Arboriculture & Urban Forestry 39(5): September 2013 221 erably smaller than calculated conventionally for an un-deformed cross section (Spatz and Niklas 2013). Therefore, the diagnosis should fol- low a different route. Equation 1 can be writ- ten in Figure 4. The maximal shear strength as function of green wood density for conifers from temperate zones (triangles), deciduous trees from temperate zones (diamonds) and from tropical zones (squares). The trend-line drawn is an OLS fit to a power function (y = 0.0038 • x1.15 ). Another diagnostic tool for tree risk assessment is the pull- ing test (Milne 1991; Wessolly und Erb 1998; Bruechert et al. 2000; Brudi and Wassenaer 2001; Peltola 2006), which pro- vides a measure of the force necessary to slightly tilt and bend a tree. An inclinometer attached to the base of the trunk reads the degree to which the tree is tilted. The degree of bending is measured with strain gauges attached at various heights of the trunk to measure the strain in the stem just below the bark. Know- ing the strain e and the outer radius r, excluding the bark, the change of curvature of the trunk at the particular height, where the strain gauge is attached, can be calculated. The basic equa- tion for the mechanics of the bending process is given by: [1] M = E I r is compared to the second moment of area for a solid cross sec- tion with homogenous material, then a rough indication for the distribution of healthy wood in the cross section can be obtained. One of the uncertainties in this interpretation may be a distinct difference between tabulated values and the density measured in the tree to be assessed. Since a nearly linear relation exists between MOE and density (Figure 1), a more realistic value for MOE can be obtained by a proportional correction for the density difference: [2] MOE = MOEtabulated densitymeasured densitytabulated Another difficulty lies in the accuracy of the determination of the second moment of area according to Equation 1. As mentioned before, the values for the moduli of elasticity have a standard de- viation of around 16%. According to the law of error propagation, Figure 5. Schematic drawing representing strains in a tree bent under wind loads. Bending, and even more so shear strains, are greatly exaggerated. An arrow points to fiber buckling on the compression side. ©2013 International Society of Arboriculture where M is the bending moment applied (force times lever arm), E is the modulus of elasticity, and I is the second moment of area (Gordon 1976; Niklas and Spatz 2012b). The second moment of area is different for a cross section with a hollow, than for a cross section with solid wood. Taking a value of E from a reliable table, an effective second moment of area Ieff can be computed. If this [3] a different M = Ieff r where s is the stress in the periphery of the stem. It is particularly useful as [4] Mcrit = Ieff r moment at which failure is expected to occur (Spatz and Brüchert 2000). Realistic estimates for Ieff , as can be calculated from book tables. Thus, within error limits also given in the tables, the bending moment critical for failure to occur in the trunk of a given tree can be determined according to Equation 4. Usually, in the interpretation of a pulling test, the critical stress sound tomography profiles (Gilbert and Smiley 2004), and scrit is taken as the stress at the yield point in a stress-strain curve. Therefore, a short excursion into fracture mechanics seems use- ful to point to the relations between different strength properties. are found in the Jessome, Lavers, and USDA Wood Hand- crit where r is the outer radius, excluding the bark, and scrit maximal bending strength (Figure 2). Mcrit is the is the bending form as the following: this reduces to 14% if the density is known, an error margin, which is still too large to use in this proce- dure for more than an indication of the degree of hollowness. The evaluation of the degree of hollow- ness in a tree from pulling tests alone is even more complicated for so-called hazard trees (Mattheck and Breloer 1994) with small t/r ratios, where t is the width of the remaining healthy wood and r is the outer radius. If ovalization of the cross section under bending loads is taken into account, it could be shown that the second moment of area Ieff is consid- e s s
September 2013
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