BIOMECHANICS. - Equilibrium shape of a tree branch.

Bernard Schaeffer

The shape of tree branches has been computed using the theory of flexure of beams. Numericalcomputation is used and takes into account large displacements and growth. The coupling betweenelastic bending and growth results in a branch shape having an inflection point connected with apermanent deformation.

ResultsThe study of stresses in living materials has been the subject of many papers. Forexample the stresses due to anisotropic growth [1], influence of stresses on growth[2], and even viscoelasticity [3] or plasticity. The influence of gravity on growthis known under the names of gravimorphism [4] or geotropism [5]. Theproportions of trees have been found to be limited by elastic criteria to standunder their own weight [6]. Growth stresses and particularly those due to reactionwood play an important role in the mechanics of trees [8], but this paper isfocused essentially on the physical influence of gravity. Growth stresses arepredominant in the trunk of a vertical tree, where gravitational forces creatingnearly hydrostatic pressures may be neglected [8]. In branches, where bending isdominant, stresses due to gravitational forces are much larger than in the trunk. Inthis paper, growth stresses will be considered as a distinct phenomenon that willnot be taken into account in a first approximation.

A forest tree that has been shifted from its normal upright position during a stormwill probably grow vertically again. Reaction wood may force the tree to anupright position again, but bending strain was not found to affect radial xylemgrowth in Douglas-fir [5]. Old branches have an inflection point and theircurvature remaining almost unchanged when cut, their deformation is permanent.It is often not possible to straighten these branches without breaking them. This isnot due to plastic or viscoelastic deformation but to the combination of growthand bending due to the weight increase of the branch.

In the absence of gravity a branch would probably grow straight in the directionof maximum light. With growth rings of constant thickness and a constant yearlyincrease in length, the shape of a branch would be a cone. Trees may beconsidered as structures made of beams (dead branches) subjected to gravity.Classical Strength of Materials considers a beam with a given shape, applies aload and calculates the resulting shape, slightly different from the original one forthick beams. Thin beams may have large deflections resulting in non lineardisplacement, though still elastic. To apply elasticity to a tree branch, it isnecessary to know its shape before any calculation. The classical theory of thebending of beams shows that the curvature of the bent beam has a constant sign,there is no inflection point. Elasticity is adequate to calculate the deflection of abranch when the changes in thickness and length are negligible as for a branchtemporarily loaded with snow or fruits.

When the load is permanent, viscous or plastic deformation may occur, but thecoupling between the simultaneous increase of the load and the thickening of thebranch is more important. A branch bends continuously with time, even if itthickens. Even with a constant load there would be no decrease of the deflectionwhen the branch thickens. On the contrary, a decrease of the thickness of thebranch, would also increase the deflection. The process of growth being time-dependent, the shape of a branch is a function of time. Dividing time intoinfinitely small intervals, it is possible to divide each time step in a few moresteps: growth, loading and bending. Growth produces an increase in length andthickness of the branch and therefore a small change in geometry. Using the new

geometry, the increase in load being small, linear elasticity may be used tocompute the deflection, giving another change in geometry of the branch. Thegrowth cycle may then be repeated for the next time step and so on. Because ofthe changes in thickness and length, the principle of superposition does not applydirectly. Therefore, the stress distribution through the branch is no more linearand a permanent deformation occurs even if the incremental stress distribution islinear.

Many methods have been devised by engineers to analyse the mechanicalbehaviour of structures. Numerous configurations have been solved withanalytical formulae, but they are valid only for simple geometries. With theadvent of computers, numerical methods, such as finite elements and finitedifferences have been developed to calculate complicated structures. None ofthem (at our knowledge), takes into account the growth phenomenon (occurring inthe same manner in the construction of buildings as for trees). In order to calculatethe shape of a branch, a simple finite difference method has been used to integratethe differential equation of flexure step by step along the branch and iterated in atime marching process. At each time step, thickness, length and deflection of thebranch are adjusted to take care of growth. Few input data are necessary: thethickness of the annual growth rings, the annual length increase of the branches,the growth angle, the density, the longitudinal elastic modulus of wood and theacceleration of gravity.

The results are synthesised in a picture of the whole tree: the young branches areat the top, almost straight and pointing in the direction of growth (fig. 1 to 4).

Fig. 1 - Growth direction at 80°.

Fig. 2 - Growth direction at 45°.

Fig. 3 - Horizontal growth direction.

Fig. 4 - Growth direction at 45°, growth speedtwice slower than on the other figures. Thisshape is not geometrically similar to that offigure 2 (scale effect).

Fig. 1 to 4 - Results of a numerical computation on a microcomputer. The Youngmodulus is 10 GPa, the specific mass is 500 kg/m3, the acceleration of gravity is9,81 m/s2 and the age 10 years.The thickness of the annual growth rings is 2 mmand the annual increase of the branches is 900 mm except on figure 4 where theyare respectively of 1 and 450 mm. The branch is divided into ten elements. Thecomputations are performed in ten growth cycles. Figures 1 to 3 show theinfluence of the growth angle. Figure 4 differs from fig. 3 only by the growthspeed: bending is more pronounced for large trees than for small ones. The shapesof the branches are characterised by an inflection point, not predictable with theelastic criteria of MacMahon [6].

Fig. 5 - The old branches, at the bottom of thetree, are curved and deflected downwardsuntil they touch the ground.

Fig. 6 - After having touched the ground, the branchhaving a leaning point, grow again up.

Fig. 7 - Old western cedars in the Bois de Boulogne, Paris, France.

Fig. 8 - This photograph of a four centuries old tree shows that a tree neverstraighten up after having bent under his own load (Jardin des Plantes, Paris).

Theory

Principle of the calculation

For a constant speed of growth l , the length of the branch at time t is:l t = l tThe radius r of the branches increases also at a constant velocity ,meaning that the width of the growth rings is constant. Thus, the diameterof a branch varies linearly with the curvilinear abscissa along the branchand also of the time t:

r s,t = r t � s

lwhere the dot indicates a time derivative. The moment of inertia for acircular section is:

I s,t =�

4r4 s,t

The branch is sujected to a load density per unit length:p(s,t) = � r2(s,t) � gwhere � is the density of the wood and g the acceleration of gravity. Thespeed of variation of the load is obtained by derivation:

p s,t = 2� r s,t � gThe bending moment M(s,t) due to the weight of the branch from a pointof curvilinear abscissa s to the end of the branch at time t is

M s,t =s

l t

x s,t - x s',t p s',t ds'

The shape of the branch is a function of time t and of the curvilinearabscissa s [x(s,t), y(s,t)]. The velocity of variation of the bending momentat time t and abscissa s is obtained by considering the branch as a timevarying curved beam of length l. The preceding expression of the bendingmoment is derivated:

M s,t =s

l t

x s,t - x s',t p s',t ds' +s

l t

x s,t - x s',t p s',t ds'

+ l p l t ,t x s,t - x l t ,tThe bending load being zero at the end s = l of the branch, the last term iszero:

p[l(t),t] = 0The second term takes into account the variation of the moment due tosuccessive horizontal displacements, usually neglected in civilengineering. The general formula of the bending of beams, valid in largedisplacements, gives the curvature 1/R as a function of the bendingmoment:

1R=d�ds

= MEI

where E is the elasticity modulus of wood, I the moment of inertia of thebranch section and �(s) the angle between the branch and the horizontal atthe point of curvilinear abscissa s. This formula is not valid for a growingbranch. An increase in the moment of inertia will not diminish thecurvature but the bending moment will increase, due to the weight increaseof the branch. The theory of elasticity has to be applied for smallincrements of time and curvilinear abscissa:

d� =M s,tE I s,t

ds

This equation has to be integrated twice in order to obtain the angle �:

� s,t =

0

s

d� s',tds'

ds'

This expression is zero for s = 0, therefore the built-in condition issatisfied. The slope angle of the branch is obtained by a second integration

� s,t = � s, s

l+

s

l

s

� s,t' dt'

relative to the time the branch has been growing to the length s

t = s

lwhere l is the growth velocity.In order to simplify, the deviation an. The branch is encastré in the trunkwith a deviation angle remaining the same at the end of the branch. Inother words, the branch grows at the same angle at both ends. The angle atits origin is fixed because of the bult-in but at its end the slope decreasesbecause of the increase of the mass of the branch.

� s, s

l= � 0,0

The coordinates of the points defining the shape of the branch arecalculated with a last integration:

x s,t =s

s

sin � s',t ds'

y s,t =s

s

cos � s',t ds'

Numerical solutionAn analytic solution of this problem is not possible but it is easy to solve itby finite differences. One proceeds by iterations each one corresponding toa growth cycle. A new branch is calculated from the former as a curvedbeam of shape and size known from the perceding cycle [11].

References[1] R. R. ARCHER, F.E. BYRNES, On the Distribution of Tree GrowthStresses - Part I: An Anisotropic Plane Strain Theory, Wood Sci. Technol.,8 (1974) pp. 184-196.

[2] FENG-HSIANG HSU, The influences of Mechanical loads on theform of a growing elastic body, J. Biomech. 1 (1968) p. 303-311.[3] J.L. NOWINSKI, Mechanics of growing materials, Int. J. Mech. Sci.,20 (1978) p. 493-504.[4] J. CRABBE, H. LAKHOUA, Ann. Sci. Nat., 12ème série, t. 19 (1978)p. 125.[5] R.M. KELLOG, G.L. STEUCEK, Mechanical stimulation and xylemproduction of Douglas fir, CAPPI Forest Biology Wood Chemistry Conf.,Madison, 1977, pp 151-157.

[6] T. MACMAHON, Size and Shape in Biology, Science, 179 (1973),pp. 1201-1204.[7] M. FOURNIER, Déformations de maturation, contraintes "decroissance" dans l'arbre sur pied, réorientation et stabilité des tiges, Sém."Développement architectural, apparition de bois de réaction et mécanique

de l'arbre sur pied", Montpellier, 20/1/1989, Ed. LMGMC, USTLMontpellier.[8] P.P. GILLIS, Theory of Growth Stresses, Holzforschung, Bd. 27, Heft6 (1973) pp. 197-207.[9] S.P. TIMOSHENKO, Théorie de la stabilité élastique, Dunod, Paris,1966[10] M. FOURNIER, Mécanique de l'arbre sur pied: maturation, poidspropre, contraintes climatiques dans la tige standard, Thèse, INPL, Nancy,1989.

[11] B. SCHAEFFER, Rhéologie des propergols en cours depolymérisation, Industrie Minérale, N° spécial Rhéologie, T. IV (1977)N° 5, pp. 225-234.

[12] B. SCHAEFFER, Forme d'équilibre d'une branche d'arbre. C.R. Acad.Sci. Paris, t. 311,série II, pp 37-43, 1990.