Classical Euler Bernoulli Beam Research Articles

Dynamics of the Euler — Bernoully beam with distributed hysteresis properties

In this paper, we present a new mathematical approach to the analysis of a beam with distributed hysteresis properties. These hysteresis characteristics are described by two methods: phenomenological (Bouc — Wen model) and constructive (Prandtl — Ishlinskii model). The equations for beam are developed using the well-known Hamilton method. We investigate the dynamic response of a hysteresis beam under various external loads, including impulse, periodic and seismic loads. The results of numerical simulations show that the hysteresis beam exhibits differently to external influences as compared to the classical Euler-Bernoulli beam. In particular, under the same external loads, the vibration amplitude and energy characteristics of the hysteresis beam are lower than those of the classical one. These findings can be useful for buildings developers in the design of external load resistant buildings and structures

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.

Loaded Euler-Bernoulli beam with the distributed hysteresis properties

In this article, we propose a new perspective mathematical model of the beam with the distributed hysteresis properties. Hysteresis properties are formalized within two approaches: phenomenological (Bouc-Wen model) and design (Prandtl-Ishlinskii model). The equations for the beam vibrations are obtained using the well-known Hamilton approach. The dynamical response of the beam with distributed hysteresis is considered under various types of external load, such as impulse, periodic, and a seismic load. Numerical simulations show that the hysteresis beam is more “resistant” to external loads than the classical Euler-Bernoulli beam. Particularly, with the same types of the external load, the amplitude of oscillations of the hysteresis beam as well as its energy characteristics are lower than those of the classical one. These results may find some applications in the field of the design of earthquake-resistant constructions and buildings.

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.

Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

Abstract We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales ( a ( t ) x Δ 2 ( t ) ) Δ 2 = b ( t ) f ( x ( t ) ) + c ( t ) , {\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right), such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour x ( t ) = y ( t ) + o ( 1 t β ) x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right) . The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.

A closed-form solution of forced vibration of a double-curved-beam system by means of the Green's function method

Double-curved-beam systems (DCBSs) are usually seen in many engineering fields. Compared to straight double-beam systems, DCBSs are more efficient in noise and vibration control. This paper aims to obtain a closed-form solution of steady-state forced vibration of a DCBS. The classical Euler-Bernoulli beam model is employed in this work to model vibration equations of the DCBS. Green's function and Laplace transform methods are successively used to obtain the fundamental solution of vibration equations of the DCBS. The fundamental solution is the general solution and can be used for any boundary conditions. In the numerical result section, the present solution is verified by comparing its results with some results in references. Effects of some important geometric and physical parameters on vibration responses and interaction between the stiffness of the elastic layer and the DCBS are discussed. Results show that the DCBS can be degenerated to a straight double-beam system by setting two radii of curvature of the beams to infinity, and the DCBS can also be reduced to a double-beam system whose one upper or lower beam is a straight beam and the other is a curved beam.

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.

Folding behaviour of a deployable composite cabin for space habitats – Part 2: Analytical investigation

In the Part 1 of two companion papers, a deployable composite cabin (DCC) for space habitats was introduced and its folding behaviour was comprehensively studied. This Part 2 paper is dedicated to present an analytical model for rapidly predicting the folding behaviour of the DCC. Based on the classical Euler-Bernoulli beam model, the geometric and physical equations describing the folding behaviour of the DCC were established in this paper. By combining Maclaurin series expansion, orthogonal Chebyshev polynomials, Galerkin method and Harmonic balance method, analytical models for predicting the load-displacement curve and shape function of the DCC were presented. The analytical solution of compressive strength was derived from the principle of minimum total complementary potential energy, classical laminate theory and Tsai-Hill failure criterion. In order to validate the analytical model proposed in this paper, the prediction results were compared with experimental results and numerical simulation results in the Part 1, and the three were in good agreement. Finally, the effect of geometric parameters (i.e. arc length, axial length and thickness of the thin plate) on the folding behaviour of the DCC was further analyzed by using the analytical model.

Lamb waves in discrete homogeneous and heterogeneous systems: Dispersion properties, asymptotics and non-symmetric wave propagation

In this paper, we study Lamb waves propagating in a discrete strip, whose microstructure is represented by either a monatomic or a diatomic triangular lattice. In considering the in-plane vector problem, we derive an analytical solution for the dispersion relation of Lamb waves. Additionally, we investigate the main features of the eigenmodes of the system, which describe how the lattice strip vibrates at different frequencies. Further, we discuss how the dispersion properties depend on the number of the lattice’s rows and on the chosen boundary conditions. For heterogeneous systems, we focus the attention on the internal stop-band and on the flat bands appearing in the dispersion diagram. Different asymptotic models are employed to approximate the low-frequency behaviour of the lattice strip, starting from the classical Euler–Bernoulli beam. The effective behaviour of a lattice strip with dense microstructure is also investigated, and we present a comparative numerical analysis with the analogous continuum for which the classical Lamb wave problem is posed. The theory developed is exploited here to design a structured medium capable of manipulating wavemodes, and, through conversion and selection, generating uni-directional wave phenomena. We envisage that the present work can fill a gap in the research field related to the analytical study of dispersive waves in microstructured media, whose dynamic performance is influenced by the presence of multiple external boundaries.

New criteria for nanoscale slender beams and thin plates: Low frequency domain of flexural wave

The classical Euler–Bernoulli beam model and Kirchhoff plate model are very useful on the macroscopic scale. In the context of Eringen’s nonlocal elasticity theory, this article aims to develop the criteria of the applicability of nanoscale Euler–Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The corresponding governing differential equations for the nanoscale Timoshenko beam and Mindlin plate are derived by the Hamilton’s principle, and the dispersion equations of wave are then obtained. By applying Taylor expansion to the corresponding solutions of the dispersion equations, new criteria are developed, simultaneously taking into account the effects of nonlocal parameters and material properties. When the nonlocal parameter is set to zero, the present criteria may be readily degenerated to their macroscopic counterparts. According to the present criteria, this article systematically evaluates the existing studies in literature. Various works in literature did not consider the effect of the nonlocal parameter, and hence, failed to satisfy the application conditions of the Euler–Bernoulli beam and Kirchhoff plate models on the nanoscale. The work in this article is of scientific significance to various studies on nanostructures.

Nonlinear size-dependent modeling and dynamics of nanocrystalline arc resonators

The adequate modeling of the micro/nano arc resonators' dynamics is vital for their successful implementation. Here, a size-dependent model, wherein material structure, porosity, and micro-rotation effects of the grains are considered, is derived by combining the couple stress theory, multi-phase model, and the classical Euler–Bernoulli beam model, aiming to characterize the frequency tunability of micro/nano arc resonators as monitoring either the axial load or the electrostatic force for the first time. The arc dimensions are optimized to show various phenomena in the same arc, namely snap-through, crossing, and veering. The first three natural frequencies are monitored, showing the size dependency on the frequency tuning, snap-through/back, and pull-in instability as shrinking the scale from micro- to nano-scale. Significant changes in the static snap-through and pull-in voltages and the resonance frequencies were shown as scale shrinks. A dynamic analysis of the resonator's vibration shows a dramatic effect of the size-dependency as shrinking dimensions around the veering zone.

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.

INVERSE PROBLEM FOR THE TIME-FRACTIONAL EULER-BERNOULLI BEAM EQUATION

In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.

Coupled thermoelastic nonlocal forced vibration of an axially moving micro/nano-beam

Since axially moving material systems were introduced in the 1960s, many different axially moving material systems, such as axially moving strings, beams, and plates have been studied for many decades. In recent years, axially moving micro/nano-beams in micro/nano-electro-mechanical systems become popular. Since thermal effects of micro/nano-beams cannot be ignored, coupled thermoelastic vibrations of axially moving micro/nano-beams are important scientific problems. This work analytically investigates coupled thermoelastic forced vibration and the heat transfer process of an axially moving micro/nano-beam. The small-scale effect is considered not only in the vibration equation but also in the heat transfer equation of the micro/nano-beam. Eringen nonlocal elasticity theory and the classical Euler-Bernoulli beam model are used to model the vibration equation of the axially moving micro/nano-beam. The Type III Green-Naghdi heat transfer model is used to establish the heat transfer equation of the micro/nano-beam. The vibration and heat transfer equations are coupled to each other. Green functions, Laplace transform, and eigenfunction expansion are key mathematical tools used to solve the coupled equations. In the numerical example, by comparing the present solutions with results of some published articles, the correctness of the present solutions is verified. It is found through numerical analyses that the fundamental natural frequency of an axially moving micro/nano-beam decreases with its axial velocity, small-scale parameter, and height-to-length ratio. Heat rotation phenomena are first found in this study. The heat rotation phenomena discovered show synchronization between the displacement and temperature. The rotation direction of the temperature can be controlled by changing the axial velocity, height-to-length ratio, and heating position.

Numerical Solution of Bending of the Beam with Given Friction

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

Beam Theory of Thermal-Electro-Mechanical Coupling for Single-Wall Carbon Nanotubes.

The potential application field of single-walled carbon nanotubes (SWCNTs) is immense, due to their remarkable mechanical and electrical properties. However, their mechanical properties under combined physical fields have not attracted researchers’ attention. For the first time, the present paper proposes beam theory to model SWCNTs’ mechanical properties under combined temperature and electrostatic fields. Unlike the classical Bernoulli–Euler beam model, this new model has independent extensional stiffness and bending stiffness. Static bending, buckling, and nonlinear vibrations are investigated through the classical beam model and the new model. The results show that the classical beam model significantly underestimates the influence of temperature and electrostatic fields on the mechanical properties of SWCNTs because the model overestimates the bending stiffness. The results also suggest that it may be necessary to re-examine the accuracy of the classical beam model of SWCNTs.

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.