Showing papers on "Timoshenko beam theory published in 2012"

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Abstract: Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.

Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory

Abstract: Thermoelectric-mechanical vibration of the piezoelectric nanobeams is first investigated in this paper based on the nonlocal theory and Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton principle. The differential quadrature (DQ) method is employed to determine the natural frequencies of the piezoelectric nanobeams with different boundary conditions. The influences of the nonlocal parameter, temperature change, external electric voltage and axial force on the thermoelectric-mechanical vibration characteristics of the piezoelectric nanobeams are discussed in detail. It is found that the nonlocal effect is significant for the natural frequencies of the nanobeams. This study also reveals that the natural frequencies of the nanobeams are quite sensitive to the thermoelectric-mechanical loadings. The results should be relevant to the design and application of the piezoelectric nanodevices.

Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model

Abstract: Double-walled carbon nanotubes (DWCNTs) are being investigated for use as latent materials for drug carriers. However, the surface effects cannot be ignored when drugs or other functional materials, such as nickel or silver, adhere to the surface of the outer tube of a DWCNT. In this paper, the vibrational frequency of DWCNTs, while accounting for surface effects, is studied using the nonlocal Timoshenko beam model. The influence of the surface elasticity modulus, residual surface stress, nonlocal parameter, axial half-wave number and aspect ratio are investigated in detail. The results show that the vibrational frequency is significantly affected by the surface material, nonlocal parameter, vibration mode and aspect ratio. In short DWCNTs on condition of higher vibrational modes, the influences of the surface and nonlocal effects on vibration are more pronounced.

Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading

Abstract: An exact, closed-form solution is obtained for the nonlinear static responses of beams made of functionally graded materials (FGM) subjected to a uniform in-plane thermal loading. The equations governing the axial and transverse deformations of FGM beams are derived based on the nonlinear first-order shear deformation beam theory and the physical neutral surface concept. The three equations are reduced to a single nonlinear fourth-order integral–differential equation governing the transverse deformations. For a fixed–fixed FGM beam, the equation and the corresponding boundary conditions lead to a differential eigenvalue problem, while for a hinged–hinged FGM beam, an eigenvalue problem does not arise due to the inhomogeneous boundary conditions, which result in quite different behavior between clamped and simply supported FGM beams. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for thermal post-buckling or bending deformation is obtained as a function of the applied thermal load. The exact solutions explicitly describe the nonlinear equilibrium paths of the deformed beam and thus are able to provide insight into deformation problems. To show the influence of the material gradients, transverse shear deformation, in-plane loading, and boundary conditions, numerical examples are given based on exact solutions, and some properties of the post-buckling and bending responses of FGM beams are discussed. The exact solutions obtained herein can serve as benchmarks to verify and improve various approximate theories and numerical methods.

A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory

Abstract: The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation.

Geometrically non-linear generalised beam theory for elastoplastic thin-walled metal members

Abstract: This paper presents the formulation and validation of a geometrically and physically (J2 plasticity) non-linear Generalised Beam Theory formulation, intended to calculate accurate non-linear elastoplastic equilibrium paths of thin-walled metal bars and associated collapse loads. This formulation extends previous work (Goncalves and Camotim, 2011) [1] by including the geometrically non-linear effects. The plate-like bending strains are assumed to be small (as in all GBT formulations), but the membrane strains are calculated exactly. Both stress-based and stress resultant-based GBT approaches are developed and implemented in a 3-node beam finite element. The stress-based formulation is generally more accurate, but the stress resultant-based formulation makes it possible to avoid numeric integration in the through-thickness direction of the walls. In order to show the potential of the proposed formulation and resulting finite element, several numerical results are presented and discussed. For validation purposes, these results are compared with those obtained with standard 2D-solid and shell finite element analyses.

A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory

Abstract: In this study, a micro scale non-linear Timoshenko beam model based on a general form of strain gradient elasticity theory is developed. The von Karman strain tensor is used to capture the geometric non-linearity. Governing equations of motion and boundary conditions are derived using Hamilton's principle. For some specific values of the gradient-based material parameters, the general beam formulation can be specialized to those based on simple forms of strain gradient elasticity. Accordingly, a simple form of the microbeam formulation is introduced. In order to investigate the behavior of the beam formulation, the problem of non-linear free vibration of a simply-supported microbeam is solved. It is shown that both strain gradient effect and that of geometric non-linearity increase the beam natural frequency. Numerical results reveal that for a microbeam with a thickness comparable to its material length scale parameter, the effect of strain gradient is higher than that of the geometric non-linearity. However, as the beam thickness increases, the difference between the results of the classical beam formulation and those of the gradient-based formulations become negligible. In other words, geometric non-linearity plays the essential role on increasing the natural frequency of a microbeam having a large thickness-to-length parameter ratio. In addition, it is shown that for some microbeams, both geometric non-linearity and size effect have significant contributions on increasing the natural frequency of non-linear vibrations.

Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories

Abstract: The present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results.