72 Kane et al.: Forces and Stresses Generated During Rigging Operations the trunk, however, dissipate energy, which accounts for small- er strains measured 1 m above ground. Future investigations should attempt to measure deflection near the rigging point. It was not surprising that mass was by far the best predic- tops and pieces. The equation for tops (•) was TM (r2 = 0.64); the equation for pieces () was TM Figure 4. Measured (TM ) vs. predicted (TP ) tension in the rope for = 0.19*TP 0.55). Tension in the rope was predicted from Equation 1 in the text. = 0.21*TP tops and pieces, tension in the rope was only 12% and 19%, respectively, of predicted tension from equation 1 (Figure 4). Physical Parameters and Stress Tops were more massive and fell a greater distance before being stopped by the lowering rope than other pieces (Table 2). Fall fac- tor, however, was greatest for tops and the last piece taken from each tree (piece 4) (Table 2). Tops were also longer than all pieces (Table 2). Trunk stress was greater for tops than the first two piec- es removed after the top, but there were no other differences in stress among pieces (Table 2). None of the independent variables was a significant predictor of stress (data not presented), and the overall model was rather weak (adjusted R2 = 0.12, P = 0.0350). DISCUSSION The law of conservation of energy stipulates that energy is nei- ther created nor destroyed, but rather, changed from one form to another. In the case of a piece of wood removed from a tree, it has potential energy by virtue of its height. When it falls, the potential energy is converted to kinetic energy, which then does work in one of three ways: elongating the rope (axial strain en- ergy in the rope), deflecting the trunk (bending strain energy in the tree), and displacing air. Of the three, rope elongation appears to be less significant, in light of the multiple regres- sion models. Neither of the other types of work was quantified rigorously, but drag on pieces as they fell was arguably negli- gible. For tops, which still had foliated branches, the effect would have been greater, as others have observed (Mayhead et al. 1975, Detter 2008). Video images of the trunks clearly illus- trate their violent oscillations, so it does not seem unreasonable to expect that pieces did considerable work to deflect trunks. This observation, however, was not fully supported by the data, as stress generated by tops and all but the smallest pieces was similar. This disparity was an artifact of experi- mental procedure, as stress was measured on the trunks 1 m above ground. Beam mechanics predicts that the deflection of a tapered, cantilevered beam is proportional to the cube of its length (Soltis 1999), so the longer trunk would deflect more when loaded at the rigging point. Large deflections higher up ©2009 International Society of Arboriculture -264 (r2 -1821 = tor of force at the block and tension in the rope for both tops and pieces, since it influences potential energy. It was initially surprising, however, that predictions of force at the block and tension in the rope were different for tops and pieces. Since trunks were longest and most slender when tops were removed, elementary beam mechanics predicts that trunk deflection will be greatest for tops because deflection is proportional to the cube of the length and inversely proportional to the fourth power of diameter of a beam (Lardner and Archer 1994). Video observa- tions also support this idea, which reflects a greater proportion of potential energy of tops doing work to deflect the trunk. Thus, a smaller proportion of potential energy can do work to elongate the lowering rope. The findings that tops had similar values for force at the block and tension in the rope to other pieces (except for piece 1) but the smallest normalized values for force at the block and tension in the rope also supports this reasoning. Drag on the top as it fell may also have slowed its descent, and while this effect was not quantified, Detter (2008) observed reduced ve- locity of removed pieces when they still had branches attached. Other variables expected to influence force at the block and tension in the rope (e.g., notch depth and angle, distance of fall, and length of rope in the system) were less influential. Fall factor did not predict force at the block or tension in the rope because equation 1 is most appropriate when the rope is elastic. Ropes used in rock climbing [for which equation 1 was originally derived (Pavier 1998)] are designed to stretch in order to reduce tension in the rope as it decelerates and stops a falling climber. Typical elon- gation in a rock climbing rope is 20%–30% during a fall, whereas the elongation of Stable Braid is only 1.1% at 10% of tensile strength (Samson 2008). Elongation tends to be bi-modal, with greater elongation when the rope is new and less elongation after the rope has been broken in (McLaren 2006). Since the rope used in the present study was not new, it is doubtful that elongation ex- ceeded 1.1%. However, some elongation would occur due to tight- ening of knots and slippage of the slings that held the block and port-a-wrap in place on the trunk. Indeed, for two tops, the force at the block was great enough to strip bark from the trunk as the sling and block slid about 0.5 m down the trunk. Incidentally, the latter may have also been a source of energy loss (due to friction). Distance of fall did not predict force at the block and tension in the rope for pieces because all pieces were the same length, an artifact of experimental design. Since tops and pieces behaved differently, it was not prudent to perform a multiple regres- sion with tops and pieces together, because it was not possible to separate the other effects from fall distance. For tops, how- ever, the paradoxical finding that distance of fall was inversely proportional to force at the block and tension in the rope may also be attributed to greater trunk deflection as the top released from the trunk, but careful measurements of trunk deflection will have to be made before this attribution can be confirmed. Depth and angle of the notch were also expected to influence force at the block and tension in the rope because shallower and deeper notches presumably cause the cut piece to release sooner from the trunk, which means they would have greater velocity (and thus would require greater stopping force). The data do not generally support this intuition, and while depth of the notch was
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