Basic Aspects of Mechanical Stability of Tree Cross-Sections
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Basic Aspects of Mechanical Stability of Tree Cross-Sections The mechanical stability of a tree trunk against bending loads caused by wind depends on the strength and condition of its wood as well as on the size and shape of its cross-section. A basic understanding of these aspects can help when evaluating tree strength loss due to decay within the scope of tree risk assessments. Diameter and Stability When a tree is impacted by a wind force, its cells in the trunk on the windward (wind-exposed) side are stretched, while those on the leeward (wind-sheltered) side are compressed (Figure 1). Furthermore, the tree crown’s own weight has to be added as well, which results in an even higher compression load on the leeward cells and a correspondingly lower tension on the opposite side. To describe the tree’s ability to withstand such Figure 1. Wind bending loads, the so from the left leads called “moment of resistto compression ance,” represented by load on the right side of the stem the symbol “W” (for the and tension on German word Widerthe left. standsmoment) is used. It characterizes the mechanical stability of a cross-section as far as it depends on size and geometrical shape. The resistance moment of a circular cross-section with a diameter D can be summarized with a simple formula: This formula helps us understand the effect of diameter on stability: if the diameter is doubled, for instance, the moment of resistance increases eightfold, since (2D)3 = 8D?. Likewise, if trunk diameter grows one percent, its moment of resistance rises by about three percent, since (1.01D)3Ϸ1.03D3. Therefore, an annual tree growth ring width of 0.2 in (5 mm) within a tree trunk crosssection of 20 in (500 mm) diameter (so one percent increase on each side) denotes a stabilization of a tree trunk cross-section of about 6 percent. In this manner, a sound tree gains stability by its annual ring growth on a yearly basis, ignoring potential changes in the tree crown’s surface or wind load and internal damages. Wrapped Curves Around Cross-Sections To precisely calculate the load-carrying capacity of a tree trunk cross-section, often called its “strength” or “stability,” one would have to know the individual characteristics of each cell and the stability of its compounds. This is hardly feasible, neither in terms of practical work with trees nor with scientific measurement. Therefore, it seems appropriate to adopt a simpler and more relative approach. We can do this by calculating the moment of resistance for loading capacities from 1° to 360° (all wind directions), then dividing each result by the maximal value to produce a relative stability percentage. Those numbers would normally be represented as a linear graph, with the x-axis representing wind direction and the y-axis the percentage of maximum strength/stability. For better understanding, we can also wrap such a graph around the trunk cross-section to better visualize this effect. A circular trunk cross-section shows, therefore, a constant stability towards wind loads from all directions. Consequently, the curve of the calculated relative stability percentages (moment of resistance) runs along the 100 percent level, creating a perFigure 2. The green curve indicates relative fect circle on strength of the cross-section against wind load the wrapped as revealed by the moment of resistance. graph (Figure 2). Influence of Trunk Profile This situation changes with different cross-sectional shapes. Trees growing between buildings that are located on their northern and southern side (so wind load is limited to the west or east), for example, mostly develop oval trunk cross-sections (Figure 3). In such a case, calculation of relative stability reveals the consequences of this mechanical impact on tree growth: a tree trunk with an E-W diameter of Figure 3. The weaker the cross section the 40 in (1 m) and a more the green curve bulges into the direction N-S diameter of of the corresponding wind flow. 28 in (0.7 m)

retains only approx. 50 percent of maximum stability when exposed to wind load from the north or south. The relative stability curve around the trunk “bulges” accordingly (Figure 3), where the bulges represent drops in strength. As a consequence, if the buildings south Figure 4. The arrow indicates the wind flow of the tree were direction where the cross section is the weakest. demolished and the tree suddenly exposed to wind from that direction, the possibility of the tree’s failure would be greatly increased. Trunk flares at the lower trunk cannot only be used to estimate the primary...

Basic Aspects of Mechanical Stability (continued) damaged crosssection for all wind directions. The outward bulge (=relative decrease in stability) on the side opposite from the decay corresponds to the comparatively higher reduction of tensile strength for wind coming from the side where the decay is located. Therefore, relaFigure 7. Theoretically, 90 percent loss of tive strength loss of a radius equals approximately 65 percent loss trunk cross-section of moment of resistance. depends not only on the extent of decay, but also – and above all – on its location. This is relevant for expert...

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