Abstract
This study aimed to analyse the critical height of a column whose weight varies vertically in order to obtain a simple scaling law for a tree where the weight distribution considered. We modelled trees as cantilevers that were fixed to the ground and formulated a self-buckling problem for various weight distributions. A formula for calculating the critical height was derived in a simple form that did not include special functions. We obtained a theoretical clarification of the effect of the weight distribution of heavy columns on the buckling behaviour. A widely applicable scaling law for trees was obtained. We found that an actual tree manages to distribute the weight of its trunk and branches along its vertical extent in a manner that adequately secures its critical height. The method and findings of this study are applicable to a wide range of fields, such as the simplification of complicated buckling problems and the study of tree shape quantification.
Highlights
This study aimed to analyse the critical height of a column whose weight varies vertically in order to obtain a simple scaling law for a tree where the weight distribution considered
To derive a simple scaling law such as Greenhill’s, we considered the use of simple fractions to express the critical heights
To clarify the effect of the weight distribution of trees on their critical height, the critical height for self-weight buckling was formulated for cylindrical models with various weight distributions and the critical height equations that include the influence of branch weight was derived for the first time
Summary
This study aimed to analyse the critical height of a column whose weight varies vertically in order to obtain a simple scaling law for a tree where the weight distribution considered. Previous studies found that the bamboo reduces self-weight by incorporating a hollow structure, such that its buckling resistance is effectively improved by adjusting the node interval and the vascular bundle distribution. Based on those prior results, this study focused on trees, which are self-standing plants similar to bamboo. Trees in the wild naturally acquire appropriate heights and concomitant levels of mechanical stability that are adapted to harsh natural environments. This implies that mechanical strategies for avoiding self-buckling are incorporated in the forms adopted by trees. This scaling law has been applied widely in forest science and e cology owing to its simplicity. von Karman and B iot solved the governing differential equation by using a series solution and derived a formula for the critical height, which is almost equivalent to Greenhill’s equation (C ≈ 2.0)
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