Classical Euler Bernoulli Beam Research Articles

Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams

A simplified second strain gradient Euler–Bernoulli beam theory with two non-classical elastic coefficients in addition to the classical constants is presented. The governing equation and the associated classical and non-classical boundary conditions are derived with the aid of variational principles. The simplified second strain gradient theory is governed by an eighth-order differential equation with displacement, slope, curvature and triple derivative of displacement as degrees of freedom. This theory can be reduced to the first strain gradient and classical Euler–Bernoulli beam theories. Analytical solutions for static behaviour, free vibration and stability analyses are presented for different boundary conditions and length scale parameters. Using the numerical Laplace transform, a spectral element is developed for dynamic analysis of a cantilever beam subjected to a Gaussian pulse. Further, spectrum and dispersion relations are derived to study wave propagation characteristics. The gradient effects on the structural response are assessed and compared with the corresponding first strain gradient and classical beam theories. Observations show that the second strain gradient theory exhibiting stiffer behaviour in comparison to the first strain gradient and classical theories. The beam deflection decreases whereas frequencies and buckling load increase for increasing values of the gradient coefficient in comparison to the first strain gradient and classical theories. The forced response for a finite beam reveals a decrease in the amplitude and a shift to smaller time values with an increase in the value of length scale parameter. Additionally, the second strain gradient beam shows a dispersive behaviour, and for a given frequency the wavenumber decreases and the phase speed increases with an increase in the length scale parameter as compared to the first strain gradient beam theory.

Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags

The aim of this article is to investigate the thermal and mechanical vibration properties of functionally graded microbeams. The governing system of equations is formulated on the basis of classical Euler–Bernoulli beam model incorporating the generalized dual-phase lag model of thermoelasticity rather than the conventional steady-state Fourier heat conduction. In our analysis, it is also assumed that the mechanical and thermal properties such as modulus of elasticity, density and coefficient of thermal conductivity change through the thickness by an exponential law distribution, with the exception of Poisson’s ratio. The effects of system parameters on different field quantities have been depicted for initial stress, material gradient index and pulses duration . The numerical results are compared with benchmark findings, for the sake of verification. Presented results demonstrate the considerable effects of temperature and the material gradients on the dynamic behavior of microscopic structures. Finally, a comparison is conducted with the results discussed in the literature to justify the quality of the present technique.

Electricity-structure-fluid coupled modelling and experiment of underwater flexible structure with partially distributed macro fiber composites

Dynamic oscillating behavior of the flexible structure immerged in viscous fluids has attracted growing attention and been widely used in various practical applications. A general electricity-structure-fluid coupled model for the forced dynamic responses of a cantilever immersed in fluids, with partially distributed macro fiber composite, is proposed in this paper. Based on the classical Euler–Bernoulli beam theory, the first mass-normalized mode shape of the cantilever with partially bonded macro fiber composite is determined using assumed mode method. The attachment of the macro fiber composite actuators stiffens the macro fiber composite-bonded portion of the cantilever. The established mode shape matches perfectly with experimental results. Considering the macro fiber composite actuator as a set of representative elements connected in parallel, the internally actuation moment provided by the macro fiber composite actuators is obtained. The hydrodynamic load caused by the surrounding fluids, decomposed into the added mass and hydrodynamic damping parts, is also added to the theoretical model in the frequency-domain form. The predicted in-air and underwater dynamic behaviors of the flexible beam are consistent with the experimental results at different auction levels. Thus, the obtained general electricity-structure-fluid coupled model can be used to predict the forced dynamic responses of flexible structure with partially bonded actuators immersed in fluids.

Local gradient Bernoulli–Euler beam model for dielectrics: effect of local mass displacement on coupled fields

The size-dependent behaviour of a Bernoulli–Euler nanobeam based on the local gradient theory of dielectrics is investigated. By using the variational principle, the linear stationary governing equations of the local gradient beam model and corresponding boundary conditions are derived. In this set of equations the coupling between the strain, the electric field and the local mass displacement is taken into account. Within the presented theory, the process of local mass displacement is associated with the non-diffusive and non-convective mass flux related to the changes in the material microstructure. The solution to the static problem of an elastic cantilever piezoelectric beam subjected to end-point loading is used to investigate the effect of the local mass displacement on the coupled electromechanical fields. The obtained solution is compared to the corresponding ones provided by the classical theory and strain gradient theory. It is shown that the beam deflection predicted by the local gradient theory is smaller than that by the classical Bernoulli–Euler beam theory when the beam thickness is comparable to the material length-scale parameter. The obtained results also indicate that the piezoelectricity has a significant influence on the electromechanical response in a dielectric nanobeam. The presented mathematical model of the dielectric beam may be useful for the study of electromechanical coupling in small-scale piezoelectric structures.

Exact solutions of bending deflection for single-walled BNNTs based on the classical Euler–Bernoulli beam theory

Abstract At the nanolevel, a classical continuum approach seems to be inapplicable to evaluate the mechanical behaviors of materials. With the introduction of scale parameter, the scale effect can be reasonably described by the modified continuum theory. For boron nitride nanotubes (BNNTs), the scale effect can be reflected by the curvature and the dangling bonds at both ends, mainly the former for a slender tube. This study aims to achieve a good capability of classical Euler–Bernoulli theory to directly predict the bending behaviors of single-walled BNNTs without introducing scale parameters. Elastic properties of BNNTs involving the scale effect have been first conducted by using an atomistic-continuum multiscale approach, which is directly constructed based on the atomic force field. The well-determined hexagonal boron nitride sheet is inherited in the present study of single-walled BNNTs which can be viewed as rolling up a boron nitride sheet into a seamless hollow cylinder. Euler–Bernoulli theory solution of bending deflection on the basis of the present thickness is found to be much closer to the atomistic-continuum simulation results than the commonly used interlayer space. Case studies with different tubular lengths, radii and constraints are investigated, and from which the yielded scattered scale parameters in modified continuum theories are discussed.

Medium-Frequency Vibration Analysis of Timoshenko Beam Structures

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

A theoretical model for analysis of ionic polymer metal composite sensors in fluid environments

By the past two decades IPMCs have been intensively studied because of their special capabilities for actuation and sensing.This paper presents a theoretical physics based model for analyzing the behavior of IPMC sensors in fluid environments. The mechanical vibration of the IPMC strip is described by the classical Euler–Bernoulli beam theory. The model also takes in to account the physical properties of the surrounding fluid. The resulting model is an infinite-dimensional transfer function that relates the input tip displacement to the output sensing current. Further the original model is reduced to a finite-dimensional one, for pure sensing applications of IPMC sensors such as structural health monitoring. The proposed model is verified using existing experimental data. Then the effect of various parameters is investigated. The acoustics physics interface in COMSOL Multiphysics software is used for coupled modal analysis of the IPMC strip. It is shown that the effect of surrounding fluid cannot be neglected.

Modified Timoshenko beam model for bending behaviors of layered materials and structures

Layered materials and structures (LMS), such as van der Waals two-dimensional (2D) layered materials and nacre-like layered structures, often exhibit highly anisotropic mechanical properties, i.e., strong in in-plane directions but weak in out-of-plane direction. Despite the strong anisotropy in their mechanical properties, Timoshenko beam model (TBM) is usually used to describe the bending deformation of LMS. We note, however, that there are two fundamental issues in using TBM to describe LMS: First, the stiffness of LMS approaches zero when the interlayer shear modulus G approaches zero; and second, the first derivative of deflection becomes discontinuous at the point of concentrated force. Clearly, both are not true for LMS. In this work, by introducing the bending energy of monolayer into the potential energy of TBM, we develop a modified Timoshenko beam model (MTBM), which is able to not only address these two issues, but also correctly predict the bending stiffness of LMS without any fitting parameters. Our analysis shows that the bending behaviors of LMS are determined by a dimensionless parameter λL, where L is the length of the beam and λ=kGA∕D0+kGA∕(nDbend), where, kGA and D0 are, respectively, the shear and bending rigidity of the beam cross-section, Dbend is the bending rigidity of monolayer, and n is the number of layer. When λL→0, the MTBM degenerates to the multi-beam model with bending stiffness of nDbend; while it degenerates to the TBM whenλL→∞. Furthermore, if kGA is much larger than D0, both MTBM and TBM degenerate to the classical Euler–Bernoulli beam model. We further perform molecular dynamics simulations, finite element simulations and experiments to validate the MTBM. Based on the MTBM, a couple of interesting applications of LMS are also demonstrated. Hence, the MTBM presented here captures the necessary intrinsic deformation modes of LMS and provides an accurate tool for the prediction and optimization of the mechanical properties of LMS.

A DSC Regularized Dirac-Delta Method for Flexural Vibration of Elastically Supported FG Beams Subjected to a Moving Load

This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.

The propagation of bending waves in a beam, the material of which accumulates damage during its operation

The linear and nonlinear formulations of a self-consistent material model are developed based on the equation of the bending vibrations of the beam and the kinetic equation of damage accumulation in its material. The beam is considered endless. Such idealization is permissible if optimal damping devices are located at its boundaries, that is the parameters of boundary fixing are such that disturbances acting on it will not be reflected. This allows us to exclude boundary conditions from the beam model and consider the vibrations propagating along the beam as traveling bending waves. As a result of analytical studies and numerical modeling, it was found that the damage of the material involves a frequency-dependent attenuation and significantly changes the nature of the dispersion of the phase velocity of a bending elastic wave. Note that in a classical Euler-Bernoulli beam there is one dispersion branch for bending waves at any frequency value, whereas for a beam with accumulated material damage in the entire frequency range there are two dispersion branches, characterizing wave propagation, and two dispersion branches, characterizing its attenuation. The problem of the formation of intense bending waves of a stationary profile is considered in the framework of a geometrically nonlinear model of a damaged beam.. It is shown that such essentially non-sinusoidal waves can be either periodic or solitary (localized in space).The dependencies relating the parameters of the waves (amplitude, width, wavelength) with the damage to the material are determined. It is shown that the amplitude of the periodic wave and the amplitude of the solitary wave increase with increasing material damage parameter, while the length of the periodic wave and the width of the solitary wave decrease with increase of this parameter.

Improved Study on the Fluctuation Velocity of High-Speed Railway Catenary Considering the Influence of Accessory Parts

To improve the prediction accuracy of the wave propagation velocity of high-speed railway catenary, a method for identifying the fluctuation velocity of catenary (FVC) considering the influence of the accessory parts of catenary (APC) is proposed. Based on the classical Bernoulli-Euler beam vibration model, the single Euler beam model under ideal conditions is improved considering the influence of the inertial forces exerted by accessory parts on the catenary. The calculation expression of FVC considering the influence of APC is deduced according to the theory of elastomer vibration mechanics. The key parameters $\alpha $ and $\beta $ that affect the FVC are obtained, and the relation curves between the fluctuation velocity and the key parameters $\alpha $ and $\beta $ are analyzed respectively. Through the fluctuation velocity test for the actual catenary, the theoretical fluctuation velocity of contact wire considering the influence of APC are compared and verified with the measured fluctuation velocity. Research results show that the relative error between the theoretical fluctuation velocity obtained from the modified fluctuation velocity expression of contact wire considering the influence of APC and the measured fluctuation velocity is less than 10%. Compared with the fluctuation velocity theoretical formula based on the string/cable and Euler beam theory, the modified expression of the fluctuation velocity of contact wire (FVCW) is more accurate in identifying the FVCW, and the recognition accuracy is increased by 3.75%, which verifies the accuracy of the recognition method and provides a more accurate theoretical basis of fluctuation velocity for designing the structural parameters of catenary.

Boundary stabilization of systems of high order PDEs arising from flexible robotics

Stabilization of a system of coupled PDEs of the fourth-order by means of boundary control is investigated. The considered setup arises from the classical Euler-Bernoulli beam model, and constitutes a generalization of flexible mechanical systems. A linear feedback controller is proposed, and using an abstract formulation based on operator semigroup theory, we are able to prove the well-posedness and the stability of the closed-loop system. The performances of the proposed controller are illustrated by means of numerical simulations.

Multi-Beams modelling for high-rise buildings subjectedto static horizontal loads

In general, the study of a high-rise building's behaviour when subjected to a horizontal load (wind or earthquake) is carried out through numerical modelling with finite elements method. This paper proposes a new, original approach based on the use of a multi-beams model. By redistributing bending and axial stiffness of horizontal elements (beams and slabs) along vertical elements, it becomes possible to produce a system of differential equations able to represent the structural behaviour of the whole building. In this paper this approach is applied to the study of bending behaviour in a 37-storey building (Torre Pontina, Latina, Italy) with a regular reinforced concrete structure. The load considered is the wind, estimated in accordance with Italian national technical rules and regulations. To simplify the explanation of the approach, the wind load was considered uniform on the height of building with a value equal to the average value of the wind load distribution. The system of differential equations' is assessed numerically, using Matlab, and compared with the obtainable solution from a finite elements model along with the obtainable solutions via classical Euler-Bernoulli beam theory. The comparison carried out demonstrates, in the case study examined, an excellent approximation of structural behaviour.

A rigorous analytical solution of interfacial stresses and overall stiffness of beam structures bonded with partially covered plates

Beam structures with partially covered plates have been widely used in macro members and micro devices, such as concrete beams with fiber-reinforced plastic (FRP) plates, aircraft structures with FRP patches and smart beams with actuators/sensors. Based on the two-parameter elastic foundation model and the classical Euler–Bernoulli beam model, a rigorous analytical solution of interfacial stresses and overall stiffness of plated beams is obtained, and influences of the interface and plate properties are presented. The limitation of the assumption of the same curvature of host beam and plate, which was widely adopted in previous studies, has been clarified. And the present solution is applicable to a wider range of interface and plate properties, especially for large values of plate thickness, plate modulus and interfacial stiffness, as well as small values of plate length. The saturation phenomenon of overall stiffness is observed in some cases, and critical values of the interface and plate parameters corresponding to the saturation limit of overall stiffness can be referred for design. The results can provide guidance for the optimal design of widely-used plated beams to prevent premature interfacial debonding and acquire better overall performance, such as higher stiffness with FRP plates or shape control with actuators, by manipulating the interface and plate properties.

Dynamic stiffness approach to vibration transmission within a beam structure carrying spring–mass systems

The dynamic stiffness method is developed for the dynamics of a beam structure carrying multiple spring−mass systems. Based on classical Bernoulli–Euler beam theory, three types of vibrations, namely, bending, longitudinal and torsional motions, are formulated in terms of dynamic stiffness matrix. Similar to finite element technique, the local dynamic stiffness matrices for individual beams and spring−mass systems are assembled into global dynamic stiffness matrices so as to address vibration transmission from machines to flexible beamlike foundations. Using our proposed method, the vibration transmission within a beam frame, carrying multiple spring−mass systems is addressed. Through numerical analysis, the calculated vibration responses agree well with those from finite element method, which demonstrates that dynamic stiffness formulation has great potential in modeling the dynamics of built-up beam structures that supports machinery, especially in characterizing vibration isolation due to wave reflections, wave conversions, and other underlying mechanisms.

Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting

With the attention focused on harvesting energy from the ambient environment for nanoscale electronic devices, electromechanical coupling effects in materials have been studied for many potential applications. Flexoelectricity can be observed in all dielectric materials, coupling the strain gradients and polarization, and may lead to strong size-dependent effects at the nanoscale. This paper investigates the flexoelectric energy harvesting under the harmonic mechanical excitation, based on a model similar to the classical Euler–Bernoulli beam theory. The electric Gibbs free energy and the generalized Hamilton’s variational principle for a flexoelectric body are used to derive the coupled governing equations for flexoelectric beams. The closed-form electromechanical expressions are obtained for the steady-state response to the harmonic mechanical excitation in the flexoelectric cantilever beams. The results show that the voltage output, power density, and mechanical vibration response exhibit significant scale effects at the nanoscale. Especially, the output power density for energy harvesting has an optimal value at an intrinsic length scale. This intrinsic length is proportional to the material flexoelectric coefficient. Moreover, it is found that the optimal load resistance for peak power density depends on the beam thickness at the small scale with a critical thickness. Our research indicates that flexoelectric energy harvesting could be a valid alternative to piezoelectric energy harvesting at micro- or nanoscales.

Generalized finite element formulation for efficient first-order plastic hinge analysis

This article presents generalized finite element formulation for plastic hinge modeling based on lumped plasticity in the classical Euler–Bernoulli beam. In this approach, the plastic hinges are modeled using a special enrichment function, which can describe the weak discontinuity of the solution at the location of the plastic hinge. Furthermore, it is also possible to insert a plastic hinge at an arbitrary location of the element without modifying its connectivity or adding more elements. Instead, the formations of the plastic hinges are achieved by hierarchically adding more degrees of freedom to existing elements. Due to these features, the proposed methodology can efficiently perform the first-order plastic hinge analysis of large-frame structures. A generalized finite element solution technique based on the static condensation scheme is also proposed in order to reduce the computational cost of a series of linear elastic problems, which is in general the most time-consuming portion of the first-order plastic hinge analysis. The effectiveness and accuracy of the proposed method are verified by analyzing several representative numerical examples.

An analytical solution for static problems of curved composite beams

Abstract An analytical solution is presented for the determination of deformation of curved composite beams. Each cross-section is assumed to be symmetrical and the applied loads are acted in the plane of symmetry of curved beam. In-plane deformations are considered of composite curved beams. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli-Euler beam theory. The curvature of beam is constant and the internal forces in a cross-section is replaced by an equivalent forcecouple system at the origin of the cylindrical coordinate system used. The internal forces are expressed in terms of two kinematical variables, which are the radial displacement and the rotation of the cross-sections. The determination of the analytical solutions of the considered static problems are based on the fundamental solutions. Linear combination of the fundamental solutions which are filling to the given loading and boundary conditions, gives the total solution. Closed form formulae are derived for the radial displacement, cross-sectional rotation, nomral and shear forces and bending moments. The circumferential and radial normal stresses and shear stresses are obtained by the integration of equilibrium equations. Examples illustrate the developed method.

An approximate solution for the inverted four-point bending test in symmetric specimens

An approximate solution is derived for the interfacial energy release rate of the inverted four-point bending test. The analysis builds on a previous model developed by one of the authors for an orthotropic edge-cracked layer subject to arbitrary generalized end forces. Contact forces exerted from the upper on the lower layer of the edge-cracked portion behind the delamination tip are introduced. Their value is chosen such that the two layers undergo the same deflection. The effects of both shear deformations along the layers and friction at the point of contact are taken into account within beam theories and approximate Coulomb model with a prescribed friction coefficient. The delamination energy release rate is derived for homogeneous and symmetric specimens. A parametric analysis is performed on varying the mechanical parameters of the model to analize the influence of shear deformations and friction. The results show that both shear deformations and friction affect the value of the energy release rate for short/intermediate interfacial cracks. For very long interfacial cracks the delamination energy release rate tends to a constant limit value which corresponds to that obtained within classical Euler-Bernoulli beam theory in absence of friction.

Deformation of rotating two-layer curved composite beams

Abstract An analytical solution is presented for the determination of the deformation of rotating two-layer composite beams. The direction of axis of rotation is vertical and the speed of rotation is constant. The axis of rotation is in the plane of symmetry of curved beam. The source of the in-plane deformation is the stationary rotation of the curved beam. The plane of the curvature is the symmetry plane of the curved beam for its material, geometrical and supporting properties. Assumed form of the displacement field meets the prescriptions of the classical Euler-Bernoulli beam theory. Examples illustrate the applications of the presented analytical solution.