Vibration and stability analysis of a cracked beam with varying thickness and constant width under axial loading

This study presents a combined finite element and component mode synthesis (FEM-CMS) approach for the vibration and stability analysis of a cracked tapered Euler-Bernoulli beam with linearly varying thickness, constant width, and a single open edge crack subjected to axial compressive loading. The crack is modeled as a massless rotational spring through a local flexibility matrix derived from fracture mechanics principles, and the cracked beam is treated as an assembly of subcomponents coupled by the crack stiffness matrix. Both clamped-free and pinned-pinned boundary conditions are considered, and parametric investigations are conducted over crack depth ratios (a/b=0.2-0.8), crack locations (Lc/L=0.1-0.9), thickness ratios (t2/t1=0.5-0.9), and normalized axial load levels (P/Pcr=0-0.9). The results demonstrate that crack-induced reductions in both buckling capacity and natural frequencies are strongly governed by the spatial correspondence between the crack location and the region of maximum bending moment of the associated mode shape, and that the boundary condition fundamentally alters the spatial distribution of crack sensitivity. For the clamped-free beam, a deep crack (a/b=0.8) near the fixed end reduces the critical buckling load by approximately 66%; for the pinned-pinned beam, the most critical position shifts to midspan, where a similar crack produces a 70% reduction. The applied axial load amplifies crack-induced frequency reductions in a strongly nonlinear manner at high load ratios, particularly in the fundamental vibration mode. The present findings provide a systematic quantitative framework for vibration-based damage assessment in non-uniform beam structures subjected to compressive loading.