Optimal structural features in trees and their application ... › Secure › elibrary › papers...

Optimal structural features in trees and theirapplication in engineering

D. Pasini & S.C. BurgessDepartment of Mechanical Engineering, University of Bristol, UniversityWalk, Bristol, BS81 TR, UK

Abstract

This paper describes how the different optimal structural features improve thestructural performance of trees, A case study of some mature trees in the UK hasbeen carried out to give specific examples of structural performance. The paperdiscusses the extent to which the structural features of trees are used inengineering such as the design of bridges and heavy mechanical equipment,Whilst most of the optimal features are already used to some extent, there isscope for greater use of the features,

Glossary

Leeward sheltered from the wind, Windward exposed to wind.Tap root: the main root of a plant, running deep into the ground.Sinker root: root penetrating deeply in the soil. Shallow root; superficial root,

1 Introduction

The main function of a tree is to develop a canopy of leaves to capture sunlight.In order to support the foliage layer, the tree builds a structure using the leastamount of bio-material. The competition with other plants and the change indirection of sunlight during the day, encourages the tree to grow toward thesource of natural light,

The tree builds a structure which is only sufficiently strong to withstand theforces encountered during its history, The tree contains several differentstructural features which combine to produce a very efficient overall structure,whose constituent elements are shown in Figure 1, While the foundation

© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved.Web: www.witpress.com Email [email protected] from: Design and Nature, CA Brebbia, L Sucharov & P Pascola (Editors).ISBN 1-85312-901-1

4 Design and Nature

structure anchors the tree to the soil, the primary structure grows vertically inelevation to support the branching system (secondary structure). The branchesgrow laterally and uprightly from the main support to carry the widest surface ofleaves, the foliage canopy. The interface between foundation, primary andsecondary structure, constitutes the structural joints, which connect all thedifferent load-carrying elements,

In this paper, the efficient structural features of trees are examined. Theinteraction and the influence of wind and self-weight on the configuration andadaptation of the structural morphology are illustrated. A review of the optimalstructural features employed by the tree is surveyed dealing first with the generallevel and proceeding to the each structural zone and concluding to the level ofmaterial. Parallels and comparisons between trees and engineering structures aredrawn to offer understanding which can be instructive in many areas of design.

2 Loading conditions

Of the external loading experienced by the tree during its life, the wind is one ofthe most significant. Under the action of the aerodynamic force of the wind onthe leaf canopy, the main trunk and each structural element act as a cantilever,The wind loading is approximately given by:

F=~.5~CdAfv2

where p air density (kg/m3), cd drag coefficient, Af frontal area (m2), v windvelocity (mhec2).

Since the wind velocity increases from the ground upward, as shown inFigure 2, the tree experiences an increased loading as the tree increases size. Theaerodynamic forces usually come from all directions and produce bendingmoments which have the highest values at the junctions between the differentstructural elements, The largest bending moment experienced by the tree occursat the foundation-trunk j oint.

The self-weight of the tree is another important source of loading. It exertscompressive force in the trunk if the trunk is vertical. Self-weight producesbending moments in any section of the trunk or branches which are not vertical.As a free-standing cantilever fixed at the base, the tree can buckle under its ownweight. The critical load is given by:

z EIP=?r —

4L2where E Young’s modulus (N/m2), 1 second moment of area, L length of thecolumn.

3 Adaptive growth

Most of the efficient features[1]. This is because the treeloading,

employed by the tree occur by adaptive growthdevelops a structure in response to mechanical


Design and Nature 5

Trees undergo structural adaptation by reinforcing regions exposed to higherstresses, especially compressive stresses, Since wood is placed faster in areaswhere higher stresses occur, the structural benefit is that the stress concentrationis reduced and the use of living material is optimised during the growth.

The natural adaptive response has the effect to produce optimaldevelopmental and morphological changes in the overall form. Optimisation offorms occurs in the longitudinal and transversal shape of each constituentelement and in their junctions.

4 Efficient features of the structural form

There are several optimal features in the form of the tree:●

●

●

●

●

Hierarchical structure. The tree develops a hierarchical structure tosupport a large foliage canopy. Each structural element has a precisefi,mction and position within the tree. Each part resists the forces transmittedby the subordinate elements and transferred its own load on the elementswhich precede in the hierarchical scale, Hierarchical structures are tolerantto damage and present decreasing sizes of the elements at each hierarchicallevel according to the magnitude of applied forces.

Flexibility. A flexible structure can adjust its form when loaded, thusreducing the severity of load, A flexible structure also improves toughness.

Convergent form. The lateral branches, which assume an equilibriumposition at a given angle to gravity, converge into one vertical support. Sincethe compression forces are concentrated into the trunk and transmitted intothe soil, one foundation structure is required.Balanced forces. The secondary structure presents a balanced distributionof its elements around the main support. The benefit of developing anequilibrated form is shown in Figure 2, Self-weight and wind on each side ofthe tree induce roughly equal but opposite bending and torsion moments onthe trunk.

Optimised proportions. The tree grows its height in relation to the trunkdiameter. As the tree grows, the self-weight increases with the cube of thedimensions whereas the buckling load increases only with the square. If atree, during its growth, should increase the dimensions of its structureproportionally, then it would eventually fail under its own weight, The tree,on the contrary, perceives the increased stresses due to self-weight andresponds by growing the dimensions of the trunk faster than the height,However, the diameter of the trunk appears to be over designed if only theself-weight is considered [2]. This is because the tree optimises theproportions of its form also in response to the stimuli induced by the wind.


6 Design and Nature

5 Efficient features of the structural constituents

5.1 Foundation structure: anchorage

The foundation structure provides the anchorage of the tree to the ground. Theforces induced by external loads and self-weight at the base must be transferredinto the soil,

Soil and roots provide the anchorage to prevent the tree from uprooting underwind action, Whereas number, direction, sizes and shapes of the roots aredeveloped in response to the applied forces, the soil cannot show any adaptiveresponse. The components, which resist the turning moment [3,4], are shown inFigure 3, The most significant is the resistance offered by the windward roots intension. Another important factor is the weight of the root-soil plate, while theresistance of the soil and of the compressive leeward roots have a marginalcontribution.●

●

●

Stiff root system. The tree builds a stiff root system in two ways. In the firstcase, shallow lateral roots are developed only horizontally and are attachedto a tap root, In the second, deep sinkers grow downward from the lateralroots [5],

Oriented distribution of roots, The number and the position of the rootsare structurally important to promote an efficient anchorage. For example, asymmetric distribution of roots is appropriate if the wind comes from anydirection. However, it is quite common that habitat conditions impose thetree to adapt to unidirectional wind [6]. In this scenario, the tree allocatesroots in the direction of rocking particularly on the lee side, becauseperpendicular roots are subject to torsion and, hence, offer little resistance touprooting,

Optimised shape of roots. The shape and the sizes of the roots are alsodeveloped by adaptive growth. When a poorly anchored root is repeatedlyswayed back and forth, the tree develops an optimised shape [7], as shownFigure 4. New material is added mainly in the areas where tensile andcompressive stresses are induced by the loading regime.

5.2 Foundation-primary structure interface

The joint between primary structure and root system experiences very largebending moment induced by the wind action. The trunk transmits tension andcompression forces to the windward and leeward sides of the anchoragerespectively.● Reinforced joint. Since a high concentration of stresses occur at this

junction, the tree is stimulated to increase the strength of its structure bygrowing wood cells at the interface of the foundation with the primarystructure, As a result, the tree enlarges the base of the trunk relative to therest of the trunk,

● Buttress. In tropical rainforests, trees minimised the use of living materialby developing a system of buttresses [3,8], The sinker roots brace the trunklike angle brackets, as shown in Figure 5. This occurs because new wood is


Design and Nature 7

added along the force flow to reduce the stress level. At this junction, theforce flow proceeds from the trunk towards the windward and leewardsinkers which transmit tension and compression forces into the soil. Thefimction of this buttress is, therefore, to prevent the snap of lateral roots withthe trunk,

5.3 Primary structure: trunk

Efficient features in the primary structure are tapering and pre-stressing of thetrunk.● Tapered trunk. The tapering of the trunk occurs by adaptive growth to the

applied forces. The direction and magnitude of self-weight and wind loadingdetermines the types of internal forces which occur in the trunk. Firstly, thecompressive load due to self-weight is vertical and induces axial stressuniformly distributed in each section of the trunk, Secondly, the bendingload, due to wind loading, causes uneven distribution of tensile andcompressive stress at the windward and leeward sides of the trunkrespectively, While multidirectional wind produces maximum compressiveand tensile stresses at the extremities of a cross-section all around itsperiphery, the highest stresses induced by a unidirectional wind occur onlyin the one direction of bending. The tree responds to these stimuli bygrowing extra cells in areas of highest stresses as shown in Figure 6a and 6b.These cells produce annual growth rings, which are concentric in the trunk.Over a period of time the net result of the growth rings is to produce atapered trunk as shown in Figure 6c, One advantage of decreasing taperingthe trunk is that tree can grow faster by having little redundant material.

● Preloading. Tree trunks have preloading in their outer fibres, which causesresidual tensile stresses in these areas, When the tree is subjected toaerodynamic loading, bending stresses are superimposed on the residualstress, This is a beneficial feature because it means that when the trunk issubjected to bending moments, the net compressive stresses are less than thetensile stresses in magnitude. Since the compressive strength of wood islower than the tensile strength, the preloading significantly improves thestrength of the tree,

5.4 Primary-secondary structure interface

The trunk-branches junction must provide a strong support to upright theposition of the branch to the sunlight. Strength is produced by reinforcement ofmaterial,● Reinforced joint. At branch joints, branches experience the highest value of

bending moment induced by self-weight and wind loading. The tree isstimulated to strengthen the underside of the junction, where highconcentration of compressive stress occurs, as shown in Figure 7 and 8.


8 Design and Nature

5.5 Secondary structure: branching system

The tree achieves an efficient use of material in the secondary structure bygrowing tapered branches and deep sections as shown in figure 8,b Tapered branches. The dimensions of the branches along the longitudinal

profile are adjusted in response to the magnitude of the applied loads.

● Deep sections. In the case of a horizontal branch, bending loads come fromtwo sources, Firstly, bending loads are provided in nearly all directions dueto the wind loading. Secondly, bending loads due to the self-weight areproduced in the downward direction of the branch. In the case of wind, asthe branch is scaled up, the wind loading increases with the square of thedimensions and the induced bending moment at the branch junction with thethird power. Since the section modulus of the branch increases with thecube, the branch does not need to change in shape in order to cope withwind loading, However, in the case of self-weight loading, as the branch isscaled up, the self-weight increases with the cube of the increase in scaleand the bending moment at the branch-bunk j oint increases with the fourthpower. During the branch growth, self-weight is the dominant load whichacts only downwards, thus the regions of the branches where highestcompressive stress occurs are at the bottom. Since the compression strengthof wood is lower than the tensile stress, the regions of the branch to be fasterreinforced are those in compression, The tree responds to these stimuli bygrowing more cells at base of the branch. As a consequence, the growthrings are not concentric as for the trunk and, in the case of large branches,can be roughly rectangular in order to shape more efficient sections,

5.6 Secondary structure-foliage canopy interface

The node of the twig with the leaf has two main fimctions, Firstly, the upper sideof the leaf must be kept outward in order to capture the sunlight and, thus, wind-bending moments must be withstood, Secondly, the drag forces, which aretransferred by the leaf to the tree and then to the root, have to be minimised inorder to reduce the uprooting of the tree.

. Multipurpose joint. This junction presents a groove along the length of theupper side of the leaf stem [9], as shown in Figure 9, The structural benefitof this groove is that the leaf can easily twist in order to reduce the winddrag, without loosing resistance to bending.

5,7 Material

. Microstructure Wood has a prismatic cellular structure as shown in Figure10. Some woods such as palm and bamboo have a hexagonal microstructure

[10], but other trees have five sided of four sided cells, Individual cells

typically have a size of the order of 100 ~m. The cells are joined end to endso that a grain structure runs parallel to the stem or trunk.The micro-structure of wood has a number of advantages:


Design and Nature 9

i. it reduces the density of the material thus placing material i?n-theraway fi-om the neutral axis, so increasing the structural efficiency

ii. it can help improve resistance against global buckling of the trunk byreducing the slenderness ratio.

.m. it can make the material tougher by introducing discontinuitiesiv. it can provide other finctions such as water take-up in woodWoods have a wide range of densities from about 150 kg/m3 for balsa woodto 1000 kg/m3 for lignum. The tensile strength of wood along the grainrange is from 40 to 100 MPa, and it is roughly threefold the compressionstrength. This is because the thin cell walls of the microstructure can buckleunder axial load.

● Strong basic material. The main component of the cell wall in wood iscellulose, Since density and yield strength of cellulose are 1500 Mg/m3 and1000 MPa respectively, one cellulose fibre has a very high strength toweight ratio,

6 Similarities with engineering structures

Since natural and human structures are subjected to the same physical laws, it isnot surprising that most of the features of a tree are already used in engineering.Engineers have learnt to design structures which resemble those exploited bynature from the beginning of time. Efficient structural features used inengineering are:●

●

●

●

●

●

Structural hierarchy. A good rule for design a structure is to organise theelements in hierarchical scale. In this structural arrangement, superfluousparts are eliminated and each constituent element is employed for onespecific ii.mction, which is to resist one internal force, Since the material canbe used to its limit, the element sizes are decreased and a weight reductionof the overall structure is achieved.Balanced forces. In bridges and crane structures, the symmetry of geometryand/or forces produces balanced forms, as shown in Figure 11,

Convergent form. In the case of ship rigging, since the horizontal beamsconverge to the same vertical support, only one mast is required, Theconvergent form shown in Figure 12 is balanced,

Reinforced joint. The reinforcement of joints is common in structuraldesign. For example, Figure 13 shows two structural joints: a foundationjunction where the rib and the holding-down bolts act as the buttresses andthe sinker root, and a column-beam connection strengthened by stiffener andhaunch,

Shaped sections, Although the I section shown in Figure 14 does not matchthe elegance of the I shaped root of Figure 4, it is widely and successfullyused in engineering structures for economical reasons.

Tapered elements. Adjusting the longitudinal profiles to the internal forcesinduced by the applied load, is another way to improve structural efficiency,Figure 15 shows examples of tapering profiles in the case of column,cantilever and beam structures,


1() Design and Nature

● Prestressing. Prestressing is extensively used for prefabricated structures asshown in Figure 16, Whereas the tree prestresses wood in all directions ofthe trunk section to increase the weaker compressive strength, engineersusually combine two materials, such as steel and concrete, to achievesprestressing in the only direction of bending. The technology employed forconcrete structures, such as cable stay for bridges, large roofs for buildings,geotechnics works and offshore structures, is also applied to masonry andcomposite material.

There are some features of trees which are not commonly used by engineers.Multipurpose joints, microstructure and adaptive structures need moreunderstanding and research to be applying to engineering cases.

7 Conclusion

Trees have a large number of optimal structural features. Since the structure of atree develops adaptively, features are well optimised for the particularenvironment which the tree is in, This paper has examined the efficient featuresof the tree, Analogies with engineering structures have been discussed in order togain understanding which can stimulate innovative solutions in many areas ofdesign.

References

1.2.3.

4.

5.

6.

7<

8.

9,10

Mattheck C, Trees: the mechanical design, 1991, Springer- Verlag, Berlin.McMahon T. A, Size and shape in biology, 1973, Science 179, 1201-1204Coutts, M. P. Root architecture and tree stability, 1983 Plant andsoil,71,171-88.Coutts, M. P. Components of tree stability in Sitka spruce on peaty, 1986,gley soil. Forestry, 59, 173-97.Stokes, A., Fitter, A.H., Coutts, M. P. Responses to young trees to wind:effects on roots growth, in Wind and trees, 1995,Cambridge UniversityPress.Coutts, M. P, Development of the structural root system in Sitka spruce.1983, Foresty, 59, 1-16.Wood, C. J, Understanding wind forces on trees, in Wind and trees,1995,Cambridge University Press.Ennos, A, R. Development of butress in reainforest trees: the influence ofmechanical stress, in Wind and trees, 1995 ,Cambridge University Press,Vogel, S. Cats’ Paws and catapults, 1998, Penguin books,Gibson L,J., Ashby, M, F., Karam G,N., Wegst U. and R. Sherliff Themechanical properties of natural materials. 1995, Proc. Sot. Lend. 450,141162.


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Figure 1 The constituent elements of a tree.

Gravity

1

Side B

Wind

Gravity

1

Figure 2 The balanced form enables the bending, M, induced by the self weight,and the torsion, T, caused by the wind, to be nearly equal and opposite.


12 Design and Nature

Wind action

‘“jjj~z j

Windward–5

Leeward%+“-r-

oots Soilresistance roots

Figure 4 Foundation structure: theFigure 3 Foundation structure: the modified shape of a shallow root undercomponents of the anchorage system, a regime of repeated strain.

In tension Sinker root In

4

Figure 5 Trunk-foundation interface: the buttress system strengthens theiunction.

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Figure 6. a) Multidirectional wind, b) Unidirectional wind. c) Tapered trunk inhe case of multidirectional wind,


14 Design and Nature

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Figure 11 Balanced form in bridge andmaterial handling structures.

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a) b)

Figure 13 a) Foundation joint,


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Figure 15 Tapered forms: a) column,b) portal, c)cantilever.

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LFigure 16 a) Pre-tensioned beam,b) I section, c) deck brid e section.


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