70 Kane et al.: Forces and Stresses Generated During Rigging Operations strains (ε) in the trunk were measured normal and parallel to the applied load. The static elastic modulus (E) of the trunk was determined from stress and strain using Hooke’s Law: [4] E = σ / ε. The value of E reflects the entire tree, including rota- tion of the root plate as the tree bends. This value was likely less than E values of the wood of the tree, but more consis- tent with a tree in situ. Strains measured during rigging opera- tions, were converted into stresses using each tree’s empirically- determined E value and a re-arranged version of equation 4: [5] σ = Eε. Two stress values were collected for each top and piece, one measured parallel and one measured normal to the direction of fall. Measured strains reflect the dynamic modulus of elasticity, which, for timber is typically 5%–10% greater than the static elastic modulus (Bodig and Jayne 1993). Thus, the actual rig- ging-induced stress was likely 5%–10% greater than presented. The percentage of the trunk cross-section that could be de- cayed was calculated by mathematically reducing the cross-sec- tion (i.e., reducing its second moment of area) until the maximum rigging-induced stress in the trunk equaled the strength of the tree. Strength of the tree was taken as 80% of the lower confidence inter- val of the average green wood strength of red pine from the Wood Handbook (Green et al. 1999). Previous studies have demonstrat- ed that the strength of trees is approximately 80% of the strength of wood samples (Fons and Pong 1957; Kane and Clouston 2008). Felling notches were assigned randomly with respect to width and depth. For tops, notches were classified as wide (i.e., >60º) or narrow (i.e., <30º); and as deep (i.e., notch depth > 50% of diameter at the cut) or shallow (i.e., notch depth < 30% of diameter at the cut). In practice, notch depth and angle var- ied from the intended classification, so measured values were used in the analysis. For pieces, the classifications were the same, except that bypass cuts were also tested. Bypass cuts were considered to have an angle of 0º and depth was not con- sidered in the analysis. The theoretical principle behind test- ing notches was that notch width and depth would presumably affect the time it took the hinge to break. This, in turn, would affect both the time of free fall and how much of the potential energy of the piece was converted to strain energy in the trunk. Using PROC REG of the SAS statistical analysis software (v. 9.1, SAS Institute, Cary, NC), two multiple linear regression models were built to determine which independent variables best predicted both force at the block and tension in the rope. The first model considered the following independent variables, chosen on the basis of their physical or practical importance: mass of piece, fall distance (which is twice the distance from the center of gravity of the piece to the sheave of the block), length of rope in the rigging, notch depth, and notch angle. A second model mim- icked the first, but considered fall factor instead of its individual components. Preliminary investigation of scatter plots revealed 1) mass strongly influenced force, and 2) the forces generated by tops and pieces exhibited different relationships with mass. The latter observation was tested using dummy variables and simple linear regression to compare slopes and intercepts of the best-fit lines for forces generated by tops and pieces as predicted ©2009 International Society of Arboriculture from mass. As a result of the preliminary analyses, both mul- tiple linear regression models were repeated twice: first, tops and pieces were analyzed separately; second, force was divided by mass (the quotient is acceleration) and mass was not considered in the regressions. The MAXR option (SELECTION=MAXR) was used to add components to each model in a stepwise fashion. Multiple linear regression was also used to investigate whether pertinent physical parameters such as mass, height of piece above ground, fall distance, length of rope, (fall fac- tor was analyzed in a subsequent model as with the models to predict force), diameter at breast height, notch angle, or notch depth influenced stress, but tops and pieces were not analyzed separately. An analysis of variance (ANOVA) was used to de- termine whether force at the block (and force divided by mass), tension in the rope (and tension divided by mass), stress, strain, mass and length of pieces, and fall distance varied among piec- es. Simple linear regression was used to investigate the rela- tionship between 1) predicted force from equation 1 and mea- sured force, and 2) force at the block and tension in the rope. RESULTS Force at the block and tension in the rope increased at a greater rate with increases in mass for pieces than for tops (Figure 2). Fall factor was not a significant predictor of force at the block, tension in the rope, or the normalized versions of those mea- sures (i.e., after they were divided by the appropriate mass), so the remaining results refer only to the regression models that included distance of fall and rope length as separate vari- ables. For all trees, the mean modulus of elasticity (standard deviation in parentheses) was 4,674 (1,440) MPa [678 (209) ksi] when measured in the direction of probe 1 and 4,569 (993) MPa [663 (144) ksi] for probe 2, a difference of 2.2%. Figure 2. Prediction of force at the block by mass of the piece; the equation for pieces () was force = 89.2*mass+1644 (r2 the equation for tops (•) was force = 57.4*mass+84 (r2 = 0.80); = 0.94). Al- though the intercepts for the two prediction equations were not significantly different, the slopes were (P < 0.0001). Slopes and intercepts for the prediction of rope tension from mass were also different between tops and pieces. Force at the Block For pieces, mass was the best and only significant predictor of force at the block, and the regression model was fairly robust
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