were removed from the first-order branch, until the first-order branch measured 2.54 cm in diameter. This loca- tion was chosen due to time constraints in the field and previ- ous research has suggested that branch tips behave as a single branch (Dahle 2009). Secondary branch length, diameter, de- parture angle, point of attachment along first-order branch (POA), mass, and CG were recorded. Second-order branches were recorded as arising from either the left or right side of the first-order branch, looking towards the tip. Mass was mea- sured within two hours of branch harvest with leaves in place using an Ohaus CD11 bench scale, measured to the nearest 10 g. Data was not recorded for branches weighing less than 10 g. Lateral balance (Bi ), an estimate for torsional mo- ment applied by a second-order branch at the proxi- mal end of a first-order branch, was calculated (Formula 2). Overall balance (Tbal calculated using Formula 3 and percent imbalance of the esti- mated torsional moments (TPct ) along a first-order branch was ) were calculated using Formula 4. of a first-order branch is determined with the following for- mula. Numeric subscripts denote a measurement obtained from 1) first-order branch or 2) second-order (lateral) branch. Secondary-order branch lateral load (Bi [2] B1 = G × M2 × Sin(A) × CG2 × POA1 Figure 1. The number of first-order Tilia cordata branches and the percent imbalance of the estimated lateral moments for the first order branch. ©2012 International Society of Arboriculture ) acting at the base Dahle and Grabosky: Lateral Branch Imbalance where A = branch attachment angle, Bi N (m/kg)2 = lateral branch force, , M = lateral branch mass (kg), and POA = distance from lateral branch to first-order point of origin (cm). [3] [4] BRight BLeft CG = center of gravity (cm), G = gravitational constant 6.674 × 10-11 Overall balance (Tbal) of the first-order branch is as follows: TBal = ∑ ∑− Percent imbalance of the first-order branch: TBal / ( TPct = ∑ ∑+ BRight BLeft ) Data were analyzed in SAS 9.2 (SAS Institute), except stan- dardized major axis (SMA) regression which used SMART 2.0 (Falster et al. 2006; Warton et al. 2006). Ordinary least squared (OLS) were run using Proc Reg, and Proc Univariate was used to verify normality of the variable and residuals. All variables were determined to be approximately normal, or ad- justed using a Log10 transformation. All statistics used α = 0.05. RESULTS AND DISCUSSION Mean branch length, diameter, slenderness, BM, and TPct for all first- and second-order branches are noted in Table 1. The aim was to select first-order branches that could be called structural branches, and the shortest first-order branch sampled was 370 cm. Dahle and Grabosky (2010) suggested that 300 cm is the point at which branches growing on large shade trees, such as maple (Acer), transition from sun branches (solar collectors) to struc- tural branches. Mean length for first-order branches in the current study was 477 cm (Table 1), and it appeared that these branch- es had transitioned to a structural role as nine of the first-order branches had second-order (laterals) that were longer than 300 cm. Table 1. Mean ± SD for mean branch length, diameter, mass, and percent imbalance (TPct Tilia cordata branches. Variables Length (cm) Diameter (cm) Mass (Kg) TPct N First-order branches 476.9 ± 60.3 6.1 ± 1.1 8.7 ± 4.3 0.43 ± 0.24 15 Second-order branches 180.5 ± 86.3 2.0 ± 0.9 0.6 ± 0.8 - 110 ) for all first- and second-order
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