42 Avalos and Sánchez: Evaluation of Failure Criteria in Branch Members ing to the 1% percentile may be used to obtain the Xmin To estimate the X mean value, denoted by X̅ , it and Zmin is calculated that the mean value of the test results be obtained under pure bending only. Experi- mentally, it is obtained as X̅ = 75.5 MPa, simi- larly for pure torsion, obtained as Z̅ = 11.2 MPa. Table 1 summarizes the statistics for the tests under pure bending and for the pure torsion case. A tension test was performed to complete the set of loads applied to the branch (see Figure 9). The test showed a peak stress value at approxi- mately 24.15 MPa, which corresponded to Point 1 of the Figure 9. In this event, during the test, Table 1. Maximum, minimum, mean, and standard devia- tion of the stress values. Min psi Max psi Mean psi (MPa) Pure moment Pure torsion 7774 (53.6) 600 (4.1) (MPa) 16739 (115.4) 1985 (13.7) (MPa) 10942 (75.5) 1621 (11.2) Std. psi (MPa) 2365 (16.3) 385 (2.7) values for the Tsai-Hill criterion. a cracking sound occurred, indicating that not all fibers have the same flexibility, and that the more rigid fibers were carrying a higher load than the more elastic fibers. After the peak, some rigid fibers were broken and the more elastic fibers started carrying higher load. This implies that the effective area is less than the nomi- nal area. The final stress value of 46.2 MPa, as shown in Figure 9 as Point 2, is lower than the pure bending stress value, with mean stress of 75.5 MPa. Thus, it can be concluded that the ten- sion stress calculated with nominal area does not provide the true stress along the cross section. DISCUSSION In the presence of torque loads, a considerably lower bending moment was required to break the branch. Torsion affects branch resistance against applied loads in two main aspects. First, torsion displaces the fibers relative to each other, requir- ing lower stress values than are needed to break the fibers aligned across the branch axis. This is con- firmed when calculating the strength due to shear (Z value), which is considerably lower than the strength needed to cut the fibers (X value). Second, torsion reduces branch flexibility, which results in lower deflection and higher bending moment as compared to a very flexible branch with both tor- sion and bending applied with the same transverse force, similar to the effect of wind across the branch. The mean ultimate stress values obtained for pure Figure 9. For the tension tests, there were three changes in the slope of load versus deformation as fibers breaks, starting from the stiffer ones to the flexible ones. Point 1 corresponds to failure of the most rigid fibers and Point 2 corresponds to the final stress upon failure of the whole branch. stress value found on the branch of the Fraxinus uhdei tree and should thus be considered as a ref- erence limit for risk assessment of the branch. In a more rigorous approach, the maximum value that the branch can withstand is probabilistic, represented by a probability density function obtained from the recollected experimental data. Zmin ©2014 International Society of Arboriculture bending X̅ = 75.5 MPa, and the mean stress values for pure torsion Z̅ = 11.2 MPa, where the ratio of the mean value of pure bending X̅ over the mean value for pure torsion Z̅ gives a value of 6.74, con- sistent with calculated values reported out of 178 species (see Niklas and Spatz 2010). Note that the ratio using the Von Mises criterion is fixed to the value of 1.73 (square root of three), a value much lower than what was observed experimentally. The minimum stress values Xmin = 53.6 MPa and = 4.1 MPa represent the minimum maximum
January 2014
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