Forced vibration analysis of cracked continuous multi-span functionally graded nanobeams

The paper analyses the forced vibration of cracked continuous multi-span functionally graded nanobeams based on the nonlocal elasticity theory (NET), Euler-Bernoulli beam theory (EBT), and the dynamic stiffness method (DSM). The NET accounts for the nanoscale size effect of the structure. A crack model using three springs with stiffness dependent on the crack depth is proposed. The equations of motion are derived using Hamilton's principle, NET, and EBT. Various boundary conditions are formulated from the weak form of the equations of motion, thereby overcoming the nonlocal paradox. The reliability of the proposed method is verified through comparison with published results. The effects of geometric, material, nonlocal, and crack parameters on the forced vibration behaviour of continuous multi-span nanobeams are analysed in detail.