ACCURACY OF FORMULAS USED TO ASSESS STRENGTH LOSS DUE TO DECAY IN TREES, THE

Journal of Arboriculture, Nov 2004 by Kane, Brian C P, Ryan, H Dennis P III

Abstract. There are four formulas that arborists in the United States often use to assess the probability of tree failure. Although they are commonly referred to as "strength loss formulas," three of the formulas (Wagener 1963; Coder 1989; Smiley and Fraedrich 1992) actually estimate the loss in stem moment of inertia (I^sub STEM^) to evaluate the probability of failure. The formulas estimate the loss in I^sub STEM^ by modeling the stem and decay cross-sectional areas as concentric circles. For many trees with decay, however, neither the stem nor the decay cross-sectional areas resemble concentric circles, which may limit the formulas' accuracy. The fourth formula (Mattheck and Breloer 1998) is based on the buckling strength of a cylinder; it also offers a measure of the probability of stem failure. To test how well the formulas estimate the loss in I^sub STEM^, we compared each formula's estimates for loss in I^sub STEM^ to the actual loss in I^sub STEM^ that we calculated using the parallel axis theorem, an engineering technique. Although the parallel axis theorem provides the actual loss in I^sub STEM^, it cannot be used in practice because an image of the tree's cross-section is needed to apply it. Significant differences existed between two formulas' (Wagener 1963; Coder 1989) estimates of loss in I^sub STEM^ and the actual value. Each of the formulas misclassified some trees as to whether they exceeded the formula's action threshold. When we calculated the actual loss in I^sub STEM^ for those trees, however, it was less than 33%. We present representative stem cross-sections for which the formulas did not accurately represent loss in I^sub STEM^.

Key Words. Tree hazard; tree risk assessment; stem moment of inertia; decay; mechanical stress.

Arborists and urban foresters often assess the risk of tree failure, and through experience and research, have developed guidelines to improve the assessment procedure. Annually, however, tree failures continue, damaging property, injuring people, and leading to costly insurance claims and, sometimes, litigation. Therefore, it is vital that arborists continue to refine risk assessment, incorporating new research data in the procedure. One tool in tree risk assessment is the use of formulas to estimate stem strength loss due to decay, which, in turn, estimates the probability of stem failure. The formulas, which are based solely on the cross-sectional geometry of the tree stem and decay area, are based on engineering beam mechanics and had been previously tested only with ex post facto studies and observation (Kane et al. 2001). Considering the large number of variables that affect tree failure and the risk of damage that can result from failure, a reevaluation of the formulas seemed appropriate.

Because of the way I is calculated (see Appendix 1), as tree diameter increases, the moment of inertia of the stem (I^sub STEM^) increases exponentially. In other words, a 4 cm (1.6 in.) diameter stem has a moment of inertia (i.e., resists bending) four times greater than a 2 cm (0.8 in.) diameter stem, even though the diameter is only twice as large. Another consequence of the way I is calculated is that the outer wood fibers of the stem contribute exponentially more to I^sub STEM^ than the inner wood fibers. Furthermore, unless the stem cross-section is symmetrical, the direction in which a force acts on the tree will also affect I^sub STEM^. We can use the analogy of bending a wooden ruler to illustrate this fact. Grasp the ruler with both hands, fingers on one edge and thumbs on the opposite edge. Try to bend the ruler into an are by exerting a downward force with the fingers and an upward force with the thumbs; the ruler does not move much. Conversely, grasp the ruler with both hands, fingers on the top side of the ruler (side with unit markings) and thumbs on the underside. It is much easier to bend the ruler by applying a downward force with the fingers and an upward force with the thumbs.

Throughout the paper, we will refer to the axis about which I is calculated; it is the axis perpendicular to the applied force that I resists (in Figure 1, the neutral axis is the axis about which I is calculated). It is easy to imagine a tree cross-section that is not perfectly circular; for such a tree, I^sub STEM^ would be different depending on the direction in which a force acted (for further explanation, see Appendix 1).

When decay is present, I^sub STEM^ is reduced and the degree to which it is reduced depends on the size and location of the decay area. Larger areas of decay reduce I^sub STEM^ exponentially more than smaller areas of decay. When decay occurs off-center in the stem cross-section, I^sub STEM^ is also reduced exponentially, even if the area of decay remains the same. This is so because the outer wood fibers are removed. For the same reason, cavities also reduce I^sub STEM^ exponentially. We explain these concepts in more detail in Appendices 1 and 2. For the derivation of I and its significance to beam mechanics, refer to Niklas (1992), or Beer and Johnston (1988).