Classical Euler Bernoulli Beam Research Articles

Postbuckling characteristics of nanobeams based on the surface elasticity theory

In the present paper, an attempt is made to numerically investigate the postbuckling response of nanobeams with the consideration of the surface stress effect. To accomplish this, the Gurtin–Murdoch elasticity theory is exploited to incorporate surface stress effect into the classical Euler–Bernoulli beam theory. The size-dependent governing differential equations are derived and discretized along with various end supports by employing the principle of virtual work and the generalized differential quadrature (GDQ) method. Newton’s method is applied to solve the discretized nonlinear equations with the aid of an auxiliary normalizing equation. After solving the governing equations linearly, to obtain each eigenpair in the nonlinear model, the linear response is used as the initial value in Newton’s method. Selected numerical results are given to show the surface stress effect on the postbuckling characteristics of nanobeams. It is found that by increasing the thickness of nanobeams, the postbuckling equilibrium path obtained by the developed non-classical beam model tends to the one predicted by the classical beam theory and this anticipation is the same for all selected boundary conditions.

Effect of Graphene Layers on Static Pull-in Behavior of Bilayer Graphene/Substrate Electrostatic Microactuators

A closed-form solution is obtained for the pull-in instability of curved multilayer graphene/substrate microcantilever electrostatic actuators. The first-order fringing-field correction and the interlayer shear between neighboring graphene layers (GLs) and between the graphene and the substrate are incorporated into the analytical model. In the solution procedure, the governing fourth-order differential equation of variable coefficients is converted into a Fredholm integral equation. The resulting equation is solved for the static pull-in voltages by adopting the first natural mode of the cantilever beam as a deflection shape function. The influence of GLs on the pull-in voltages of the electrostatic microactuators is investigated. It is found that laying 10, 30, and 60 GLs on top of the substrate results in increases of about 95%, 190%, and 295%, respectively, in the pull-in voltage of the straight bilayer graphene/substrate electrostatic microactuators. It is also observed that the classical Euler-Bernoulli beam theory fails to predict the pull-in voltages of the multilayer graphene/substrate electrostatic microactuators, showing that the pull-in voltage is highly affected by the graphene interlayer shear.

Strain gradient beam element

The classical continuum theory is neither able to accurately model the mechanical behavior of micro/nano-scale structures nor capable of justifying the size-dependent behavior observed in these structures; so the non-classical continuum theories such as the strain gradient theory have been emerged and developed. In order to enable the finite element method (FEM) to more accurately deal with the problems in micro/nano-scale structures, a size-dependent Euler–Bernoulli beam element is developed based on the strain gradient theory. Compared to the classical Euler–Bernoulli beam element, the nodal displacement vector of the new Euler–Bernoulli beam element has an additional component, i.e. the nodal curvature, associated with the additional kinematic parameter existing at the boundaries of strain gradient beams. The mass and stiffness matrices of the new non-classical beam element are derived based on the Galerkin's method. In some examples, it is shown that how the new element can be employed to solve a real-case problem and the results are compared to the analytical and available experimental data as well as the results obtained by employing the classical beam elements. It is observed that there is a good agreement between the experimental and the strain gradient based FEM results while the difference between the experimental and the classical FEM results is significant. In addition, it is indicated that the new beam element can successfully capture the size-dependency and the structures modeled by this element show stiffer behavior than those modeled by the classical beam element. Moreover, by setting some material length scale parameters to zero the new beam element is able to recover the results of the classical theory and the modified couple stress theory (another non-classical continuum theory).

Prediction of compressive post-buckling behavior of single-walled carbon nanotubes in thermal environments

In the present investigation, the axial buckling and post-buckling configurations of single-walled carbon nanotubes (SWCNTs) are studied including the thermal environment effect. For this purpose, Eringen’s nonlocal elasticity continuum theory is implemented into the classical Euler–Bernoulli beam theory to represent the SWCNTs as a nonlocal elastic beam model. A closed-form analytical solution is carried out to analyze the static response of SWCNTs in their post-buckling state in which the axial buckling load is assumed to be beyond the critical axial buckling load. Common sets of boundary conditions, named simply supported–simply supported (SS–SS), clamped–clamped (C–C), and clamped–simply supported (C–SS), are considered in the investigation. Selected numerical results are given to represent the variation of the carbon nanotube’s mid-span deflection with the applied axial load corresponding to various nonlocal parameters, length-to-diameter aspect ratios, temperature changes, and end supports. Moreover, a comparison between the post-buckling behaviors of SWCNTs at low- and high-temperature environments is presented. It is found that the size effect leads to a decrease of the axial buckling load especially for SWCNTs with C–C boundary conditions. Also, it is revealed that the value of the temperature change plays different roles in the post-buckling response of SWCNTs at low- and high-temperature environments.

Analysis of wave propagation in micro/nanobeam-like structures: A size-dependent model

By incorporating the strain gradient elasticity into the classical Bernoulli-Euler beam and Timoshenko beam models, the size-dependent characteristics of wave propagation in micro/nanobeams is studied. The formulations of dispersion relation are explicitly derived for both strain gradient beam models, and presented for different material length scale parameters (MLSPs). For both phenomenological sizedependent beam models, the angular frequency, phase velocity and group velocity increase with increasing wave number. However, the velocity ratios approach different values for different beam models, indicating an interesting behavior of the asymptotic velocity ratio. The present theory is also compared with the nonlocal continuum beam models.

Capillary collapse of a micro-double cantilever beam: a rigorous study

High-aspect-ratio microstructures have been found, in the literature, to collapse due to capillary forces of liquids. In this paper, mathematical models are developed to study the collapse of a microstructure represented by a double cantilever beam (DCB) with a liquid droplet located at the free end. Formulations are presented using the classical Bernoulli–Euler beam theory as well as an analysis that accounts for geometrical nonlinearity. The models introduce rigorous coupling between the DCB deformation, the capillary forces, and the meniscus position, and have predicted interesting nonlinear behaviors that previous models could not. Parameters governing the capillary collapse of the DCB are identified, and their influence is discussed. A single dimensionless number that controls the condition for collapse is proposed and validated against numerical results. Comparison between the linear and nonlinear beam analyses shows that linear analysis generally suffices for the description of capillary collapse of microstructures.

A fundamental numerical and theoretical study for the vibrational properties of nanowires

Based on the molecular dynamics (MD) simulation and the classical Euler-Bernoulli beam theory, a fundamental study of the vibrational performance of the Ag nanowire (NW) is carried out. A comprehensive analysis of the quality (Q)-factor, natural frequency, beat vibration, as well as high vibration mode is presented. Two excitation approaches, i.e., velocity excitation and displacement excitation, have been successfully implemented to achieve the vibration of NWs. Upon these two kinds of excitations, consistent results are obtained, i.e., the increase of the initial excitation amplitude will lead to a decrease to the Q-factor, and moderate plastic deformation could increase the first natural frequency. Meanwhile, the beat vibration driven by a single relatively large excitation or two uniform excitations in both two lateral directions is observed. It is concluded that the nonlinear changing trend of external energy magnitude does not necessarily mean a non-constant Q-factor. In particular, the first order natural frequency of the Ag NW is observed to decrease with the increase of temperature. Furthermore, comparing with the predictions by Euler-Bernoulli beam theory, the MD simulation provides a larger and smaller first vibration frequencies for the clamped-clamped and clamped-free thin Ag NWs, respectively. Additionally, for thin NWs, the first order natural frequency exhibits a parabolic relationship with the excitation magnitudes. The frequencies of the higher vibration modes tend to be low in comparison to Euler-Bernoulli beam theory predictions. A combined initial excitation is proposed which is capable to drive the NW under a multi-mode vibration and arrows the coexistence of all the following low vibration modes. This work sheds lights on the better understanding of the mechanical properties of NWs and benefits the increasing utilities of NWs in diverse nano-electronic devices.

Theoretical and numerical investigation of bending properties of Cu nanowires

Based on the AFM-bending experiments, a molecular dynamics (MD) bending simulation model is established which could accurately account for the full spectrum of the mechanical properties of NWs in a double clamped beam configuration, ranging from elasticity to plasticity and failure. It is found that, loading rate exerts significant influence to the mechanical behaviours of nanowires (NWs). Specifically, a loading rate lower than 10m/s is found reasonable for a homogonous bending deformation. Both loading rate and potential between the tip and the NW are found to play an important role in the adhesive phenomenon. The force versus displacement (F–d) curve from MD simulation is highly consistent in shapes with that from experiments. Symmetrical F–d curves during loading and unloading processes are observed, which reveal the linear-elastic and non-elastic bending deformation of NWs. The typical bending induced tensile-compressive features are observed. Meanwhile, the simulation results are excellently fitted by the classical Euler–Bernoulli beam theory with axial effect. It is concluded that, axial tensile force becomes crucial in bending deformation when the beam size is down to nanoscale for double clamped NWs. In addition, we find shorter NWs will have an earlier yielding and a larger yielding force. Mechanical properties (Young’s modulus and yield strength) obtained from both bending and tensile deformations are found comparable with each other. Specifically, the modulus is essentially similar under these two loading methods, while the yield strength during bending is observed larger than that during tension.

Modal characteristics of cross-ply laminated smart beams using various beam theories

Analytical models and solutions are obtained for the free vibration of cross-ply laminated beams with extension piezoelectric actuators. The piezoelectric actuators are bonded along the beam surface. The models are based on the higher-order shear deformation beam theory (HOBT), first-order shear deformation beam theory (FOBT) and the classical Euler-Bernoulli beam theory (CBT). The equations of motion and associated boundary conditions are derived for the beam models using Hamilton's principle. The state-space approach is used to find accurate natural frequencies and mode shapes for arbitrary combinations of boundary conditions. The exact analytical solutions obtained are illustrated numerically in a number of figures and tables revealing the effects of the number of layers, the actuators' thickness, the length to thickness ratio and the character of boundary conditions on the non-dimensional frequencies. Finally comparison of frequencies obtained by CBT, FOBT and HOBT is made to assess the importance of shear deformation.

Size Dependent Pull-in Phenomena in Electro-Statically Actuated Micro-Beam Based on the Modified Couple Stress Theory

A modified continuum model of electro-statically actuated micro-beam is presented based on the modified couple stress theory. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Bernoulli-Euler beam theory. The governing equation of the micro-beam is derived based on the Hamilton’s principle, which accounts for the effects of the moderately large deflection, the residual stress and the fringing electrostatic field. The numerical analysis of mechanical characterization is performed by the Analog Equation Method (AEM). The effects of the couple stress on the static and dynamic responses, pull-in voltage and pull-in time are discussed.

A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory

In the present paper, the free vibration characteristics of nanobeams embedded in an elastic medium are investigated. Inclusion of size effects is considered in the analysis by incorporating Eringen’s nonlocal elasticity continuum into the classical Euler–Bernoulli beam theory. To include the surrounding elastic medium, the Pasternak elastic foundation model is utilized, including shear deformation of the elastic medium. A high-order compact finite difference method (CFDM) is employed for sixth-order discretization of the nonlocal beam model to obtain the fundamental frequencies of nanobeams corresponding to three commonly used boundary conditions, namely simply supported–simply supported, clamped–clamped, and clamped–free. Numerical results are presented to indicate the accuracy of the method based on the sixth-order discretization for predicting the vibrational response of embedded nanobeams subject to various boundary conditions.

Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies

This paper deals with the accurate evaluation of complete three-dimensional (3D) stress fields in beam structures with compact and bridge-like sections. A refined beam finite-element (FE) formulation is employed, which permits any-order expansions for the three displacement components over the section domain by means of the Carrera Unified Formulation (CUF). Classical (Euler-Bernoulli and Timoshenko) beam theories are considered as particular cases. Comparisons with 3D solid FE analyses are provided. End effects caused by the boundary conditions are investigated. Bending and torsional loadings are considered. The proposed formulation has shown its capability of leading to quasi-3D stress fields over the beam domain. Higher-order beam theories are necessary for the case of bridge-like sections. Various theories are also compared in terms of shear correction factors on the basis of definitions found in the open literature. It has been confirmed that different theories could lead to very different values of shear correction factors, the accuracy of which is subordinate to a great extent to the section geometries and loading conditions. However, an accurate evaluation of shear correction factors is obtained by means of the present higher-order theories.

Effect of flowing medium for a simply supported pipe subjected to impulse loading

The primary aim of this study is to investigate the effects of a flowing medium on the transient response of a simply supported pipe subjected to dynamically applied loading. The importance of this study is manifested in numerous applications such as oil and gas transportation where dynamic loading can be the result of an accident. The classical Bernoulli-Euler beam theory is adopted to describe the dynamic behaviour of an elastic pipe and a new governing equation of a long pipe transporting gas or liquid is derived. This governing equation incorporates the effects of inertia, centrifugal and Coriolis forces due to the flowing medium. This equation can be normalised to demonstrate that only two non-dimensional parameters govern the static and dynamic response of the system incorporating a pipe and flowing medium. The transient response of this system is investigated based on a standard perturbation approach. Further, a numerical method utilising the finite difference method is developed and applied to investigate the dynamic response of a simply supported pipe. It is demonstrated that the previous dynamic models which largely ignore the internal flow effects and interactions between the flow and structure normally produce a large error and are inapplicable to the analysis of many practical situations. One interesting effect identified in the numerical study is that at certain flow ratio the system becomes dynamically unstable and any, even very small, external perturbation leads to a growing unstable dynamic behaviour. Such behaviour, which is called pipe whip, is well known to everyone who waters a garden using a flexible long hose.

On advantages of derivation of exact dynamical stiffness matrices from boundary integral equations

Dynamical stiffness matrices are broadly used for solving problems in wave propagation in elastic structures. For elastic waveguides that support both propagating and evanescent waves, the standard use of transfer matrices to obtain their dynamical stiffness matrix may cause problems in computations. In this letter to the editor, an advanced way to derive dynamical stiffness matrices is proposed and exemplified in the classical Bernoulli-Euler beam theory.

Active buckling control of beams using piezoelectric actuators and strain gaugesensors

In this paper, a finite element model incorporating active control techniques has beendeveloped to stabilize the first two buckling modes of both a simply supported and acantilevered beam. The goal is to increase the corresponding beam buckling loads by usingpiezoelectric actuators along with optimal feedback control.The uniform beams are bonded with two pairs of segmented piezoelectric actuators at thetop and bottom. Resistive strain gauges are attached to the centres of the actuators assensors. Measurements are taken using these, to estimate the system states. The beamsare simply supported or cantilevered and subjected to a slowly increasing axialcompressive load. Finite element formulations based on the classical Euler–Bernoullibeam theory and linear piezoelectric constitutive equations for the actuators arepresented. The associated reduced-order modal equations and the state-spaceequations are derived for the design of a standard linear quadratic regulator (LQR).The finite element analysis and the active control simulation results are consistent withboth theoretical analysis results and experimental data. The designed full state feedbackLQR controller is shown to be successful in stabilizing the first two buckling modes ofthe beams. Also the control simulation shows that the present optimally locatedsegmented actuator pairs along the beam are more effective for buckling control.

Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation

Abstrac t Carbon nanotubes (CNTs) are used for a variety of technological and biomedical applications. Fluid flows inside CNTs have become an attractive research topic in recent years. In the present study, the vibration and instability analysis of a CNT resting on a linear viscoelastic Winkler foundation are investigated based on the classical Euler–Bernoulli beam model. The effect of internal moving fluid is characterized by two parameters, the steady flow velocity and the mass density of the fluid. The finite element method is used to discretize the equation of motion, and the resonant frequencies are obtained by solving a quadratic eigenvalue problem. The lowest four frequencies are determined for different boundary conditions. The effects of the modulus and the damping factor of the linear viscoelastic Winkler foundation and the fluid velocity on the resonance frequencies are also examined. The validity of the present analysis is confirmed by comparing the results with those obtained from the literature.

A bending theory for beams with vertical edge crack

In this paper a linear continuous theory for bending analysis of beams with an edge crack perpendicular to the neutral plane subject to bending has been developed. The model assumes that the displacement field is a superposition of the classical Euler–Bernoulli beam's displacement and of a displacement due to the crack. It is assumed that in bending the additional displacement due to crack decreases exponentially with distance from the crack tip. The strain and stress fields have been calculated using this displacement field and the bending equation has been obtained using equilibrium equations. Using a fracture mechanics approach the exponential decay rate has been calculated. There is a good agreement between the analytical results from solving the differential equation of cracked beam and those obtained by finite element method.

Analytical modelling of the transient dynamics of pipes with flowing medium

The primary purpose of this work is to present an innovative approach to investigate the effect of a flowing medium on the transient response of a simply supported pipe due to impulsively applied loading. The importance of this problem is manifested in numerous applications in the field of transportation and power energy. In this paper we describe a simplified mathematical model of the pipe-filled-medium interactions adopting the classical Bernoulli-Euler beam theory. Further we develop an analytical procedure based on the perturbation method to analyse the dynamic response of the system due to an impulse loading and consider particular cases of the applied loading.

Simple Mechanical Model of Curved Beams by a 3D Approach

Starting from three-dimensional 3D continuum mechanics, a simple one-dimensional model aimed at analyzing the whole static behavior of nonhomogeneous curved beams is proposed. The kinematics is described by four one-dimensional unknown functions representing radial, tangential, and out-of-plane displacements of the beam axis, which are due to flexures and extension, and the twist of the cross section due to torsion. The flexural and axial displacements fit with the classical Euler-Bernoulli beam theory of straight beams, and nonuniform torsion is also considered. The relevant elastogeometric parameters have been determined, and the system of governing equilibrium equations is obtained by means of the principle of minimum potential energy. Finally, the general theory is illustrated with examples.

Hybrid wave/mode active control of bending vibrations in beams based on the advanced Timoshenko theory

A hybrid approach to active control of bending vibrations in beams based on the advanced Timoshenko theory is described in this paper. It combines elements of both wave and mode approaches to active control and is an attempt to improve on the performance of these approaches individually. As is well known that the classical Euler–Bernoulli beam model considers only the lateral inertia and the elastic forces caused by bending deflections, and the effects of rotary inertia and shear distortion are neglected. As a result, the theory is not valid for higher frequencies, typically when the transverse dimensions are not negligible with respect to the wavelength. In the proposed hybrid approach based on the advanced Timoshenko model, wave control is first applied at one or more points in the structure. It is designed on the basis of the local behavior of the structure and is intended to either absorb vibrational energy or add damping, especially at higher frequencies. Then modal control is applied, being designed on the basis of the modified global equations of motion of the structure-plus-wave-controller. Because the higher order modes are relatively well damped, hybrid control improves the model accuracy and the robustness of the system and gives better broadband vibration attenuation performance. Numerical results are presented.