Showing papers on "Timoshenko beam theory published in 2009"

A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration

Abstract: In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM

Abstract: Nonlocal elasticity theory is a popular growing technique for the mechanical analyses of MEMS and NEMS structures. The nonlocal parameter accounts for the small-size effects when dealing with nano-size structures such as single-walled carbon nanotubes (SWCNTs). In this article, nonlocal elasticity and Timoshenko beam theory are implemented to investigate the stability response of SWCNT embedded in an elastic medium. For the first time, both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of the (SWCNT) with the surrounding elastic medium. A differential quadrature approach is utilized and numerical solutions for the critical buckling loads are obtained. Influences of nonlocal effects, Winkler modulus parameter, Pasternak shear modulus parameter and aspect ratio of the SWCNT on the critical buckling loads are analyzed and discussed. The present study illustrates that the critical buckling loads of SWCNT are strongly dependent on the nonlocal small-scale coefficients and on the stiffness of the surrounding medium.

Timoshenko beam model for buckling and vibration of nanowires with surface effects

Abstract: In this paper, surface effects on the axial buckling and the transverse vibration of nanowires are examined by using the refined Timoshenko beam theory. The critical compression force of axial buckling and the natural frequency of nanowires are obtained analytically, in which the impacts of surface elasticity, residual surface stress, transverse shear deformation and rotary inertia have been included. The buckling and vibration behaviour of a nanowire is demonstrated to be size dependent, especially when its cross-sectional dimension reduces to nanometres. The surface effects with positive elastic constants tend to increase the critical compression force and the natural frequency, especially for slender nanowires, while the shear deformation lowers these values for stubby nanowires. This study may be helpful to accurately measure the mechanical properties of nanowires and to design nanowire-based devices and systems.

On the stability of damped Timoshenko systems - Cattaneo versus Fourier law

Abstract: We consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While exponential stability under the Fourier law of heat conduction holds, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier’s law by changing to the Cattaneo law causes a loss of the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property.

Gradient elasticity and flexural wave dispersion in carbon nanotubes

Abstract: Higher-order elasticity theories have recently been used to predict the dispersion characteristics of flexural waves in carbon nanotubes (CNTs). In particular, nonlocal elasticity and gradient elasticity (with unstable strain gradients) have been employed within the framework of classical Euler-Bernoulli or improved Timoshenko beam theory to capture the dynamical behavior of CNTs. Qualitative agreement with the predictions of related molecular-dynamics (MD) simulations was observed, whereas the MD results departed significantly from those obtained with classical elasticity calculations. The present contribution aims to alert that the aforementioned higher-order models may yield questionable results for the higher wave numbers. As an alternative, gradient elasticity (with stable strain gradients), by also incorporating inertia gradients for dynamical applications, is used in combination with both Euler-Bernoulli and Timoshenko beam theories and shown to describe flexural wave dispersion in CNTs realistically for the small-to-medium range of wave numbers, i.e., the range for which MD results are available.

A Refined Zigzag Beam Theory for Composite and Sandwich Beams

Abstract: A new refined theory for laminated composite and sandwich beams that contains the kinematics of the Timoshenko Beam Theory as a proper baseline subset is presented. This variationally consistent theory is derived from the virtual work principle and employs a novel piecewise linear zigzag function that provides a more realistic representation of the deformation states of transverse-shear flexible beams than other similar theories. This new zigzag function is unique in that it vanishes at the top and bottom bounding surfaces of a beam. The formulation does not enforce continuity of the transverse shear stress across the beam s cross-section, yet is robust. Two major shortcomings that are inherent in the previous zigzag theories, shear-force inconsistency and difficulties in simulating clamped boundary conditions, and that have greatly limited the utility of these previous theories are discussed in detail. An approach that has successfully resolved these shortcomings is presented herein. Exact solutions for simply supported and cantilevered beams subjected to static loads are derived and the improved modelling capability of the new zigzag beam theory is demonstrated. In particular, extensive results for thick beams with highly heterogeneous material lay-ups are discussed and compared with corresponding results obtained from elasticity solutions, two other zigzag theories, and high-fidelity finite element analyses. Comparisons with the baseline Timoshenko Beam Theory are also presented. The comparisons clearly show the improved accuracy of the new, refined zigzag theory presented herein over similar existing theories. This new theory can be readily extended to plate and shell structures, and should be useful for obtaining relatively low-cost, accurate estimates of structural response needed to design an important class of high-performance aerospace structures.

Non-linear Modeling and Analysis of Solids and Structures

Abstract: Preface 1. Introduction 2. Non-linear bar elements 3. Finite rotations 4. Finite rotation beam theory 5. Co-rotating beam elements 6. Deformation and equilibrium of solids 7. Elasto-plastic solids 8. Numerical solution techniques 9. Dynamic effects and time integration References Index.

Non-linear dynamics of a drill-string with uncertain model of the bit-rock interaction

Abstract: The drill-string dynamics is difficult to predict due to the non-linearities and uncertainties involved in the problem. In this paper a stochastic computational model is proposed to model uncertainties in the bit-rock interaction model. To do so, a new strategy that uses the non-parametric probabilistic approach is developed to take into account model uncertainties in the bit-rock non-linear interaction model. The mean model considers the main forces applied to the column such as the bit-rock interaction, the fluid-structure interaction and the impact forces. The non-linear Timoshenko beam theory is used and the non-linear dynamical equations are discretized by means of the finite element method. (C) 2009 Elsevier Ltd. All rights reserved.

Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials

Abstract: Free vibration and elastic buckling of beams made of functionally graded materials (FGMs) containing open edge cracks are studied in this paper based on Timoshenko beam theory. The crack is modeled by a massless elastic rotational spring. It is assumed that the material properties follow exponential distributions along beam thickness direction. Analytical solutions of natural frequencies and critical buckling load are obtained for cracked FGM beams with clamped-free, hinged-hinged, and clamped-clamped end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, total number of cracks, material properties, beam slenderness ratio, and end supports on the free vibration and buckling characteristics of cracked FGM beams.

Energy decay rate of the thermoelastic Bresse system

Abstract: In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions.

Sensitivity Analysis of Atomic Force Microscope Cantilever Made of Functionally Graded Materials

Abstract: The purpose of this paper is the enhancement of the AFM sensitivity through the selection of an optimized FGM micro cantilever beam. In this paper, resonant frequencies and sensitivities of first two modes of micro cantilever which is made of functionally graded materials are investigated and a relationship is developed to evaluate the sensitivity of FGM micro cantilever. Effect of volume fraction of materials and surface contact stiffness on the resonant frequencies and sensitivities are studied. The rectangular FGM beam is modeled by an Euler-Bernoulli beam theory. It is assumed that beam is made of a mixture of metal and ceramic with properties varying through the thickness following a simple power law of n. This variation is a function of the volume fraction of the beam material constituents. The interaction between AFM tip and surface is modeled by a linear spring which expresses the contact stiffness. Results show that, increasing the ceramic volume fraction increases the resonant frequencies of both modes 1 and 2. When contact stiffness is small, for both modes, as ceramic volume fraction increases, sensitivities decreases, while for large contact stiffness, as ceramic volume fraction increases the sensitivities will be increased. Results also show that at each contact stiffness, there is a unique value of n at which the sensitivity is maximized. Using these values for n, the high quality and high contrast images can be obtained.Copyright © 2009 by ASME

Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes.

Abstract: This paper presents an assessment of continuum mechanics (beam and cylindrical shell) models in the prediction of critical buckling strains of axially loaded single-walled carbon nanotubes (SWCNTs). Molecular dynamics (MD) simulation results for SWCNTs with various aspect (length-to-diameter) ratios and diameters will be used as the reference solutions for this assessment exercise. From MD simulations, two distinct buckling modes are observed, i.e. the shell-type buckling mode, when the aspect ratios are small, and the beam-type mode, when the aspect ratios are large. For moderate aspect ratios, the SWCNTs buckle in a mixed beam-shell mode. Therefore one chooses either the beam or the shell model depending on the aspect ratio of the carbon nanotubes (CNTs). It will be shown herein that for SWCNTs with long aspect ratios, the local Euler beam results are comparable to MD simulation results carried out at room temperature. However, when the SWCNTs have moderate aspect ratios, it is necessary to use the more refined nonlocal beam theory or the Timoshenko beam model for a better prediction of the critical strain. For short SWCNTs with large diameters, the nonlocal shell model with the appropriate small length scale parameter can provide critical strains that are in good agreement with MD results. However, for short SWCNTs with small diameters, more work has to be done to refine the nonlocal cylindrical shell model for better prediction of critical strains.

Assessment of timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics

Abstract: In this paper, we study the flexural vibration behavior of single-walled carbon nanotubes (SWCNTs) for the assessment of Timoshenko beam models. Extensive molecular dynamics (MD) simulations based on second-generation reactive empirical bond-order (REBO) potential and Timoshenko beam modeling are performed to determine the vibration frequencies for SWCNTs with various length-to-diameter ratios, boundary conditions, chiral angles and initial strain. The effectiveness of the local and nonlocal Timoshenko beam models in the vibration analysis is assessed using the vibration frequencies of MD simulations as the benchmark. It is shown herein that the Timoshenko beam models with properly chosen parameters are applicable for the vibration analysis of SWCNTs. The simulation results show that the fundamental frequencies are independent of the chiral angles, but the chirality has an appreciable effect on higher vibration frequencies. The SWCNTs is very sensitive to the initial strain even if the strain is extremely small.

Hybrid Modeling of Ball Screw Drives With Coupled Axial, Torsional, and Lateral Dynamics

Abstract: It has been a common practice to assume that the torsional and axial dynamics are totally decoupled from the lateral dynamics of the screw when modeling ball screw drives. However, experiments show that there is a considerable coupling between them, which could adversely affect the positioning accuracy and fatigue life of the drive. In this paper, the lateral dynamics of the screw is explicitly incorporated into the hybrid finite element model of ball screw drives. The ball screw is modeled by Timoshenko beam elements, and the balls, joints, bearings, and fasteners are modeled as pure springs. Rigid components are modeled as lumped masses. The proposed screw-nut interface model, which includes the effects of lateral vibrations, is shown to predict the coupling between axial, torsional, and lateral dynamics of ball screw drives. The effects of this dynamic coupling on the positioning accuracy of the drive are also presented with experimental proof The proposed model provides a more realistic platform for a designer to optimize the drive parameters for high speed-high acceleration machine tool applications, where the ball screw vibrations limit the fatigue life of the mechanism, bandwidth of the servo systems, and positioning accuracy of the machine.

Static Analysis of a Functionally Graded Beam under a Uniformly Distributed Load by Ritz Method

Abstract: Static analysis of a functionally graded (FG) simply-supported beam subjected to a uniformly distributed load has been investigated by using Ritz method within the framework of Timoshenko and the higher order shear deformation beam theories. The material properties of the beam vary continuously in the thickness direction according to the power-law form. Trial functions denoting the transverse, the axial deflections and the rotation of the cross-sections of the beam are expressed in trigonometric functions. In this study, the effect of various material distributions on the displacements and the stresses of the beam are examined. Numerical results indicate that stress distributions in FG beams are very different from those in isotropic beams

A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams

Abstract: In the present paper, we present a continuum mechanics based derivation of Reissner’s equations for large-displacements and finite-strains of beams, where we restrict ourselves to the case of plane deformations of originally straight Bernoulli–Euler beams. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all of the generalized strains, which were originally introduced by Reissner at the beam-theory level. Our derivations thus circumvent the problem of needing to determine constitutive relations between stress resultants and generalized strains by physical experiments. Instead, constitutive relations at the stress–strain level can be utilized. Subsequently, this is exemplarily shown for a linear relation between Biot stress and Biot strain, which leads to linear constitutive relations at the beam-theory level, and for a linear relation between the second Piola–Kirchhoff stress and the Green strain, which gives non-linear constitutive relations at the beam theory level. A simple inverse method for analytically constructing solutions of Reissner’s non-linear relations is shortly pointed out in Appendix I.

Surface stress effects on the bending properties of fcc metal nanowires

Abstract: The major purpose of this work is to investigate surface stress effects on the bending behavior and properties of $⟨100⟩/{100}$ gold nanowires with both fixed/fixed and fixed/free boundary conditions. The results are obtained through utilization of the recently developed surface Cauchy-Born model, which captures surface stress effects on the elastic properties of nanostructures through a three-dimensional, nonlinear finite element formulation. There are several interesting findings in the present work. First, we quantify the stress and displacement fields that result in the nanowires due to bending deformation. In doing so, we find that regardless of boundary condition, the stresses that are present in the nanowires due to deformation induced by surface stresses prior to any applied bending deformation dominate any stresses that are generated by the bending deformation unless very large $(\ensuremath{\approx}5%)$ bending strains are applied. In contrast, when the stresses and displacements induced by surface stresses prior to bending are subtracted from the stress and displacement fields of the bent nanowires, we find that the bending stresses and displacements do match the solutions expected from bulk continuum beam theory, but only within the nanowire bulk, and not at the nanowire surfaces. Second, we find that the deformation induced by surface stresses also has a significant impact on the nanowire Young's modulus that is extracted from the bending simulations, where a strong boundary-condition dependence is also found. By comparing all results to those that would be obtained using various linear surface-elastic theories, we demonstrate that a nonlinear, finite deformation formulation that captures changes in both bulk- and surface-elastic properties resulting from surface stress-induced deformation is critical to reproducing the experimentally observed boundary-condition dependence in Young's modulus of metal nanowires. Furthermore, we demonstrate that linear surface-elastic theories based solely on the surface energy erroneously predict an increase in Young's modulus with decreasing nanowire size regardless of boundary condition. In contrast, while the linear surface-elastic theories based upon the Gurtin and Murdoch formalism can theoretically predict elastic softening with decreasing size, we demonstrate that, regardless of boundary condition, the stiffening due to the surface stress dominates the softening due to the surface stiffness for the range of nanowire geometries considered in the present work. Finally, we determine that the nanowire Young's modulus is essentially identical when calculated via either bending or resonance for both boundary conditions, indicating that surface effects have a similar impact on the elastic properties of nanowires for both loading conditions.