Classical Euler Bernoulli Beam Theory Research Articles

Vibration analysis of Euler-Bernoulli beams in multiple steps and different shapes of cross section

Stepped beams with elastic end supports have been extensively investigated due to their importance in structural engineering fields, including active structures, structural elements with integrated piezoelectric materials, shaft-disc system components, turbomachinery blades and many other structural configurations. Considering the importance of the use of discontinuous structures in the engineering, the authors propose a mathematic modeling which advantage is that the results are independent of the degree of mesh refinement. The analysis is based on the classical Euler-Bernoulli beam theory. In comparison with the published literature on the transverse vibration of single cross section change beams, there are relatively few works covering beam vibration when there is more than one change in the beam cross section. In the present study, the natural frequencies and the mode shapes of beams with variable geometry or material discontinuities are investigated. The mode shapes of a beam with multiple step changes in cross section are discussed theoretically and experimentally. Numerical results obtained by Euler-Bernoulli beam theory are compared with experimental results.

The Application of Mesh-Free Method in the Numerical Simulation of Beams with the Size Effect

The mesh-free method is employed to implement the numerical simulation of the bending behavior of beams with the size effect. On the basis of the classical Bernoulli-Euler beam theory, two higher-order strain components are involved in the beam model. The intrinsic bulk length and the directional surface length components are introduced into the constitutive relationship to describe the size effect, and the variation of the total potential is provided. The moving-least square approximation is used to construct the shape function and its second- and third-order derivatives, and the choice of the scaling factor is discussed in detail. A mesh-free scheme is built to implement numerical simulation, in which the higher-order strains are directly approximated with the nodal components due to the higher-order continuity of the shape function. The convergence of method is illustrated in virtue of an example of the simply supported beam, and the effect of the intrinsic bulk length and the directional surface length components are studied.

Effect of higher-order deformation gradients on buckling of single-walled carbon nanotubes

A single-walled carbon nanotube (SWCNT) is treated as a tubular Bernoulli–Euler beam and the higher-order deformation gradients are involved to establish a higher-order multiscale beam model. The higher-order Cauchy–Born rule is used to calculate the deformation of bond vectors in the representative cell and the strain energy density is equivalent to the energy per unit surface area, calculated from the Brenner potential. On the basis of the classical Bernoulli–Euler beam theory, the second-order deformation gradients with respect to the axial direction are also considered. The total strain energy is expressed as an integral equation, in which all parameters are obtained by calculating the constitutive response around the circumference. The physical meaning of these parameters is discussed in detail. The global buckling of SWCNTs is studied, and the critical buckling force is obtained as the analytical formula for different boundary conditions. The critical buckling force is plotted against the tube chirality, tube radius and tube length, and it is discovered that the contribution of the higher-order terms rapidly becomes large when the tube radius or length is small enough.

Flexural buckling of simply-supported thin-walled columns with consideration of membrane shear deformations: Analytical solutions based on shell model

In this paper, flexural buckling of pin-ended thin-walled columns is discussed. The classical formulae for the critical force are based on a beam model. The simplest formulae use the classical Euler–Bernoulli beam theory, but solutions based on the shear-deformable beam theory are also known. In the presented research alternative formulae are derived. The column is modeled as a set of flat plane elements, and the in-plane membrane shear deformations are explicitly considered. The derivations can be carried out in various, slightly different ways, leading to different formulae. The derived critical force formulae are discussed through theoretical considerations and numerical studies.

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.

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.

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.

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.

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.

Dynamic response of microcantilever-based sensors in a fluidic chamber

The dynamic response of a microcantilever beam used as a transducer in a biomechanical sensor for molecule detection is analyzed. The cantilever motion is modeled using the classical Euler-Bernoulli beam theory coupled to the Reynolds equation of lubrication accounting for the hydrodynamic interactions. The beam is oscillated by an electrostatic harmonic force in close proximity to a rigid wall in an analysis chamber. It is shown that the dynamic response of the beam can be accurately predicted by a simple equivalent damped harmonic oscillator whose quality factor has been estimated as a function of the beam-wall gap. As the gap is reduced the quality factor is reduced and damping becomes more and more important. A transduction relationship between the frequency shift and the mass of molecules adherent on the beam is proposed which accounts for the beam-fluid-substrate interactions.

A linear model for the static and dynamic analysis of non-homogeneous curved beams

A simple one-dimensional mechanical model is presented to analyse the static and dynamic feature of non-homogeneous curved beams and closed rings. Each cross-section is assumed to be symmetrical and the “resultant loads” are acted in the plane of symmetry. The internal forces in a cross-section are replaced by an equivalent force–couple system at the origin of the cylindrical coordinate system used. The equations of motion and the boundary conditions are expressed in terms of two kinematical variables. The first kinematical variable is the radial displacement of cross-sections and the second one is the rotation of the cross-sections. Each of them depends on the time and the polar angle. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli–Euler beam theory. Rotary inertia is included in the equations of motion. Natural frequencies for simply supported laminated composite curved beams and non-homogeneous circular rings are obtained by exact solutions. The application of the model presented is illustrated by examples.