Arboriculture & Urban Forestry 40(1): January 2014 bination of the torsion and bending moment loads on the branch failure is not considered. Research that considers both bending and tor- sion in branch failure is a more realistic approach, as both are present in the tree in natural condi- tions. The branch’s own weight, inertial forces, wind interaction with leaves and branches, and possible forces due to an external element like a rope attached to a branch, or an external boundary to the branch, like that of a building or wall, rep- resent different scenarios where both torsion and bending might be present. To avoid unpredictable branch failure, it is important to identify if tor- sion is present in the tree branches, determine its effect on the structural integrity of the branch, and finally, identify which values characterize the ulti- mate stress value the branch can withstand to forces. The current investigation originated aſter visu- ally comparing branch cross sections that failed naturally against those that failed in a controlled environment. Three cases were reproduced: pure bending, pure torsion, and combined. The section of the branches that failed in real conditions did not match with the characteristics of what pure bending moment generates in the failure zone. In this paper, researchers present a study of failure in branches by calculating the internal stresses due to external forces of bending moment and torsion at the point of failure; this is achieved by probing a destructive test to the branch under both loads. Furthermore, this study aims to evaluate two different failure criteria, and to explain the dis- tinction between the applied forces classified as axial force, bending moment, and torsion. MATERIALS AND METHODS Of the variables to consider in most structural failure criteria, the stress tensor contains up to six independent stress values. If the structural element is considered an axial element, like a beam, which is a reasonable assumption for a branch, then re- searchers can consider three types of forces than can be applied, namely axial, bending, and torsion. Axial forces consist in pulling or compressing the branch, where the force direction is parallel to the branch core as shown in Figure 1a. Bending mo- ment is present from the branch’s own weight as well as when there is an external weight on the branch (e.g., snow) and whose direction is perpen- 37 dicular to the branch cross section, as depicted in Figure 1b. Lastly, torsion, being a consequence of torque, is directed around the branch axis direc- tion as seen on Figure 1c, where the torsion sym- bol is expressed as a line along the rotation axis. The primary forces that break a branch consist of two element forces. One force is perpendicu- lar to the branch, generating bending moment, the second is torque acting on the axis of the branch, creating torsion by the subsequent smaller branches that generate a force at distance by being attached to the main branch, as seen in Figure 2. Shear stresses appear in a cantilever beam when a force perpendicular to the beam axis is applied (Fig- ure 1b); this perpendicular force is commonly known in strength of materials as shear force, which gener- ates a shear stress at the neutral fiber of the branch (the neutral fiber is where there are no longitudinal strains, for a symmetric section it occurs at the geo- metric centroid). However, this shear force does not induce shear stress at the cross section of the branch where the maximum stress is located, due to the fact that bending occurs, thus, the shear stress does not need to be considered (Ferdinand et al. 2011) The stress tensor for the critical point when the branch is subjected to a bending moment and torsion is shown in Figure 3 and is labeled as Point 1. At said point, the maximum stress Figure 1. (a) Axial load acting along the axis of the branch, b) bending moment tends to rotate the branch around the z axis, and c) torsion tends to rotate the branch around the x axis. ©2014 International Society of Arboriculture
January 2014
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