Segmented impact fracture of toppling truncated cones and tall trees

Critical dynamic stresses and segmented fracture lengths are predicted for a tall, initially upright, truncated right circular cone as it rotates about its base point and as the cone tip impacts the ground plane. This cone can represent a tall masonry structure such as a tapered stone tower or a brick chimney which has a foundation failure; or the trunk of a tree which topples in a windstorm. In any case, ground impacts can induce sequential fractures along the coneʹs length, resulting in multiple in-line cone segments that come to rest on the ground plane. A dynamic analysis for the coneʹs bending moment and stress distributions during impact is used to predict the primary fracture points for the broken segment lengths. The dynamic bending stresses controlled the failures since they were found to be about two orders of magnitude higher than the longitudinal stresses caused by the self-weight and the centrifugal acceleration of a rotating cone. In a case study, this dynamic analysis is shown to correlate well with observed fracture patterns of fallen segments for a dead tree toppled by wind. An unexpected secondary fracture point observed in this case study is also explained using the dynamic theory.