Observations of the softening phenomena in the nonlocal cantilever beams

A longstanding puzzle of nonlocal cantilever models is that they do not predict the dynamic softening behaviors of beams compared with the classical beam models. This puzzle exists and is not well solved in the past several years. In this paper, we revisit and make our first attempt to address this issue. By using the weighted residual approaches, the nonclassical force resultants and boundary conditions are obtained. Based on the nonclassical boundary conditions, closed-form solutions for the vibration and buckling problems of the nonlocal Euler-Bernoulli cantilever beams and Timoshenko cantilever beams are derived. Numerical results show that the softening behaviors of cantilever beams can be captured in the nonlocal Euler-Bernoulli beam theory and Timoshenko beam theory. In addition, the differences of the frequencies predicted by the proposed models are increasing larger than those given in the literature as the nonlocal parameter increases, demonstrating clearly the prominent effect of nonclassical boundary conditions on the dynamic behaviors of beams. The asymptotic analysis is constructed to unveil the underlying mechanism of dynamic behaviors of the beams. The numerical results of the analytical solutions obtained in this work may serve as benchmarks for future studies of the dynamic behaviors of composite structures.