The Right Load Distribution: The Axiom of Uniform Stress and Tree Shape

Abstract

The fact that a tree trunk is thick at the bottom and thin at the top may seem so obvious to us that we are unlikely to enquire about a deeper mechanical significance. And yet this tapering of the trunk is probably the simplest and most obvious kind of component optimization which our instructor, the tree, can reveal. As the main loading of the tree is the wind load, acting transversely to the trunk and inducing a bending moment (Fig. 36), it is naturally important whether the crown of a tree is localized high up on the trunk, as is often found in trees in a dense stand (Fig. 36A), whether it increases rather linearly from top to bottom, as is often the case in free-standing conifers (Fig. 36B), or whether, as shown in example (C) in Fig. 36, the foliage is distributed almost uniformly over the length of the trunk. The latter is achieved quite well in Lombardy poplars. In Fig. 36, the relevant lengths are shown as `h', it being assumed here that the distribution of the wind pressure is proportional to the projected crown area. The bending stresses on the trunk surface can be calculated from these distributions with Eq. (3), assuming a circular trunk cross-section. Now, assuming that the tree trunk is a component optimized over millions of years of evolution and satisfies the axiom of uniform stress, we only need to put $σ$max = $σ$1 = constant in Eq. (3) and we obtain a relationship D(h), i.e. between diameter and measured distance h, as explained in Fig. 36.