Euler-Bernoulli Model Research Articles

Overcoming stretching and shortening assumptions in Euler–Bernoulli theory using nonlinear Hencky’s beam models: Applicable to partly-shortened and partly-stretched beams

This paper addresses the challenges of the Euler–Bernoulli beam theory regarding shortening and stretching assumptions. Certain boundary conditions, such as a cantilever with a horizontal spring attached to its end, result in beams that partly shorten or stretch, depending on the spring stiffness. The traditional Euler–Bernoulli beam model may not accurately capture the geometrical nonlinearity in these cases. To address this, nonlinear Hencky’s beam models are proposed to describe such conditions. The validity of these models is assessed against the nonlinear Euler–Bernoulli model using the Galerkin method, with examples including cantilever and clamped-clamped configurations representing shortened and stretched beams. An analysis of a cantilever with a horizontal spring, where stiffness varies, using the nonlinear Hencky’s model, indicates that increasing horizontal stiffness stiffens the system. This analysis reveals a transition from softening to linear behavior to hardening near the second resonance frequency, suggesting a bifurcation point. Despite the computational demands of nonlinear Hencky’s models, this study highlights their effectiveness in overcoming the inherent assumptions of stretching and shortening in Euler–Bernoulli beam theory. These models enable a comprehensive nonlinear analysis of partly shortened or stretched beams.

An efficient numerical method for the quasi-static behaviour of micropolar viscoelastic Timoshenko beams for couple stress problems

A viscoelastic Timoshenko micro-beam is investigated using micropolar viscoelasticity theory. The governing equations are derived by applying the principle of virtual displacements, and can be used in a variety of problems. The equations are then reduced to the special case of a homogeneous uniform beam and solved for quasi-static elastic and viscoelastic cases. The viscoelastic material is modeled via standard linear solid model. For the elastic case, a closed-form solution is reached, but for the viscoelastic case, the inverse Laplace transform is calculated numerically by utilizing De Hoog's algorithm. It is shown that the proposed model can capture the size effects, and converges to the classical Timoshenko beam if the beam is thick enough. The model also converges to the micropolar Euler-Bernoulli model if the thickness of the beam is small. The role of distributed couple stress is also investigated in this paper. The results indicate that in a cantilever beam, the distributed couple stress has a high influence on the deflection of the beam, micro-rotation, stress components and couple stress. However, in a beam resting on three supports, it only affects the micro-rotation and the shear stress component in transverse direction (σxz). Further investigation shows an extremum in some of the results over time, illustrating the importance of taking viscoelastic behaviour into account.

On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems

Port-Hamiltonian systems were recently extended to include implicitly defined energy and energy ports thanks to a (Stokes-)Lagrange subspace. Here, we study the equivalent port-Hamiltonian representations of two systems with damping, written using either a classical Hamiltonian or a Stokes-Lagrange subspace. Then, we study the Timoshenko beam and Euler-Bernoulli models, the latter being the flow-constrained version of the former, and show how they can be written using either a Stokes-Dirac or Stokes-Lagrange subspace related by a transformation operator. Finally, it is proven that these transformations commute with the flow-constraint projection operator.

Extending the analysis of the Euler–Bernoulli model for a stochastic static cantilever beam: Theory and simulations

In this paper, we present a comprehensive probabilistic analysis of the deflection of a static cantilever beam based on Euler–Bernoulli’s theory. For the sake of generality in our stochastic study, we assume that all model parameters (Young’s modulus and the beam moment of inertia) are random variables with arbitrary probability densities, while the loads applied on the beam are described via a delta-correlated process. The probabilistic study is based on the calculation of the first probability density function of the solution and the probability density of other key quantities of interest, such as the shear force, and the bending moment, which are treated as random variables too. To conduct our study, we will first calculate the first moments of the solution, which is a stochastic process, and we then will take advantage of the Principle of Maximum Entropy. Furthermore, we will present an algorithm, based on Monte Carlo simulations, that allows us to simulate our analytical development computationally. The theoretical findings will be illustrated with numerical examples where different realistic probability distributions are assumed for each model random parameter.

On the Separate Assessment of Structural Effects on the Simple Beam Deflection in the Light of Fractional Calculus

Euler-Bernoulli (EB) and Timoshenko-Ehrenfest (TE) theories model simple beams under linear constraints. But even keeping these constraints, specific structural effects in real applications compromise the accuracy of the models, such as stress concentration due to force reactions on the support contacts or bucking, for example. Both EB and TE solutions assume planar cross sections and structural stability, and therefore do not address those particular effects; the interest in using them is to explore the conditions under which shear effects are significant or not. Numerical solutions such as the ones obtained from the Finite Element Method (FEM) reach structural effects quite well, depending on the complexity of the problem and degree of refinement. However, although accurate, the numerical solutions do not distinguish whethera particular effect is on charge or not; they implicitly encompass all of them as a whole. To diagnose them for simple beams, analytical solutions such as EB or TE can be employed as long as they noticeably differ from the corresponding affected structures modeled in a FEM environment, the latter taken as reference. To measure this disagreement one can either directly compare each resulting deflection profile or, alternatively, compare the analytical solutions EB or TE against the corresponding fractional calculus ones fed with theFEM data. The second case, focus of this study, provides a better measurement, to our understanding, as the fractional order informs the magnitude of the effect not depending on absolute values, which allows a fairer evaluation. Each structure effect must be isolated so that to obtain the individually consequent order deviation from the original integer value. A separate assessment using fractional calculus is proposed in this work to, first, evaluate the effects of stress concentration on the support contacts, and, second, to preparethe modeling to potentially reveal another structural effect. Consistent results have been achieved.

SIMPLE INCREMENTAL APPROACH FOR ANALYSING OPTIMAL NON-PRISMATIC FUNCTIONALLY GRADED BEAMS

This paper presents a simple incremental approach of analysing the static behaviour of functionally graded tapered beams. This approach involves dividing the non-uniform beam into segments with uniform cross-sections, and using two separate finite element models to analyse the structural behavior of slender beams (Euler-Bernoulli model) and deep beams (Timoshenko beam theory). The material properties of the beam vary according to a power law distribution through the thickness, resulting in smooth variations in the mechanical properties. The finite element system of equations is obtained using the principle of virtual work. Detailed information on the shape functions and stiffness matrix of the beam is provided, and the numerical results are evaluated and validated using data from the literature. The comparison demonstrates that the response of the functionally graded tapered beams is accurately assessed by the proposed approach. Additionally, the effects of material distribution, boundary conditions, and tapering parameter on the deflection behavior are presented. Results show that an increase in the power law index increases the flexibility of the functionally graded tapered beams, resulting in higher deflection. Furthermore, lower tapering parameters also result in higher deflection. Compared to other boundary conditions, clamped-clamped boundary conditions demonstrate the best performance in terms of maximum deflection.

Free vibration analysis of Timoshenko pipes with fixed boundary conditions conveying high velocity fluid

Sever vibration will be induced due to the high flow velocity in the pipe. When the flow velocity exceeds the critical value, the static equilibrium configuration of the pipe loses its stability, and the vibration properties change accordingly. In this paper, the free vibration characteristics of the pipe with fixed-fixed ends are revealed in the supercritical regime. Based on the Timoshenko beam theory, the governing equations of the nonlinear vibration near the non-trivial static equilibrium configuration are established. The influences of system parameters on equilibrium configuration, critical velocity, and free vibration frequency is analyzed. The effects of supercritical velocity in different ranges on the natural frequencies are revealed. In addition, the comparison with the Euler-Bernoulli pipe model shows that the differences in critical velocity, equilibrium configuration, and frequency are still significant even the length-diameter ratio is large. The increase of the flow velocity reduces the difference of non-trivial static equilibrium configurations, but eventually aggravates the difference of natural frequencies. Within a certain supercritical velocity range, the vibration difference between the two pipe models is small, beyond this range, the vibration difference increases significantly.

Three-dimensional vibration investigation of the thin web gear pair based on Timoshenko beam

In the aerospace transmission system, thin web gears are appreciated for reducing weight. Due to the force excitation from the helical gear, except the in-plane vibrations, gears are also vulnerable to out-of-plane vibrations which may cause system instability. However, few papers investigated how the out-of-plane vibrations influence the system in current years. This study analyzes both the in-plane and out-of-plane vibration modes of thin web gear based on the Timoshenko theory. The governing equations of three-dimensional displacements are derived from Hamilton’s principles. The results show that instability will likely happen in out-of-plane vibration and the existence of discrete springs will deteriorate the instability. Additionally, frequency veering is a common phenomenon in a non-axisymmetry gear system. A comparison between the Timoshenko and Euler–Bernoulli models indicates little differences are founded at zero speed, but a significate gap comes up at high rotation speed. Lastly, a Timoshenko gear pair model with a coupling mesh stiffness matrix is presented. The natural frequencies divers from those with independent discrete springs fixed to the ground. The results indicate the importance of the coupling mesh stiffness matrix in system vibration analysis. This work is introductive in the thin web gear transmission system design.

Nonlinear Eigen frequencies of a functionally graded porous nano-beam with respect to the coulomb and Casimir forces

A new mathematical model of a porous functionally graded micro/nano-beam under the action of the Casimir force and the Coulomb force is constructed in this paper. The construction of the mathematical model takes into account the Euler-Bernoulli kinematic model. The dimensional-dependent parameter is taken into account by modified moment theory of elasticity. The variational, differential equations, boundary and initial conditions are derived from Ostrogradsky-Hamiltonian variational principle. The problem of non-linear natural oscillations of a beam under the action of force Casimir and Coulomb force is solved. The influence of the Casimir and Coulomb forces on the nonlinear eigenfrequencies of the micro/nano-beam is shown.

Predictive analysis toward the identification of cracks in functional gradient beam structures using an optimization algorithm based on the transit search technique

Machine learning techniques can be used for the prediction of cracks in beam type structures. Indeed, these techniques present an important management aspect which consists in developing a maintenance decision model, which can anticipate future failure trends in order to improve the maintenance decision process. The prediction of open cracks on the edges of beams is a problem often encountered in industry and can be detected by considering that the crack is simulated by means of a rotating spring, whose stiffness can be identified by the size of the crack. This article proposes two different techniques for crack detection in Euler-Bernoulli model functional gradient beams. The first technique generates frequency contours from a three-dimensional plot of the crack position and size, and the intersection of distinct mode contours predicts the crack location and size. The second technique uses a meta-heuristic optimization, which is inspired by astrophysics and based on well-known exoplanet discovery methods, to determine crack size and position concurrently. The weighted sum of the squared errors between the measured and computed natural frequencies is utilized to design the objective function in this second strategy. The results reveal that the two crack size and location prediction techniques agree quite well.

Analiticity of the Type III Thermoelastic Euler Bernoulli Model with Tip

Analiticity of the Type III Thermoelastic Euler Bernoulli Model with Tip

A modal-based approach for modelling the cantilever FBG accelerometer with the presence of tip mass and its sensitivity analysis

The modal-based approach for modelling cantilever-type FBG accelerometer (FBG-MM), which has been recently published, does not include the presence of tip mass and sensitivity study since it mainly focusses on the feasibility of the Euler-Bernoulli model onto a cantilever FBG accelerometer. The adaptability of tip mass into the modal model of the cantilever FBG accelerometer, namely FBG-MMTP, is presented in this manuscript, which can precisely predict the response and sensitivity of the accelerometer. The newly presented model is compared to the experimental results for different sizes of tip mass and the range of excitation frequencies less than its resonant frequency. Within the transmission range, there is an excellent agreement between the time response of wavelength shift calculated using FBG-MMTP and the experimental results. The sensitivity obtained from both FBG-MMTP and experimental results is roughly comparable, with around 25% discrepancies, which are thought to be due to imprecision in the physical dimension of tip mass and beam, as well as measurement errors.

Hybrid Bragg-locally resonant bandgap behaviors of a new class of motional two-dimensional meta-structure

Bragg scattering and local resonance are two fundamental mechanisms for bandgap (BG) formation of phononic crystals (PCs) and acoustic metamaterials (AMs). In this paper, a new class of motional two-dimensional (2D) hybrid Bragg-locally resonant (LR) meta-pipe model is developed, and the BG interaction behaviors of such a meta-structure are explored. The pipe is axially composed of alternate materials, and is periodically encircled with dual-layer rings, which are used to simultaneously trigger Bragg scattering and local resonance. Meanwhile, the pipe conducts an axially spinning motion, and a steady fluid flows inside the pipe. Compared to static periodic structures, the orthogonal traveling waves due to spin bring about a 2D meta-structure, and the spinning local resonators yield an additional centrifugal effect on the pipe. The results reveal the formation of hybrid Bragg-LR BGs in such a motional meta-structure, and further demonstrate their complicated evolutions with the location, number and geometry of the local resonators as well as the pipe material. The impacts of motional properties on the hybrid BGs are discussed, and they are compared with the behavior of Euler-Bernoulli model. This study provides a more in-depth interpretation for the interaction of Bragg and LR BGs, which is especially beneficial to the vibration and noise reduction of rotors and fluid-transporting devices.

Surface ground vibration of a tunnel embedded in soil layer of cold regions to a moving point load

The vibration characteristics of tunnel buried in seasonal frozen soil is considered as a two-dimensional model in which a moving unit vertical load is applied to an infinite beam. The beam is regarded as the Euler Bernoulli model and the unsaturated frozen soil is treated as a poroelastic medium. The analysis of the influence of dynamic response of tunnel in cold region on the ground surface in frequency-wavenumber domain can be obtained through the conservation of momentum, mass and energy of unsaturated frozen soil, as well as the use of boundary conditions. The analytical solution in transformed domain is obtained by Fourier transform. The critical velocity of frozen soil in cold region under different saturation and frozen ratio is obtained by means of the dispersion curve, and the influence of its change on the critical velocity is also analyzed. The effects of changes in parameters of seasonal frozen soil on the dynamic response are studied. It indicates that the typical parameters like saturation, frozen ratio, and critical velocity etc. in unsaturated frozen soil have a great influence on the vertical and horizonal displacement caused by the load. In addition, this paper raises several unresolved issues to highlight the need for further research in this rapidly expanding field.

Experimental verification research of pipeline deflection deformation monitoring method based on distributed optical fiber measured strain

Distributed fiber optic sensing is suitable for long-distance water pipeline deformation monitoring. However, the distributed optical fiber measures the structural strain, which is difficult to reflect the deflection deformation of the pipeline directly. Hence, it is necessary to study how to transform the measured structural strain of distributed optical fiber into deflection deformation of pipeline. In this paper, the integral method and the Long Short-Term Memory neural network (LSTM) are used to realize the transformation from the measured strain of the optical fiber to the deflection of the pipeline. Based on the Euler-Bernoulli elastic foundation model, a set of test equipment was designed to simulate the deformation monitoring of underground water pipelines, and two optical fiber strain-deflection conversion methods were verified. The results show that the LSTM intelligent transformation model can effectively and accurately transform the strain-deflection of pipelines, and is suitable for practical engineering.

Coupling effects with vibration-based estimation of individual bolt tension in multi-bolt structures

Recent work with various single-bolt experimental setups has shown promising results for applying a vibration-based bolt tension estimation procedure. To estimate tension, the novel procedure models a bolt’s transverse vibrations with a single Euler–Bernoulli beam. The present work investigates the applicability of that novel approach when a structure contains more than one bolt. The principal question is if the vibrational response obtained from excitation of a specific bolt, mounted in a structure with other similar bolts, can still roughly be modeled as an independent single-bolt system, or if the other bolts affect the vibrational response in an undesirable way. To determine if the novel single-bolt approach is applicable, it is necessary to gain insight into possible coupling phenomena. For this, a two-bolt problem is investigated, modeled both as a coupled two-beam Euler–Bernoulli model and numerically with FEM software, and with experimental testing of two real bolts mounted in the same structure. It is found that strongly coupled bending vibrations, with in-and out-of-phase modes, only occur when tension and boundary stiffness in the two bolts are very close to identical. When the bolts have different boundary parameters, the coupling is weak, and the two bolts can roughly be treated as two independent single-bolt systems. In both situations, it is actually possible to estimate bolt tension and boundary stiffness based on measured transverse natural frequencies using a single-bolt beam model.

Dynamic displacement response analysis of bridge in the impact vibration of slight and frequent vehicle bump

To study the impact of vehicle moving on bridge, the bump occurring under the determined limit conditions of vehicle from no bump to the beginning of bump was named as slight and frequent vehicle bump. With a 1/4 simplified vehicle model and the Euler-Bernoulli bridge model, the required minimum impact height for the bump and the effect of bump impact on the vertical dynamic mid-span displacement response of bridge were figured out in different vehicle speed, axle load and wheel diameter. The simulation analysis by ANSYS software, with HSFLD242 unit simulating air inside the tire and MASS21 simulating the car weight held by tires, verified the results of the minimum impact height. The effect of bump impact on the vertical dynamic mid-span displacement response of bridge was also verified by observation using the TCQN-6A bridge deflection detector on the mid-span dynamic deflection of a 25-metre simply supported beam bridge.

ИССЛЕДОВАНИЕ КОЛЕБАНИЙ БАЛКИ ИЗ ФУНКЦИОНАЛЬНО-ГРАДИЕНТНОГО МАТЕРИАЛА С УЧЕТОМ ЗАТУХАНИЯ

The problem of oscillations of an inhomogeneous beam made of a functionally gradient material in the framework of various deformation models - the Euler-Bernoulli model and the Timoshenko model in the presence of attenuation, which is described within the framework of the concept of complex modules for a standard viscoelastic body, is considered. The left end of the beam is pinched, and at the right end there is a concentrated moment oscillating with a certain frequency. The solution is built in two ways. In the first of them, on the basis of asymptotic analysis, a solution was constructed in the low-frequency domain, and a complete coincidence of solutions for the models under consideration in the low-frequency domain (up to the first resonance) for any laws of inhomogeneity was shown. In the second case, the influence of rheological factors on the amplitude-frequency characteristics of an inhomogeneous viscoelastic beam in the frequency range up to the third resonance was analyzed using the targeting method. A comparative analysis of the frequency response of two models for different laws of heterogeneity is carried out. Frequency response for various laws of inhomogeneity are presented, the movement of resonant frequencies depending on the dimensionless relaxation time is studied.

Sensitivity Identification of Low-frequency Cantilever Fibre Bragg Grating Accelerometer

Vibration response of low-frequency cantilever fibre Bragg grating (FBG) accelerometer produced by Euler–Bernoulli model (namely FBG-MM model) is found to be frequency-dependent, unsimilar to SDOF model. Therefore, the sensitivity of the cantilever FBG accelerometer could not be identified using polynomial or basic fitting methods. This paper presents the use of cascade-forward backpropagation neural network (CFB) to predict the sensitivity of the cantilever FBG accelerometer in a "black box", which refers to the behaviour of the deep neuralnetwork. The inputs of the network are maximum base accelerations and forcing frequencies, which was set between 20 and 90 Hz (below than the first fundamental frequency of the proposed FBG accelerometer), while the output is the wavelength shift. The validation results show that the wavelength shift predicted by the trained CFB demonstrates good agreement with the FBG-MM, with the input parameter within the range of that used in training process. In addition, results also show that the trained CFB would be invalid if the input parameter is out of the range of that used in training process. In real acceleration measurement, since the forcing frequency is unknown beforehand, the trained CFB must be re-trained by considering the maximum base accelerations are embedded with forcing frequencies.

Nonlocal Timoshenko modeling effectiveness for carbon nanotube-based mass sensors

Carbon nanotubes (CNTs) have been considered for a wide range of nanotechnology applications due to their unique mechanical, thermal, and electrical properties. One of the most notable nanotechnology applications is the frequency shift-based carbon nanotube mass sensor, which has motivated the development of reliable reduced-order models. As such, this research effort focuses on a carbon nanotube-based mass sensor that utilizes Timoshenko beam theory to extend the usability of these models to short and stout structures. Additionally, Eringen's nonlocal elasticity is considered to account for certain nanoscale phenomena. To the authors' knowledge, this is the first time a CNT-based mass sensor modeled with the nonlocal Timoshenko beam theory with a complex-shaped deposited particle has been studied. Considering Timoshenko beam theory and Eringen's nonlocal theory, the Hamilton's principle is utilized to derive the governing equations of motion, boundary conditions, and continuity conditions accounting for a single deposited nanoparticle. Then, it is shown that the obtained frequency shift of the sensor is dependent on the rotary inertia, shear effects, nonlocal parameter, and particle mass, geometry, and location. The variation of each of these parameters leads to a holistic characterization of the proposed system. The discrepancies and limits of applicability between Timoshenko and Euler-Bernoulli models are deeply explored and discussed, particularly for short and stout structures. It is shown that the Euler-Bernoulli model leads to an overprediction of the frequency shifts compared to Timoshenko beam theory, especially for higher modes. The nonlocal Timoshenko-based model is valuable because short and stout structures were found to be preferred for mass sensing applications, since they lead to a higher sensitivity. Other researchers can utilize these findings for the design, modeling, and analysis of nanoscale sensors and resonators.