Equilibrium shape of a tree branch.

BIOMECHANICS. - Equilibrium shape of a tree branch.

The shape of tree branches is computed using the theory of flexure of beams. Numerical computation is used and takes into account large displacements and growth. The coupling between elastic bending and growth results in a branch shape having an inflection point connected with a permanent deformation.

The study of stresses in living materials has been the subject of many papers. For example the stresses due to anisotropic growth [1], influence of stresses on growth [2], and even viscoelasticity [3] or plasticity. The influence of gravity on growth is known under the names of gravimorphism [4] or geotropism [5]. The proportions of trees have been found to be limited by elastic criteria to stand under their own weight [6]. Growth stresses and particularly those due to reaction wood play an important role in the mechanics of trees [8], but this paper is focused essentially on the physical influence of gravity. Growth stresses are predominant in the trunk of a vertical tree, where gravitational forces creating nearly hydrostatic pressures may be neglected [8]. In branches, where bending is dominant, stresses due to gravitational forces  are much larger than in the trunk. In this paper, growth stresses will be considered as a distinct phenomenon that will not be taken into account in a first approximation.
A forest tree that has been shifted from its normal upright position during a storm will probably grow straight again. Reaction wood may force the tree to an upright position again, but bending strain was not found to affect radial xylem growth in Douglas-fir [5]. Old branches have an inflection point and their curvature remaining almost unchanged when cut, their deformation is permanent. It is often not possible to straighten these branches without breaking them.
In the absence  of gravity a branch would probably grow straight in the direction of light. With growth rings of constant thickness and a constant yearly increase in length, the shape of a branch would be a cone. Trees may be considered as structures made of beams (dead branches) subjected to gravity. Classical Strength of Materials considers a beam with a given shape, applies a load and calculates the resulting shape, slightly different from the original one for thick beams. Thin beams may have large deflections resulting in non linear displacement, though still elastic. To apply elasticity to a tree branch, it is necessary to know its shape before any calculation. The theory 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 a branch when the changes in thickness and length are negligible as for a branch temporarily loaded with snow or fruits.
When the load is permanent, viscous or plastic deformation may occur, but the coupling between the simultaneous increase of the load and the thickening of the branch is more important. A branch bends continuously with time, even if it thickens. Even with a constant load there would be no decrease of the deflection when the branch thickens. On the contrary, a decrease of the thickness of the branch, would also increase the deflection. The process of growth being time-dependent, the shape of a branch is a function of time. Dividing time into infinitely small intervals, it is possible to divide each time step in a few more steps: growth, loading and bending. Growth produces an increase in length and thickness 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 to compute the deflection, giving another change in geometry of the branch. The growth cycle may then be repeated for the next time step and so on. Because of the changes in thickness and length,  the principle of superposition does not apply directly. Therefore, the stress distribution through the branch is no more linear and a permanent deformation occurs even if the incremental stress distribution is linear.
Many methods have been devised by engineers to analyse the mechanical behaviour of structures. Numerous configurations have been solved with analytical formulae, but they are valid only for simple geometries. With the advent of computers, numerical methods, such as finite elements and finite differences have been developed to calculate complicated structures. None of them (at our knowledge), takes into account the growth phenomenon (occurring in the same manner in the construction of buildings as for trees). In order to calculate the shape of a branch, a simple finite difference method has been used to integrate the differential equation of flexure step by step along the branch and iterated  in a time marching process. At each time step, thickness, length and deflection of the branch are adjusted to take care of growth. Few input data are necessary: the thickness of the annual growth rings, the annual length increase of the branches, the growth angle, the density, the longitudinal elastic modulus of wood and the acceleration of gravity.
The results are synthesised in a picture of the whole tree: the young branches are at the top, almost straight and pointing in the direction of growth. The old branches, at the bottom of the tree, are curved and deflected towards the bottom. Figures 1 to 3 show the influence of the growth angle. Figure 4 differs from fig. 3 only by the growth speed: bending is more pronounced for large trees than for small ones. The shapes of the branches are characterised by an inflection point, not predictable with the elastic criteria of MacMahon [6].

Fig 1: Small grouth angle

Fig 2: 45° growth angle.

Fig. 3: 90° growth angle.

Fig. 4: Old tree with branches grow up after touching earth.