Shape of Tree Stems_uniform Stress Hypothesis

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Tree Physiology 14,49-620 1994 Heron Publishing-V ictoria, Canad a

Shapeof tree stems- a re-examination of the uniform stresshypothesisJOHN MORGAN and MELVIN G. R. CANNELL*1 Departm ent of Civi l Engineering and Bui lding Science, University of Edinburgh, EdinburghEH9 3JL, Scotland, UK2 Insti tute of Terrestrial Ecology, Bush Estate, Penicuik, M idlothian EH26 O QB, Scotland, UKReceived Apri l 8,1993SummaryThe transfer m atrix method of structural analysis was used to examine the hypothesis that tree stem s growto a shape that tends to equalize the average bending plus axial stre sses o which they are subjected alongtheir length. The method and computational procedures were checked by comparing compu ted height-diameter profi les with those calculated using elementary stress theory for trees with simple forcedistributions in the crown. M easured height-diameter profi les for trees were then taken from thel iterature and shown to be wel l-f it ted by profi les calculated to give uniform stress along the stem s, usingthe mo st real ist ic average forces and force distributions within the crown s. At high wind speeds, theheight-diameter profile giving uniform stress was more tapered than the profile giving uniform stress atlow w ind speeds. The profile giving uniform stress was similar o ver the normal range of average windspeeds of 2 .5 to 10.0 m SC (at the top of the canopy). But a tree that had grown to give uniform stressalong i ts stem in an average wind of 5 m s- showed markedly decreased stress with height at windspeeds of about 15 m s-t or more, and increased stre ss with height (to the crown base) at wind speeds ofabout 1.25 m S C or less. The fact tha t tree stems develop shapes in response to average condit ions, butshow varying stress distributions in extreme condit ions, ma y help to explain some of the apparentevidence for non-uniform stress distribution in the l i terature. In general, our analysis supports the abovehypothesis for the stem region above the butt sw el l .Keyw ords: axial stress, bending stre ss, force, structural analysis, transfer m atrix m ethod, wind speed .

IntroductionWhen trees bend in the wind, they experience maximum bending stress at the outersurface of their stems, close to the cambium. One of the most enduring theories intree biology is that the cambium produces new wood in such a way as to equalize thedistribution of stress along the outer surface of the stem (Metzger 1893, Larson 1965,Assmann 1970, Wilson and Archer 1979, Mattheck 1990, 1991). In other words,cambial growth tends to be greatest in regions of highest stress, and least in areas oflowest stress. This theory is sometimes referred to as the mechanistic theory ofconstant stress: but it is more exact to cal l it the mechanical theory of uniformstress because the stress distribution is not constant over time (or constantly uniform)when trees are subject to variable wind forces.

There is some difference of opinion about the valid ity or universality of theuniform stress hypothesis. Early foresters observed that tree stems assume a shapethat gives approximately uniform stress because the stem taper below the crown of

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50 MORGAN AND CANNELL

many trees approximates a cubic paraboloid (Assman 1970). Others have beenwill ing to accept the uniform stress hypothesis (Petty and Swain 1985, Dean andLong 1986) and Mattheck (1990, 1991) has assumed that it applies over the entirewoody structure of trees. However, West et al. (1989) found that the stress distribu-tion along the stems of Eucalyptus regnans F. Muell. depended on the wind velocitygradient within the canopy. In another analysis, Milne and Blackburn (1989) foundthat Picea sitchensis (Bong.) Can: stems tended to have a region of maximum stress,which occurred closest to the ground in stems with the greatest taper, and whichcoincided with the region where stem breakage often occurs in strong winds.

Experimental and observational studies lend considerable support for the view thatcambial growth is stimulated by continual high mechanical stress (force/area) orpossibly, the associated strain (fractional change in length) (Zimmermann andBrown 1971). Most notably, several tree bending and guying experiments haveshown that cambial growth is greatest where most bending occurs (Jacobs 1954,Larson 1965, Valinger 1990). The ways in which mechanical forces are translatedinto increased cell growth are unknown, although piezoelectric forces in crystallinesubstances, such as cellulose, and a redistribution of growth hormones may beinvolved (Jaffe 1973, Mitchell et al. 1977, Hunt and Jaffe 1980, Worrall 1980).

In this paper, we used the transfer matrix method of structural analysis, previouslyused to examine the growth of branches (Morgan and Cannel1 1987, Cannel1 andMorgan 1988, 1989, 1990), to examine the evidence for uniform stress along treestems. We offer a slightly modified hypothesis (see below) and show that some ofthe apparently conflicting evidence in the literature can be resolved by realizing that:(1) the profile of stem diameter with height is a poor test of the uniform stress

hypothesis because bending stress is inversely related to the cube of stemdiameter-consequently small differences in measured diameters associatedwith trunk irregularities give large differences in calculated stress; and

(2) the distribution of stress along a stem depends on the wind force; the stress maybe uniformly distributed in a steady wind close to the average wind speed, butwill not be uniformly distributed at high or low wind speeds.

HypothesisIf tree stems thicken preferentially each year in regions where mechanical weaknessoccurs (i.e., in areas of highest bending plus axial stresses), the stem shapes (height-diameter profiles) observed will be the time-averaged response to the stressesproduced by variable forces during the life of the tree. In other words, tree stemsgrow to a shape that tends to equalize the average bending plus axial stresses to whichthey are subjected along their length. The average bending plus axial stress willdepend on the history of the tree crown and the average wind speed over the life ofthe tree.

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SHAPE OF TREE STEMS-UNIFORM STRESS HYPOTHESIS 51

Method

Elementary theoryTree stems may be considered to be cantilever beams fixed at a point above the regionof butt swell. Elementary theory states that the longitudinal stress, (3, in beams ofcircular cross section with bending moment, M, is given by:

where d is the beam diameter (Timoshenko 1953, provided that beam diameter ismuch less than beam length, and the deflection is small. The height-diameter profileof tree stems having uniformly distributed stress (0) below and within the crown canbe calculated using this elementary theory, in which the horizontal (wind) force, F,is distributed within the crown in a simple way.

Consider the two simple cases in Figure 1. Equation 1 implies that, for stress to beuniformly distributed along the stem, the cube of the diameter at any height must beproportional to the distance from that height (h) to the center of pressure of thehorizontal wind force on the crown (II,). That is, to give uniform stress along the

0.0 0.2 0.4 0.6 0.8 1.0Relative Diameter

Figure 1. Theoretical (lines) and calculated (points) height-d iamete r profiles of stems with uniform stressalong their length. (a) A uniform distribution of wind force, and (b) a triangular distribution of windforce. The theoretical relationships (lines) were based on text Equatio ns 2 and 4 for case (a), andEquatio ns 2 and 6 for case (b). The points were calculated indepe ndently using the transfer matrixmethod. H = tree height; H , = length of crown, h = height, h, = height within crown, H , = height to thecenter of pressure from the wind force, F, acting at point g (see text).

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52

stem below the crown:

andM=F(Hg-h)

MORGAN AND CANNELL

(2)therefore

d3 = (Hg - h) .For case (a) in Figure 1, the bending moment within the crown, which has a

uniformly distributed force, is given by:

M=&(Hc-hc)2, c (3)

where F is the force on the crown, H, is the length of the crown, and h, is the heightabove the base of the crown. It follows that, to give uniform stress along the stemwithin the crown in case (a):

d3 0~ (Hc - /z,)~or

$/2 oc (Hc - hc) .For case (b) in Figure 1, the bending moment within the crown, which in this

instance has a triangular distribution of force, is given by:M=F3H2 WC- Q3 .

C

It follows that, to give equal stress along the stem within the crown in case (b):d3 0~ (H, - Q3 (6)

ord= (H,-A,).

Equations 2, 4 and 6 were used to calculate the theoretical tree height-diameterprofi les assuming that the crown occupied the top third of the tree (Hc = H/3). These


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relationships are shown by the continuous lines in Figure 1, in which heights aregiven relative to tree height (H) and diameters are given relative to the diameter atthe base of the tree (ignoring butt swell).Transfer matrix methodPreviously, we showed how the distribution of forces within cantilever beams ofcomplex shape, with small or large deflections, can be calculated by dividing thebeam into small sections and using a matrix equation to relate the conditions at theends of each section (shear force, V, bending moment, M, angle, and deflection) tothe distributed load, p (or any other load), given the length, diameter and Youngsmodulus of the material (Figure 2) (Morgan and Cannel1 1987, Morgan 1989, seealso Milne and Blackburn 1989). In studies on branches, we used iterative proceduresto calculate the diameters required to support self weights, with given deflections,angles, taper and Youngs moduli (Cannel1 et al. 1988, Cannel1 and Morgan 1990).It is equally possible to specify the diameters and to calculate the resulting distribu-tion of forces, or to specify that stress is uniform along the length of the beam withgiven forces and to calculate the resulting diameter-length profile.

Our computer program was checked by determining the diameter-height profilesunder uniform stress conditions for the two simple cases shown in Figure 1. Thecalculated values are shown as points and give an exact fit to the theoreticalexpectation (Figure 1).

Figure 2. A section of a deflected stem showing the forces acting upon it. V = shear force, N = axial force,M = bending moment, W = weight, p = distributed force, 8 = angle. Subscripts L and R refer to left andright. (See also Morgan and Cannel1 1987, Morgan 1989).


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In the analyses reported below, the vertical force distribution on the crowns wastaken directly from published values, or was assumed to be triangular, approximatingthe leaf area density distribution of many tree species (e.g., Landsberg and James1971, Kellomaki et al. 1980, Massman 1982, West et al. 1989, Wang et al. 1990).Also, the effects of self weight, W, and of left and right axial forces, NL and NR,respectively, were incorporated into the transfer matrix equations as described byMorgan (1989) (Figure 2). Self weight affects the stress distribution in two ways:first, as the stem deflects, it causes an additional bending moment along the stem;and second, it produces an axial force (compression along the axis of the stem) and,therefore, an axial stress. The axial stress is generally small compared with thebending stress, but it does mean that the total bending at any point along a stem is nolonger exactly proportional to d3 (Equations 1 and 2), but is proportional to thediameter to a power greater than three. Also, when self-weight is included, the stemshape giving uniform stress depends on the extent to which the stem is deflected fromvertical, and hence, on the average force due to the wind.

Measured and calculated height-diameter profilesOther workers have measured the height-diameter profi les of tree stems, estimatedthe force distribution on the crowns and then calculated the distribution of stress inthe stems (West et al. 1989, Milne and Blackbum 1989). Our approach was to usethe force distr ibution on the crowns to calculate the height-diameter profile that gavea uniform stress distribution. We then compared the measured height-diameterprofiles with those calculated for uniform stress (Figures 3 and 4).

Ten trees of Picea sitchensis (Gardiner 1989)The points in Figure 3 are stem diameters at different heights of ten trees ofP. sitchensis, reproduced from Gardiner (1989), normalized to a breast height diam-eter of 1.5 m (i.e., the diameter at breast height is 1.0 in Figure 3). The trees weregrowing in Scotland at 3600 stems ha-, and averaged 15 m in height, 0.165 m inbreast height diameter, 162 kg stem fresh weight, 50 kg branch fresh weight, withYoungs modulus 6.6 GPa, and a drag force of 1587 N at 20 m s- wind speed. Thetransfer matrix method was used to calculate the height-diameter profile givinguniform stress based on three assumptions. First, the crown was assumed to bedistributed over the top half of the tree, as suggested by other measurements (Coutts1986). Secondly, the vertica l distribution of the force was assumed to be triangular,as shown in Figure 3, approximating the near-normal distribution of leaf area withincrowns of 19 si/c lrens is of similar age, as mcasurcd !)y Norman and Jarvis (1974).Thirdly, it was assumed that the average wind speed experienced by the trees was5 m s-l , based on wind speed averages for the UK uplands (Shellard 1976), and thatdrag was proportional to the square of wind speed.

The continuous line in Figure 3 shows the calculated height-diameter profile forthe condition of uniform stress (bending plus axial stress) along the stem. There was


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SHAPE OF TREE STEMS-UNIFORM STRESS HYPOTHESIS 551.0K..0.6

E.Fr

Within crownH

B;Id

Relative Diameter

Figure 3. Measu red (points) and calculated (line) profiles of relative s tem diameter (diameter at breastheight = l.O), and relative ste m height for Pica sitche nsis. The measured values are for 10 trees , fromGardiner (1989). The calculations were made using the transfer matrix method for the condition ofuniform stres s along the ste m, using the horizontal (wind) distribution show n in the inset diagram, a windspeed of 5 m SC, and the weight distribution and other tree properties given by Gardiner (1989).

SD [ (a) Eucalyptus regnans

25 -

(b) Pica sitchensls

0 4 8 12 16 20 24

Stem diameter (cm)Figure 4. Measu red (points) and calculated (lines) height-diameter profiles (above butt swell) for singletrees of (a) Eucalyptusregnans and (b) Piceasitchensis. The measured values are from We st et al . (1989,their Figure la) and Milne and Blackb um (1989, their Figure 4). The calculated lines were derived usingthe transfer matrix method for the condition of uniform stress up the stem s, using the horizontal forcedistributions show n in the inset diagrams and other conditions as stated in the text. The open points arediame ters in the region of butt s well.


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good agreement between the measured height-diameter profile and the profilecalculated for uniform stress.Single tree of Eucalyptus regnans (West et al. 1989)The points in Figure 4a are stem diameters at different heights within a singledominant 20-year-old tree of E. regnans growing in a thinned stand in Tasmania(West et al. 1989, their Figure la). West et al. (1989) also reported the verticaldistribution of crown and stem fresh weights (their Figure lb), and the verticaldistribution of force on the crown (their Figure 3). The latter was used directly in thetransfer matrix calculations, taking the force distribution illustrated in Figure 4a(with their kr = 0.048, kZ = 0.021, see West et al. 1989). The overall force on the crownwas 20 kg (about 200 N).The continuous line in Figure 4a shows the calculated height-diameter profile forthe condition of uniform stress (bending plus axial stress), which deviated from themeasured values only in the region of butt swell. Different basal stem diameters weretried, ignoring the region of butt swell, and the line shown is that giving the best fitof calculated to measured values.Single tree of Picea sitchensis (Milne and Blackburn 1989)The points in Figure 4b are the stem diameters at different heights within a single22-year-old tree of P. sitchensis growing in a stand at 3800 stems ha- in Scotland(Milne and Blackbum 1989, their Figure 4). The wind drag forces on l-m thick layersof foliage within the canopy were supplied by Mime (personal communication) andwere based on measured leaf areas, drag coefficients given by Landsberg and James(197 l), a wind speed of 15 m s- at the top of the canopy and a measured wind profilewithin the canopy (Milne and Blackbum 1989). Figure 4b shows the distribution offorce on the crown.

The continuous line in Figure 4b shows the calculated height-diameter profile forthe condition of uniform stress (bending plus axial stress), which, as for E. regnans,deviated from the measured values only in the region of butt well. As for E. regnans,different basal stem diameters were tried, ignoring the region of butt swell. Theoptimal basal diameter was 20.4 cm which corresponded to a uniform stress of0.34 MPa (see below).Sensitivity qf the test of equal stress to measured diametersBecause bending stress is inversely proportional to the cube of stem diameter(Equation 1) (and bending plus axial stress is inversely proportional to the diameterto a power greater than three), a small change in diameter brings about a large changein stress. Conversely, many different stress distributions lead (in the transfer matrixmethod) to very similar predictions for the height-diameter profile. Consequently,the goodness of fit of the lines to measured diameters in Figures 3 and 4 was not asensitive test of the hypothesis of uniform stress distribution in stems.

Figure 5 shows the calculated height-diameter profile for a tree of P. sitchensis,with the same force parameters as for Figure 3 (Gardiner 1989), with either uniform

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1.0 rHeight

0.2 -

Lb!- stress-10% Equal +10xrtrerr

0 0.2 0.4 0.6 0.6 1.0Relative diameter

Figure 5. Calculated relative height-diameter profiles of stems of P. sitchensis assuming uniform stressalong the stem (open squares and line) or allowing stress to increase or decrease by 10% from top tobottom. The sol id points refer to a 10% increase in stress (thinner stem ), star points refer to a 10%decrease in stress (thicker stem ). Calculations were made assum ing the same forces, with stem and treeproperties as in Figure 3 (Gardiner 1989).

stress, or with stress increasing or decreasing by 10% from the top to the bottom ofthe tree. The 10% change in stress caused a change of only 5% in relative stemdiameter. Measurements of diameters along single stems often vary by 5%, owing toeccentricity, or irregularities in the bark, as shown by the scatter of points in Figures4a and 4b. Thus, the stress values calculated by West et al. (1989) and Milne andBlackbum (1989) from measured diameters may be subject to an error of about 10%.

Conversely, the apparently good fit of the lines for uniform stress to measureddiameters in Figures 3 and 4 provided evidence that stress was only approximatelyuniform (i.e., to within 10%).

Wind speeds and stress distribution in tree stemsHeight-diameter profiles at different wind speedsThe transfer matrix method can be used to calculate the height-diameter profiles thatgive uniform stress in trees subject to different average forces corresponding todifferent average wind speeds. Figure 6 shows the profiles for such trees ofP. sitchensis, based on the same assumptions as for Figure 3 (Gardiner 1989), andassuming drag was proportional to the square of the wind speed.The most notable result was that the height-diameter profile was not sensit ive toa change in average wind speed over the range 2.5 to 10.0 m s-l. Many of the worldsforests will experience average wind speeds over their lifetime within this 4-foldrange, as will trees growing at different spacings. Thus, it would appear thatdifferences in crown dimensions and properties that affect the vertical force distribu-


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MORGAN AND CANNELL

0.0 0.2 0.4 0.6 0.8 1.0Relative diameter

Figure 6. Calcul ated relative heig ht-diam eter profiles of stems of P. sitchensis assuming uniform stressalong the stem with different constant horizontal wind speeds. Calculations were made assuming thesame tree properties as in Figure 3 (Gardiner 1989). The profiles giving uniform stress at high windspeeds signify greater stem taper. The profiles giving uniform stress at wind speeds 2.5 to 10.0 m s-r aresufficiently simila r to be represented by one thick li ne.

tion probably have a greater impact on stem taper than differences in average windspeeds per se.

At high wind speeds (20 m s-l), the height-diameter profile giving uniform stresswas more tapered (Figure 6). At such high wind speeds, the stem deflected more than50% of its length, which resulted in a relative reduction in the bending moment alongthe stem, permitting a relative decrease in diameter with uniform stress. At low windspeeds (< 5 m SC) the opposite was true, and the diameter required to maintainuniform stress changed relative ly little with increase in height. A s the wind speeddecreased, the bending moment, and therefore, the tensile stress, became small andthe axial stress, due to self-weight, became significant. In the limiting case, wherewind speed was zero, there was no bending moment and the stress that remaineduniform with height was solely the compressive stress due to self weight.Stress distributions at different wind speedsIf we assume that most trees grow in areas with an average wind speed of 5 m s-land produce stems with uniform stress, we can calculate the stress distribution thatdevelops when those trees are temporarily subject to higher or lower wind speeds.Figure 7 shows the stress distributions in such a tree of P. sitchensis, based on thesame assumptions as for Figure 3 (Gardiner 1989).

When wind speed was increased from 2.5 to 10.0 m SC, the stress at the base ofthe tree was calculated to increase from 0.6 to 11.3 MPa. However, the stress


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0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Relative stress

Figure 7. Calcula ted stress distribution in a tree of P. sirchensis which has a height-dia meter profilegiving uniform stress in a steady wind of 5 m ss. (This is the same pr ofile as shown by the line inFigure 3.) The stress distributions are shown for wind speeds of 20, 10,5,2.5 and 1.25 m s-. Each stressdistribution is scaled to a value of 1.0 at the base of the tree, which corresponds to 39.0, 11.3, 2.7, 0.6and 0.1 MP a in wind speeds of 20, 10, 5, 2.5 and 1.25 m ss, respectively. Calculations were madeassuming the same tree properties as in Figure 3 (Gardiner 1989).

distribution remained approximately uniform up the stem (Figure 7). As stated above(Figure 6), the heightdiameter profile for uniform stress was similar for forcesproduced by wind speeds over this range. The reason is that there are opposingforces. As a stem deflects, the effective lever arm of the wind force in the crowndecreases with height, which reduces the relative bending stress with height, but, atthe same time, the weights of the crown and stem produce an increase in relativestress with height. These two forces w ill cancel out until the deflection becomes large(> 20% of stem length).

At high wind speeds (20 m s-l), the absolute stresses in the stem were large (seelegend to Figure 7) but the deflection of the trees caused a substantial decrease inrelative stress from the bottom to the top of the stem. Conversely, at a wind speed of1.25 m s- (light breeze), the absolute stresses were very small, but the relative stressincreased from the bottom of the stem to below the crown (Figure 7).

The change in stress distribution with change in wind speed may help to explainsome of the observations of non-uniform stress in the literature. The points anddashed line in Figure 8 reproduce the stress values calculated by Milne and Black-bum (1989) for the tree of P. sitchensis that is illustrated in Figure 4b (from theirFigure 5). Milne and Blackbum (1989) based their calculations on a wind speed of15 m s- at the top of the canopy. As mentioned above, our calculation of diametersgiven in Figure 4b was based on a wind speed of 5 m s-l, and assumed uniform stressof 0.34 MPa. When our tree (with diameters giving 0.34 MPa uniform stress) was

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60

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MORGAN AND CANNELL

2

0 1 2 3 4 6stress (MPa)

Figure 8. Stress distribu tions in the tree of P. sitchensis whose height-diameter profile is shown inFigure 4b. Broken line = stress values calculated by Mim e and Blackbum (1989, their Figure 5) basedon measured diameters and a wind speed of 15 m s- at the top of the canopy. The open point is thecalculated stress in the region of butt swell. Curved line = stress distribu tion produced by a wind speedof 15 m s-, m a tree with the same height-diameter profile as shown in Figure 4b, that is, with a shapegiving uniform stress (0.34 MPa) in a wind speed of 5 m s-l.

subjected to a wind speed of 15 m SC (at the top of the canopy) the estimated stressincreased to 3 MPa at the base of the stem, but decreased with height, giving a closefit to Mime and Blackbums values (Figure 8).

West et al. (1989) reported that the stress distribution in E. regnans trees deviatedfrom uniform when the wind speed, and hence, the force on the crown, decreasedfrom the top to the bottom of the crown. However, it would appear that their statedwind speeds were arbitrary, because the overall force on the crown corresponding toa stated wind speed (at the top of the canopy) of 12 m s-l was given as only 200 N,which we estimated to give a deflection of only 0.35 m at the top of a 28 m tall tree(with diameters giving equal stress, shown in Figure 4a). Evidence for non-uniformstress was then based on a stated wind speed of only 5 m s-, corresponding to a totalforce of perhaps 60 N, which we estimated to give a deflection of only 0.08 m. If atree with diameters giving uniform stress in response to a total force of 200 N (as inFigure 4a) is then subjected to a force of only 60 N, we estimated that the stressincreased non-linearly by over 50% from the base to the top of the stem. We suggestthat the use of very low force values may also account for the fact that West et al.(1989) found that the bending moment was proportional to the diameter to the powerof only 2.3 (l/O.43 in their Table l), whereas our analysis, based on a 200 N force,gave a power value of 3.1.

DiscussionIn general, our analysis supported the hypothesis stated at the beginning of the paper,


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SHA PE OF TREE STEMS-UNIFORM STRES S HYPOTHES IS 61

at least to an approximation (say 10%) for the stem region above butt swell. Threesalient points have come from our analysis.(1) Tree stems seem to grow to equalize the average stress (bending plus axial) attheir outer surface. This apparent goal is a time-averaged response to a

variable force. At any time, the actual s tress distribution may deviate fromuniform, because of lags that occur between changes in crown dimensions (andhence, of forces on the tree) and changes in stem diameter.

(2) The stress distribution in a tree stem is not constant over time, but changes as theforce on the crown changes. The stem shape that develops in response to averageforces wil i show non-uniform stress distributions if measurements or calcula-tions are made based on very low or high force values.(3) Average wind speeds over the 4-fold range of 2510.0 m s- give similar forcedistributions in stems, which consequently have similar stem shapes under thecondition of uniform stress. We conclude that most of the differences observedin the taper of tree stems (e.g., at different spacings) may be related more todifferences in their crown dimensions than to differences in the average windspeeds that they experience.

The transfer matrix method of structural analysis is clearly a powerful method forexamining relationships between force distributions and the dimensions of trees.Stress can be calculated from diameter measurements (Dean and Long 1986, Milneand Blackbum 1989, West et al. 1989), or measured diameters can be compared withthose calculated using assumptions about the distribution of stress (this paper).Nevertheless, either way, the test of the uniform stress hypothesis relies on diametermeasurements that are subject to error and give rise to a 10% uncertainty about thedistribution of stress. It is unlike ly that further progress can be made until methodsare developed to measure stress in tree stems directly. Meanwhile, if we accept theuniform stress hypothesis as a good approximation (e.g., Mattheck 1990, 1991), thetransfer matrix method can be used to constrain models of assimilate partitioning intrees (Cannel1 and Dewar 1993), and to predict stem shapes for any given tree size,weight and force distribution in the crown that may result from genetic or silv icul-tural manipulation.AcknowledgmentsWe are grateful to Dr. R. Mil ne for making data availa ble for the construction of Figures 4b and 8, andfor discussions that stimula ted further analysis of his data. We are also grateful to Dr. R.C. Dewar for hiscritical review of an early manuscript.

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Shape of Tree Stems_uniform Stress Hypothesis

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