A High-order FEM Formulation for Free and Forced Vibration Analysis of a Nonlocal Nonlinear Graded Timoshenko Nanobeam based on the Weak Form Quadrature Element Method

M Trabelssi (University of Carthage & University of Tunis, Tunisia), S El-Borgi (Texas A&M University at Qatar, Qatar) & MI Friswell (Swansea University)

Archive of Applied Mechanics, Vol. 90, No. 10, October 2020, pp. 2133-2156


The purpose of this paper is to provide a high order Finite Element Method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the Weak form Quadrature Element Method (WQEM). This formulation offers the advantages and flexibility of the Finite Element Method (FEM) without its limiting low-order accuracy. The nanobeam theory accounts for the von Karman geometric non-linearity in addition to Eringen's nonlocal constitutive models. For the sake of generality a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first or second-order interpolation functions. Hamilton's principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the Differential Quadrature Method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems.

Paper Availability

This material has been published in the Archive of Applied Mechanics, Vol. 90, No. 10, October 2020, pp. 2133-2156. The article is published as open access.

Link to paper using doi: 10.1007/s00419-020-01713-3

Archive of Applied Mechanics on SpringerLink