Tree Physiol 1999 Neild 141 51

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Summary This paper describes a nondestructive method fordistinguishing root flexibility from stem flexibility in livingtrees. It is used here for Sitka spruce ( Picea sitchensis (Bong.)Carrire.), but is applicable to any species where the main stemis normally straight and near-vertical. Well-known engineeringequations permit the calculation of deflected shape for a verti-cal cantilever with arbitrary distribution of mass and bending

stiffness, when subjected to a lateral force. The equations areused to calculate stem deflections of four Sitka spruce trees forwhich the stem and branch mass distribution and stem taperhave been measured.

Free parameters in the mathematical model are a nominalvalue of Youngs Modulus E (assumed uniform and isotropicover the cross section and height of the tree stem) and aroot-anchorage stiffness k . The former allows the stem tocurve, whereas the latter represents the flexibility of the rootsand allows the stem to tilt elastically at ground level. For eachof the four trees, the calculated deflection curve is comparedwith actual deflections measured when the living tree is pulledby a rope at a specified point. By adjusting both E and k ,iteratively, a best fit solution is obtained. This provides asimple and effective way to determine both stem stiffness androot hinge stiffness from a single experiment on a living tree.

Keywords: growth, Picea sitchensis, root-anchorage stiffness,swaying, tree bending, Youngs Modulus of elasticity.

Introduction

Trees are flexible. When subjected to a force due to the wind,or to a force applied artificially, they bend. What is less obviousis that the bending is not merely a curving of the stem. The baseof the stem inclines also, because the roots flex so that the

attachment to the ground is not rigid. The rotation is assumedto be about the point where the stem centerline intersects theground plane and the elasticity of it is described here by theterm root-anchorage stiffness.

This hinge-like movement is quite distinct from the roothinge that has been described by Coutts (1986). He was de-scribing a hinge line near the edge of the root-plate aboutwhich the entire tree and root-plate system rotates as the treeuproots and falls over. The present trees are not about to fallover and the entire root-plate remains firmly anchored in theground. However, within the solidity of the root plate, thevarious individual roots do flex slightly as the tree sways.

The stiffness of a linearly elastic system is defined as theratio of force to deflection; the deflection being measured atthe point of application of the force and in the direction of theforce. A similar definition is applicable if a moment causes arotation or curvature. In either case, the reciprocal of stiffnessis flexibility. Flexibility is the ratio of deflection to force, or of rotation to moment.

In the case of beams that bend to a curved shape when amoment is applied, the bending stiffness may be described as:

Bending momentCurvature

= EI , (1)

where E is Youngs Modulus of elasticity for the material,(assumed to be homogeneous) and the member being bent hasa cross section with second moment of area I about the neutralaxis (see section on Mathematical model for tree bending).

The purpose of this paper is to highlight the fact that theswaying of trees is affected not just by the bending curvatureof the stems, but also by flexibility of the root-anchorage atground level. This affects all trees, not just those that are poorlyrooted.

If a horizontal force P (Newtons) is applied to a vertical treeat a height zP (m) above the ground, the corresponding moment

M 0 at the base is:

M 0 = P zP. (2)

In response to this moment, the base of the stem tilts througha small angle (radians) as though hinged at ground level. Thestiffness k (Newton meters per radian) of the root anchorage is:

k = M 0

. (3)

If the base of a tree rotates through a small angle , then thedeflection at height z will be increased by zabove the valueassociated with stem curvature alone.

Milne and Blackburn (1989) pulled eight Sitka spruce(Picea sitchensis (Bong.) Carrire.) trees laterally with a winchand compared the measured curve of stem deflection withtheoretical calculations. This led to estimates of an effective(nominally uniform) value of Youngs Modulus E . In makingtheir estimates, Milne and Blackburn found it necessary tomake an allowance for stem inclination at ground level, but did

Estimating stem and root-anchorage flexibility in trees

S. A. NEILD and C. J. WOOD Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.

Received February 25, 1998

Tree Physiology 19, 141--151 1999 Heron Publishing----Victoria, Canada


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not relate this to a stiffness. Such inclinations had previouslybeen measured by Fraser and Gardiner (1967) and also byHintikka (1972) who used clinometers on Scots pine ( Pinussylvestris L.).

We have also made comparisons between measured andcalculated tree-stem deflections; however, on this occasion,

ground level inclination is included by incorporating a non-in-finite root-anchorage stiffness into the mathematical model.Field data are from tree-pulling experiments described byGardiner (1992 a ). The authors are indebted to Dr. Gardinerand the Forestry Commission for the raw data associated withthat paper but not previously published in detail. Reference tothis material is noted as Gardiner 1992 b.

The Gardiner experiments were performed on only fourtrees. Therefore the authors make no claim that the results aredefinitive, but only that the method works and gives meaning-ful results. The purpose of this paper is to set out the analysisin a form that is easy to understand, and the computational

procedure in sufficient detail to enable others to perform simi-lar calculations whenever suitable data become available.

Data from Gardiner tree pull tests

Force and deflection measurements

Under calm conditions, four trees were each pulled by a hori-zontal rope attached to the stem at a measured height. Plumblines were also attached to the stem at four heights to indicatehorizontal displacements. The force and deflection data aresummarized in Table 1, for four test trees, G1, G2, G3 and G4.

Assuming small deflections and conventional elastic behav-ior, it is demonstrated that when (and only when) the pullingrope is aligned horizontally, the deflection at every point on theelastically curving stem will be a linear function of the ropetension. Therefore for each tree, and at each of the four meas-urement heights, we have fitted a straight line to the load-de-flection data by linear regression. The zero load intercept of the

Table 1. Summary of measured loads and displacements (Gardiner 1992 a ).

Tree G1 H (m) zp (m) Force P (N) Regression

13.9 10 0 280 476 530 280 0 Intercept Slope z (m) z / h Deflection x (cm) x0 d x /d P

10 0.7194 9.0 66.2 106.5 118.0 69.2 8.5 9.150 0.2068 0.5755 6.5 41.5 70.5 74.4 45.5 7.5 7.117 0.1306 0.4317 3.0 22.0 38.5 41.2 25.0 3 3.037 0.0734 0.2878 1.5 9.5 16.5 18.0 10.4 2 1.631 0.031

Tree G2

H (m) zp (m) Force P (N) Regression13.2 10 0 221 368 579 368 0 Intercept Slope

z (m) z / h Deflection x (cm) x0 d x /d P

10 0.7576 23.5 62.4 85.5 126.0 90.0 25.6 24.235 0.1748 0.6061 18.8 43.5 59.0 83.4 60.5 21.2 19.801 0.1096 0.4545 14.0 27.6 36.2 50.2 37.9 15.2 14.464 0.0614 0.3030 11.0 17.5 20.6 27.8 22.0 11.9 11.351 0.028


13.9 10 0 118 221 329 221 118 0 Intercept Slope z (m) z / h Deflection x (cm) x0 d x /d P

10 0.7194 6.5 56.3 106.6 149.6 98.3 49.3 6.0 4.473 0.439

8 0.5755 5.0 29.6 60.7 92.7 66.6 35.3 5.0 3.525 0.2696 0.4317 4.0 19.6 36.7 52.3 34.0 16.7 4.0 2.796 0.1474 0.2878 3.5 10.2 18.5 26.9 20.0 11.8 2.5 2.800 0.073


10.2 8 0 83 132 83 0 Intercept Slopez (m) z / h Deflection x (cm) x0 d x /d P

8 0.7843 17.3 110.7 153.2 116.3 27.1 23.892 1.0206 0.5882 13.0 65.7 90.0 69.2 18.8 16.839 0.5774 0.3922 12.2 36.3 47.5 38.3 15.5 14.284 0.2622 0.1961 9.0 16.2 19.5 16.7 9.8 9.533 0.079

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regression line is taken as the displacement origin, and theslope gives the flexibility in cm per Newton.

The data in Table 1 are presented graphically in Figure 1 sothat the validity of the assumed linearly elastic behavior maybe examined. From these graphs it may be observed that eachtree shows a degree of hysteresis as the load is first increasedand then decreased. However, if measurements are taken forincreasing and decreasing loads and averaged, then the linear-ity is good and justifies the use of a single slope to representthe flexibility for the entire experiment.

It is apparent from Figure 1D that the response of Tree G4is less accurately linear than the others, becoming stiffer at thehighest load. The explanation is that, at the highest load, theinclination of the upper portion was 63.2 cm in 2 m (seeTable 1), which is too large for linear behavior.

Tree dimensions, weights and elastic properties.

Dimensions and weights from the Gardiner (1992 b) experi-ments are shown in Table 2. This includes the stem and branch

masses ms and mb separately for each meter of stem height. Forthe present calculations, it is necessary to interpolate these datato produce tables of mass per unit height and diameter, withintervals sufficiently close to allow accurate numerical integra-tion. Therefore the Gardiner data were hand-plotted and re-digitized to produce a 50-point table for each tree. A sample(Tree G1) is shown in Table 3.

In Table 3, the local stem mass ms and branch mass mb aresummed to produce a distribution of total mass m per unitheight. This is expressed as a ratio m / mr using as a reference

value mr, the mass per unit height at one tenth of tree height h.This corresponds approximately to the breast height referencethat is frequently used in forestry.

A corresponding dimensionless ratio d / d r is used to describethe variation of stem diameter. The primary use of d is in thesecond moment of area I in the bending equation (Equation 1).For a circular section, I is given by:

I = d 4

64. (4)

This allows a convenient description of the variation of bending stiffness EI , because if a reference stiffness EI r isdefined, also at 0.1 h, then:

EI EI r

= d d r

4

. (5)

In suggesting that the local bending stiffness is proportionalto the fourth power of the diameter, it is implied that the valueof Youngs Modulus E , (the ratio of tensile stress to tensilestrain in a direction along the fibers), is uniform over the crosssection of the stem and also uniform over the height of the tree.This is certainly not true. There are differences in propertiesbetween young green wood near the periphery and the olderwood in the core (e.g., Zobel and van Buijtenen 1989). Thismeans that strictly E should be written as E (r , z). The problemcan be partially side-stepped if we consider a section-effective

Figure 1. Measured tree stem deflections for increasing load.

ESTIMATING STEM AND ROOT-ANCHORAGE FLEXIBILITY IN TREES 143

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modulus E ( z). For a circular cross section of diameter d , therelationship with the local modulus E (r , z) is given by:

E ( z) = 64d 4 E 0

d 2(r , z)r 3dr . (6)

The use of E ( z) still allows a formally correct statement of local curvature at any height. It then remains only to considerhow this effective modulus varies with height.

Blackburn (1997) attempted to measure this variation of Youngs Modulus by relating measured strain in the outer

fibers to the moment of an applied horizontal force at a particu-lar height. The Blackburn analysis ignores any radial variationof elastic properties and thus, although without an overt defi-nition, yields the present section-effective modulus.

Blackburn also ignores the self-weight of the overhangingtree-mass (the third term in Equation 7). Consequently, boththe bending moment and the associated value of E ( z) areunder-estimated with a progressively increasing error at lowerheights. Thus, although Blackburn (1997) concludes that E ( z)has a maximum, it is possible that if the weight term werecorrectly included, his data might in fact yield a constant valueover the lower stem or even a monotonic decrease with height.

The present analysis ignores this possible variation and usesa single value of E for the whole-tree stem. In this respect, themodel is open to improvement should convincing data for thevariation in E ( z) become available.

Mathematical model for tree bending

Equation for local bending moment

A straight vertical tree of height h and mass per unit length m( z)is pulled by the tension P of a rope as shown in Figure 2. Therope is attached at height zP and is inclined downward at anangle . The force P bends the tree to a curved shape and as itdoes so, the overhanging weight increases the bending effectso that the deflection is greater than that due to P alone.Writing the deflection at any height z as x( z), we seek anexpression for the bending moment M () at a particular height caused by all forces acting above ; i.e., arising from that partof the stem drawn with solid lines in Figure 2.

This bending equation must take account of: (a) the horizon-tal component of P acting at height zP (while < zP); (b) thevertical component of P which has an overhang displacement

xP -- x() (while < zP); and (c) the sum of moments due to allweight elements m, g, d z, between height and height h, eachwith a relative overhang displacement x( z) -- x().

Table 2. Summary of measured tree dimensions and weights (Gardiner 1992 b).

z (m) Tree G1 Tree G2 Tree G3 Tree G4

h = 13.9 m ms mb h = 13.2 m ms mb h = 13.9 m ms mb h = 10.2 m ms mbd (cm) (kg m 1) (kg m 1) d (cm) (kg m 1) (kg m 1) d (cm) (kg m 1) (kg m 1) d (cm) (kg m 1) (kg m 1)

0.0

0.51.0 18.9 23.40 18.0 19.44 14.8 14.93 10.2 7.091.3 17.7 20.52 17.6 18.59 14.8 14.93 9.7 6.411.52.0 16.7 18.27 17.0 17.34 13.5 12.42 8.8 5.282.53.0 16.0 16.77 15.7 14.79 12.8 11.17 8.2 4.583.54.0 15.0 14.74 14.9 13.32 13.0 11.52 7.6 3.944.5 0.40 0.405.0 14.2 13.21 14.0 11.76 11.8 9.49 7.2 3.535.5 1.43 2.93 4.226.0 12.9 10.90 13.4 10.77 11.1 8.40 6.2 2.626.5 3.97 3.54 4.93 3.537.0 12.3 9.91 12.1 8.79 10.3 7.23 4.9 1.64

7.5 2.10 7.19 8.22 3.218.0 11.0 7.93 11.3 7.66 9.4 6.02 3.9 1.048.5 0.23 5.62 2.80 2.489.0 9.8 6.29 9.6 5.53 8.1 4.47 2.2 0.339.5 6.90 4.50 3.45 1.4210.0 8.0 4.19 8.0 3.84 6.9 3.25 1.1 0.0810.5 6.07 8.03 3.45 0.7711.0 5.8 2.20 5.7 1.95 5.4 1.9911.5 3.13 6.54 2.1512.0 3.9 1.0012.5 0.4513.0

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Including all these factors, we write:

M () = P cos [ zP ] + P sin [ xP x()]

+ m

h

( z)g x( z) x() d z. (7)

In this equation, terms within square brackets are counted aszero if negative, because the load P causes bending of the stemonly below the point of application.

Equation for curvature

Provided that angular deflections are small, the bending mo-ment M (), causes a local curvature in the tree stem at height

Table 3. Sample tree description for computations (interpolated from Table 2) for Tree G1.

Stem diameter Stem mass Branch mass h = 13.90 m d r = 0.1750 m m r = 20.06 kg m1

d (cm) M s (kg m1) mb (kg m

1) z / h d / d r EI / EI r m / m r

0.70 0.03 1.00 0.040 0.000 0.0020.85 0.05 0.98 0.049 0.000 0.0021.30 0.11 0.96 0.074 0.000 0.006

1.80 0.21 0.00 0.94 0.103 0.000 0.0112.34 0.36 0.02 0.92 0.134 0.000 0.0192.85 0.53 0.45 0.90 0.163 0.001 0.0493.35 0.74 0.90 0.88 0.191 0.001 0.0823.90 1.00 1.66 0.86 0.223 0.002 0.1324.48 1.31 2.55 0.84 0.256 0.004 0.1935.07 1.68 3.71 0.82 0.290 0.007 0.2695.60 2.05 4.70 0.80 0.320 0.010 0.3376.22 2.53 5.44 0.78 0.355 0.016 0.3986.83 3.06 5.95 0.76 0.390 0.023 0.4497.41 3.60 6.36 0.74 0.423 0.032 0.4967.98 4.17 6.67 0.72 0.456 0.043 0.5408.62 4.87 6.89 0.70 0.493 0.059 0.5869.13 5.46 6.91 0.68 0.522 0.074 0.6179.57 6.00 5.65 0.66 0.547 0.089 0.581

9.93 6.46 1.83 0.64 0.567 0.104 0.41310.25 6.88 0.25 0.62 0.586 0.118 0.35610.51 7.24 0.25 0.60 0.601 0.130 0.37310.88 7.75 0.79 0.58 0.622 0.149 0.42611.37 8.47 1.41 0.56 0.650 0.178 0.49211.80 9.12 2.10 0.54 0.674 0.207 0.55912.11 9.61 2.82 0.52 0.692 0.229 0.61912.37 10.02 3.42 0.50 0.707 0.250 0.67012.50 10.23 3.79 0.48 0.714 0.260 0.69912.56 10.33 3.89 0.46 0.718 0.265 0.70912.78 10.70 3.35 0.44 0.730 0.284 0.70013.10 11.24 2.43 0.42 0.749 0.314 0.68113.55 12.03 1.57 0.40 0.774 0.359 0.67813.89 12.64 0.97 0.38 0.794 0.397 0.67814.20 13.21 0.50 0.36 0.811 0.434 0.683

14.42 13.62 0.00 0.34 0.824 0.461 0.67914.60 13.96 0.32 0.834 0.484 0.69614.81 14.37 0.30 0.846 0.513 0.71615.13 14.99 0.28 0.865 0.559 0.74715.45 15.64 0.26 0.883 0.608 0.77915.76 16.27 0.24 0.901 0.658 0.81115.97 16.71 0.22 0.913 0.694 0.83316.20 17.19 0.20 0.926 0.734 0.85716.40 17.62 0.18 0.937 0.771 0.87816.56 17.96 0.16 0.946 0.802 0.89516.72 18.31 0.14 0.955 0.833 0.91317.01 18.95 0.12 0.972 0.893 0.94517.50 20.06 0.10 1.000 1.000 1.00018.35 22.06 0.08 1.049 1.209 1.10019.35 24.53 0.06 1.106 1.495 1.223

20.25 26.86 0.04 1.157 1.793 1.33921.00 28.89 0.02 1.200 2.074 1.44021.80 31.13 0.00 1.246 2.408 1.552




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proportional to the applied force P through a value of x* whichis a function of height but not of load.

This expectation justifies the use of linear regression toobtain the slope of the deflection curves in Table 1. It followsalso that, in the present computational procedure, x* may bedefined in terms of a flexibility ( x x0)/ P which is actually thevalue of that regression slope (see Table 4).

Computation procedure

Equations 7a and 9a are solved iteratively for each of the fourtrees. In each case, data for the dimensionless mass distributionm*(z) and stiffness distribution EI *(z), are used in tabular formas in Table 3. After an initial guess for the 50-point deflection

curve x*(z), the bending moment M *(z) is calculated by Equa-tion 7a, followed by a refined estimate of x*(z) from Equa-tion 9a.

Adequate convergence requires 5--10 cycles. The presentiteration was terminated at 20 cycles. In many repetitions of this iterative procedure, the dimensionless stem stiffness K Sand root hinge stiffness K R are used as free parameters. Theirvalues are refined progressively by trial-and-error to obtain abest fit between the calculated x*(z) curve and the measureddeflection points (Figures 3--6). During this procedure, thereducing error is monitored by means of a sum of squareddifferences at the experimental data points. The final opti-mized values for K S and K R are listed in Table 5. For compari-

Table 4. Dimensionless parameters and reference values.

Dimensionless parameters

Vertical position z = zh

Vertical position = h

Local diameter d = d d r

Local mass/unit height m = mmr

Local bending stiffness EI = EI E I r

= d 4

Rope force F = Pmrgh

(Force scaled by nominal tree weight)

Bending moment M = M Ph

(Moment scaled by rope force times tree height)

Stem stiffness K S = EI r

mrgh 3 (Moment scaled by nominal tree weight times height)

Root stiffness K R = k

mrgh2 (Moment scaled by nominal tree weight times height)

Slope of linear x versus P displacement curve 1 x = mrg( x x0)P

Reference values

mr = Mass per unit length at breast height (0.1 h)

d r = Stem diameter at breast height (0.1 h)

EI r = Bending stiffness at breast height (0.1 h)

I r =d r464

1 The inclusion of x0 in the definition of x* is in recognition that the regression curves in Figure 1 show a zero error offset that is not counted aspart of the bending deflection. It is emphasized that the slope of the displacement curve is used rather than a displacement for any particularload.




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Figure 3. Computed stem shapes for best fit with field measurementsfor Tree G1.




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son, optimum values of K S are also computed with K R held atinfinity to represent a rigid root anchorage (Table 5, figures inparentheses).

Discussion

Improvement in prediction by allowing root-anchorage flexibility

Figures 3 to 6 show comparisons between measured and cal-culated deflections for Trees G1 to G4. In each case, the bestelastic hinge fit is compared with the best rigid hinge fit, toshow the improvement when optimization of root anchoragestiffness is allowed. In every case, when root flexibility isallowed, estimated stem stiffness is found to increase, demon-strating that any estimate of stem stiffness will be too low if allof the deflection is assumed to be due to stem bending alone.

In each case also, with root stiffness as a free parameter, theimprovement in the prediction of deflected shape is illustratedby an decrease of up to two orders of magnitude in the squarederror sum.

Validation of small-deflection theory

The method used here to calculate the stem deflection assumesthat the curvature is small (see section equation for curvature).We checked that this assumption is reasonable for the deflec-tions measured in the pull tests by comparing stem deflectionscalculated using the present approximate method with an alter-native calculation that remains valid for large deflections.

A large-deflection calculation method for tapered cantile-

vers was published by Morgan (1989) and in almost identicalform by Morgan and Cannell (1987); it was used by Milne andBlackburn (1989). Using an independent derivation very simi-lar to that described by Morgan (1989), Neild (unpublisheddata) has extended conventional small-deflection theory tocompute large bending deflections by cumulative summationof lateral and angular displacements of short beam elementsend-to-end. This is referred to here as the Elemental Deflec-tion Method.

In Figure 7, comparisons of computed stem-bending shapesare shown for the largest deflections of Tree G1 and separatelyfor the largest deflections of Tree G4. The agreement for Tree

G1 is within about 2% and it justifies the use of the presentmethod rather than a more complicated computation. How-ever, for Tree G4, the error in the linear prediction is 7%. This

is because Tree G4 was pulled to a deflection beyond the rangeof linear theory.

It would extend the range of accuracy to higher deflectionsif this large-deflection approach were used for all of the com-parisons described here. By the same argument it would bemore accurate still to use the full power of modern FiniteElement computational modelling, but we have not consideredthat. The reason for retaining classical small deflection theoryis that the equations and the associated computation are easyto understand and use. The choice is consistent with the pri-mary aim of this paper, which is to make a physically compre-hensible technique readily available to those who may wish touse it.

Sensitivity to errors in deflection/load gradient.

In Figure 1, all of the measured deflection curves show somedegree of hysteresis and, in the case of Tree G4, there is also avisible nonlinearity due to overload. These factors contributeto uncertainty about the values taken for the deflection/loadslope----an uncertainty that, being of systematic rather thanrandom origin, is not quantified by the conventional statisticalindicators of regression analysis.

To test the sensitivity of the K S and K R results to errors of thistype, the iterative stiffness estimation procedure for Tree G4

Figure 7. Comparison of stem shape predictions with Elemental De-flection Method.

Table 5. Best-fit values for stem ( K S) and root-hinge (K R) stiffness.Optimum values computed with K R held at infinity to represent a rigidroot plate are in parentheses.

Tree mr (kg m1) h (m) K S K R Curve fit error

G1 20.06 13.9 0.490 Rigid 4.00E-04

(0.565) (11.97) (1.76E-05)G2 18.74 13.2 0.784 Rigid 1.67E-04

(0.882) (19.79) (4.73E-06)G3 13.94 13.9 0.358 Rigid 3.07E-04

(0.372) (30.40) (2.38E-04)G4 7.36 10.2 0.513 Rigid 4.78E-03

(0.675) (4.59) (4.72E-05)




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was repeated using the data presented in Table 6 as G4A. Thisis simply the Tree G4 data from Table 1 with the largest loadpoint removed so that the deflection/load slopes are higher(averaged over the four measurement locations) by 7.4%.

The matched stem shape for G4A is again excellent, asshown in Figure 8. Table 7 shows that the estimated stemstiffness K S is lowered by 7%, but the estimated root stiffness

is not significantly altered. This confirms that the nonlinearitycaused by over-pulling in this case is associated with excessivestem curvature only and not with failure of the root anchorage.

A recommendation for tree-winching experiments.

A significant simplification is achieved if the pulling or winch-ing experiments are organized with the applied force exertedhorizontally and not inclined downward. This is because thebending response for small deflections is linear if the angle is zero in Equation 7. Therefore to avoid unnecessary compli-cation in the analysis and interpretation of pull tests, a horizon-tal pull is recommended.

Peculiarities of individual trees

Before the peculiarities of individual trees can be discussed, itis necessary to consider what might be expected of an idealizednormal tree. For example, if all normal trees were geometri-cally similar ( h d r) and made of the same homogeneousmaterial, then variations in mechanical and elastic charac-teristics would be easy to correlate with increasing size. Inparticular, the dimensionless stem and root anchorage stiffness

parameters K S( EI r / (mrgh 3)) and K R(k / (mrgh 2)), might both beexpected to vary in proportion to 1/ h because I r d r4, mr d r 2

and k d r3.This does not happen, indicating that such a simplistic

model of normality is not realistic. Because the four study treesare not geometrically similar in that d r / h is different for eachone, K s varies between 0.4 and 0.9 over the four trees, and K Rtakes values between 5 and 30 for Trees G1 and G3.

If peculiarity is meaningful, then it might be said thatTree G4 is rather poorly rooted with a K R of only 4.6. This isevident in Figure 6 where the stem has a far larger tilt at groundlevel than the other three trees. In contrast, Tree G3 has thestiffest root anchorage but has a very thin flexible stem for itsheight and weight.

Adaptive growth

Variation among individual trees prompts a comment andsome questions on adaptive growth. Many authorities describethe driving influences of plant growth in terms of two distinctbut not inconsistent principles. The first is economy in the useof nutrients in the competition for light. The second is thatstress-bearing material develops where strain is greatest inorder to reduce that strain. This leads to shapes (e.g., stem

taper) that provide uniformity of stress in surface fibers.These factors mean that tree growth is responsive in a large

measure to the details of local environment and there is noparticular reason why any two trees of the same species shouldbe geometrically similar. However, the present study promptsa curiosity about the relationship between the development of root anchorage stiffness and stem stiffness. The latter seemsrelatively easy to explain in terms of adaptive response to windand other mechanical loads; however, the former is not sosimple because roots have a foraging function as well as afixing function.

It is a foraging response that the roots of Picea sitchensis dieoff when they encounter the water table and cannot go deep if

the soil is waterlogged. There is graphic evidence (Wood 1991)that individual roots develop increased bending strength inresponse to long-term repeated bending movement. This is thefixing aspect. Fixing is the one objective that cannot be com-pletely met by adaptive growth. Beyond the limits of the rootplate there is only soil, its strength may depend on wetness butit cannot adapt as a living system to give more strength wheremore is needed.

It is only the size of the root-plate that is influenced by theadaptive growth mechanism of the tree. Therefore, it is tempt-ing to speculate on the interaction between foraging and fixingand to wonder, for example, whether a tree in soil devoid of

Figure 8. Computed stem shapes for best fit with field measurementsfor Tree G4A.

Table 7. Comparison of stem ( k S) and root hinge ( k R) stiffness esti-mates for Trees G4 and G4A (cf. Table 3).

Tree Max P M r H K S K R Curve(N) (kg m 1) (m) fit error

G4 132 7.36 10.2 0.675 4.59 4.72E-05G4A 83 7.36 10.2 0.629 4.58 3.46E-05

Table 6. Field data for Tree G4 omitting largest load (cf. Table 1).

Tree G4Ah (m) zp (m) Force P (N) Regression

10.2 0 0 83 83 0 Intercept Slope z (m) z / h Deflection x (cm)

8 0.7843 17.3 110.7 116.3 27.1 22.200 1.0956 0.5882 13.0 65.7 69.2 18.8 15.900 0.6184 0.3922 12.2 36.3 38.3 15.5 13.850 0.2812 0.1961 9.0 16.2 16.7 9.8 9.400 0.085

150 NEILD AND WOOD



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nutrients might, in its search for food, develop roots that aremore extensive and therefore stronger than those of a tree innutrient-rich soil.

Conclusions

Comparisons among measured deflection curves for four SitkaSpruce trees and classical linearized (small-deflection) beam-bending theory have highlighted the fact that a significantlybetter prediction of the deflected shape of a tree is obtained if the flexibility of the root anchorage is taken into account aswell as the bending flexibility of the stem. By optimizing thefit between measured and calculated deflection curves, simul-taneous estimates are achieved for stem stiffness and root-hinge stiffness. The calculation method takes account of themeasured dimensions and weights of each tree and includesthe additional bending moments due to gravity as the treeinclines from the vertical.

Although the data sample (four trees tested including onegiving suspect data) is too small to support a general quantita-tive conclusion about the root-hinge stiffness of Sitka spruce,we can say that, in the trees examined, the root hinge contrib-uted between 5 and 15% of the flexibility. The variation re-flects actual physical differences among the trees.

Acknowledgment

The authors thank Dr. B.A. Gardiner of the Forestry Commission,Northern Research Station, Roslin for supplying the raw data on whichhis 1992 paper was based. Some details are reproduced here for thefirst time and are referenced as Gardiner 1992 b.

References

Blackburn, G.R.A. 1997. The growth and mechanical response of trees

to wind loading. PhD. Thesis. Manchester University, 173 p.Coutts, M.P. 1986. Components of tree stability in Sitka spruce on

Peatey Gley soils. Forestry 59:173--198.Fraser, A.I. and J.B.H. Gardiner. 1967. Rooting and stability in Sitka

spruce. For. Comm. Bull. 40, HMSO, London, 28 p.Gardiner, B.A. 1992 a . Mathematical modelling of the static and

dynamic characteristics of plantation trees. In MathematicalModelling of Forest Ecosystems. Eds. J. Franke and A. Roeder.Sauerlnders Verlag, Frankfurt, pp 40--61.

Hintikka, V. 1972. Wind induced root movements in forest trees.Commun. Inst. For. Finland 76:1--56.

Milne, R. and P. Blackburn. 1989. The elasticity and vertical distribu-tion of stress within stems of Picea sitchensis . Tree Physiol. 5:195--205.

Morgan, J. 1989. Analysis of beams subjected to large deflections.Aeronautical J. Nov. 1989:356--360.

Morgan, J. and M.G.R. Cannell. 1987. Structural analysis of treetrunks and branches. Tree Physiol. 3:365 --374.

Wood, C.J. 1991. Understanding wind forces on trees. In Wind andTrees. Eds. M.P. Coutts and J. Grace. Cambridge University Press,pp 133--164.

Zobel, B and van Buijtenen, J.P. 1989. Wood variation, Its causes andcontrol. Springer-Verlag, Heidelberg, 363 p.

AppendixProof of the second moment-area Theorem (for Equation 9).

Examine the integral S defined by S = (0

z

z )d2 x

d2d.

In this integration z is not a variable, so we write:

S = z d2 x

d2d

0

z d2 x

d2d

0

z

.

The first integral term gives the slopes at the limits = z andat = 0:

S = z d xd ( = z)

z d xd ( = 0)

0

z

d d xd .

The second integral is evaluated by parts, also between thelimits = 0 and = z:

S = z d xd ( = z)

z d xd ( = 0)

z d xd ( = z)

+ 0

z d xd

d.

Complete the final integration and note also that the first andthird terms are equal and opposite:

S = z d xd ( = 0)

+ x( z) x(0).

Finally, within the original definition of S, note that by

the Euler equationd2 x

d2 = M

() EI () :

0

z M ()( z ) EI ()

d + z d xd ( = 0)

= x( z) x(0).

To apply this theorem to the present case, we define that atthe base of the tree ( = 0), the deflection x(0) shall be zero andthat the slope [d x /d]( = 0) shall be the root hinge rotation .




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Tree Physiol 1999 Neild 141 51