Arboriculture & Urban Forestry 38(4): July 2012 from 7% to 83%. Since more than 60% percent of the first-order branches were imbalanced by 40% or more (Figure 1), it appears that the branches were likely preloaded with a torque. While this limited data set may not be representative for all tree branches it does suggest that normal development can lead to branches that are preloaded to one side, at least in branches with alternate at- tachment. The branches in this study were growing in all four cardinal compass directions (N, S, E, and W), and so research- ers did not feel as though wind or light was a factor in overall branch imbalance. The only pruning on these trees was to raise the canopy for lawn mowing. The test branches did not appear to have any pruning scars and were located above the bottom of the canopy and thus were deemed unaffected by any canopy raising. Interestingly, twelve of the fifteen branches had the greater to the left side of the branch, while three had greater moments Overall, the first-order branches were not found to be bal- anced, as the mean percentage for TPct was 43%, with a range log branch length (r2 (r2 on the right. Although it is unknown as to why this occurred it does not appear to be directly related to the solar position, as the branches were growing in all cardinal directions. This may be a function of growth in this particular species, yet more research is needed to determine if this apparent pattern is consistent across species or phyllotaxy (opposite versus alternate). OLS regres- sion did not identify a relationship between TPct TPct and either the = 0.01, N = 15) or the log branch diameter = 0.01, N = 15). In total, the results suggest that the first-order branch allometry may not be directly affected by a lateral load- ing imbalance, although more research is needed to confirm this. Modulus of rigidity, strain, or resulting shear stress were not measured in these branches, and therefore whether the apparent unbalance leads to the development of strains in the branch wood cannot be addressed. Branches most likely adapt to moderate strains during yearly radial growth, and therefore overall shear stress due to self-loading may not be a concern. Further research might be warranted to determine the levels of shear strain and stress during branch development and ascertain if selective re- moval during pruning operations leads to an increase in overall shear stress either during the short or long-term. The removal of a few branches on the less loaded side might further tip the im- balance resulting in a branch with a higher risk of failure dur- ing a loading event, such as wind, snow, or ice. Alternatively, reduction pruning could reverse the loading of an imbalanced branch, which could increase the risk of tissue delamination. It must be pointed out that the loading imbalance identified in this study may only be valid for branches that have recently transitioned to structural branches. Imbalance may increase or possibly decrease as branch development continues. Addition- ally, the branches in this study displayed alternate branch at- tachment; it is not known if these findings will be similar to all alternate branching trees or to those with opposite branching. It has been pointed out that trees build wood in a man- ner consistent with a need to protect against bending, which results in high flexural:torsional rigidity (Vogel 1995; Vogel 1996). Such adaptations have been invoked to suggest protec- tion against failure in bending is more important than failure in torsion (Vogel 1995). Yet it appears that normally devel- oped branches can be preloaded with a considerable torsional component. The inclusion of torsion may be important when investigating branch failure, especially in failures that occur during dynamic loading events. Researchers should consider 143 including torsion in biomechanical models of branches and tree crowns before assuming an insignificant role in analysis. The log of branch mass was significantly related to log branch length (log BM = -6.48 + 2.69 × log length, r2 = 0.83, N = 124) and log branch diameter (r2 = 0.95, N = 124, Fig- ure 2), both equations were determined using a SMA regres- sion. This study confirms what previous research has shown in that branch diameter can be a good predictor of branch mass (Grabosky et al. 2007). If arborists are interested in estimating pruning dose, one method could be to use the branch diameter to estimate branch mass for a given species. Additionally, a sig- nificant SMA log-log relationship was identified between mass and center of gravity (log mass = -5.54 + 2.74 log center of grav- ity, r2 = 0.83, N = 124). As a relationship exists between mass and center of gravity, two important components of a branch’s mechanics, future researchers may wish to investigate how the removal of a lateral branch or a branch subordination alters not only the overall mass of a given branch but the center of grav- ity and thus potentially the biomechanical makeup of the branch. While no relationship was found between the log of a branch’s center of gravity and TPct (r2 = 0.015, N = 15), significant OLS = 0.76), yet a stronger relationship was found with log branch length (r2 = 0.95, Figure 3). This suggests that it is pos- relationships were identified between a branch’s center of grav- ity and the branch’s diameter and length. The relationship with branch diameter was good (log CG = 1.59 + 0.90 × log diam- eter, r2 sible to estimate the center of gravity for a given branch with a high degree of confidence, which could be useful during failure testing exercises. Researchers would only need to remove a lim- ited number of branches to derive a quick equation to provide an accurate estimate for the center of gravity and mass that could be used when modeling self-loading and lateral balance in branches. It appears that development can lead to unbalanced branch- es. While it is not known if the unbalanced nature of branches will be found beyond T. cordata branches, this knowledge might be an important consideration during pruning opera- tions. Research studies should consider whether it is appropri- ate to include torsion when utilizing the normal stress equation to predict how branches will behave during loading trials. Ad- ditionally, if research trials utilize a single attachment point, this data suggests that it is possible to develop a strong predictive equation for estimating where a branch’s center of gravity lies. Figure 2. Log-log SMA relationship between branch mass and di- ameter for first and second order Tilia cordata branches. Log BM = -1.29 + 2.82 log diameter, r2 = 0.95, N = 124. ©2012 International Society of Arboriculture
July 2012
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