Classical Euler Bernoulli Beam Theory Research Articles

Dynamics and Stability of Double-Walled Carbon Nanotube Cantilevers Conveying Fluid in an Elastic Medium

The paper concerns the dynamics and stability of double-walled carbon nanotubes conveying fluid. The equations of motion adopted in the current study to describe the dynamics of such nano-pipes stem from the classical Bernoulli–Euler beam theory. Several additional terms are included in the basic equations in order to take into account the influence of the conveyed fluid, the impact of the surrounding medium and the effect of the van der Waals interaction between the inner and outer single-walled carbon nanotubes constituting a double-walled one. In the present work, the flow-induced vibrations of the considered nano-pipes are studied for different values of the length of the pipe, its inner radius, the characteristics of the ambient medium and the velocity of the fluid flow, which is assumed to be constant. The critical fluid flow velocities are obtained at which such a cantilevered double-walled carbon nanotube embedded in an elastic medium loses stability.

MODAL BEHAVIOUR OF LONGITUDINALLY PERFORATED NANOBEAMS

Nano-electro-mechanical systems (NEMS) require perforated beams for structural integrity. Hole sizes, hole numbers, and scale effects need to be modelled appropriately in their design. This paper presents a new finite element model to investigate the modal behaviour of longitudinally perforated nanobeams (LPNBs) using the classical Euler–Bernoulli beam theory. A symmetric array of holes arranged parallel to the length direction of the beam with equal spacing was assumed for the perforation. The non-local Eringen’s differential form was used to incorporate the nanoscale sizes. The accuracy of the proposed model was verified by comparing the obtained results with the available analytical solutions for fully filled nanobeams. The effects of aspect ratios, non-local parameters, boundary conditions, and perforation characteristics on the modal behaviour of LPNBs were investigated. The non-local parameter reduced the natural frequency owing to a decrease in the stiffness of the structures. However, the perforation filling ratio led to higher values of the fundamental frequency. Furthermore, compared with other boundary conditions, clamped–clamped boundary conditions demonstrated the best performance in terms of the maximum frequency.

Room-temperature strength of the interfacial bond between silica inclusions and iron

The strength of the interface between iron (Fe) and individual spherical silica (SiO2) inclusions of 5 ± 1 µm diameter is measured by means of in-situ micro-cantilever bending tests conducted under displacement control within a scanning electron microscope, using a wedge indenter tip. Attention is paid to ensure that reported data are not affected by either the electron beam during in-situ testing or by focused ion beam milling, used to produce the cantilevers. For the latter, a simple yet effective strategy, validated using finite element modeling, is devised to shift the peak interfacial stress towards the beam center, thereby relocating the fracture initiation position away from potential FIB-induced artefacts at the beam edge. All bend beams tested fractured along the Fe/SiO2 interface, with no visible evidence of silica microcracking or iron plastic deformation. The rapidly cooled iron matrix features a submicron ferrite grain size, causing its yield strength to exceed 1 GPa. Using classical Euler-Bernoulli beam theory, the estimated Fe/SiO2 interfacial tensile strength yields values ranging from 0.9 to 2.3 GPa, proving that SiO2 inclusions are strongly bonded to the Fe matrix at room-temperature. There is, thus, potential for engineering inclusion-containing Fe-based alloys that, under monotonic loading conditions, do not compromise the mechanical integrity of the ferrous matrix.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

Size dependent pull-in instability of functionally graded microbeams using a finite element formulation

The size dependent static pull-in instability of functionally graded (FG) microbeams in micro-electromechanical systems (MEMS) is studied, considering the influence of the axial force. The material properties of the microbeams are varied in the beam thickness by a power-law function, and they are calculated by the rule of mixture. To account for the microsize effect, the classical Euler-Bernoulli beam theory is employed in combination with the modified couple stress theory to describe the microbeams deformation. Based on Von Kármán nonlinear relationship, a beam element is derived and employed to establish the discretized governing equation for the microbeams. Newton-Raphson iterative procedure is adopted to compute frequencies and pull-in voltages for the microbeam with clamped ends. Numerical result reveals that the pull-in voltage is increased by the increase of the power-law exponent and the microscale parameter. The effects of the material distribution, the axial force as well as the microstructural parameter on the pull-in instability of the FG microbeams are investigated in detail.

Design and Validation of Symmetrical Elastic Elements in Series Elastic Actuator

Abstract In recent years, the demand for robot joint compliance increased with more complex human–robot interaction scenarios. Series elastic actuators (SEAs) are extensively utilized in multiple fields with abilities to provide accurate force control and energy storage. As the main flexible component in SEA, the accurate modeling of the stiffness of the elastic element is essential. However, the existing stiffness models based on classical Euler–Bernoulli beam theory contain large errors with the actual situation, which increases the difficulty of design. In this paper, a typical elastic element is analyzed by the finite element method to investigate its stiffness properties with different geometric parameters. A more accurate stiffness model is proposed for designing elastic elements. The stiffness model is validated by designed experiments, with a fitting accuracy of 98.27%, which significantly exceeds the stiffness model based on classical beam theory. The proposed stiffness model can be applied to design elastic elements that meet specific stiffness requirements.

Enhanced thermal buckling resistance of folded graphene reinforced nanocomposites with negative thermal expansion: From atomistic study to continuum mechanics modelling

Materials with tunable negative thermal expansion (NTE) are of crucial importance in engineering applications. Currently, the route to achieving large tunability for the coefficient of thermal expansion (CTE) in metal matrix composites (MMCs) is very limited. This paper develops folded graphene (FGr) structures to engineering the CTE of MMCs by embedding the FGr into a copper (Cu) matrix through the molecular dynamics (MD) method. Moreover, the present work investigates the thermal buckling resistance capability of the NTE material composite beam using the classical Euler-Bernoulli beam theory and Ritz method. Atomistic studies based on the MD simulations show that the CTE of FGr/Cu nanocomposite can be effectively tuned by mechanical prestress and temperature, especially within the range from 200 K to 400 K. By using the origami-patterned FGr designed herein, the maximum negative and positive CTEs can reach approximately −95.42 × 10−6 K−1 and 56.32 × 10−6 K−1 under the prestress of 1000 MPa at 300 K and 200 K, respectively. More importantly, the quantitative relationship of the temperature- and prestress-dependent material properties of the FGr/Cu composites are built. Based on the built material model, the FGr/Cu composite beam exhibits considerably improved thermal buckling performance than the pristine graphene/Cu beam.

An Exact Elasticity Solution for Monoclinic Functionally Graded Beams

In this study, an elasticity solution is presented for monoclinic functionally graded beams subject to a transverse pressure distributed sinusoidally. Monoclinic material properties are assumed to vary exponentially throughout the thickness of the beam’s layers. An analytical formulation based on the classical Euler–Bernoulli beam theory is also derived for comparison purposes of simply supported monoclinic functionally graded beams. In benchmark examples, the numerical results of normal stresses, transverse shear stress, as well as axial and vertical displacements are presented. The effect of material grading, fiber angle, and beam length to thickness ratio on the stress and displacement distributions is comprehensively investigated. The proposed elasticity-based analytical solution and presented numerical results can be used for verification or comparison purposes of numerical procedures.

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.

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.

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.

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.

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.

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.

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.