Classical Euler Bernoulli Beam Theory Research Articles

The vibration of nanobeam resting on elastic foundation using modified couple stress theory

In this paper, the vibration of nanobeams resting on the Winkler foundation is proposed using the modified couple stress theory. Hamilton’s principle is utilized to construct the governing equations. The size effect of the nanobeam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can capture it because it includes material length scale parameter that a newly developed model has. Once the material length scale parameter is assumed to be zero, the classical Euler-Bernoulli beam theory equation is obtained. Multiple scale method is employed to obtain the result. Simply supported boundary condition is used to study natural frequencies. The influence of material length scale parameter and the Winkler elastic foundation parameter on the fundamental frequencies of the nanobeam is investigated and tabulated. Also, in the present study, Poisson’s ratio is taken as constant. Nanobeam resting on the Winkler foundation which is simply supported is analyzed to illustrate the size effects on the free vibration. Numerical results for the simply supported nanobeam indicate that the first fundamental frequency calculated by the presented model is higher than the classical one. Moreover, it is obtained that the size influence is more substantial for higher vibration modes. The results indicate that the significant importance of the size influences the analysis of nanobeams. The vibration of nanobeam exhibits a hardening spring behavior, and the newly developed models are the beams stiffer than according to the classical beam theory. Modified couple stress theory tends to be more helpful in describing the size-dependent mechanical properties of nanoelectromechanical systems (NEMS).

Edgewise Bending Vibration Analysis of a Rotating Sandwich Beam with Magnetorheological Elastomer Core

The vibration of a rotating sandwich beam with magnetorheological elastomer (MRE) as a core between two elastic layers is theoretically analyzed in this paper. This study is focused on the bending vibration along the edgewise direction of a sandwich beam of rectangular cross-section, which, to the best of our knowledge, has not been addressed yet. The classical Euler–Bernoulli beam theory is used to model the dynamic behavior of the elastic layers. In the modeling, the effect of the MRE layer is considered by incorporating its shear strains and the inertia due to shear deformation and bending motion. The governing equations of motion of the rotating sandwich beam are derived by using the Ritz method and the Lagrange’s equations. The effects of the applied magnetic field, core layer thickness, rotational speed, setting angle and hub radius on the natural frequencies and the corresponding loss factors are investigated parametrically. The results show the significant effect of the magnetic field intensity and the MRE layer thickness on the modal characteristics of the MRE sandwich beam.

A Variation Analysis Method for Linkage Mechanism with Consideration of Joint Clearance and Deformation

As typical transmission mechanisms, linkage mechanisms are widely applied in mechanical engineering field, and accuracy prediction of them has received much more attentions especially from some high-precision application fields. For a certain linkage mechanism system, external forces will also cause linkage deformations and deterministic oriented-drift of joint clearances, which become additional variation sources besides component tolerances. In order to perform a more accurate prediction of linkage mechanism, a variation analysis method is proposed in this study, in which component tolerances, joint clearances and deformations are considered. With an equivalent method, positional tolerance and joint clearances are taken into the present variation analysis model, and serve as circular tolerances. Based on classical Euler-Bernoulli beam theory, linkage deformations are disposed as equivalent deviations and contribute to assembly deviations. A decomposing method of percentage contribution, which contains different variation sources, is presented as well. A case study of four-bar linkage mechanism is illustrated to validate the accuracy of the present method by corresponding FEA simulation and experimental test. Moreover, a case study of a three-loop mechanism is also analyzed for the accuracy and percentage contribution of different variation sources with the present method.

Flutter of multi-cracked laminated composite beams subjected to a non-conservative compressive load

Free vibration and buckling instability of laminated composite beams and flutter instability of an isotropic beam with different defects have been investigated before, however, still dynamic stability of laminated composite beam with weak section subjected to follower forces has not considered. This paper presents a simple semi-analytical solution to the flutter instability of a laminated composite beam with multi open-edge cracks subjected to a tip follower force. Using classical lamination theory and Euler-Bernoulli beam theory, equation of motion of the beam is derived analytically and characteristic eigenvalue equations of the cracked laminated composite beam are obtained. A Matlab® code is used to solve these equations numerically. After validating the results, a numerical study is performed to investigate the effect of crack depth and stacking sequences on the dynamic stability of the cracked beam.

Geometrically nonlinear beam analysis of composite wind turbine blades based on quadrature element method

A new nonlinear beam theory is proposed for the analysis of composite wind turbine blades. The beam theory is developed by extending classical Euler–Bernoulli beam theory to a generalized Timoshenko beam. Mechanics-based variables are used to describe finite rotation such that the problems of the sequence dependence or spatially discontinuity of rotational variables can be avoided. Furthermore, nonlinear beam theory is implemented using the weak-form quadrature element method. Numerical examples of both non-rotating and rotating beams are given and the comparison with analytical and finite element results shows high computational accuracy and efficiency of the proposed nonlinear quadrature element. A simple parametric study of a virtual 5-MW wind turbine blade shows that bend-twist coupling due to both material anisotropy and geometrical nonlinearity affects the dynamic performance of the blade significantly.

Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams

This paper is pertinent to the analytical solutions for vibration analysis of initially stressed Nonlocal Euler-Bernoulli nano-beams. In order to take into account of small length scale effect, this vibration problem formulation is depending upon both nonlocal Euler-Bernoulli and also Eringen’s nonlocal elasticity theory. The boundary conditions and governing equation are obtained by use of Hamiltonian’s principle. These equations are solved analytically with different initial stresses (both compressive and tensile) and boundary conditions. The effect of small length scale and the initial stress on the fundamental frequency are investigated. The solutions obtained are compared with the ones depending upon both classical Euler-Bernoulli and Timoshenko beam theory to comprehend the responses of nano-beams under the effect of initial stress and small scale in terms of frequencies for both theories. The results supply a better declaration for vibration analysis of nano-beams which are short and stubby with initial stress.

Comparison of small scale effect theories for buckling analysis of nanobeams

Theories which consider small scale effect have a great importance on analysis in micro and nano scale. In present paper, three kind of nanotubes (Carbon Nanotube (CNT), Boron Nitride Nanotube (BNNT), and Silicon Carbide Nanotube (SiCNT)) are analyzed in case of buckling on two parameters elastic foundation. Three different small scale theories (Nonlocal Elasticity Theory (NET), Surface Elasticity Theory (SET), and Nonlocal Surface Elasticity Theory (NET&SET)) are applied to calculate the buckling loads. Also Classical Euler-Bernoulli Beam Theory (CT) is used to see the effect of small scale effective theories. Comparative results are given for simply supported nanotubes in figures.

Nonlinear analysis of reinforced concrete plane frames exposed to fire using direct stiffness method

This article presents a framework based on the direct stiffness method for nonlinear thermo-mechanical analysis of reinforced concrete plane frames subjected to fire. It accounts for geometric nonlinearity, material nonlinearity, and nonlinear thermal gradients and incorporates two-way coupling between thermal and structural analyses. Force deformation relations are derived from classical Euler–Bernoulli beam theory and are expressed in terms of temperature-dependent stability and bowing functions. This is one of the unique features of proposed framework and allows a coarser spatial discretization to be used as opposed to full finite element–based approaches (such as SAFIR [registered trademark of the software SAFIR developed at the University of Liege]). The cross sections of the structural members are discretized with two-dimensional meshes for thermal analysis while structural analysis utilizes a line element based on direct stiffness method. Equivalent bending and axial rigidities of this line element are computed using several fibers along the length of the member, passing through the nodes of the two-dimensional mesh used for thermal analysis. The total strain at each fiber is decomposed into mechanical, thermal, creep, and transient thermal components. A discrete damage parameter is introduced at fiber level to ensure irreversibility of crushing and cracking in accordance with relevant constitutive laws. Five numerical examples are presented to demonstrate the accuracy and efficacy of the developed framework with respect to theoretical solutions, experimental observations, and some of the existing macro- and micro-finite element–based approaches. It is found that the developed framework can predict the response of reinforced concrete structures very well.

Effect of temperature-dependent physical properties on nanobeam structures induced by ramp-type heating

The classical Euler-Bernoulli beam theory based upon the nonlocal theory of thermoelasticity is constructed and employed to investigate the vibrating nanobeam due to a ramp-type heating. The thermal conductivity and modulus of elasticity are taken as temperature-dependent functions. The generalized theory of thermoelasticity with Dual-Phase-Lags (DPLs) model is also employed to solve the problem. Analytical and numerical techniques based on Laplace transforms and their inversions are used to calculate the vibrational deflection and temperature of the nanobeam. The effects of the PLs and ramping-time parameters on the lateral vibration, the temperature, the displacement and the flexure moment of the nanobeam are investigated. Some plots have been presented for comparison purposes to estimate the effects of the nonlocal parameter on the field quantities. For the sake of completeness a comparison is also made between the present results and those obtained in the case of temperature-independent thermal and mechanical properties.

Investigation of adding fins to external stores for improving the flutter characteristics of a wing/store configuration

In this paper, the effect of aerodynamic damping produced by store rear horizontal fins on the aeroelastic instability of a wing with a large underwing store is investigated in an incompressible flow. The wing is modeled by using the classical Euler–Bernoulli beam theory while the unsteady strip aerodynamic theory based on the Wagner function is used to simulate the aerodynamic field. The underwing store is modeled as a concentrated mass attached to the wing by a rigid pylon and has rear horizontal fins. The numerical results of the developed generic and simple model are compared with available results, and an excellent agreement is observed. It is found that the aerodynamic damping of the store produced by its rear fin can compensate the flutter boundary of the wing-store configuration and therefore for more accurate analysis, in addition to the store weight and location, the effect of store fin must also be considered.

Stability and initial post-buckling of a standing sandwich beam under terminal force and self-weight

Beams (columns) subjected to axially distributed load (e.g., self-weight) are commonly treated using the classical Euler–Bernoulli beam theory, which ignores the transverse shear effect. Adopting the Engesser and Haringx shear theories, respectively, we study in this article the stability and initial post-buckling of sandwich (or laminated composite) beams under terminal force and axially distributed load. Nonlinear governing equations are derived from geometrical compatibility, equilibrium of forces, and moments. The critical buckling load, modal shapes of deformation, and shear force together with bending moment at buckling can be obtained by using the Galerkin’s method in terms of trigonometric functions, and the initial post-buckled configuration of the beam is determined employing the shooting method. Predictions based on the Engesser theory agree with finite element simulation results, while the Haringx theory overestimates the buckling load. The effects of transverse shear and various different end constraint conditions on static buckling and initial post-buckling of sandwich beams are systematically explored.

Mathematical model for the bending of plastically anisotropic beams

A mathematical model for the bending of a plastically anisotropic beam simply supported at both ends and subjected to a constant moment is considered. A differential equation with variable coefficients is derived for the beam curvature. The yield points of the beam material under tension and compression are assumed to be known. The elastoplastic bending of the beam with allowance for the strength-different (SD) effect is considered. The classical Bernoulli–Euler beam theory and the ideal plasticity model are used construct the mathematical model, and the problem is solved analytically. The solutions obtained for a classical isotropic beam and an SD beam are compared, and the contribution of the SD effect is analyzed. The problem is solved completely, and its results can be used to study bending under different loading.

THEORETICAL OF DYNAMIC LOADING AGAINST WATER-FILLED SIMPLY SUPPORTED PIPES

The primary aim of this study had been to investigate the effects of water-filled flow 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 transportations, where dynamic loading can be the result of an accident. The classical Bernoulli-Euler beam theory was adopted to describe the dynamic behavior of an elastic pipe and a new governing equation of a long pipe transporting gas or liquid was derived. This governing equation incorporated the effects of inertia, centrifugal, and Coriolis forces due to the flowing water. This equation can be normalized to demonstrate that only two non-dimensional parameters governed the static and the dynamic responses of the system incorporating a pipe and flowing water. The transient response of this system was investigated based on a standard perturbation approach. Moreover, it had been demonstrated that the previous dynamic models, which largely ignored the internal flow effects and interactions between the flow and the structure, normally produced a large error and are inapplicable to the analysis of many practical situations. One interesting effect identified was that at certain flow ratio, the system became dynamically unstable and any, even very small, external perturbation led to a growing unstable dynamic behavior. Such behavior, which is called pipe whip, is well-known to everyone who waters a garden using a flexible long hose.

Fractional Euler–Bernoulli beams: Theory, numerical study and experimental validation

In this paper the classical Euler-Bernoulli beam (CEBB) theory is reformulated utilising fractional calculus. Such generalisation is called fractional Euler-Bernoulli beams (FEBB) and results in non-local spatial description. The parameters of the model are identified based on AFM experiments concerning bending rigidities of micro-beams made of the polymer SU-8. In experiments both force as well as deflection data were recorded revealing significant size effect with respect to outer dimensions of the specimens. Special attention is also focused on the proper numerical solution of obtained fractional differential equation.

Two-noded zigzag beam element accounting for shear effects based on an extended Euler Bernoulli theory

We present a new 2-noded beam element based on the refined zigzag theory and the classical Euler–Bernoulli beam theory for the static analysis of composite laminate and sandwich beams. The proposed element is able to take into account distortion effects due to shear elastic strains and can predict delamination. The element has four degrees of freedom per node. A C1 cubic Hermite interpolation is used for the vertical deflection while a C0 linear interpolation is employed for the other kinematics variables. The stiffness matrix and the load vector are calculated in explicit form using exact integration. The element is free from shear locking as confirmed with numerical tests on a wide range of the slenderness ratios. Numerical results show the ability of the EEBZ2 element to reproduce accurately the vertical deflection along the beam length and complex zigzag distributions of the axial displacement and the stresses across the thickness. Delamination effects are modeled by incorporating of an additional zigzag function corresponding to the kinematics of a zero thickness layer where delamination occurs. An example showing the capability of the new EEBZ2 element for accurately reproducing delamination effects is presented.

Postbuckling of Unknown-Length Nanobeam Considering the Effects of Nonlocal Elasticity and Surface Stress

The objective of this paper is to study the postbuckling behaviors of an unknown-length nanobeam combined with small-scale effects. The concept of variable-arc-length elastica is firstly applied on the problem of nanobeams. The span length is not changed while the arc length is varied increasingly. The nanobeam is on a clamped support at one end, while the other end is an overhanging part through a frictionless slot subjected to axial compression. At this end, the nanobeam is movable only in a horizontal direction. The governing equation is developed by the moment–curvature relationship based on the classical Euler–Bernoulli beam theory, including the effects of nonlocal elasticity, residual surface stress, and both combined effects. The shooting–optimization technique with two-point boundary condition is employed to solve the differential equations in this problem. The results, including nonlocal elasticity, reveal that nanobeams have decreased structural stiffness; meanwhile, the residual surface tension and both combined effects have increased strength. The postbuckling loads decrease as the arc length of nanobeams is increased. The equilibrium configurations are close to an anti-loop for very large deflections. The friction force at the nanoslot is also considered.

On the Thermomechanical Response of HTPB-Based Composite Beams Under Near-Resonant Excitation

Currently, there is a pressing need to detect and identify explosive materials in both military and civilian settings. While these energetic materials vary widely in both form and composition, many traditional explosives consist of a polymeric binder material with embedded energetic crystals. Interestingly, many polymers exhibit considerable self-heating when subjected to harmonic loading, and the vapor pressures of many explosives exhibit a strong dependence on temperature. In light of these facts, thermomechanics represent an intriguing pathway for the stand-off detection of explosives, as the thermal signatures attributable to motion-induced heating may allow target energetic materials to be distinguished from their more innocuous counterparts. In the present work, the thermomechanical response of a sample from this class of materials is studied in depth. Despite the nature of the material as a polymer-based particulate composite, classical Euler–Bernoulli beam theory, along with the complex modulus representation for linear viscoelastic materials, was observed to yield predictions of the thermal and mechanical responses in agreement with experimental investigations. The results of the experiments conducted using a hydroxyl-terminated polybutadiene (HTPB) beam with embedded ammonium chloride (NH4Cl) crystals are presented. Multiple excitation levels are employed and the results are subsequently compared to the work's analytical findings.

An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory

In the present investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) subjected to thermal environment including the effect of surface stress. The material properties of functionally graded (FG) nanobeams vary through the thickness direction on the basis of a simple power law. The geometrically nonlinear beam model, taking into account the surface stress effect, is developed by implementing the Gurtin–Murdoch elasticity theory together with the classical Euler–Bernoulli beam theory and using a variational approach. Hamilton’s principle is utilized to obtain the nonlinear governing partial differential equation and corresponding boundary conditions. After that, the Galerkin technique is employed in order to convert the nonlinear partial differential equation into a set of nonlinear ordinary differential equations. This new set is then solved analytically based on the method of multiple scales which results in the frequency–response curves of FG nanobeams in the presence of surface stress effect. It is revealed that by increasing the beam thickness, the surface stress effect diminishes and the maximum amplitude of the stable response is shifted to the higher excitation frequencies.

Size-Dependent Resonant Frequency and Flexural Sensitivity of Atomic Force Microscope Microcantilevers Based on the Modified Strain Gradient Theory

In the present study, the resonant frequency and flexural sensitivity of atomic force microscope (AFM) microcantilevers are predicted incorporating size effects. To this end, the modified strain gradient elasticity theory is applied to the classical Euler-Bernoulli beam theory to develop a non-classical beam model which has the capability to capture size-dependent behavior of microcantilevers. On the basis of Hamilton's principle, the size-dependent analytical expressions corresponding to the frequency response and sensitivity of AFM cantilevers are derived. It is observed that by increasing the contact stiffness, the resonant frequencies of AFM cantilevers firstly increase and then tend to remain constant at an especial value. Moreover, the resonant frequencies of AFM cantilevers obtained via the developed non-classical model is higher than those of the classical beam theory, especially for the values of beam thickness close to the internal material length scale parameter.

Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations

The aim of the present work is to analyze free flexural vibration and buckling of single-walled carbon nanotubes (SWCNT) under compressive axial loading based on different constitutive equations and beam theories. The models contain a material length scale parameter that can capture the size effect, unlike the classical Euler-Bernoulli or Reddy beam theory. The equations of motion of the Reddy and the Huu-Tai beam theories are reformulated using different gradient elasticity theories, including stress, strain and combined strain/inertia. The equations of motion are derived from Hamilton's principle in terms of the generalized displacements. Analytical solutions of free vibration and buckling are presented to bring out the effect of the nonlocal behavior on natural frequencies and buckling loads. The presented theoretical analysis is illustrated by a numerical example, and the results are qualitatively compared by another results.