44 Avalos and Sánchez: Evaluation of Failure Criteria in Branch Members torsion and bending moment load to predict failure at Section 1. Another case is shown in Figure 11b, where one adjacent branch is identified and labeled as Branch B, which will generate torsion to Branch A at the attachment Node 1. For this case, bending moment load is dominating; however, torsion load should be considered as well in the failure analy- sis prediction to give an accurate risk assessment. CONCLUSION The effectiveness of the Tsai-Hill failure criteria to predict the reduction of the ultimate stress value for branch failure of Fraxinus uhdei. The results clearly show that neglecting the presence of torsion, cor- responding to the frame used in the Von Mises criteria, underestimates the risk of branch failure. This analysis has not previously been addressed on healthy branches and for the Fraxinus uhdei tree. The results shown in this study can be comple- mented with literature providing maximum stress value due to bending and the shear stress due to torsion (see Niklas and Spatz 2010). Such values, and the Tsai-Hill failure criteria, should be used to predict the failure condition for any combina- tion of bending moment and torsion. Although the branches are expected to be largely domi- nated by the bending moment loads, torque loads should not be ignored in the risk assessment. Acknowledgments. The authors wish to thank Dr. Alfredo Cueva (ITESO University) for his expertise and guidance in understand- ing the details behind the theory of composite materials. The authors also wish to thank to Mrs. Alejandra Lopez (University of Arizona) and the anonymous reviewers who kindly improved the quality of the text. LITERATURE CITED Azzi, V.D., and S.W. Tsai. 1965. Anisotropic Strength of Compos- ites. Experimental Mechanics 5(9):283–288. Ennos, A.R., and A. van Casteren. 2009. Transverse stresses and modes of failure in tree branches and other beams. Proc. R. Soc. B2010 277. 1253–1258. Ferdinand, B., Jr., J. Russell, J. DeWolf, and D. Mazurek. 2011. Mechanics of Materials, McGraw-Hill. Hill, R. 1950. The Mathematical Theory of Plasticity. Oxford, Clarendon Press. Kane, B. 2007. Branch strength of Bradford pear (Pyrus calleryana var. ‘Bradford’). Arboriculture & Urban Forestry 34:101–109. Matheny, N.J., J.R., Clark, D. Attewell, K. Hillery, A. Graham, and G. Posner. 1999. Assessment of fracture moment and fracture angle in 25 tree species in the United States using fractometer. Journal of Arboriculture 25:18–23. Mattheck, C., H. Breloer, K. Bethge, W. Albrecht, and A. Zipse. 1995. Use of the Fractometer to determine the strength of wood with incipient decay. Journal of Arboriculture 21(3):105–112. Mattheck, C., H. Breloer, and K. Bethge. 1994. A guide to Fractom- eter tree assessment. Arborist News 3(2):9–12. Mattheck, C.G., and K.A. Bethge. 1993. Detection of decay in trees with the metriguard stress wave timer. Journal of Arboriculture 19(6):374–378. McCracken, F.I. 1985. Using sound to detect decay in standing hardwood trees. pp. 281–287. In: Proceedings, 5th Symposium on Nondestructive, Testing of Wood, 9–11 September 1985. Washington State University, Pullman, Washington, U.S. Miller, V.J. 1959. Crotch influence on strength and breaking point of apple tree branches. Journal of the American Society for Horti- cultural Science 73:27–32. Niklas, K.J., and H.C. Spatz. 2010. Worldwide correlation of me- chanical properties and green wood density. American Journal of Botany 97(10):1587–1594. Smiley, E.T., C.M. Greco, J.G. Williams, 2000. Brace rods for co- dominant stems: Installation location and breaking strength. Journal of Arboriculture 26:170–176. Van Casteren, A., W. Sellers, S. Thorpe, S. Coward, R. Crompton, and A.R. Ennos. 2012. Why don’t branches snap? The mechan- ics of bending failure in three temperature angiosperm trees. Tress: Structure and Function 26:789–797. Von Mises, R. 1913. Mechanik der festen Körper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys.. 1:582–592. Wang, X., and R.B. Allison. 2008. Decay detection in red oak trees using a combination of visual inspection, acoustic testing, and resistance microdrilling. Arboriculture & Urban Forestry 34(1):1–4. Wang, X., F. Divos, C. Pilon, B.K. Brashaw, R.J. Ross, and R.F. Pellerin. 2004. Assessment of decay in standing timber using stress wave timing nondestructive evaluation tools–A guide for use and interpretation. Gen. Tech. Rep. FPL GTR–147. U.S. De- partment of Agriculture, Forest Service, Forest Products Labo- ratory, Madison, Wisconsin, U.S. 12 pp. Yamamoto, K., O. Sulaiman, and R. Hashim. 1998. Nondestructive detection of heart rot on Acacia Mangium trees in Malaysia. Forest Products Journal 48:83–86. Javier Avalos (corresponding author) Department of Mechanical Engineering Western Institute of Technology ITESO Periferico Sur 8585 Tlaquepaque, Jal. Mexico 45604 Adolfo Sánchez Department of Mechanical Engineering Western Institute of Technology ITESO Periferico Sur 8585 Tlaquepaque, Jal. Mexico 45604 ©2014 International Society of Arboriculture
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