Journal of Experimental Botany, Vol. 51, No. 345, pp. 797-806,
April 2000
© 2000 Oxford University Press
Computing factors of safety against wind-induced tree stem damage
Department of Plant Biology, Cornell University, Ithaca, NY 14853, USA
Received 18 March 1999; Accepted 22 November 1999
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The drag forces, bending moments and stresses acting on stems differing in size and location within the mechanical infrastructure of a large wild cherry (Prunus serotina Ehrh.) tree are estimated and used to calculate the factor of safety against wind-induced mechanical failure based on the mean breaking stress of intact stems and samples of wood drawn from this tree. The drag forces acting on stems are calculated based on stem projected areas and field measurements of wind speed taken within the canopy and along the length of the trunk. The bending moments and stresses resulting from these forces are shown to increase basipetally in a nearly log–log linear fashion toward the base of the tree. The factor of safety, however, varies in a sinusoidal manner such that the most distal stems have the highest factors of safety, whereas stems of intermediate location and portions of the trunk near ground level have equivalent and much lower factors of safety. This pattern of variation is interpreted to indicate that, as a course of normal growth and development, trees similar to the one examined in this study maintain a cadre of stems prone to wind-induced mechanical damage that can reduce the probability of catastrophic tree failure by reducing the drag forces acting on older portions of the tree. Comparisons among real and hypothetical stems with different taper experiencing different vertical wind speed profiles show that geometrically self-similar stems have larger factors of safety than stems tapering according to elastic or stress self-similarity, and that safety factors are less significantly influenced by the ‘geometry’ of the wind-profile.
Key words: Drag, plants, safety factors, stem taper, trees, wind.
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Introduction |
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Organisms must be mechanically reliable to survive and reproduce otherwise even modest unprecedented mechanical forces can inflict severe damage or cause death (McMahon, 1973
; McMahon and Kronauer, 1976b; Wainwright et al., 1976; Holbrook and Putz, 1989; Niklas, 1992, 1994a, b). In the engineering sciences, the traditional method for numerically evaluating mechanical reliability is to compute a factor of safety, which is typically taken as the quotient of the load capability and the actual load of a structure (Weibull, 1939; Timoshenko, 1956; Volk, 1958; Spotts, 1959; Ang and Tang, 1975). The load capability is the magnitude of the force that produces observable damage; the actual (working) load is the magnitude of the force that a structure normally contends with during its functional lifetime. The quotient of these two loads equals or exceeds unity for any mechanically reliable structure. Numerous studies indicate that the factor of safety for organic structures is much higher than unity (McMahon, 1973; King, 1986; Tateno and Bae, 1990; Tateno, 1991; Mattheck et al., 1993; Niklas, 1994a).
Alternative measures of load capabilities and actual loads can also be used to compute factors of safety. The breaking and working stresses are often used as alternative measures (Keller and Spengler, 1989; Blickman et al., 1993; Kirkpatrick, 1994), especially for structures like stems or petioles (which can vary in size among conspecifics) because stress is a size-independent mechanical parameter (Wainwright et al., 1976; Niklas, 1992). The breaking stress, which may be used as a surrogate measure for load capability, is the magnitude of the externally applied force that causes permanent damage normalized with respect to the cross-sectional area through which it acts just before mechanical failure. The working stress, which serves as a convenient surrogate measure for the working load, is the force normally experienced by a structure normalized with respect to the cross-sectional area of the structure.
The breaking stresses of stems are comparatively easy to measure empirically. In contrast, the working stresses are more difficult to determine because they depend on the magnitudes of dynamic forces that stems experience, such as wind-induced drag, which can change over many orders of magnitude in a short time (Vogel, 1981; King, 1986). Perhaps, for this reason, the factors of safety that are often reported for plant stems are calculated on the basis of the stresses resulting from static loads (McMahon, 1973; Niklas, 1994a, b), although these are misleading because most healthy stems mechanically fail as a result of wind-loading rather than supporting their own weight and because the working stress of a stem is the sum of the stresses created by dynamic and static forces.
For this reason, a method for computing factors of safety based on the stresses created by wind-induced drag is presented. The method requires field measurements of wind speeds at different locations within a plant and morphometric data on stem diameter, length and location. These measurements are then used to estimate the drag forces, bending moments and stresses, and finally the factors of safety for stems differing in size and location within the plant body.
This method is illustrated for the stems of a large wild cherry, Prunus serotina (Ehrh.), tree whose size, shape and habitat permitted detailed measurements of wind speeds and stem diameter and length at different locations within the canopy and along the length of the trunk. Data were gathered when the tree was in the leafless condition. The data from this tree are also used to explore the influence of trunk taper and different wind speed profiles on stem mechanical reliability. Differences in stem taper and in the shape of the wind speed profile are shown to alter estimates of mechanical reliability significantly. Nevertheless, the data and calculations to be presented indicate that trees similar to the one studied here will either shed peripheral branches or mechanically fail near their base when subjected to excessively high wind speeds.
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Method for computing the factor of safety
To simplify mathematical derivations and the collection of data, the method for computing factors of safety assumes that stems are leafless, inflexible, and possess the same breaking stress regardless of size or location within a tree. Some of these assumptions are valid for some species (dicot stems can be either leafless or deciduous); others are contentious (even woody juvenile stems are capable of modest to large elastic deflection in high winds); and others may be generally incorrect (the breaking stresses of many woody stems tend to increase, to a limit, in a basipetal direction away from the apex). The implications of these assumptions (and their possible solutions) on estimates of factors of safety are discussed later (see Discussion).
The factor of safety for any stem in the canopy of even a large tree can be quantified provided that the magnitudes of stem working and breaking stresses are known. In turn, the magnitude of the working stress depends on the magnitude of the drag force exerted on stems. For any obstruction to fluid-flow, the drag force Df is given by the formula
 | (1) |
is the density of air, u is (local) wind speed experienced by the entire stem or some portion of it, Sp is the surface area projected toward the wind, and CD is the drag coefficient (throughout this paper, the drag coefficient was set at 1.00). Equation 1 can be used by drawing a physical analogy between a stem and a tapered beam composed of a series of circular cylindrical elements differing in diameter d and length x, and by assuming that u is sufficiently low such that Df acts perpendicular to x (Fig. 1). Under these circumstances, the drag-induced bending moment Mi acting on the base of any element i is the sum of all the moments acting on elements j distal to element i (Fig. 1), and is thus given by the formula
 | (2) |
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The maximum working stress
i of element i, is computed by noting that tensile and compressive bending stresses invariably reach their maximum intensities at the surface of the element. Assuming that each stem element is a circular cylinder with an axial second moment of area Ii=0.049
d 4i/4, it follows that i is given by the formula
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Taking the breaking stress b as a surrogate measure of the load capability of element i, the factor of safety S for element i is given by the formula
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Field and laboratory measurements
The four variables required to determined the factor of safety for each stem element i (i.e. u, d, x, and b) in a tree were measured empirically for a wild cherry (P. serotina) tree measuring 13.11 m in height and 0.401 m in diameter at 0.5 m from ground level, growing in a open site in the author’s backyard.
Wind speed was measured by placing 13 Sierra-Misco anemometers (model WSD355) connected to three ‘Easy Logger’ recording devices (model EL824-MS) at 1 m intervals along the length of the tree and 1.5 m from the surface of its main trunk. Wind speeds were recorded at each location between 2–27 November 1998, and subsequently normalized with respect the maximum wind speed U recorded at the top of the tree (where x=0). These normalized wind speeds were plotted as a function of their normalized distance from the top of the tree (x/h) to obtain a normalized wind speed profile (Fig. 2). The regression formula that best fit this wind profile was used to estimate u anywhere within the tree canopy or along trunk length for three arbitrarily selected maximum wind speeds (U=10 m s-1, 20 m s-1, and 50 m s-1).
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Stem projected surface areas were determined by measuring the diameter and length of 83 stem elements. This was accomplished by photographing different portions of the tree in a direction orthogonal to the height of the tree, projecting the negatives of these silhouette photographs onto the surface of a digipad connected to a computer equipped with the software SECTION, and manually digitizing stem outlines after correcting for the magnification factor. The location of each stem element with respect to the top of the tree was determined based on measurements taken from the same photographs. The data gathered in this manner were spot-checked by randomly selecting 30 of the 83 stem elements and directly measuring stem diameter, length and elevation on the intact tree.
The tree was then cut down near its base to gather representative branches and to cut prismatic samples of wood from its trunk to determine the breaking stress measured in bending. Each stem or wood sample was horizontally suspended between two vertical supports and loaded at its midlength ll/2 with various weights P until the specimen failed elastically to return to its original shape when unloaded. On average, a 10 minute interval separated each loading cycle. The breaking stress b was computed on the basis of the smallest force required to produce an unrecoverable (plastic) deformation in the bent specimen normalized with respect to the previously measured cross-sectional area of the specimen (i.e. as used here, the breaking stress and the yield stress are synonymous). Factors of safety were computed for 35 lateral branches differing in size and location and for 48 stem elements differing in location along the length of the tree trunk based on equation 4 and plotted as a function of the location of the specimen with respect to the top of the tree (where x=0).
Effect of stem taper and wind speed profiles on factors of safety
Factors of safety were computed for the wild cherry tree trunk without its lateral branches to explore the influence of branching on mechanical reliability of an isolated large woody stem. This was accomplished by mathematically removing the drag forces acting on the lateral branches of the real tree and recalculating the bending moments and stresses and the safety factors for the remaining 48 trunk elements.
The influence of stem taper on mechanical reliability was explored by constructing three hypothetical stems whose diameter and length scaled according to one of three models for stem taper: geometric, stress or elastic self-similarity (i.e. L
D1/1, LD1/2 or LD2/3, respectively). The drag forces, bending moments, and working stresses acting along the length of each of these three hypothetical stems were computed assuming a maximum wind speed of 10 m s-1 at the top of the stem and then compared to those calculated for the wild cherry trunk devoid of its lateral branches. Size-independent comparisons between the hypothetical stems and the wild cherry trunk were possible by normalizing the distance measured from the top of the stem x with respect to total stem length L and plotting this parameter against stem diameter measured at any distance from the top of the stem d with respect to the diameter measured at the base of the stem D.
The influence of the shape of the wind speed profile on safety factor estimates was evaluated by mathematically fabricating two wind speed profiles, calculating the drag forces, bending moments and stresses these profiles produced for the wild cherry tree, and comparing these results with those calculated for the tree on the basis of the empirically determined wind profile (shown in Fig. 2
). Both of the fabricated wind speed profiles predicted a decrease in wind speed from the top to the bottom of the tree. However, one profile described a simple linear diminution in the normalized wind speed u/U; whereas the other established a curvilinear decline: u/U=1.0–1.0(x/h) and u/U=1.01–0.47(x/h)+1.08(x/h)2–1.58(x/h)3, respectively.
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Empirically determined wind speed profile and tree taper
The maximum wind speed U measured at the top of the tree was 10.2±2.1 m s-1 (n=1896). The relationship between wind speed (recorded for each of the 13 locations within the tree) normalized with respect to this maximum wind speed u/U and the distance of each of these locations from the top of the tree normalized with respect to tree height x/h was best approximated by a third-order polynomial regression curve: u/U=1.05–3.62(x/h)+ 9.93(x/h)2–7.27(x/h) (r2=0.96; n=13) that predicted local wind speeds u would decrease, on average, to ~0.65U at x~0.25h, increase to ~0.9U at x~0.68h, and then decrease to zero at ground level where x=h (Fig. 2
). This regression curve was used to predict u within the tree’s canopy and along its trunk length for U=10, 20, 50 m s-1.
The relationship between stem length x and diameter d measured at x was log–log non-linear and best approximated by a second-order polynomial regression formula when the data were log10-transformed: log x=1.16+1.66 (log d)–0.29 (log d)2 (r2=0.992, n=83) (Fig. 3). The relationship between x and d, which was statistically indistinguishable for the 35 lateral branches and the 48 elements of the trunk, failed to comply with any of the three mathematical models traditionally used to describe stem taper (i.e. geometric, stress and elastic self-similarity; see below).
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The drag-induced bending moments and stresses (computed on the basis of empirically determined stem dimensions and wind speeds) increased basipetally toward the base of the tree (Fig. 4
). The relationship between the bending moment and the location of each element with respect to the top of the tree was nearly log–log linear; deviations from this linear trend resulted from the drag-induced bending moments created by lateral branches at the points of their attachment to the trunk (Fig. 4A). The bending stresses resulting from these moments increased from an absolute minimum value for furthest twigs, reached an absolute maximum for stems located 0.4 m from the top of the tree, decreased for older stems, and then increased once again toward the base of the trunk (Fig. 4B). This sinusoidal pattern was best approximated by a third-order polynomial regression formula that predicted the highest stress levels for the lowest portions of the trunk and for a cadre of stems located ~0.4 m from the top of the canopy.
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A mean stem breaking stress of 186±35.3 MN m-2 (n=78) was used as the numerator in the formula for the factor of safety (see equation 4). Since the breaking stress was taken as a constant, the factor of safety varied as an inverse function of stem bending stresses: stems elements located ~0.4 m from the top of the tree and near the base of the tree had the lowest factors of safety, regardless of the maximum wind speed U used to compute drag forces and the resulting bending moments and stresses (Fig. 5
). All stem elements had a factor of safety 
1 when U 
10 m s-1. The factor of safety was <1 for stems elements located ~0.4 m from the top of the tree and near the base of the tree when U=20 m s-1, and all but the most distal stem elements had factors of safety much less than unity when U=50 m s-1. This was interpreted to mean that the tree examined in this study could not withstand wind speeds greater than 20 m s-1 measured at the top of its canopy without incurring mechanical damage.
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Effect of trunk taper and wind speed profiles on factors of safety
The overall taper of the tree trunk did not agree with any of the three allometric models for stem taper (i.e. geometric, elastic and stress self-similarity). Comparisons based on normalized stem vertical profiles indicated that the taper of the tree trunk was, on average, intermediate between that expected for geometrically and elastically self-similar stems (Fig. 6). Likewise, the bending moments and stresses computed along much of the length of the tree trunk were intermediate between those expected for stems with these two tapers (Fig. 7).
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Stems tapering according to geometric self-similarity had larger factors of safety than stems tapering according to either stress or elastic self-similarity. However, stems tapering according to stress self-similarity had the larger factors of safety near their base compared to those tapering according to geometric or elastic self-similarity (Fig. 8
). The factor of safety along the length of the wild cherry trunk varied because trunk diameter (and thus cross-sectional area and bending stress) varied non-linearly. The factor of safety computed for the upper 40% of the length of the wild cherry trunk was, on average, intermediate between those expected for stems tapering according to geometric or elastic self-similarity, complied reasonably well with those predicted for a geometrically self-similar stem toward the middle third of the trunk, and was lower than that predicted for any of the three hypothetical stems near (but not at) the base of the tree trunk (Fig. 8).
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The shape of the wind speed profile also significantly altered estimates of factors of safety. For example, although a basipetal curvilinear decrease in normalized wind speed had little effect on the factors of safety computed for branches located
1 m from the top of the tree, this wind speed profile resulted in a decrease in the factors of safety computed for larger lateral branches and intermediate portions of the trunk, and increased the factor of safety near or at the base of the trunk (Fig. 9). Since the shape of a wind speed profile within the canopy of a tree is influenced by the frequency of branching, stem taper and other morphometric factors, the differences in the estimates of factors of safety computed on the basis of different wind speed profiles were interpreted to indicate that the growth habit of a tree significantly affects stem mechanical reliability even for species with woods that have similar breaking stresses.
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Caveats
The method used to compute the mechanical reliability of wind loaded stems assumes that all stems have equivalent breaking stresses and that all stems are inflexible and leafless. These assumptions, some of which hold true for the particular tree studied, necessarily obtain estimates of mechanical reliability that vary throughout the plant body solely as a function of variations in stem working stresses that, in turn, are overestimated because stems cannot reduce their drag by downwind elastic flexure and underestimated because stems lack leaves that add drag.
The proposition that all stems have the same breaking stress is unrealistic because longitudinal variations in the volume fraction of sapwood relative to heartwood (as well as other anatomical features normally attending stem growth and development) are also known to influence the mechanical properties of stems (Carlquist, 1961, 1975). Likewise, the volume fractions of ground and other tissues will influence the mechanical properties of very young stems (Speck, 1994). For the tree examined in this study, however, no statistically discernible pattern of variation among stem breaking stresses was observed, and so the mean breaking stress of stems and samples of wood drawn from older branches was used to calculate factors of safety. None the less, the younger stems of wild cherry were, on average, more flexible and less rigid (and thus had slightly higher breaking stresses) then their older counterparts, perhaps because of differences in the volume fractions of primary and secondary tissues (Speck, 1994). Since the factors of safety calculated for these stems were higher than for any other cadre of stems, the results presented here appear not to be jeopardized by the assumption of a uniform stem breaking stress.
Whether the assumptions that stems are inflexible and leafless have equivalent and opposing effects on estimates of factors of safety cannot be answered synoptically because stem flexibility and leaf size, shape, number, and tissue flexibility vary across and within species. Since the tree examined in this study was leafless by virtue of the season, data were collected and, since its leafless twigs were not observed to bend significantly when subjected to low to moderate wind speeds (i.e. 10 m s-1), the factors of safety calculated for this tree are arguably reasonable estimates for winter conditions, but are nevertheless larger than those that would be expected for the same tree during the growing season when younger stems bear leaves or fruits.
The method presented here can be modified, if needed, to account for the influence of stem flexure and the influence of leaves on drag-induced stresses. For example, tip-deflection formulas for cantilevered beams may be used to estimate reiteratively the reduction in the drag forces as stems increasingly flex downwind with higher wind speeds provided the bending stiffness of stem tissues is known or adduced a priori. Likewise, the drag forces generated by shoots can be either predicted or empirically determined and added to those calculated for their subtending leafless stems when estimating factors of safety. None the less, the following discussion is offered exclusively in the context of the general implications of a method for calculating factors of safety.
Variation in the factor of safety
Wind-induced drag forces are the most conspicuous cause of tree mechanical failure since healthy woody stems are typically capable of supporting their own weight and that of the organs they normally support (Metzger, 1893; Esser, 1946 a, b; Alexander, 1971; Banks, 1973; King and Loucks, 1978; Niklas, 1992, 1994a; Vogel, 1996). Prior studies show that the leaves and young stems of many woody species either elastically flex backward or fold upon themselves in high winds, thereby decreasing their projected surface areas with concomitant reductions in the drag forces they would otherwise impart to older subtending stems (Fraser, 1962; Hutte, 1968; Mayhead, 1973; Vogel, 1996). Branches and tree trunks nevertheless frequently snap somewhere along their length, such that trees typically do not uproot when subjected to very large wind-induced drag forces (Jones, 1956; Hutte, 1968; Wood, 1970; Runkle, 1982; Harcombe and Marks, 1983; Putz et al., 1983). This phenomenology indicates that the factor of safety is not uniform throughout the mechanical infrastructure of most trees, but rather varies such that some stems or portions of a trunk are more prone to dynamic wind-failure than others.
The complex pattern of variation in the factor of safety reported here provides further evidence for preferential stem failure because the calculations show that the lowest factors of safety occur for a constellation of peripheral branches and for the lower portions of the trunk. In contrast, calculations show that the intervening portions of the tree have significantly higher factors of safety regardless of the wind speed used to estimate drag. The mechanical failure of peripheral, younger stems will necessarily reduce the drag-induced bending moments and stresses experienced by the rest of the tree. Their forfeiture in a wind storm, therefore, can reduce the probability that the trunk will break. Since the older portions of a tree represent a large biomass investment and have the capacity to regenerate new shoots, the mechanical failure of this constellation of stems is arguably functionally adaptive. As a consequence of their normal growth and development, trees may thus establish a class of stems prone to mechanical failure during windstorms, thereby providing a margin of safety against the catastrophic failure of the tree as a whole.
This hypothesis is consistent with surveys of tree damage and death and with a limited number of studies estimating the mechanical reliability of stems based on static rather than dynamic loads. For example, in a study of Barro Colorado Island, it was reported that 70% of a total of 310 damaged trees (>10 cm dbh) snapped near, but not at, their base (of which 217 subsequently developed new shoots), whereas 25% uprooted (of which 5 produced viable shoots) and only 5% broke at their base (Putz et al., 1983). Likewise, among mesic eastern USA forest tree species, it was reported that the typical mode of failure is the snapping of trunks (67%) (Runkle, 1982). Importantly, these and other similar surveys of tree failure (Behre, 1921; Curtis, 1943) exclude from consideration trees that sustain comparatively minor damage to their lateral branches, and do not exclude trees whose trunks snapped as a result of disease (e.g. 13% of the trees with snapped trunks in the Barro Colorado Island study had heartrot; Putz et al., 1983), suggesting that the typical form of tree failure may be the loss of branches or the failure of the trunk some distance from its base, which is consistent with the most likely modes of failure predicted for the wild cherry tree examined in this study.
Additional support for the supposition that the wind-induced ‘self-pruning’ of branches provides a safety measure against the failure of the tree as a whole comes from estimates of factors of safety against static load-induced failure and from anecdotal information on tree damage after ice-storms. For example, the factor of safety against failure in bending under self-loading varied for a large Robinia pseudoacacia L. tree in much the manner as that reported in this study, indicating that R. pseudoacacia stems of intermediate age, diameter and location in the canopy are less mechanically reliable than those distal to them and much of the trunk (Niklas, 1999). Likewise, the typical mode of failure observed by the author among 213 dicot trees following an unprecedented ice-storm (in the North Country of New York State between 7–9 January 1998) was the mechanical failure of lateral branches (73%) rather than trunks at or near their base (18%).
Effect of stem taper and wind speed profile
Comparisons among stems with different taper indicate that a trunk tapering according to geometric self-similarity has, on average, a larger factor of safety against wind-induced failure compared to a trunk tapering according to either elastic or stress self-similarity. The same conclusion was reached (King and Loucks, 1978) based on an allometric (dimensional) argument. Specifically, these authors assume that drag D is proportional to branch silhouette area A and that this area is proportional to the square of branch length L (i.e. DAL2). If the relation between stem diameter d and length is governed by two opposing and balanced torques (one generated by the wind Tw, which is proportional to DL, and another generated by stresses acting within branches Ts, which is proportional to the cube of stem diameter d3), then, since Tw=Ts such that DLd3, it mathematically follows that Ld. The same conclusion is reached if branch silhouette area and drag are assumed to be proportional to the product of stem diameter and length (equation 1). Under these circumstances, DAdL and, assuming, once again, that Tw=Ts and DLd3, it then follows that dL2d3 such that Ld. Accordingly, both derivations obtain the same conclusion—geometric self-similarity Ld is the optimal stem taper for maximizing the factor of safety against wind-induced mechanical failure.
Regardless of the type of stem taper, calculations indicate that the manner in which wind speed varies within a tree canopy as a function of distance above ground level has a significant influence on where factors of safety reach their lowest levels and thus which stems are most likely to fail as a result of drag. The prediction of wind speeds within tree canopies with or without leaves is complex and few useful formulae exist for this purpose, in large part because the numerical value of important parameters, such as the attenuation coefficient, vary as a function of foliage and stem density, and ambient wind speed measured at tree height (Lumley and Panofsky, 1964; Hicks, 1971; Monteith, 1973; Campbell, 1977). The normalized profile used to predict wind speeds within the wild cherry tree canopy is probably quantitatively useful only at low to modest wind speeds such as those that were actually measured during this study (i.e. ~10 m s-1), but probably both qualitatively and quantitatively inaccurate for estimating factors of safety for the same tree experiencing much higher wind speeds. Indeed, both theory and practice suggest that the two alternative wind speed profiles explored in this study are likely to be more representative of the wind speeds occurring within a canopy when ambient wind speeds measured above a large tree 20 m s-1 (see, for example, Campbell, 1977, Fig. 4.7).
In this regard, the factors of safety computed on the basis of a simple curvilinear decrease in wind speeds within the tree canopy predict the failure of larger lateral branches or of younger more distal portions of the trunk than those predicted on the basis of factors of safety predicated on the normalized wind speed profile empirically determined for more modest, lower ambient wind speeds (Fig. 9). This is consistent with the observation reported by other authors that global tree failure generally takes the form of trunk snapping some distance above ground level as opposed to uprooting.
All of the preceding discussion assumes that trees have mechanically stable root systems, which is known not to be the case even for a healthy plant under some circumstances (see Boe, 1965; Falinski, 1978, for case studies). No attempt was made during this study to deal with the issue of how and when roots fail, but it is clear that this is a serious omission when considering the mechanics of tree failure. As noted, wind produces a torque at the base of a large tree equal to the product of drag and the height at which this force acts (King and Loucks, 1978; Vogel, 1981; Niklas, 1998), and even a healthy tree will uproot rather than break under exceptionally high winds if its branches and trunk are made of sufficiently stiff and strong wood (Putz et al., 1983). Further elaboration of the method presented here to calculate factors of safety for dynamic wind-loadings, therefore, must account for the factors predisposing trees to wind-throw.
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1 Fax: +1 607 255 5407. E-mail:kjn2{at}cornell.edu
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References |
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Alexander RM.1971. Size and shape. London: Edward Arnold.
Ang A.-S, Tang WH.1975. Probability concepts in engineering planning and design. New York: Wiley & Sons.
Banks CC.1973. The strength of trees. Journal of the Institute of Wood Science 6, 44–70.
Behre CE.1921. A study of windfall in the Adirondacks. Journal of Forestry 19, 632–637.
Blickman R, Full RJ, Ting L.1993. Exoskeletal strain evidence for a trot-gallop transition in rapidly running crabs. Journal of Experimental Biology 179, 301–321.
Boe KN.1965. Windfall after experimental cuttings in old-growth redwood. Proceedings of the Society of American Forestry 1965, 59–63.
pbell GS1977. An introduction to environmental biophysics. New York: Springer-Verlag.
Carlquist S.1961. Comparative plant anatomy. New York, NY: Holt, Reinhart & Winston.
Carlquist S.1975. Ecological strategies of xylem evolution. Berkeley, CA: University of California Press.
Curtis JD.1943. Some observations of wind damage. Journal of Forestry 41, 877–882.
Esser MHM.1946a. Tree trunks and branches as optimal mechanical supports of the crown. I. The trunk. Bulletin of Mathematical Biophysics 8, 65–74.
Esser MHM.1946b. Tree trunks and branches as optimal mechanical supports of the crown. II. The branches. Bulletin of Mathematical Biophysics 8, 95–100.
Falinski JB.1978. Uprooted trees, their distribution and influence in the primeval forest biotrope. Vegetatio 38, 175–183.
Fraser GF.1962. Wind tunnel studies of the forces acting on the crowns of small trees. Reports of Forest Research 1962, 178–183.
Harcombe PA, Marks PL.1983. Five years of tree death in a Fagus–Magnolia forest, southeast Texas, USA. Oecologia 57, 49–54.
Hicks BB.1971. The measurement of atmospheric fluxes near the surface: a generalized approach. Journal of Applied Meteorology 9, 386–388.
Holbrook NM, Putz FE.1989. Influence of neighbors on tree form: effects of lateral shading and prevention of sway on the allometry of Liquidambar styraciflua (sweet gum). American Journal of Botany 76, 1740–1749.
Hutte P.1968. Experiments on windflow and wind damage in Germany: site and susceptibility of spruce forests to storm damage. Forestry Supplement 41, 20–27.
Jones EW.1956. Ecological studies on the rain forest of southern Nigeria. IV (cont.). The reproduction and history of the Okuma Forest Reserve. Part II. The reproduction and history of the forest. Journal of Ecology 44, 83–117.
Keller TS, Spengler DM.1989. Regulation of bone stress and strain in the immature and mature rat femur. Journal of Biomechanics 22, 1115–1128.[Web of Science][Medline]
King DA.1986. Treeform, height growth, and susceptibility to wind damage in Acer saccharum. Ecology 67, 980–990.
King DA, Loucks OL.1978. The theory of tree bole and branch form. Radiation and Environmental Biophysics 15, 141–165.[Web of Science][Medline]
Kirkpatrick SJ.1994. Scale effects on the stresses and safety factors in the wing bones of birds and bats. Journal of Experimental Biology 190, 195–215.[Abstract]
Lumley JL, Panofsky HA.1964. The structure of atmospheric turbulence. New York: John Wiley & Sons.
Mattheck C, Bethge K, Schäfer J.1993. Safety factors in trees. Journal of Theoretical Biology 165, 185–189.
Mayhead GJ.1973. Some drag coefficients for British forest trees derived from wind tunnel studies. Agricultural Meteorology 12, 123–130.
McMahon TA.1973. The mechanical design of trees. Science 233, 92–102.
McMahon TA, Kronauer RE.1976. Tree structures: deducing the principle of mechanical design. Journal of Theoretical Biology 59, 443–466.[Web of Science][Medline]
Metzger K.1893. Der Wind als massgebender Faktor für das Wachtsum der Bäume. Mundener Forstliche Hefte 3, 35–86.
Monteith JL.1973. Principles of environmental physics. New York: Elsevier.
Niklas KJ.1992. Plant biomechanics: an engineering approach to plant form and function Chicago IL.: University of Chicago Press, 607.
Niklas KJ.1994a. Interspecific allometries of critical buckling height and actual plant height. American Journal of Botany 81, 1275–1279.
Niklas KJ.1994b. Plant allometry: the scaling of form and process. Chicago IL.: University of Chicago Press.
Niklas KJ.1998. The influence of gravity and wind on land plant evolution. Review of Palaeobotany and Palynology 102, 1–14.[Web of Science][Medline]
Niklas KJ.1999. Changes in the factor of safety within the superstructure of a dicot tree. American Journal of Botany 86, 688–696.
Putz FE, Coley PD, Lu K, Montalvo A, Aiello A.1983. Uprooting and snapping of trees: structural determinants and ecological consequences. Canadian Journal of Forest Research 13, 1011–1020.
Runkle JR.1982. Patterns of disturbance in some old growth mesic forests of eastern North America. Ecology 63, 1533–1546.
Speck T.1994. Bending stability of plant stems: ontogenetical, ecological, and phylogenetical aspects. Biomimetics 2, 109–128.
Spotts MF.1959. An application of statistics to the dimensioning of machine parts. Transactions of the American Society of Mechanical Engineering. Journal of Engineering for Industry 81, 317–322.
Tateno M.1991. Increase in lodging safety factor of thigmomorphometrically dwarfed shoots of mulberry tree. Physiologia Plantarum 81, 239–243.
Tateno M, Bae K.1990. Comparison of lodging safety factor of untreated and succinic acid 2,2-dimethylhydrazide-treated shoots of mulberry tree. Plant Physiology 92, 12–16.
Timoshenko SP.1956. Theory of elastic stability. Toronto: D. Van Norstrand, Comp.
Vogel S.1981. Life in moving fluids. Boston: Willard Grant Press.
Vogel S.1996. Blowing in the wind: storm-resisting features of the design of trees. Journal of Arboriculture 22, 92–98.
Volk W.1958. Applied statistics for engineers. New York: McGraw-Hill, Comp.
Wainwright SA, Biggs WD, Currey JD, Gosline JM.1976. Mechanical design in organisms. Princeton NJ: Princeton University Press.
Weibull W.1939. A statistical theory of the strength of materials. Stockholm: Royal Swedish Institute.
Wood TWW.1970. Wind damage in the forest of Western Samoa. Malaysia Forestry 33, 92–99.
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