Showing papers on "Timoshenko beam theory published in 2008"

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

Abstract: A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.

Nonlocal continuum theories of beams for the analysis of carbon nanotubes

Abstract: The equations of motion of the Euler–Bernoulli and Timoshenko beam theories are reformulated using the nonlocal differential constitutive relations of Eringen [International Journal of Engineering Science 10, 1–16 (1972)]. The equations of motion are then used to evaluate the static bending, vibration, and buckling responses of beams with various boundary conditions. Numerical results are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies of carbon nanotubes.

Surface Effect on the Elastic Behavior of Static Bending Nanowires

Abstract: The surface effect from surface stress and surface elasticity on the elastic behavior of nanowires in static bending is incorporated into Euler-Bernoulli beam theory via the Young-Laplace equation. Explicit solutions are presented to study the dependence of the surface effect on the overall Young's modulus of nanowires for three different boundary conditions: cantilever, simply supported, and fixed-fixed. The solutions indicate that the cantilever nanowires behave as softer materials when deflected while the other structures behave like stiffer materials as the nanowire cross-sectional size decreases for positive surface stresses. These solutions agree with size dependent nanowire overall Young's moduli observed from static bending tests by other researchers. This study also discusses possible reasons for variations of nanowire overall Young's moduli observed.

Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity

Abstract: Based on the Bernoulli–Euler and Timoshenko beam theories, a single-elastic beam model using nonlocal elasticity is developed for the wave propagation in carbon nanotubes (CNTs). The small-scale effect is taken into consideration in the present theory. Frequency equations and modal shape functions of Timoshenko beams structures with some typical boundary conditions are also derived from nonlocal elasticity. In addition, the applicability of the two beam models is explored by numerical simulations. The research work reveals the significance of the small-scale effect on wave propagation in single-walled CNTs.

Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction

Abstract: The free and forced vibration of a laminated functionally graded beam of variable thickness under thermally induced initial stresses is studied in this paper within the framework of Timoshenko beam theory. The beam consists of a homogeneous substrate and two inhomogeneous functionally graded layers whose material composition follows a power law distribution in the thickness direction in terms of the volume fractions of the material constituents. Both the axial and rotary inertia of the beam are considered in the present analysis. It is assumed that the beam may be clamped, hinged, or free at its ends and is subjected to one-dimensional steady heat conduction in the thickness direction before undergoing dynamic deformation. To include the effect of temperature change, the initial stress state is determined through a thermo-elastic analysis before the free and forced vibration analyses. The differential quadrature method that makes use of Lagrange interpolation polynomials is employed as a numerical solution tool to solve both the thermo-elastic equilibrium equation and dynamic equation. Numerical results are presented in both tabular and graphical forms for various laminated functionally graded beams, showing that vibration frequencies, mode shapes and dynamic response are significantly influenced by the thickness variation, temperature change, slenderness ratio, volume fraction index, the thickness of the functionally graded layer, and the end support conditions.

Surface stress effect on bending resonance of nanowires with different boundary conditions

Abstract: The influence of surface stress on the resonance frequencies of bending nanowires was studied by incorporating the generalized Young–Laplace equation into Euler–Bernoulli beam theory Theoretical solutions are presented for three different boundary conditions The overall Young’s modulus was used to study the surface stress influenced mechanical behavior of bending nanowires and a comparison was made for the overall Young’s modulus calculated from nanowires in resonance and static bending It was found that the overall Young’s modulus can be simply related to a nondimensional surface effect factor via empirical formulae

Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

Abstract: This paper is concerned with the bending problem of micro- and nanobeams based on the Eringen nonlocal elasticity theory and Timoshenko beam theory. In the former theory, the small-scale effect is taken into consideration while the effect of transverse shear deformation is accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. General solutions for the deflection, rotation, and stress resultants are presented for transversely loaded beams. In addition, specialized bending solutions are given for beams with various end conditions. These solutions account for a better representation of the bending behavior of short, stubby, micro- and nanobeams where the small-scale effect and transverse shear deformation are significant. Considering particular loading and boundary conditions, the effects of small-scale and shear deformation on the bending results may be observed because of the analytical forms of the solutions.

The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory

Abstract: This paper is concerned with the use of the nonlocal Timoshenko beam model for free vibration analysis of single-walled carbon nanotubes (CNTs) including the thermal effect. Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. The elastic Timoshenko beam model is reformulated using the nonlocal differential constitutive relations of Eringen (1972 Int. J. Eng. Sci. 10 1–16). The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the traditional Timoshenko beam model but also by the nonlocal elasticity characterizing the microstructure of CNTs in a wide frequency range up to terahertz. Numerical results are presented using the nonlocal beam theory to bring out the effect of both the nonlocal parameter and the temperature change on the properties of transverse vibrations of CNTs. The exact nonlocal Timoshenko beam solution presented here should be useful to engineers who are designing microelectromechanical and nanoelectromechanical devices.

Finite Elements in Analysis andDesign

Abstract: Article history: Received 19 October 2007 Received in revised form 22 May 2008 Accepted 23 June 2008 Available online 12 August 2008

Modeling, Nonlinear Dynamics, and Identification of a Piezoelectrically Actuated Microcantilever Sensor

Abstract: Nanomechanical cantilever sensors (NMCSs) have recently emerged as an effective means for label-free chemical and biological species detection. They operate through the adsorption of species on the functionalized surface of mechanical cantilevers. Through this functionalization, molecular recognition is directly transduced into a micromechanical response. In order to effectively utilize these sensors in practice and correctly relate the micromechanical response to the associated adsorbed species, the chief technical issues related to modeling must be resolved. Along these lines, this paper presents a general nonlinear-comprehensive modeling framework for piezoelectrically actuated microcantilevers and validates it experimentally. The proposed model considers both longitudinal and flexural vibrations and their coupling in addition to the ever-present geometric and material nonlinearities. Utilizing Euler-Bernoulli beam theory and employing the inextensibility conditions, the coupled longitudinal-flexural equations of motion are reduced to one nonlinear partial differential equation describing the flexural vibrations of the sensor. Using a Galerkian expansion, the resulting equation is discretized into a set of nonlinear ordinary differential equations. The method of multiple scales is then implemented to analytically construct the nonlinear response of the sensor near the first modal frequency (primary resonance of the first vibration mode). These solutions are compared to experimental results demonstrating that the sensor exhibits a softening-type nonlinear response. Such behavior can be attributed to the presence of quadratic material nonlinearities in the piezoelectric layer. This observation is critical, as it suggests that unlike macrocantilevers where the geometric hardening nonlinearities dominate the response behavior, material nonlinearities dominate the response of microcantilevers yielding a softening-type response. This behavior should be accounted for when designing and employing such sensors for practical applications.

Effect of small size on wave propagation in double-walled carbon nanotubes under temperature field

Abstract: The effect of small size on wave propagation in double-walled carbon nanotubes (DWCNTs) under temperature field is investigated using the Euler–Bernoulli beam model Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal thermal elastic theory The effects of temperature change and van der Waals forces between the inner and outer nanotubes are taken into account Results show the significance of the small-scale effect on wave propagation in DWCNTs and that some properties of transverse vibrations of DWCNTs are dependent on the change in temperature The results demonstrate the great potential of the proposed nonlocal beam theory in studying wave propagation in CNTs including thermal effects and also indicate the limitations of local continuum mechanics in analysis of small-scale effects The work should be useful in the design and application of nanoelectronics and nanoelectromechanical devices

Semi-inverse Method for Vibration and Buckling of Axially Functionally Graded Beams

Abstract: In this study, vibration and buckling of axially functionally graded simply supported beams is analyzed by using the semi-inverse method. Euler—Bernoulli beam theory was used in the analysis. By using a pre-specified frequency and buckling loads, variation of the Young's modulus in the axial direction is obtained in terms of the axial coordinate. It is found that the Young's modulus changes exponentially between the edges of the beam for the vibration and buckling problem.

Computational modelling of a non-viscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam

Abstract: In the design of nanotube-based fluidic devices, a critical issue is the effect of the induced vibrations in the nanotube arising from the fluid flow, since these vibrations can promote structural instabilities, such as buckling transitions. It is known that the induced resonant frequencies depend on the fluid flow velocity in a significant manner. We have studied, for the first time, the flow of a non-viscous fluid in stubby multi-walled carbon nanotubes, using the Timoshenko classical beam theory to model the nanotubes as a continuum structure. We have obtained the variations of the resonant frequencies with the fluid flow velocity under several experimentally interesting boundary conditions and aspect ratios of the nanotube. The main finding from our work is that, compared to an Euler-Bernoulli classical beam model of a nanotube, the Timoshenko beam predicts the loss of stability at lower fluid flow velocities.

Surface stress effects on the resonant properties of silicon nanowires

Abstract: The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires. We accomplish this by using the surface Cauchy–Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite element eigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free ⟨100⟩ silicon nanowires with unreconstructed {100} surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

Sound wave propagation in single-walled carbon nanotubes with initial axial stress

Abstract: This paper studies the vibrational characteristics of single-walled carbon nanotubes (SWNTs) with initial axial loading based on the theory of nonlocal elasticity. The consistent equations of motion for the nonlocal Euler-Bernoulli and Timoshenko beam models are provided taking into account the initial axial stress. The small scale effect on CNT wave propagation dispersion relation is explicitly revealed for different CNT wave numbers and diameters by theoretical analyses and numerical simulations. In addition, the applicability of the two beam models is explored by numerical simulations. The research work reveals the significance of the effects of small scale, transverse shear deformation and rotary inertia on wave propagation in short SWCNTs with initial axial loading.

Relevance of Timoshenko-beam model to microtubules of low shear modulus

Abstract: Microtubules are characterized by extremely low shear modulus that is a few orders of magnitude lower than longitudinal modulus. In this paper, the effects of transverse shearing due to low shear modulus of microtubules are investigated using a Timoshenko-beam model, with detailed comparison between the Timoshenko-beam model, classical isotropic Euler–Bernoulli beam model and a more accurate 2D orthotropic elastic shell model. It is confirmed that transverse shearing is mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler–Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the Timoshenko-beam model is found to be in good quantitative agreement with known experimental data. In particular, the present Timoshenko-beam model predicts that, because of the length dependence of flexural rigidity, microtubules of different lengths could sustain almost equal maximum axial compressive force against column buckling, a conclusion that could have some interesting consequences to the mechanical behavior of cells. These results recommend that the Timoshenko-beam model offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics.

A Compliant Bistable Mechanism Design Incorporating Elastica Buckling Beam Theory and Pseudo-Rigid-Body Model

Abstract: In this work, a new compliant bistable mechanism design is introduced. The combined use of pseudo-rigid-body model (PRBM) and the Elastica buckling theory is presented for the first time to analyze the new design. This mechanism consists of the large deflecting straight beams, buckling beams, and a slider. The kinematic analysis of this new mechanism is studied, using nonlinear Elastica buckling beam theory, the PRBM of a large deflecting cantilever beam, the vector loop closure equations, and numerically solving nonlinear algebraic equations. A design method of the bistable mechanism in microdimensions is investigated by changing the relative stiffness of the flexible beams. The actuation force versus displacement characteristics of several cases is explored and the full simulation results of one of the cases are presented. This paper demonstrates the united application of the PRBM and the buckling Elastica solution for an original compliant mechanism kinematic analysis. New compliant mechanism designs are presented to highlight where such combined kinematic analysis is required.

On the Internal and Boundary Stabilization of Timoshenko Beams

Abstract: In this paper we consider Timoshenko systems with either internal or boundary feedbacks. We establish explicit and generalized decay results, without imposing restrictive growth assumption near the origin on the damping terms.