Cracking failure of curved hollow tree trunks

Understanding the failure modes of curved hollow tree trunks is essential from both safety and conservation perspectives. Despite extensive research, the underlying mechanism that determines the cracking failure of curved hollow tree trunks remains unclear due to the lack of theoretical analysis that considers both the initial curvature and orthotropic material properties. Here we derive new mathematical expressions for predicting the bending moment, M(crack), at which the cracking failure occurs. The failure mode of a tree species is then determined, as a function of t/R and cR, by comparing M(crack) with M(bend), where t, R and c are, respectively, the trunk wall thickness, outer radius and initial curvature; M(bend) is the bending moment for conventional bending failure. Our equation shows that M(crack) is proportional to the tangential tensile strength of wood σ(T), increases with t/R, and decreases with the final cR. We analyse 11 tree species and find that hardwoods are more likely to fail in conventional bending, whereas softwoods tend to break due to cracking. This is due to the softwoods' much smaller tangential tensile strength, as observed from the data of 66 hardwoods and 43 softwoods. For larger cR, cracking failure is easier to occur in curvature-decreasing bending than curvature-increasing due to additional normal tensile force F acting on the neutral cross-section; on the other hand, for smaller cR, bending failure is easier to occur due to decreased final curvature. Our formulae are applicable to other natural and man-made curved hollow beams with orthotropic material properties. 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.