Failure mechanism of hollow tree trunks due to cross-sectional flattening.

Failure of hollow trees in urban areas is a worldwide concern, and it can be caused by different mechanisms, i.e. bending stresses or flattening-related failures. Here we derive a new analytical expression for predicting the bending moment for tangential cracking, and compare the breaking moment of various failure modes, including Brazier buckling, tangential cracking, shear failure and conventional bending failure, as a function of t/R ratio, where t and R are the trunk wall thickness and trunk radius, respectively, of a hollow tree. We use Taiwan red cypress as an example and show that its failure modes and the corresponding t/R ratios are: Brazier buckling (Mode I), tangential cracking followed by longitudinal splitting (Mode II) and conventional bending failure (Mode III) for 0 < t/R < 0.06, 0.06 < t/R < 0.27 and 0.27 < t/R < 1, respectively. The exact values of those ratios may vary within and among species, but the variation is much smaller than individual mechanical properties. Also, shear failure, another type of cracking due to maximum shear stress near the neutral axis of the tree trunk, is unlikely to occur since it requires much larger bending moments. Hence, we conclude that tangential cracking due to cross-sectional flattening, followed by longitudinal splitting, is dominant for hollow trunks. Our equations are applicable to analyse straight hollow tree trunks and plant stems, but are not applicable to those with side openings or those with only heart decay. Our findings provide insights for those managing trees in urban situations and those managing for conservation of hollow-dependent fauna in both urban and rural settings.