The Accuracy Of Formulas Used To Assess Strength Loss Due To Decay In Trees

• 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.

The formulas model the tree as a cantilevered beam and look at the cross-sectional geometry of the stem (beam), including decay, to estimate the probability of failure. To determine the probability of failure of a beam using a static engineering analysis, one needs three pieces of information: (1) the force acting on the beam, (2) the beam’s crosssectional geometry, and (3) the beam’s material properties [i.e., the material’s breaking strength, or modulus of rupture (MOR)]. The first two pieces of information can then be used with the following equation, which an engineer uses to calculate bending stress in a beam: 1

where (σ) is bending stress (a measure of force per unit area), (M) represents the applied force or bending moment, (y) is the distance from the neutral axis of bending to the edge of the beam (Figure 1), and (I) is the moment of inertia of the beam. The moment of inertia of any beam (including a tree stem) is based only on the geometry of the beam’s cross-section and essentially measures the beam’s resistance to bending. From Equation 1, it is clear that the stress calculation does not depend on the beam’s material properties. Material properties are still important, however, as the following example illustrates. Consider two trees, a red oak (Quercus rubra L.) and a white willow (Salix alba L.) with the same dbh. If the same force acts on each tree, the stress each tree endures is the same by Equation 1. Because the oak wood is stronger (i.e., has a greater MOR) than the willow wood, the willow cannot endure as much stress as, and thus will fail before, the oak.

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Figure 1.

A beam of circular cross-section to which a bending moment (M) is applied; (y) is the distance between the neutral axis of bending (which is perpendicular to M and shown as the dashed line) and an outside edge of the beam. In this case, y equals the radius of the circle. The neutral axis is also the axis about which the moment of inertia would be calculated.

Because of the way I is calculated (see Appendix 1), as tree diameter increases, the moment of inertia of the stem (I_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_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_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 arc 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_STEM would be different depending on the direction in which a force acted (for further explanation, see Appendix 1).

When decay is present, I_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_STEM exponentially more than smaller areas of decay. When decay occurs off-center in the stem cross-section, I_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_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).

Three of the strength loss formulas (Wagener 1963; Coder 1989; Smiley and Fraedrich 1992) actually estimate the percent loss in I_STEM due to decay (Kane et al. 2001). This is an important clarification, because in assessing the probability of failure, the formulas do not consider the force acting on the tree nor do they consider the tree’s wood strength. They only consider the amount of decay and stem diameter to estimate the percent loss in I_STEM and use that percentage to assess the probability of failure. Wagener’s (1963) and Coder’s (1989) formulas, however, do not account for off-center decay areas or cavities on the trunk. Instead, their formulas estimate the tree cross-section and decay areas as concentric circles, like a pipe. The Smiley and Fraedrich (1992) formula also approximates the tree crosssection and decay areas as concentric circles, and it accounts for trunk cavities. Fraedrich (1999) created a table of sound wood thicknesses based on the Smiley and Fraedrich (1992) formula. To avoid confusion, we will refer only to the Smiley and Fraedrich (1992) formula in this paper. Because the formulas consider trunk cross-sections and decay areas to be concentric circles, they may not accurately reflect the actual loss in I_STEM.

Mattheck and Breloer’s (1998) formula is based on the buckling strength of a cylinder. Developed by examining failed and standing trees with decay, it is an empirically based threshold [i.e., the ratio of sound wood thickness (t) to tree radius (R), t/R]. The t/R ratio does not give loss in I_STEM values; it only presents the threshold at which remedial action (like removal or crown reduction) should occur. Although this approach accounts for off-center decay areas, it is not directly comparable with the loss in I_STEM. Mattheck and Breloer’s (1998) formula considers cavities less than one-third of the stem circumference having no effect on failure.

Each formula also has an associated threshold for action, either percent loss in I_STEM (Wagener 1963; Coder 1989; Smiley and Fraedrich 1992) or sound wood thickness (Mattheck and Breloer 1998). In other words, when the formula returns either a percent loss in I_STEM or a t/R value that exceeds the threshold, the risk of failure merits some type of remediation. Each formula’s threshold assumes that the only stem defect is decay, and that no complicating factors exist. Wagener’s (1963) threshold is 33% loss in I_STEM. Values greater than 33% loss in I_STEM imply that the tree should be classified a hazard. Coder (1989) proposed two thresholds; between 20% and 44% loss in I_STEM, the tree is classified in the “caution” zone; with greater than 44% loss in I_STEM, the tree is classified a hazard. Smiley and Fraedrich (1992) also use 33% loss in I_STEM for a threshold. Mattheck and Breloer (1998) consider cross-sections with t/R < 0.3 a hazard. Table 1 summarizes the thresholds.

To test the accuracy of each formula’s estimates for loss in I_STEM, we compared each formula’s estimates to the actual maximum loss in I_STEM for trees with cavities and decay. We calculated the actual loss in I_STEM by applying the parallel axis theorem to the trees. Although the parallel axis theorem gives the actual loss in I_STEM, it cannot be readily applied in practice because its use requires a clear view of the stem cross-section (e.g., Figures 2 and 3). For a detailed treatment of the parallel axis theorem, see Appendix 2 or Beer and Johnston (1988).

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Figure 2.

A typical tree underestimated by the strength loss formulas (i.e., each formula’s estimate for loss in I_STEM did not consider that formula’s threshold even though the actual loss in I_MAX exceeded each formula’s threshold). Mattheck and Breloer’s (1998) t/R ratio did not underestimate this tree.

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Figure 3.

A typical tree overestimated by using the t/R ratio (i.e., t/R < 0.3, but the actual loss in I_MAX < 33%). Mattheck and Breloer (1998) noted that when t/R = 0.3, loss in I_STEM ≈ 33%.

MATERIALS AND METHODS

To compare the formulas’ estimates for loss in I_STEM to the actual loss in I_STEM, we examined 19 trees with cavities and decay [described in Kane and Ryan (2003)] and photographed a cross-section from each. We printed each photograph, measured sound wood thickness at several locations on each cross-section, and averaged them as described by Fraedrich (1999). We multiplied the average sound wood thickness for each stem by two and subtracted it from the stem diameter to obtain an approximate hollow diameter. We entered the tree and decay column diameters for each cross-section into the Wagener (1963), Coder (1989), and Smiley and Fraedrich (1992) formulas to determine each formula’s estimate for percent loss in I_STEM. From each photograph, we calculated the actual percent loss in I_STEM about two or three different axes through the cross-section using the parallel axis theorem. For each stem, we calculated the loss in I_STEM about the north–south (I_NS) and east–west (I_EW) centroidal axes. A centroidal axis is one that passes through the centroid or geometric center of the cross-section. We chose north–south and east–west centroidal axes because they were easily and repeatedly measured dimensions for each stem section and to compare the effect of decay position on loss in I_STEM. If it were visually apparent from the photograph that the actual maximum loss in I_STEM (we will refer to it as loss in I_MAX from now on) did not correspond to the loss in either I_NS or I_EW, we chose the axis about which loss in I_MAX would occur and calculated it using that axis (see Figure 6 in Appendix 3). In Appendix 3 we explain in more detail how we used the parallel axis theorem to calculate loss in I_NS, I_EW, and I_MAX.

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Figure 6.

For a roughly circular cross-section, the maximum loss in I_STEM (I_MAX) will occur about the centroidal axis parallel to a cavity opening (shown as a dashed line). The dot represents the centroid of the cross-section and the north-south (NS) and east-west (EW) centroidal axes are shown as thin lines with arrowheads. For a cross-section similar to that shown here, we calculated loss in I_STEM about the NS and EW centroidal axes, and loss in I_MAX because neither loss in I_STEM about the NS centroidal axis nor loss in I_STEM about the EW centroidal axis represented the actual loss in IMAX.

We tested for significant differences between each formula’s (Wagener 1963; Coder 1989; and Smiley and Fraedrich 1992) loss in I_STEM estimate and the actual loss in I_MAX using Wilcoxon’s signed rank test for paired data (Mendenhall and Reinmuth 1982). For each of the sample stem cross-sections, we also assessed whether each formula’s estimate of loss in I_STEM exceeded the action threshold of the formula and whether the actual loss in I_MAX value exceeded each formula’s threshold. To compare whether actual loss in I_MAX values differed from Mattheck and Breloer’s (1998) threshold, we used the thinnest remaining wall on each stem to compute the t/R ratio. When t/R = 0.3, loss in I_STEM equals approximately 33% (Mattheck and Breloer 1998), so we used 33% as the threshold to compare the t/R values to the actual loss in I_MAX values.

If a formula’s estimate for loss in I_STEM did not exceed that formula’s threshold, but the actual loss in I_MAX did exceed the threshold used with the formula, we considered the discrepancy an “underestimate.” Conversely, an “overestimate” occurred when a formula’s estimate for loss in I_STEM exceeded that formula’s threshold, but the actual loss in I_MAX did not exceed the threshold used with the formula. For each formula, we present the number (under- or overestimates) between each formula and the actual loss in I_MAX.

Because of the threshold nature of the formulas’ loss in I_STEM calculations, small variations in the results become important. For example, if the actual loss in I_MAX were 35%, the tree would be considered a hazard because it exceeded the 33% threshold. Yet, if one of the formulas determined the loss in I_STEM to be 29%, the tree would be considered “safe” (i.e., below the threshold). The loss in I_STEM values (29% and 35%) might not be statistically different, but they are different as far as determining whether an arborist would recommend remedial action for the tree. This is an important distinction because an arborist or urban forester must be aware of both a formula’s overall accuracy and its potential to misclassify a tree as a hazard.

APPENDIX 1

Essentially, the moment of inertia of a tree stem (I_STEM) reflects its resistance to bending. It is based on the shape of the stem cross-section and is calculated as the summation over the entire stem cross-sectional area (A) of each increment of stem cross-sectional area (dA) multiplied by the square of the distance (y) between the increment of area dA and the centroid (or geometric center) of the stem:

A1

From Equation A1, it is clear that to calculate I, one considers only cross-sectional geometry; material properties (e.g., wood strength) are irrelevant. Thus, one calculates I the same way for a steel beam as for a tree stem. The only difference is that as trees add girth each year, I_STEM will change because A and y will change in Equation A1) It is also clear from Equation A1 that I increases exponentially as the distance y from the centroid to an increment of area increases.

When decay is present, I_STEM decreases because there is less wood to resist a bending force. The percent loss in I_STEM due to decay can be calculated by dividing I of the decayed area (IDECAY) by ISTEM:

A2

Equation A2 is the procedure that three of the formulas (Wagener 1963; Coder 1989; Smiley and Fraedrich 1992) use to estimate strength loss, which is actually an estimate of loss in I_STEM. In Equation A2, we need to calculate two separate values of I, one for the stem cross-sectional area assuming no decay is present (I_STEM), and one for the decay area. To calculate I_DECAY, we again use Equation A1, but this time A represents the decay area and y represents the distance between an increment of the decay area and the centroid of the decay area.

Unless the stem cross-section is symmetrical, the axis about which one calculates I_STEM will affect its value (recall the ruler analogy earlier in this article). Consider two cross-sections—a circle and an ellipse (Figure 4). For the circle, I is constant regardless of the axis about which it is calculated, as long as the axis passes through the centroid (this is called a centroidal axis). For the ellipse, I depends on the centroidal axis about which it is calculated because the axes are not equal in length. In Figure 4, A3 A4 A5

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Figure 4.

Centroidal moment of inertia (I) of a circle is the same regardless of the centroidal axis about which I is calculated, because the circle is symmetrical about any centroidal axis: I_XX’ = I_YY’ = πR⁴/4. Centroidal moment of inertia (I) of an ellipse varies as a function of the centroidal axis about which I is calculated: I_XX’ = π a³ b/4; I_YY’ = π ab³/4. Centroids for the circle and ellipse occur where the XX’ and YY’ axes intersect.

where the notation is explained in Figure 4 (π ≈ 3.14). In Figure 4, only two centroidal axes are shown (XX’ and YY’), but remember that I can be calculated about any centroidal axis. Figures 4 and 5 (Figure 5 is explained in Appendix 2) show that I_STEM depends on (1) the direction in which a force acts on the tree, if its cross-section is asymmetric; (2) the stem cross-sectional shape; and (3) the cross-sectional shape and position of decay in the stem. For the ellipse in Figure 4, if we imagine a force acting perpendicular to the YY’ axis, the resistance to that force will be much greater than if the force acted perpendicular to the XX’ axis.

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Figure 5.

Two circular cross-sections representing stems. On the left, decay area (shaded circle) is a concentric circle; on the right, decay is circular but off-center. For the cross-section on the left, C is the shared centroid of the stem and decay areas. For the cross-section on the right, C_D is the centroid of the decay area, C_S is the centroid of the stem, and d is the distance between the centroids of the stem and decay areas. In each crosssection, R_I is the radius of the decay area and R_O is the radius of the stem. The parallel axis theorem accounts for off-center decay areas when calculating loss in I_STEM.

APPENDIX 3

We used the parallel axis theorem to calculate loss in I_STEM for the sample sections as follows. First, we took a 35 mm slide of each trunk cross-section. We scanned the slides, printed the images, traced the prints onto cardboard and trimmed the cardboard, leaving a scaled cardboard replica of the original cross-section. We used the cardboard cutouts to determine the centroid of the stem cross-sections using the method described by Niklas (1992, p. 135). The method uses the fact that, for homogeneous materials, the centroid of an area corresponds to its center of mass. Although wood from tree cross-sections is not homogeneous, differences in density are small enough that it is safe to approximate the centroid as the center of mass. We determined two centroidsfor each tree cross-section, one assuming that decay and hollow areas were absent (C_S) and a second assuming decay and hollow areas were present (C_H). We assumed that the mass of any decayed or discolored wood was insignificant, and eliminated it from the cardboard cutouts when determining C_H. We transferred C_H and C_S for each stem cross-section from the cardboard cutouts to the printed images. The C_S was on one printed image and the C_H on a second.

On each image, we drew north–south (NS) and east– west (EW) axes through the centroid as well as lines parallel to each axis at 6.35 mm (0.3 in.) intervals. If it were visually apparent that calculating loss in I_NS and loss in I_EW would not produce the loss in I_MAX, we calculated an additional value for loss in I_STEM about the centroidal axis that returned the loss in I_MAX for that stem section. When looking at a cross-section, it is not hard to determine which centroidal axis will return the loss in I_MAX because the outer fibers of the stem cross-section contribute exponentially more to I_STEM. On cross-sections with large cavities, for example, the loss in I_MAX would occur about the centroidal axis parallel to the widest part of the cavity (Figure 6).

Figure 7 shows a cross-section with the NS axis (dashed black line) through the centroid (large white dot) and lines parallel to the axis creating rectangles. The small white dots in the middle of each rectangle represent their centroids (C_R). We measured the length of each line, took the average of two adjacent lines (e.g., L_i and L_{i + 1}), and multiplied the average by 6.35 mm to estimate the area (A) of each rectangle:

A9

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Figure 7.

Diagram illustrating how we used the parallel axis theorem to calculate loss in I for tree crosssections. The cross-section shows the north–south (NS) centroidal axis (dashed black line) through the centroid of the cross-section (large white dot) and lines parallel to it (solid black). Small white dots indicate centroids of each rectangle. See the text for details of the calculation procedure.

We divided all distance measurements (e.g., L_i, L_i+1, and 6.35) by the image scale (IS) before calculating A and I of each rectangle.

To calculate I_NS we used the parallel axis theorem and summed I of each rectangle about the NS centroidal axis: A10

where (n) is the number of rectangles parallel to the NS centroidal axis and (i) represents a given rectangle. Equation A9 gives the area for each rectangle (A_i), (d_i) is the measured distance between C_S and C_R, and I for each rectangle (I_i) is calculated

A11

We calculated I_NS assuming the trunk sections had no decay, and then we calculated I_DECAY about the NS centroidal axis (I_{DECAY NS}) using the same procedures (Equations A9, A10, and A11). To calculate I_{DECAY NS}, we summed the rectangles that covered decay areas and used C_H. Using Equation A6, we calculated the percent loss in I_NS due to decay. We repeated the above procedures (Equations A9, A10, A11, and A6) for the EW centroidal axis, and, if necessary, for the centroidal axis that produced the loss in I_MAX. For each trunk cross-section, then, we calculated loss in I_NS (I_{DECAY NS} / I_NS), loss in I_EW (I_{DECAY EW} / I_EW), and, if necessary, loss in I_MAX.

• © 2004, International Society of Arboriculture. All rights reserved.