Elastic Euler-Bernoulli Beam Research Articles

Stability of a Multi-Span Current-Carrying Beam Resting on Periodic Supports and Exposed to an External Magnetic Field

Abstract The stability of a current-carrying beam in an external magnetic field is studied in this paper. The beam rests on periodic supports and is modeled as an Euler–Bernoulli elastic beam. Based on the linearized stability equation and propagator transfer matrix approach for a multi-span finite-length beam, the stability equation and solutions of the several boundary value problems are obtained. While the Floquet–Bloch theory is widely used in the dynamic problem of phononic and photonic structure conditions, its application to the solution of infinite length beam static stability problem is novel. The stability values of the periodic infinite beam model are in very good agreement with one of the finite-length multi-span beams.

A finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping

We analyze a finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping that undergoes vibrations under external excitation. We prove the wellposedness of the problem and regularity estimates of the exact solution to the model. We then utilize these results to prove an optimal-order error estimate of the numerical approximation assuming only the regularity of the data of the model but not that of the exact solution. Because the model exhibits its salient features that are different from those of conventional elastic Euler–Bernoulli beams, a new estimate technique is used in the analysis. We finally carry out numerical experiments to substantiate the error estimate and to investigate the dynamic response of the viscoelastic Euler–Bernoulli beam, in comparison with the conventional Euler–Bernoulli beam.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

Kinematic response of end-bearing piles in saturated soil to seismic SH-waves

This paper presents a mathematical model for the seismic analysis of end-bearing piles founded in a homogeneous saturated soil of finite thickness. Seismic loading is modelled as harmonic horizontal strong motion of the rigid bedrock, that generates vertically-propagating shear SH-waves. The transfer function of the soil layer is first established, considering soil as uniform isotropic poroelastic medium that obeys Biot’s governing equations. Accordingly, free-field and scattered soil displacements are co-evaluated, to establish a series-form expression that provides the dynamic lateral force acting on an Euler-Bernoulli elastic beam representing the pile. We employ the model to show that energy dissipation due to relative movement between the pore fluid and the solid matrix may have a noteworthy effect on the amplitude of high-frequency components of the strong motion in high permeability soils, and this in turn affects the kinematic response of piles. Finally, we use arithmetic examples to identify the soil permeability threshold, beyond which simpler single-phase soil models will produce reliable results for piles embedded in saturated two-phase soil.

Energy-Based Approach: Analysis of a Laterally Loaded Pile in Multi-Layered Non-Linear Elastic Soil Strata

Several studies have been reported in published literature on analytical solutions for a laterally loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. The present study describes a new continuum-based analysis or energy-based approach for predicting the pile displacement responses subjected to static lateral loads and moments considering the soil non-linearity. This analytical analysis treats the pile as an elastic Euler–Bernoulli beam and the soil as a three-dimensional (3D) continuum in which the non-linear elastic properties are described by a modulus degradation relationship. The principle of virtual work was applied to the energy equation of a pile–soil system in order to obtain the governing differential equation for the pile and soil displacements. An iterative procedure was adopted to solve the equations numerically using a finite difference method (FDM). The pile displacement response was obtained using the software MATLAB R2021a, and the results from the energy-based method were compared with those obtained from the field test data as well as the finite element analysis (FEA) based on the software ANSYS Workbench 2021R1. The present study investigated the effect of explicit incorporation of soil properties and layering through a parametric study in order to understand the importance of predicting appropriate pile displacement responses in a linear elastic soil system. The responses indicated that the effect of soil layers and their thicknesses, pile properties and the variation in soil moduli have a direct impact on the displacements of piles subjected to lateral loading. Hence, a proper emphasis has to be given to account for the soil non-linearity. Considering the effect of soil non-linearity, it is observed that the results obtained from the energy-based method agreed well with the field measured values and those obtained from the FEA. The results indicated a difference of approximately less than 7% between the proposed method and the FEA. The approach presented in this study can be further extended to piles embedded in multi-layered soil strata subjected to the combined action of axial loads, lateral loads and moments. Furthermore, the same approach can be extended to study the response of the soil to group piles.

A damaging beam-lattice model for quasi-brittle fracture

This work aims to propose a new damaging beam-lattice model using the Discrete Element Method paradigm dedicated to the simulation of quasi-brittle fracture under complex loadings. Enrichment of the elastic Euler–Bernoulli beam link, inspired by the cohesive zone models, is proposed to provide a damageable behavior in mixed mode and contribution of frictional behavior is not considered in this first version of the damage model. The tensile contribution on the beam link is taken into account from the first order elongation of the beam while all other contributions, i.e. bending, shear, and torsion are considered from the second-order elongation of the beam. These orders of elongation refer to beam theory, where the first elongation is induced by a force normal to the cross-section and the second is the elongation of the curvilinear length of the beam resulting from shear, bending and torsion loads. As these two kinematics do not correspond to the conventional modes I, II, and III, a deep checking step of the model is undertaken. First, mixed-mode testing on a single beam is performed to monitor the energy components dissipated in each mode and to ensure that energy dissipated in mixed mode exhibits a monotonic evolution between boundary values related to pure modes. Based on this first verification, a tensile test and a compression one are simulated on a cylinder specimen to evaluate the model capabilities to qualitatively describe the well-known characteristics of quasi-brittle fracture such as failure facies, unilateral effect, and the ratio between the compression and tensile strength. Finally, the model is used to simulate a complex crack propagation test coming from the recent international Carpiuc benchmark.

The effect of the TMD on the vibration of an offshore wind turbine considering three soil-pile-interaction models

In this paper we propose the use of the power series method and the Newmark-Beta algorithm to study the mitigation by the tuned mass damper (TMD) of an offshore wind turbine(OWT). The monopile of the OWT is taken as slender beam buried in a homogeneous soil while the tower is considered as tapered slender beam. Mathematically, both monopile and tower are modeled as elastic Euler-Bernoulli beams, with a point mass at the tower top representing the rotor nacelle assembly (RNA). First of all, the power series method is utilized to calculate the first natural frequencies of AF and CS models. The obtained results are compared with the first natural frequency of DS model obtained from FEM-Abaqus with good satisfaction. Next, the obtained mode shapes are used to establish the system of ordinary differential equations (ODE) governing the dynamic of OWT subjected to a TMD. Afterwards, the Newmark-Beta algorithm is employed to solve the ODE. Accuracy of the introduced approach is verified by setting a comparison between our results with those obtained using FEM-Abaqus. Finally, the influence of several parameters on the performance of TMD is shown in some plots.

Analytic Method for Vibration Analysis of Track Structure Induced by High-Speed Train

Nowadays, the increase in heavy freight rail transport and high-speed train (HST) operations has encouraged scientists to investigate the dynamic response of rail structures under moving load using analytically and numerically computational methods. In this study, to analyse vibration of rail structure, the rail has been modelled as continuous Euler-Bernoulli elastic beam system. The effects of some basic parameters such as track foundation elasticity modulus, rail stiffness, wheel set axle load and rail critical velocity which affect rail vibrations, on vibrations were examined in detail by considering different track foundation properties. The vibration wave amplitude increases as the train speed approaches the rail infrastructure critical speed.

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.

Instability of single-walled carbon nanotubes conveying Jeffrey fluid* *Project supported by the National Natural Science Foundation of China (Grant No. 11772162), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2019BS01004), and the Inner Mongolia Grassland Talent, China (Grant No. 12000-12102408).

We report instability of the single-walled carbon nanotubes (SWCNT) filled with non-Newtonian Jeffrey fluid. Our objective is to get the influences of relaxation time and retardation time of the Jeffrey fluid on the vibration frequency and the decaying rate of the amplitude of carbon nanotubes. An elastic Euler–Bernoulli beam model is used to describe vibrations and structural instability of the carbon nanotubes. A new vibration equation of an SWCNT conveying Jeffrey fluid is first derived by employing Euler–Bernoulli beam equation and Cauchy momentum equation taking constitutive relation of Jeffrey fluid into account. The complex vibrating frequencies of the SWCNT are computed by solving a cubic eigenvalue problem based upon differential quadrature method (DQM). It is interesting to find from computational results that retardation time has significant influences on the vibration frequency and the decaying rate of the amplitude. Especially, the vibration frequency decreases and critical velocity increases with the retardation time. That is to say, longer retardation time makes the SWCNT more stable.

Influence of geometrical uncertainties of analytical modelling on the evaluation of building deflections induced by ground movements

The objective of this paper is to investigate the influence of various geometrical uncertainties of analytical modelling affecting the evaluation of building deflections induced by tunnelling or mining subsidence. Based on Winkler model coupled with an elastic Euler-Bernoulli beam, an analytical approach is applied to evaluate the influence of major sources of uncertainties related to the Soil-Structure Interaction (SSI) geometric conditions and properties, namely, the free-field ground movement profile, the building position with respect to the settlement curvature and the variability of the structure stiffness caused by the heterogeneity of structural materials and openings (doors, windows, etc.). A comparison of results from various configurations is done to evaluate the degree to which the developed analytical approach reproduces reality. Results reveal a discrepancy in the structure response by considering these uncertainties in the entire domain of variation of the SSI parameters. For an operational use, a coefficient A is proposed; this coefficient can be used by engineers and designers when evaluating the building damage induced by the transmission of ground movements to structures taking into consideration these uncertainties.

Dynamic analysis of a pipe conveying a two-phase fluid considering uncertainties in the flow parameters

This paper is interested in the dynamics of a horizontal pipe conveying a two-phase fluid (gas and liquid), which is a problem of great regard for the oil and gas industry. The purpose of this paper is twofold. First, it introduces dimensionless coefficients that carry the information of the two-phase flow, allowing the dynamic equation to be written analogously to the equation of a single-phase fluid. Second, a probabilistic model is developed considering uncertainties in three flow parameters, (1) the flow profile factor, (2) the slip ratio and (3) the vapour quality, in order to analyse the influence of these parameters on the dynamic stability and on the frequency response of the system. The pipe is described using the linear elastic Euler–Bernoulli beam theory, and the fluid is modelled taking into account a constant tangential velocity of the flow. The coupled system is discretized by means of the finite element method, and the stochastic problem is approximated using the Monte Carlo method. The flow parameters affect greatly the system response, and different values of the critical flow velocity are obtained, depending on the level of uncertainty of the parameters.

Generalized Multi-Symplectic Numerical Implementation of Dynamic Responses for Saturated Poroelastic Timoshenko Beam

Based on the porous media theory and Timoshenko beam theory, properties of dynamic responses in fluid-solid coupled incompressible saturated poroelastic Timoshenko beam are investigated by generalized multi-symplectic method. Dynamic response equation set of incompressible saturated poroelastic Timoshenko beam is presented at first. Then a first order generalized multi-symplectic form of this dynamic response equation set is constructed, and errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum and generalized multi-symplectic local energy are also derived. A Preissmann Box generalized multi-symplectic scheme of the dynamic response equation set is presented, the discrete errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum conservation law and generalized multi-symplectic local energy conservation law are also obtained. In view of the dynamic responses of incompressible saturated poroelastic Timoshenko cantilever beam with two ends permeable and free end subjected to the step load, the transverse dynamic response process of the solid skeleton is simulated numerically, the evolution processes of solid effective stress and the equivalent moment of the pore fluid pressure over time are also presented numerically. The effects of fluid-solid coupled interaction parameter and slenderness ratio of the beam on the solid dynamic response process are revealed, as well as the effects on all generalized multi-symplectic numerical errors are checked simultaneously. From results obtained, the processes for solid deflection, solid effective stress and the equivalent moment of the pore fluid pressure approaching to their steady response values are all shortened with increasing of fluid-solid coupled interaction parameter, while the response process of solid deflection and the pore fluid equivalent moment are lengthened with increasing of slenderness ratio of the beam. Moreover, the steady value of solid deflection is much closer to the static deflection value of classic single phase elastic Euler-Bernoulli beam with increasing of the slenderness ratio. As time goes on, the solid skeleton of the beam will support all outside load, so equivalent moment of the pore fluid pressure becomes zero at last. In addition, it is presented all generalized multi-symplectic numerical errors decrease with the decreasing of parameters representing the dissipation effect for the dynamic system.

A true PML approach for steady-state vibration analysis of an elastically supported beam under moving load by a DLSFEM formulation

This paper concerns a computational implementation for solving a Moving Load (ML) problem on an infinite Euler–Bernoulli elastic beam on a Pasternak visco-elastic support. A steady-state dynamic response in convected coordinate is sought, by a numerical approach with discretization over a finite domain, implying spurious boundary reflections of non-evanescent waves. This is effectively solved by: (a) analytically formulating a new, true Perfectly Matched Layer (PML) approach, toward handling the underlying fourth-order differential problem and the corresponding far-field conditions, without adopting special boundary conditions; (b) outlining a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation, apt to provide a robust approach for the present non self-adjoint problem and to conveniently handle the jump condition in the shear force at the concentrated ML position. Consistent numerical results are illustrated and compared to an available analytical solution, showing a perfect match, with a complete removal of spurious boundary effects and a proof of theoretical a priori error estimates. Further results are produced for a case with multiple MLs. The paper shows that the present innovative DLSFEM-PML formulation is effectively suitable to numerically solve a steady-state ML problem on an infinite beam, setting up a new computational tool in such a challenging mechanical context.

Bond graph modeling of a typical flapping wing micro-air-vehicle with the elastic articulated wings

In the present research, a new comprehensive model of a flexible articulated flapping wing robot using the bond graph approach is presented. The flapping kinematics of a two-section wing is introduced via the bond graph based approach on a hybrid mechanism providing amplitude and phase characteristics. The aerodynamic quasi-steady approach equipped with stall correlation is utilized according to the reduced flapping frequency and the angle of attack ranges. The local flow velocity and the wing position are calculated in both wing and body coordinates taking into account rotation and translation of the wing different parts. Estimation of the effective angle of attack is performed by calculating the instantaneous torque distribution on both wing sections. Aeroelastic modeling is employed in which the wing structure is assumed as an elastic Euler–Bernoulli beam at the leading edge with three linear torsional modes. In this novel integrated bond graph model, computation of the performance indices including the average lift and thrust, consumed and produced powers by flapping and mechanical efficiency are presented. Due to existence of the numerous geometric and kinematic parameters in articulated flexible flapping wing, such a model is essential for design and optimization. Consequently, an example of a typical parametric study and the results validation are carried out. It is indicated that the sensitivity of the bird performance to relative change in design variables would increase for out of phase flapping, second part stiffness, flapping amplitude, frequency and velocity respectively. It is interesting that by employing the reverse-phase flapping which is possible only via articulated wings, the maximum efficiency could be achieved. In addition, it is shown that adjusting the wing torsional stiffness is a crucial item in design of passive flapping robots. The key advantage of the two-section flapping wing is depicted as the controlling capability of the angle of attack in the outer part of the wing. Finally, the improved version of the bird is being addressed by approximately 15% progress in propulsive efficiency.

Simplified probabilistic evaluation of the variability of soil-structure interaction parameters on the elastic transmission of ground movements

An efficient probabilistic approach is presented to consider the uncertainty related to the evaluation of the transmission of ground movements to structures considering the influence of the variability of the soil-structure interaction (SSI) parameters (namely the building stiffness EI/B, the soil modulus k and the structure length L). These movements can be produced by tunnels excavation, sinkholes, mining subsidence, etc. This probabilistic approach aims to quantify the relative importance of the variability of the main SSI parameters on the structure response via a sensitivity analysis (SA). The structure response is evaluated using analytical and numerical models. The analytical model is applied using Mathematica where the structure is modeled by an elastic Euler-Bernoulli beam and the soil by a combination of elastic springs as per the Winkler model. The numerical model is developed using Plaxis 2D coupled with the programming tool Python for a significant number of numerical simulations to validate the analytical results obtained. The objective is to evaluate the impact of the parameters variability which are assumed to be of a random nature. To avoid time and computation difficulties, a simplified analytical meta-model that reflects the structure response is proposed. Based on this meta-model, the SA is performed using three selected procedures, one local and two global, namely, one-way analysis, Sobol and McKay respectively. Results reveal the contribution of each input parameter on the variability of the response using various sampling techniques, such as random Monte-Carlo Simulations (MCS) and Latin Hypercube Sampling (LHS). Accordingly, the effect of EI/B increases with the increasing values of the relative stiffness (ρ*); however, the effect of k increases when ρ* decreases. Results also reveal the influence of the variability of the main SSI parameters on the rate of the ground movement that is transmitted to structures; confidence intervals are set and can be used by engineers and designers when evaluating the induced building damage in response to the transmission of ground movements taking into consideration these uncertainties.

Euler–Bernoulli theory accurately predicts atomic force microscope cantilever shape during non-equilibrium snap-to-contact motion

We prove that the Euler–Bernoulli elastic beam theory can be reliably used to describe the dynamics of an atomic force microscope cantilever during the far from equilibrium snap-to-contact event. In conventional atomic force microscope operation, force-separation curves are obtained by post-processing voltage versus time traces produced by measuring one point on the cantilever close to the hanging end. In this article, we assess the validity of the Euler–Bernoulli equation during the snap-to-contact event. The assessment is based on a direct comparison between experiment and theory. The experiment uses Doppler vibrometry to measure displacement versus time for many points along the long axis of the cantilever. The theoretical algorithm is based on a solution of the Euler–Bernoulli equation to obtain the full shape of the cantilever as a function of time. The algorithm uses as boundary conditions, experimentally obtained information only near the hanging end of the cantilever. The solution is obtained in a manner that takes into account non-equilibrium motion. Within experimental error, the theory agrees with experiment indicating that the Euler–Bernoulli theory is appropriate to predict the cantilever kinematics during snap-to-contact. Since forces on the tip can be obtained from the instantaneous shape of the cantilever, this work should allow for computation of tip-sample forces during the snap-to-contact event from a conventional force-distance measured input.

Simplified Formulation of Mechanics of Structure Genome

This paper presents a simplified formulation of the mechanics of a structure genome (MSG). The MSG is a unified theory for multiscale constitutive modeling for all types of composite structures. It is generalized from the previous research works based on the variational asymptotic method (VAM), including the variational asymptotic beam sectional analysis, the variational asymptotic plate and shell analysis, and the variational asymptotic method for unit cell homogenization. In this paper, the original complex formulation of the MSG is simplified and the principle of minimum information loss underpinning the MSG is formalized. The simplified formulation of the MSG is applied to construct the linear elastic Euler–Bernoulli beam model, the Kirchhoff plate model, and three-dimensional Cauchy continuum model. The connections of the MSG to previous VAM-based theories are clearly pointed out.

Dynamics of a beam on a bilinear elastic foundation under harmonic moving load

The present paper is concerned with the numerical modelization of the transient dynamic response of a simply supported Euler–Bernoulli elastic beam resting on a Winkler-type foundation, under the action of a transverse concentrated load, moving at a constant velocity along the beam, displaying an harmonic-varying magnitude in time. The elastic foundation, assumed as homogeneous in space, behaves according to a bilinear constitutive law, characterized by two different stiffness coefficients in compression and in tension. A finite element method approach coupled with a direct integration algorithm is developed for efficiently tracing the nonlinear dynamic response of the beam-foundation system. An original automated procedure is set, as being apt to resolve all required space/time discretization issues. Extensive parametric numerical analyses are performed to investigate how the frequency of the harmonic moving load amplitude and the ratio between the foundation’s moduli in compression and in tension affect the so-called critical velocities of the moving load, leading to high transverse beam deflections. Analytical interpolating expressions are proposed and fitted for the achieved two-branch critical velocity trends. The present outcomes shall reveal potential practical implications in scenarios of contemporary railway engineering, especially in terms of lowering down the admissible high-speed train velocities, as for structural requirement or preventing potential passenger discomfort.

Finite element model of ballasted railway with infinite boundaries considering effects of moving train loads and Rayleigh waves

This paper proposes a three-dimensional model incorporating finite element (FE) meshes with infinite element (IE) boundaries for ballasted railways. Moving train loads are simulated with sliding motions of moving elements which have hard contact feature at the interface with supporting rails. Dynamic responses of ballasted railway under different train speeds are investigated in time domain and frequency domain to identify the predominant frequency and critical speed. Rayleigh wave (R-Wave) propagation is simulated using the combined FE-IE model to determine the velocity of R-Wave in the layered embankment model and its relationship with the critical speed of the ballasted railway. The proposed model is successfully validated against the results of Euler-Bernoulli Elastic Beam (E-BEB) model.