Vibration of nonlocal strain gradient functionally graded nonlinear nanobeams using a novel locally adaptive strong quadrature element method

The primary objective of this paper is to propose a novel method to derive Differential Quadrature Method matrices with several degrees of freedom at the boundaries that can be used to build Strong Quadrature Elements to solve fourth and higher-order equations of motion. The proposed method, referred to as Locally adaptive Strong Quadrature Element Method, is applied to higher-order equations of motion for nonlinear graded Timoshenko and Euler-Bernoulli nanobeams formulated using the Second Strain Gradient Theory or the Nonlocal Strain Gradient Theory. To limit the formulation complexity, the proposed approach is based on the regular formulation of the differential quadrature method combined with custom-built transfer matrices. Moreover, it does not require a different formulation for fourth and sixth-order equations and can be extended beyond sixth-order equations. Validation was carried out using examples from the literature as well as data obtained using the classical Locally adaptive Quadrature Element Method. Both linear and nonlinear frequencies were evaluated for a large number of configurations and boundary conditions. The proposed approach resulted in good accuracy and a convergence speed comparable to the conventional Locally adaptive Quadrature Element Method.