Vibration of nonlocal Timoshenko beams

Abstract

This paper is concerned with the free vibration problem for micro/nanobeams modelled after Eringen’s nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using Hamilton’s principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The vibration solutions obtained provide a better representation of the vibration behaviour of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.

Buckling of Functionally Graded Nanobeams Based on the Nonlocal New First-Order Shear Deformation Beam Theory

“…The nondimensional buckling load defined is defined as: because a e 0 should be smaller than 2 nm for carbon nanotubes as described by Wang and Wang [9]. Table 1 shows that the present buckling loads agree very well with the solutions of Simsek [8] and the solutions of Aydogdu [10,11] for first order shear deformation beam theory. To investigate the significance of using functionally graded materials on the buckling of nanobeams, the FG nanobeams have the following material properties: The obtained results are compared with those reported by [Simsek] based on nonlocal Timoshenko beam theory.…”

A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams

“…The nonlocal Euler-Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) first proposed by Peddieson et al [5] and Wang [6], respectively, were adopted by many researchers to investigate bending [7][8][9], buckling [10][11][12], and vibration [13][14][15] responses of carbon nanotubes. A complete development of EBT and TBT was presented by Reddy and Pang [16] who provided the analytical solutions for the deflection, buckling load, and natural frequency of nanobeams with various boundary conditions.…”

Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory

“…To maintain the confidence in results obtained based on nonlocal constitutive equations, the dimensionless linear frequencies (ωl 2 ρA/EI 2 ) of a straight (η = 0) SWCNT are listed in Table 2 for different values of the nonlocal parameter, and various boundary conditions. For possibility of comparison, the SWCNT properties are chosen as follows: radius r = 0.339 nm, tube thickness h = 0.066 nm, Young's modulus E = 5.5 TPa, Poisson's ratio υ = 0.19, and shear correction factor K s = 0.563 [37]. As expected, increasing the nonlocal effect causes a decrease in dimensionless frequency.…”