Closed-form dynamic stiffness formulation for exact modal analysis of tapered and functionally graded beams and their assemblies

Abstract The paper proposes a closed-form dynamic stiffness (DS) formulation for exact transverse free vibration analysis of tapered and/or functionally graded beams based on Euler–Bernoulli theory. The novelties lie in both the DS formulation and the solution technique. For the formulation, the developed DS is applicable to a wide range of non-uniform beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This fills the gap of existing closed-form DS element library which is generally limited to linearly tapered/functionally graded beams. For the solution technique, an elegant and efficient  J 0 count of tapered element is proposed to apply the Wittrick-Williams (WW) algorithm most effectively. The investigation sheds lights on the so-called  J 0 count challenge of the algorithm for other DS elements. The above two novelties make exact and highly efficient modal analysis possible for a wide range of tapered and/or functionally graded beams, without resorting to series solution, numerical integrations or refined mesh discretization. Results for a particular case show excellent agreement with published results. Moreover, we investigate the effects of the taper/functional gradient rate/index and boundary conditions on the free vibration behaviour. Benchmark solutions are provided for individual beams as well as beam assemblies.