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Bending, buckling and free vibration of Timoshenko beam-based plane frame via FEM with nonlocal integral model

Yuan Tang and Hai Qing

Vol. 18 (2023), No. 3, 355–374
Abstract

A finite element method is developed to study the size-dependent static bending, elastic buckling and free vibration of Timoshenko beam-based plane frame using two-phase local/nonlocal integral model. The explicit expression for the stiffness matrix, geometric stiffness matrix and mass matrix is derived. Coordinate transformation is employed to obtain the stiffness matrix, the geometric stiffness matrix and mass matrix for a plane frame element oriented around an arbitrary angle. The effects of the nonlocal parameters on static bending, elastic buckling and free vibration of Timoshenko beams as well as a clamped square frame are studied numerically under different boundary and loading conditions. The bending deflections obtained are validated against existing analytical and asymptotic results in literature. The numerical results show clearly that, with the application of the two-phase local/nonlocal integral model, a consistent softening effect can be obtained for both Timoshenko beams and plane frames.

Keywords
nonlocal integral model, finite element formulation, Timoshenko beam, plane frame, mechanical response
Milestones
Received: 4 August 2022
Revised: 20 December 2022
Accepted: 5 February 2023
Published: 5 May 2023
Authors
Yuan Tang
State Key Laboratory of Mechanics and Control of Aerospace Structures
Nanjing University of Aeronautics and Astronautics
Nanjing
China
Hai Qing
State Key Laboratory of Mechanics and Control of Aerospace Structures
Nanjing University of Aeronautics and Astronautics
Nanjing
China