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Section of Plant Biology, Cornell University, Ithaca, New York 14853
Received for publication June 23, 1998. Accepted for publication October 8, 1998.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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w. These two dimensionless safety factors were determined for a total of 420 shoot segments comprising much of the aboveground biomass of a Robinia pseudoacacia (Fabaceae) tree measuring 18.7 m in height and 1347 kg in mass, and 0.46 m in diameter (40 yr old) at 1.2 m from the ground.
An S-shaped trend was observed when each of the two factors of safety was plotted as a function of stem age. Each factor decreased from a local maximum for the most distal (peripheral) stems in the canopy to a local minimum value for stems 
10 yr old; each factor increased again to another local maximum for stems 11–18 yr old, and then decreased steadily toward the base of the trunk. This trend was the result of the allometric relationships among stem diameter, length, biomass, and material properties (stiffness and strength) with respect to stem age. Although they were disproportionately more slender than their older counterparts, peripheral stems were sufficiently stiff and strong to sustain the stresses resulting from their weight and that of foliage without deflecting under these loads, yet they were sufficiently flexible to easily bend and thereby presumably provide a mechanism to reduce the drag forces acting on the entire tree. In contrast, the internally imposed mechanical forces acting on progressively older stems increased at a greater rate than the observed rate of increase in stem stiffness, strength, or diameter. The probability of mechanical failure, which must be considered from a demographic perspective (i.e., an age-dependent phenomenon), thus increased from older branches to the base of the trunk. Reports of similar allometric trends based on interspecific comparisons among diverse dicot species comply with the allometry observed for the R. pseudoacacia tree and suggest that the S-shaped trend for the factor of safety holds for stems differing in age drawn from individual trees and for the trunks of conspecifics differing in age drawn from a dense population.
Key Words: biomechanics • dicot trees • Fabaceae • factor of safety • plant allometry • Robinia pseudoacacia • stem stiffness and strength
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; McMahon and Kronauer, 1976
; Niklas, 1992
). The result is a basipetal increase in the magnitude of self-loading from the tips of the canopy to the base of the trunk. This load distribution is mechanically accommodated by ontogenetic changes in the girth of successively older stems as a consequence of the accumulation of secondary wood, which is the stiffest plant tissue relative to its bulk density known (i.e., the highest density-specific stiffness; see Wainwright et al., 1976
; Vincent, 1990
; Niklas, 1992
). Since the mechanical stress experienced anywhere along the length of a branch equals the bending force of the distal biomass divided by the cross-sectional area measured at any point and since the accumulation of secondary tissues in stems increases both the force and the cross-sectional area of stems, the issue is whether the rate of increase in the force resulting from stem growth is sufficiently scaled to the rate of increase in the cross sectional area such that mechanical stresses do not exceed the strength of branch tissues. Since even very old trees support their own weight, there is good reason to believe that the ontogenetic changes in stem girth and strength due to the accumulation of wood in progressively older stems are scaled to provide a sufficient factor of safety against mechanical failure under normal loading conditions (King and Loucks, 1978
; Dean and Long, 1986
; King, 1986
; Mattheck, 1992
; Sterck and Bongers, 1998
). However, tree trunks and old branches mechanically fail when subjected to large external loads, even though the younger branches they support remain mechanically intact (Putz et al., 1983
; Mattheck, 1992
; Coutts and Grace, 1995
). This indicates that the factor of safety against catastrophic mechanical failure may not be exceedingly large for older stems and that the factor of safety may vary as a function of stem age and position. Indeed, the failure of tree trunks and older branches, which is often seen in ice storm conditions (personal observation), suggests that the factor of safety may vary in complex ways throughout the mechanical superstructures of trees.
The concept and application of a factor of safety are well known in the engineering and biological sciences, which recognize that any load-bearing structure must cope with unprecedented loads, yet not at the expense of an uneconomical investment in materials that increases the magnitude of self-loading (Volk, 1958
; Spotts, 1959
; Leiser and Kemper, 1973
; McMahon, 1973
; McMahon and Kronauer, 1976
; Wainwright et al., 1976
; King, 1981
). Factors of safety provide convenient quantitative expressions of the trade-off between the probability that a structure will fail when excessively loaded and the construction costs required to prevent this from happening. Several alternative but biologically equally realistic ways exist to compute factors of safety, e.g., the quotient of the maximum length to which the stem can grow before it elastically deflects under its own weight (the critical buckling height) and the actual length of the stem, and the quotient of the bending stress that will cause the stem to rupture (the rupture modulus) and the working stress of the stem. Either or both of these quotients can provide the necessary means to evaluate the extent to which the factor of safety varies in the superstructure of a tree. In theory, each quotient must equal or exceed unity because each stem must de minimus mechanically support itself as well as externally applied mechanical loads (e.g., wind pressure or the accumulation of snow or ice) whose anticipated effects require a factor of safety greater than one.
In this paper, I demonstrate that the factor of safety varies significantly within the superstructure of a tree. This demonstration is based on data gathered from a representative Robinia pseudoacacia tree, which was cut into 420 of its constituent support members (i.e., twigs, branches, and trunk segments) differing in taper and mechanical properties (stiffness and strength). For each of these members, the numerical values of two factors of safety were calculated: the quotient of the critical buckling height and the actual length of each stem, and the quotient of the rupture modulus and the working stress of each stem. These two factors, which are based on the criterion that elastic failure is intolerable, yield a similarly shaped trend along the length of the entire tree. This trend is discussed in the context of the hypothesis that tree development and growth impose constraints on the ability to maintain elastic stability. These constraints result in the steady erosion of factors of safety that may favor the survival of young and adolescent trees and that preprogram the death of older trees by means of mechanical failure of old branches or entire trunks when excessive loads are externally applied. The influence of root anchorage on factors of safety is only discussed briefly in this context.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Jaffe, 1973
; Shukla and Ramakrishnan, 1986
; Canham et al., 1990
; Kohyama and Hotta, 1990
; Tateno, 1991
; Ackerly, 1996
). This was considered desirable for mechanical testing because twigs and portions of older branches would, on average, have larger slenderness ratios (length/diameter) than those of their counterparts from a tree growing in an open site (Niklas, 1992
) and because the factors of safety to be calculated are based on the assumption that the stems had vertical or near vertical orientations (see below). The tree was dissected into segments (using the nodes of divergent branches as natural division points) to provide specimens differing in size and age for mechanical tests (see below). The length, diameter, and mass of each segment were measured and "mapped" onto a sketch of the tree for future reference. Segment diameter D was determined from the minimum and maximum for the proximal cut end of each segment (many stems had elliptical cross sections whose major axes were oriented in the vertical direction with respect to ground level); segment length L was determined as the sum of its length plus the lengths of the segments distal to it on the longest pathway to the most distal point on the branch (Fig. 1). Segment mass M was determined as the sum of the mass of the segment plus the mass of all segments distal to it. Negligible amounts of mass were lost due to cutting branches and the trunk. The mass of each segment was subsequently converted in its corresponding force F (i.e., M in kg x 9.8067 N/kg = F). The age of each segment A (in years) was determined on the basis of terminal bud scars on young stems or the number of concentric growth layers ("growth circles") on the surfaces of transverse cuts through older stem segments.
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To determine E, each segment was placed between two V-shaped vertical supports and loaded with varying amounts of sand that were slowly poured into a container attached by a cord around the segment's effective mid-length 
/2 at which point a 
-shaped needle was inserted. The horizontal portion of the needle was sighted against a metric ruler before and after loading each segment to measure the mid-length vertical deflection 
(in units of metres) resulting from the externally applied load P (the mass-force of the sand, container, and cord expressed in units of newtons). The load was then removed to determine whether the segment elastically recovered its original posture ( = 0); for those that did, E (in units of newtons per square metres) was computed using the formula
is the length (in units of metres) of the segment measured between the two V-shaped vertical supports and I is the second moment of area, which is given by the formula I = 0.01563
D4 for a circular cross section (Niklas, 1992
). Three representative cross sections taken through the middle and both ends of each segment were used to determine a mean D (Wainwright et al., 1976
).
MR was measured by repeating the bending test protocol but with loads sufficient to produce a permanent, plastic deformation (i.e., > 0). This was accomplished by reloading each segment with a mass-force that was initially 10% larger than that used to measure E, gradually increasing this load in increments of 5% of the original value and removing each incrementally larger load until the segment failed to elastically rebound (i.e., > 0 was permanent). For most woody stems, mechanical failure was indicated by a "snapping" sound and a sudden increase in . The smallest load PR that resulted in this phenomenon (or in > 0 after a younger specimen was unloaded) was used to compute MR by means of the formula PR /0.785D2 (i.e., MR has units of stress, newtons per square metres).
In the case of segments for which D > 0.03 m, E and MR were measured for milled beams of wood with a square cross section. The slenderness ratio (length/diameter) of each beam was 
20. This protocol was required because the theory of elastic behavior assumes that the aspect ratio of a specimen tested in bending 20 and because segments with D > 0.03 m were often too short to satisfy this assumption. Each wooden beam was tested in the manner described above to measure E and MR, however, I was computed using the formula I = 0.0833s4, where s is the dimension of side of the square cross section.
Because the outer and presumably less stiff and strong stem tissues were removed during the milling process (e.g., phellem), the values of E and MR reported for them here are likely overestimates of the stiffness and load-bearing capabilities of the stem segments from which the beams were drawn. However, this bias was not considered severe because, if anything, it would work against the decrease seen in the factors of safety observed toward the base of the tree (see Fig. 5).
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w (Niklas, 1992
). The critical buckling height of each specimen was computed from the formula 
is the bulk density of the segment for which E was determined, and D is here the diameter measured at the proximal end of the segment. [This formula holds for vertical support members (Greenhill, 1881
; McMahon, 1973
) and, for this reason, the 420 stems examined here were selected on the basis of their vertical or near vertical orientation with respect to the main axis of the tree. The numerical value of C was computed on the basis of stem taper distal to where D was measured.] For Eq. 2 to be dimensionally consistent, the bulk density, which was initially determined in units of kilograms per cubic metres was converted into units of newtons per cubic metres (i.e., the expression E/ in Eq. 1 has units of metres); was calculated by dividing the mass of the segment by the volume of water it displaced. The working stress w used to compute MR /w was calculated by dividing the cumulative force F measured distal to each segment (see Fig. 1) by 0.7854 D2 (i.e., w has units of stress, N/m2).
Statistical analyses
The goal of this study was to determine how the factor of safety varied within the superstructure of the R. pseudoacacia tree. If they occur, differences in the numerical value of the factor of safety, denoted generically here as S, can occur as a result of differences in any of the morphological or biomechanical properties measured for segments, and so no independent variable legitimately exists against which the dependent variable S can be regressed. Nevertheless, the ordinary least squares regression model was used to determine statistical trends in the data and the extent to which paired variables were correlated because the objective was to quantify trends and not to predict the values of variables based on the values of other variables. Stem D and A were selected as the independent variables in most cases, although a number of statistical trends reported here have a biomechanical property as their independent variable.
Cluster analyses were used to determine the age-distributions of segments (e.g., Fig. 2). For A < 18 yr, the number of stem segments was sufficient to report legitimate means and standard errors for S. Statistical analyses of segments A > 18 yr, however, indicated an insufficient number to provide statistically legitimate means and standard errors. On average, the segments that could be assigned to the A > 18 yr age-class had diameters large enough to produce two rectangular beams of wood, each of which could be tested in bending. Thus, although the number of segments A > 18 yr was comparatively small, the number of wood specimens for which mechanical properties and the factor of safety could be calculated was doubled to report statistically legitimate means and standard errors (see Fig. 5B).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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10 yr.
The relationships between Young's elastic modulus and the rupture modulus with respect to force exerted by the cumulative biomass measured distal to stem diameter were likewise best approximated by a second-order polynomial regression curve when all stem age-classes were considered: log E = 1.26 + 0.08 log F + 0.003 (log F)2 (N = 488, r2 = 0.821) and log MR = 1.71 + 0.08 log F + 0.003 (log F)2 (N = 488, r2 = 0.608) (Fig. 3). [The sample size N = 488 reported for these regression curves is larger than that of the number of stem specimens (N = 420) because two samples of wood were tested from each stem in the older age class (N = 136 samples of wood from N = 68 stems).] Other than for their y-intercepts, these two regression curves were statistically indistinguishable, indicating that the Young's elastic modulus and the rupture modulus varied in nearly the same way with respect to the cumulative stem mass-force measured at D regardless of stem age. However, the statistical trends observed for E and MR were age dependent and log-log linear for young and old stems. For stems A <10 yr, log E = 1.11 + 0.030 log F (N = 352, r2 = 0.509) and log MR = 1.54 + 0.028 log F (N = 136, r2 = 0.255). For stems A 10 yr, log E = 1.21 + 0.049 log F (N = 352, r2 = 0.907) and log MR = 1.66 + 0.051 log F (N = 136, r2 = 0.586). Statistical comparisons between the slopes and standard errors of these log-log linear regression curves indicated that the Young's modulus and the rupture modulus increased at a faster rate with respect to the mass-force for older stems compared to their younger counterparts. This phenomenology was explicable in terms of the accumulation of wood in older stems, which is stiffer and stronger than any other tissue. The Young's elastic modulus and the rupture modulus had a log-log linear relationship for stems representing all age-classes: log E = 2.18 + 0.77 log MR (N = 488, r2 = 0.723) (Fig. 4). The statistical trends between these two biomechanical properties were not age dependent. Although the rate of increase in Young's elastic modulus (stiffness) with respect to the rupture modulus (a measure of strength) was indistinguishable between young and old stems, the slope of the allometric relation between these two mechanical properties indicated that an increase in stem strength resulted in a disproportionately smaller increase in stem stiffness.
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w and stem diameter D (N = 488, r2 = 0.957). No statistically significant correlation was observed between the factor of safety computed on the basis of Hcrit/L and D (N = 420, r2 = 0.074) (Fig. 5A). However, when the means and standard errors of each of the two factors of safety were computed for stems differing in age and plotted against stem age, statistically significant trends were observed for each factor (Fig. 5B). In general, twigs and juvenile stems A 10 yr had higher factors of safety compared to stems A <10 yr. The factor of safety increased for stems between 11 and 18 yr old and then steadily decreased toward the base of the tree, reaching a minimum factor of safety 2.8. Thus, in general, there were two inflection points in the trend between the two safety factors and stem age: one at 10 yr and another at 20 yr (Fig. 5B).
The age-dependent changes observed for the factor of safety S were associated with changes in stem taper. This was conveniently visualized when log10-transformed data for normalized stem length (L measured at 1.2 m = 18.75 m) were plotted against the log10-transformed data for normalized stem diameter (D measured at 1.2 m = 1.1 m), and the resulting curvilinear relationship was compared with the log-log linear regression curves predicted by the geometric, elastic, and stress self-similarity models for the mechanical design of trees, which obtain slopes 
equal to 1/1, 2/3, and 1/2, respectively, when normalized D and L are log10–transformed (Fig. 6). Regression analyses indicated that stems A 
10 yr, for which S and A were inversely proportional, tapered roughly according to a 3/2 power rule (which complied with none of the three contending models; = 3.4, r2 = 0.93), while stems A >10 yr, for which S initially increased but subsequently declined to a minimum value with increasing A, tapered roughly according to a 2/3 power rule (which complied with the elastic self-similarity model; = 0.69, r2 = 0.97).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Jacobs, 1954
; Maggs, 1960
; Larson, 1965
; Brunig, 1976
; Holbrook and Putz, 1989
; Tateno, 1991
). Likewise, interspecific differences in these features are well documented (e.g., Whittaker and Woodwell, 1968
; Horn, 1971
; King, 1981
, 1986
; Niklas, 1992
, 1994
). It is thus unwise to propose a general allometric model to account for the trend in the factor of safety (based on two criteria of failure) reported for a single R. pseudoacacia specimen growing in a protected site unless this trend is confirmed by observations of a larger number of conspecifics or phyletically diverse dicot tree species growing in different habitats. Substantiating evidence for the trend exists. Comparisons among 27 other R. pseudoacacia trees differing in age and growth sites show that sapling trunks and the peripheral stems of mature trees have equivalent taper and that both undergo an allometric transition from scaling exponents that substantially exceed unity to those that comply with elastic self-similarity as they grow in size (Niklas, 1995
). The morphometric data reported here for stem taper are thus not atypical for R. pseudoacacia. These data are also similar to those reported by Bertram (1989)
who showed that the peripheral stems of Acer saccharinum were disproportionately more slender than their older counterparts and that a transition from geometric to elastic self-similarity occurs as stems accumulate secondary plant tissues. Likewise, the allometry of tree height with respect to trunk diameter reported by Thomas (1996a
, b
) for phyletically diverse tropical tree species indicates a transition from very slender taper during early growth to tapers that eventually comply with elastic self-similarity.
Previous studies have also shown that younger R. pseudoacacia stems are less stiff and strong than their older counterparts (Niklas, 1997a
–
c
). This change in the material properties of stems is the consequence of ontogenetic changes in the relative proportions of primary and secondary tissues in comparatively young stems and changes in the relative proportions of sap- and heartwood as older stems mature. Although the relative proportions of these tissues were not measured during the present study, the relationship among stem diameter, stiffness, and strength observed was indistinguishable from that previously reported (Niklas, 1997a
–
c
).
The trend for the factor of safety within the mechanical superstructure of a R. pseudoacacia tree is in general agreement with those observed by other workers. King (1986)
found that the safety factor for static buckling increased with trunk diameter for old A. saccharum trees but that this trend was accompanied by an increased occurrence of trunk failure due to high wind. This trend accords with that reported here because it suggests that the margin for safety against catastrophic failure when externally applied loads are high decreases as trees mature. Bongers and Sterck (1998)
likewise report that the factor of safety decreases with the size of smaller tropical tree species and then increases for relatively larger trees (diameter at breast height >0.3 m). A similar trend is reported for both temperate (King, 1981
, 1986
) and tropical forest tree species (Claussen and Maycock, 1995
). Using critical buckling height as the criterion for failure, Sterck and Bongers (1998)
more recently showed that the factor of safety can be greater for small as well as large specimens (diameter at breast height <0.1 m and >0.4 m, respectively) and reaches a minimum value for conspecifics of intermediate size (diameter at breast height 0.3 m; see Sterck and Bongers, 1998
, fig. 2b). Putz et al. (1983)
found that wood properties were the most important factors determining the type of tree death (snapping vs. unrooting) based on a census of trees damaged by an intense tropical storm affecting Barro Colorado Island (Republic of Panama). Uprooted trees tended to have denser, stiffer, and stronger wood than snapped trees. Nevertheless, the most common form of death was trunk snapping (out of 310 fallen trees, 70% snapped, 25% uprooted, and 5% broke off at ground level). These data are consistent with a decrease in the factor of safety toward the base of the trunk of mature trees.
However, because most of these studies make the assumption that the material properties of stem tissues are age independent and because all of these studies are based on inter- or intraspecific comparisons rather than comparisons among stems drawn from within a tree canopy, the extent to which previously reported trends in safety factors comply with that presented here is difficult to evaluate. Nevertheless, the advocacy of a general allometric model for ontogenetic changes in the safety factors in dicot tree stems may not be premature, especially when seen in the light of the conservative nature of primary and secondary stem growth and development, which holds across most dicot species (Esau, 1965
; Gifford and Foster, 1988
). While this model could provide insights into the maximum overall height and biomass individual trees could achieve before they mechanically fail under their own weight or some externally applied mechanical force, such a model must also admit to latitude in these limits owing to the consequence of growth responses to local environmental conditions and phyletic differences in the relative growth rates and hydraulic and nutritional requirements of different species. Likewise, although differences in the structural and biomechanical properties of the stems differing in age result from the same fundamental developmental program underwriting primary and secondary growth, differences occur because the age-class of stems for which the transition from primary to secondary growth occurs and the relative proportions and biomechanical properties of the different kinds of tissues present in stems differing in age influence the allometry of mechanical failure. A demographic as well as phyletic perspective therefore must be advocated to reflect the fact that stem age-classes can achieve variation in the factor of safety but nevertheless obtain a general trend of declining margins of mechanical safety toward the base of old trees.
Such an allometric model rests on the recognition that the peripheral, more juvenile stems of a mature dicot tree are disproportionately more slender than their older more proximal counterparts and that the stiffness and strength of stems can fail to increase at a rate sufficient to cope with the mechanical stresses imposed on progressively older stems by younger cohorts of shoots despite a basipetal increase in stem diameter attending the build up of secondary tissues. These features, which are evident in the data from the tree studied here, are summarized by the S–shaped trend in the factor of safety going from the peripheral to basal structural elements of trees. This trend indicates that the peripheral elements in the tree canopy are sufficiently stiff and strong to cope with their static loads, but are also the most flexible structural members and thus can bend and deflect in the wind, thereby reducing the drag force imposed on subtending older branches and the tree as a whole owing to their slenderness (see Vogel, 1989
, 1992
, 1995
). The slenderness ratio of peripheral stems (length/diameter) increases with respect to stem diameter, reaching a limit of 30 (Fig. 7A). This allometry may relate to a reduction in light availability in the upper canopy with increasing tree size, attended by a greater investment in stem length with respect to diameter compared to older stems for which further elongation is impossible (Sterck and Bongers, 1998
). But, from a biomechanical perspective, two mutually nonexclusive scenarios for the allometry of peripheral stem shape are self-evident (Fig. 7B). First, peripheral stems may initiate growth in length with diameters sufficient to insure a high factor of safety against elastic deflection, and, second, the transition from primary to secondary growth can increase the stiffness of peripheral stems (without a noticeable diminution of the slenderness ratio) such that continued stem elongation does not result in elastic deflection. The first of these scenarios mirrors the growth of arborescent palm species, for which Rich et al. (1986)
show that the factor of safety is initially high but steadily declines because of the pachycaulous growth habit and the inability to increase stem diameter with age. The second scenario accords with measurements of the stiffness and strength of primary vascular tissues and sap- and heartwood (see Niklas, 1997a
–
c
). It is worth nothing that the high but variable factors of safety calculated for peripheral, young twigs may reflect the fact that these elements in a tree's mechanical infrastructure tend to experience large and variable bending moments resulting from their weight and that of the leaves they bear. These stems also likely experience high drag forces because of their location in the tree canopy.
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Since the biomechanical properties and morphology of stems are correlated with stem age, the S-shaped pattern for the factor of safety reflects the demography for the probability of survival, both for the stems comprising the superstructure of an individual tree and, by inference, for the survival of conspecifics differing in age drawn from the same population. Stems of equivalent age from the same tree have nearly equivalent probabilities of mechanical failure under the same loading conditions. Likewise, the trunks of trees belonging to the same age-class drawn from the same population of conspecifics likely have probabilities of failure equivalent to those of stems of the same age-class from an individual tree provided that similar local environmental factors occur. This inference is based on the supposition that the ontogenetic changes in the taper and mechanical properties attending the growth and development of tree branches are nearly identical to the ontogenetic changes attending the growth of the trunks of conspecifics. If so, then the demography of shoot failure observed for stems drawn from within a tree's superstructure can be nearly the same as that of trunk failure observed for individuals drawn from the same population of conspecifics, provided that the local environments experienced by the individual branches of a mature tree and by adolescent conspecifics growing under crowded conditions are similar.
Any general allometric model for the probability of mechanical failure must take notice of the influence of root anchorage on the factor of safety. This omission is important because variations in the factor of safety within the aerial portions of trees may have little or no effect on the probability of the catastrophic mechanical failure, provided that root systems fail first. Field studies and experiments indicate that even large trees can uproot before their trunks break when trees are excessively wind loaded (Putz et al., 1983
; Crook, Ennos, and Banks, 1997
). Large trees with very stiff and strong wood tend to uproot in high winds rather than snap at their base, although considerable interspecific variation in the mode of failure exists (see Putz et al., 1983
). The condition of the soil is also an important consideration; uprooting is more likely to occur when soils are wet because of shearing at the root–soil interface (Niklas, 1992
). Unfortunately, considerably less is known about the mechanics of root vs. stem failure. Generalizations about the relative susceptibility of trunk vs. root failure are thus less forthright but essential to our understanding of tree biomechanics.
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FOOTNOTES |
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Brunig, E. F. 1976 Tree forms in relation to environmental conditions: an ecological viewpoint. In M. G. R. Cannell and F. T. Last [eds.], Tree physiology and yield improvement, 139–156. Academic Press, London.
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