Showing papers on "Timoshenko beam theory published in 2010"

A micro scale Timoshenko beam model based on strain gradient elasticity theory

Abstract: A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.

Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory

Abstract: Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear free vibration characteristics of SWCNTs.

Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory

Abstract: In the present study, forced vibration of a simply supported single-walled carbon nanotube (SWCNT) subjected to a moving harmonic load is investigated by using nonlocal Euler–Bernoulli beam theory. The time-domain responses are obtained by using both the modal analysis method and the direct integration method. The effects of nonlocal parameter, aspect ratio, velocity and the excitation frequency of the moving load on the dynamic responses of SWCNT is discussed. For comparison purposes, free vibration frequencies and static deflections of the SWCNT subjected to a point load at the midpoint are obtained and compared with previously published studies. Good agreement is observed. The results show that dynamic deflections of the SWCNT increase with increase in the nonlocal parameter, which means that dynamic deflections based on the local beam theory are underestimated, and the effect of nonlocal parameter is dependent on the aspect ratio. Furthermore, load velocity and the excitation frequency play an important role on the dynamic behavior of the SWCNT.

An analytical study on the nonlinear vibration of functionally graded beams

Abstract: Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Karman geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.

A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory

Abstract: This paper presents a new and general approach for the calculation of cross-section deformation modes in thin-walled beams, to be used in the framework of generalized beam theory (GBT). The proposed approach subdivides and hierarchizes the cross-section deformation modes by employing several kinematic hypotheses. This makes it possible to discard a priori specific types of deformation modes and consequently reduce the number of cross-section degrees-of-freedom. The approach is applicable to arbitrary (with open and closed parts) polygonal cross-sections with external and internal constraints and allows for the a posteriori inclusion of particular deformation modes (e.g., shear deformation modes in a part of the cross-section). Although only GBT applications are dealt with, the deformation modes obtained may be straightforwardly incorporated in other thin-walled beam formulations that include cross-section deformation.

Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model

Abstract: The hybrid nonlocal Euler-Bernoulli beam model is applied for the bending, buckling, and vibration analyzes of micro/nanobeams. In the hybrid nonlocal model, the strain energy functional combines the local and nonlocal curvatures so as to ensure the presence of small length-scale parameters in the deflection expressions. Unlike Eringen's nonlocal beam model that has only one small length-scale parameter, the hybrid nonlocal model has two independent small length-scale parameters, thereby allowing for a more flexible and accurate modeling of micro/nanobeamlike structures. The equations of motion of the hybrid nonlocal beam and the boundary conditions are derived using the principle of virtual work. These beam equations are solved analytically for the bending, buckling, and vibration responses. It will be shown herein that the hybrid nonlocal beam theory could overcome the paradoxes produced by Eringen's nonlocal beam theory such as vanishing of the small length-scale effect in the deflection expression or the surprisingly stiffening effect against deflection for some classes of beam bending problems.

Continuum Models Incorporating Surface Energy for Static and Dynamic Response of Nanoscale Beams

Abstract: Nanoscale beams are commonly found in nanomechanical and nanoelectromechanical systems (NEMS) and other nanotechnology-based devices. Surface energy has a significant effect on nanoscale structures and is associated with their size-dependent behavior. In this paper, a general mechanistic model based on the Gurtin-Murdoch continuum theory accounting for surface energy effects is presented to analyze thick and thin nanoscale beams with an arbitrary cross section. The main contributions of this paper are a set of closed-form analytical solutions for the static response of thin and thick beams under different loading (point and uniformly distributed) and boundary conditions (simply-supported, cantilevered, and clamped ends), as well as the solution of the free vibration characteristics of such beams. Selected numerical results are presented for aluminum and silicon beams to demonstrate their salient response features. It is shown that classical beam theory is not accurate in situations where the surface residual stress and/or surface elastic constants are relatively large. An intrinsic length scale for beams is identified that depends on beam surface properties and cross-sectional shape. The present work provides a convenient set of analytical tools for researchers working on NEMS design and fabrication to understand the static and dynamic behavior of nanoscale beams including their size-dependent behavior and the effects of common boundary conditions.

Timoshenko beam model for static bending of nanowires with surface effects

Abstract: In this paper, surface effects on the elastic behavior of static bending nanowires (NWs) are studied by using a comprehensive Timoshenko beam model. For NWs with different boundary conditions, explicit solutions are derived to study the combined effects of residual surface stress, surface elasticity and shear deformation on the effective stiffness and Young's modulus of the NWs. The stiffness is found to be size-dependent and this dependence is more significant for slender NWs. Residual surface stress tends to increase or decrease the stiffness of NWs depending on their boundary conditions, and the shear deformation always makes the NWs softer compared with the results of the Euler–Bernoulli beam model. The solutions agree well with experimental measurement in predicting the Young's modulus, especially for NWs with small length-to-thickness ratio. This work might be helpful for characterizing the mechanical properties of NWs and the design of nanobeam-based devices.

Thermal buckling analysis of functionally graded material beams

Abstract: Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.

A higher-order theory for static and dynamic analyses of functionally graded beams

Abstract: The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.

Influences of the surface energies on the nonlinear static and dynamic behaviors of nanobeams

Abstract: A modified continuum model of the nanoscale beams is presented by incorporating surface elasticity in this paper. The classical beam theory is adopted to model the bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. On the basis of this modified beam theory the governing equation of the nanobeam is derived where the effect of the geometry nonlinearity is also considered. The Galerkin method is used to give a reduced-order model of the problem. Beams made from two materials: aluminum and silicon are chosen as examples. The numerical results show that the mechanical buckling and free vibration phenomena of nanobeams are size-dependence. The effects of the surface energies on the critical axial force of buckling, post-buckling and linear free vibration frequency are discussed. Finally, the amplitude frequency response is given numerically through the incremental harmonic balanced method.

Free Vibration Analysis of Carbon Nanotubes Based on Shear Deformable Beam Theory by Discrete Singular Convolution Technique

Abstract: In this study, free vibration analysis of carbon nanotubes is investigated based on Timoshenko beam theory. Discrete singular convolution (DSC) method is used for free vibration problem of numerical solution of carbon nanotubes. Numerical results are presented and compared with that available in the literature. It is shown that reasonable accurate results are obtained.

A Timoshenko beam model for cantilevered piezoelectric energy harvesters

Abstract: Piezoelectric bimorph cantilevered beams are often used as energy harvesting devices. These devices are desired for, among other applications, remote sensing and animal tracking due to their potential for eliminating the need for battery replacement. Existing models of piezoelectric bimorph cantilevered beams have proved to describe the dynamics of slender beams at high frequencies accurately. In this paper, a Timoshenko model of transverse piezoelectric beam vibration is developed to address these limitations. Exact expressions for the voltage, current, power, and tip deflection of the piezoelectric beam are derived. Subsequently, several case studies are presented that examine the frequency response of vibration-based energy harvesters using this model. It is shown that the predicted responses converge towards previously derived Euler–Bernoulli beam models under certain limiting conditions. The Timoshenko model shows that the Euler–Bernoulli model severely over-predicts the tip displacement and consequently the power transduction of a cantilevered piezoelectric bimorph at low length-to-width aspect ratios.

Modeling and analysis of microtubules based on a modified couple stress theory

Abstract: Based on a modified couple stress theory, a new Timoshenko beam model is established to address the size effect of microtubules (MTs) in this paper. The bending equation and the buckling equation are derived from the minimum total potential energy principle. Results obtained from the present model show that length dependence of MTs is related not only to shear effect but also to size effect, and the size effect is coupled in the shear effect, which means that the phenomenon of length dependence will disappear when the shear effect is neglected. Moreover, when very long MTs are considered, the persistence lengths are related to the internal material length scale parameter, which is different from the conclusions obtained from the classical and previous nonlocal beam models. The effect of the internal material length scale parameter on the buckling wavelength, the buckling growth rate and the buckling amplitude of the MTs is also discussed in this paper, and a comparison between present and previous results is presented.

Vibration and instability of a viscous-fluid-conveying single-walled carbon nanotube embedded in a visco-elastic medium

Abstract: In this study, for the first time, the transverse vibrational model of a viscous-fluid-conveying single-walled carbon nanotube (SWCNT) embedded in biological soft tissue is developed. Nonlocal Euler–Bernoulli beam theory has been used to investigate fluid-induced vibration of the SWCNT while visco-elastic behaviour of the surrounding tissue is simulated by the Kelvin–Voigt model. The results indicate that the resonant frequencies and the critical flow velocity at which structural instability of nanotubes emerges are significantly dependent on the properties of the medium around the nanotube, the boundary conditions, the viscosity of the fluid and the nonlocal parameter. Detailed results are demonstrated for the dependence of damping and elastic properties of the medium on the resonant frequencies and the critical flow velocity. Three standard boundary conditions, namely clamped–clamped, clamped–pinned and pinned–pinned, are applied to study the effect of the supported end conditions. Furthermore, it is found that the visco-elastic foundation causes an obvious reduction in the critical velocity in comparison with the elastic foundation, in particular for a compliant medium, pinned–pinned boundary condition, high viscosity of the fluid and small values of the nonlocal coefficient.

Modeling and characterization of MEMS-based piezoelectric harvesting devices

Abstract: Vibrational piezoelectric harvesting devices (PHD) provide an autonomous power source for various types of sensors, actuators and MEMS devices. There have been several examples of vibrational energy harvesters published in the literature over the years. However, for many applications the generated power is not yet sufficient. In this paper, a physical model for predicting the generated electric power from piezoelectric harvesting devices is introduced. The model is based on estimating the total charge generated on a piezoelectric material when it is subjected to mechanical strain as a result of bending at the fundamental resonance frequency. Based on Euler–Bernoulli beam theory, the strain can be determined in terms of the beam deflection at purely mechanical excitation. The proposed model extends the current state of the art by consideration of the strain distribution due to the presence of an extended mass volume at the end of the beam. The constitutive equations of piezoelectricity in the sensing mode correlate the strain and the induced charge in the piezoelectric element. Using the device design parameters and the beam deflection as inputs, the power output can be calculated. The results of the model were experimentally verified for MEMS-based PHDs. The model was found to give an accurate prediction of the electrical parameters under various damping conditions. After model validation, a subsequent device optimization has been made to improve the power generation.