Euler-Bernoulli Beam Research Articles

Simultaneous determination of Young's modulus and residual stress via dynamic analysis of nanobridge exhibiting beam-to-string behaviors using a genetic algorithm

The mechanical properties of nanostructures, particularly Young's modulus and residual stress, exert significant influence on the performance and reliability of nanomechanical devices. However, conventional methods prove inadequate for evaluating these properties due to the impact of fabrication processes on nanostructures. This study addresses this challenge by proposing a dynamic model that considers the transition between “beam-like” and “string-like” characteristics under residual stress, a factor overlooked in previous research. Through analysis of nanobridge vibration modes and utilization of genetic algorithms (GAs) to reverse-engineer Young's modulus and residual stress from resonant frequencies, we introduce a novel and precise methodology. In contrast to the conventional Euler-Bernoulli beam theory, our proposed model integrates the system's kinetic and potential energies for resonant frequency calculation. This energy-based approach, grounded in physical principles, excels at capturing dynamic changes in structural behavior. Particularly in the critical “transition region” where the resonator shifts from “beam-like” to “string-like” behavior, our method adeptly accounts for energy conversion processes, resulting in more accurate resonant frequency estimations. By comparing with existing published data, our proposed method achieves a maximum error of 5.87 % in the determination of Young's modulus and residual stress of nanostructures.

A phase-field model for the brittle fracture of Euler–Bernoulli beams coupling stretching and bending

Damage gradient models seek to simulate fracture mechanics by modulating the material stiffness. Within this framework, a singular scalar field representing damage is commonly utilized to globally decrease elastic energy. However, when considering structural models like beams and plates, this approach often fails to adequately capture important aspects due to the interaction between stretching and bending contributions. In this work, we propose a model for planar Euler–Bernoulli beams that incorporates the following features: firstly, we utilize two phase-field damage parameters to describe material damaging, specifically addressing the ‘erosion’ occurring above and below the original, undamaged beam; secondly, we assume a simple linear dependence of the axial stress field on the thickness coordinate, along with linear dependence on the axial force and bending moment. By appropriately identifying the constitutive response, our model effectively considers the coupling between stretching and bending induced during through-the-thickness damage, demonstrating good agreement with 2D observations.

Analytical solutions for joints in an infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to time-lag loads

This paper investigates the analytical method for the infinite beam with joint parts resting on a foundation under time-lag loads. The ground is simulated as a viscoelastic foundation. The existence of joint parts in beams is explained by introducing virtual forces expressed through the Dirac delta function. Integral transformation simplifies the dynamic governing equations for standard beam sections and joint parts into an algebraic equation. The convolution theorem converts the solutions into the time domain. The analytical solutions for dynamic analysis of standard beam sections and joint parts are obtained conveniently in the frequency domain. The proposed analytical method is verified by comparing it with the traditional methods and also validated by numerical examples with time-lag loads. Parametric discussions study the effects on the dynamic response of joint parts and standard beam sections, such as the amplitude and frequency of time-lag loads and coefficient of bending stiffness reduction of joint parts. The parametric analyses emphasize the dramatic effect of critical parameters on the discontinuous deformation of beams. The damage caused by critical values of some factors is emphasized. The severe effects of discontinuous deformation induced by dynamic loads in the safety of beam structures are revealed in this paper. The differences in dynamic responses between the beam and joint part must be monitored to ensure the safety of whole beam structures.

A new buckling model for thin-walled micro-beams based on modified gradient elasticity: Coupling effect and size effect

More and more thin-walled micro-beams, including the I-section thin-walled micro-beams, are applied in MEMS. The buckling of a thin-walled micro-beam exhibits some different phenomena from that of a large scale thin-walled beam, such as the size effect, and the coupling effects caused by strain gradient. To investigate the influence of the coupling effects and size effect on the buckling of thin-walled micro-beams, a new buckling model is proposed. Based on modified gradient elasticity (MGE), the governing equations and boundary conditions of the new model are derived from the variational principle. The governing equations and boundary conditions can be simplified to the classical Vlasov's theory, the MGE Bernoulli–Euler beam model, MGE Bernoulli–Euler beam model considering bi-directional flexure respectively. Solved by the feature expansion method, the size effects of the thin-walled micro-beam can be captured by the new model, that is the buckling load increases with internal length scales. Besides the classical flexural–torsional coupling effect, two new coupling effects, such as the higher-order flexure–flexure coupling and the higher-order flexural–torsional coupling, are also captured by the new model. These two new high-order coupling effects of the new model reduce the critical buckling load of thin-walled micro-beams. The higher-order flexural–flexural coupling effect is far greater than the other two coupling effects, and the higher-order flexural–flexural coupling effect is on the same order of magnitude as the effect of MGE (size effect), with the former being negative and the latter positive. The size effect and higher-order coupling effect cannot be ignored and need to be considered carefully for the buckling of thin-walled micro-beams.

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.

Free vibration for the thermo-magneto-electro-elastic (TMEE) nanobeams based on modified couple stress theory

Size-dependent vibration of thermo-magneto-electro-elastic (TMEE) nanobeams accounting for the thermal entropy is implemented. Modified couple stress theory is adopted, Euler-Bernoulli beam model and strain gradient theory are adopted. Free vibration of TMEE nanobeams is demonstrated, dispersion equation is depicted, and natural frequencies are evaluated. Effects of various parameters such as volume fraction, size-dependent parameter, thickness and length of the nanobeam are investigated. The conducted results perfectly match with the ones in existing literature, and the newly invented governing equation is rather compact, which can be easily used to efficiently estimate the natural frequencies of the TMEE nanobeams.

A new method for rapidly capturing the strength and full nonlinear response of partially interacting steel–concrete composite beams

A semi-analytical procedure is presented for predicting the complete flexural response of partially interacting steel–concrete composite beams up to failure. The governing equation of the Euler–Bernoulli beam theory is solved wherein concrete, steel and the shear connectors joining the concrete slab to the steel beam are assumed to have nonlinear stress-deformation relationships. The adopted constitutive relationship for the connectors allows for partial or full composite action. The solution is applicable to beams and one-way slabs subjected to concentrated or uniform load and/or their combination. The governing equation is numerically solved by satisfying the equilibrium and compatibility requirements along the member. For the reinforced concrete part of the composite beam, a nonlinear moment–curvature relationship is developed that accounts for concrete nonlinearity in compression and for cracking and tension-stiffening in tension as well as for steel reinforcement nonlinearity. The steel profile is assumed to have a bilinear elasto–plastic strain-hardening moment–curvature relationship. Comparison of the proposed model results with the corresponding experimental load–deflection curves and interfacial shear–slip curves of several beams tested by others shows good agreement. The relative simplicity, efficiency and easy application of the present solution make it possible to accurately predict the failure load, interfacial slip and full nonlinear response of partially interacting composite beams.

RBF neural network disturbance observer-based backstepping boundary vibration control for Euler–Bernoulli beam model with input saturation

The main objective of this paper is to address the issue of vibration control for a class of Euler–Bernoulli beam systems that are subject to external disturbances and input saturation. The proposed controller differs from other backstepping methods in that it employs a radial basis function (RBF) neural network to accurately estimate boundary disturbances and incorporates the hyperbolic tangent function to ensure input constraints. The nonlinear partial differential equation (PDE) model is initially derived based on Hamilton’s principle to capture the dominant dynamic characteristics of the flexible beam. In the framework of the Lyapunov direct approach, an adaptive RBF neural network-based law is subsequently designed to estimate the state-related boundary disturbances. The backstepping approach is then developed to propose sufficient conditions for ensuring the stability and convergence of closed-loop systems subject to input saturation. Finally, the effectiveness and superiority of the proposed methodology are further demonstrated by comparing the simulation results of constrained backstepping controllers.

Mechanical responses of shallowly buried pipelines to adjacent deep excavations considering heavy rainfall influence

Climate change has resulted in a high global incidence of heavy rainfall, severely damaging the urban infrastructure. The analytical method used to predict the mechanical response of pipelines under the influence of rainfall influence could serve as a valuable design basis in the preliminary stage. This study developed an analytical solution based on the stratified Green-Ampt model and displacement-controlled two-stage method. The first stage evaluates the greenfield soil displacement at pipeline elevation by employing the Mindlin solution and considering the superposed effects of excavation (unloading) and rainfall infiltration. The second stage predicts the mechanical responses of the pipeline by simplifying the pipeline–soil interaction into a continuous Euler–Bernoulli beam resting on a Winkler foundation model. The analytical solution was calculated using the finite difference method and verified against a published study. Finally, parametric studies were conducted to investigate the rainfall– excavation–pipeline interaction, including the rainfall intensity and duration, maximum lateral wall deflection, and pipeline location. The results showed that a 48-8-mm/h rainfall increased the maximum displacement and maximum bending moment of the pipeline by 6.54% and 7.93%, respectively. The prediction results provide practical guidance for designing deep excavations adjacent to pipelines buried under heavy rainfall conditions.

DYNAMIC RESPONSE OF ELASTICALLY-SUPPORTED BRIDGE UNDER OPPOSITE MOVING HIGH-SPEED LOADS

This paper explores the design parameters for elastically supported (ES) bridges, particularly for short to medium under the influence of high-speed trains. It introduces a non-dimensional mathematical framework that conceptualizes the ES as a Euler-Bernoulli beam supported by elastic bearings at its ends and subjected to evenly spaced moving loads. The framework highlights various resonances and speed ratios for cancelling out specific effects based on the stiffness parameter and the distance between loads. The analysis indicates that most of the maximum displacement occurs near the beam's midpoint. Additionally, it shows that lower resonance speeds coupled with higher stiffness parameter values can result in significant amplification of displacement response due to cumulative forces acting on the system. This amplification is a consequence of the cumulative forces acting on the system. Hence, this research provides valuable insights for designers and engineers to optimize foundation design, thereby reducing structure-borne vibrations transmitted to the ground.

DYNAMIC RESPONSE OF ELASTICALLY-SUPPORTED BRIDGE UNDER OPPOSITE MOVING HIGH-SPEED LOADS

This paper explores the design parameters for elastically supported (ES) bridges, particularly for short to medium under the influence of high-speed trains. It introduces a non-dimensional mathematical framework that conceptualizes the ES as a Euler-Bernoulli beam supported by elastic bearings at its ends and subjected to evenly spaced moving loads. The framework highlights various resonances and speed ratios for cancelling out specific effects based on the stiffness parameter and the distance between loads. The analysis indicates that most of the maximum displacement occurs near the beam's midpoint. Additionally, it shows that lower resonance speeds coupled with higher stiffness parameter values can result in significant amplification of displacement response due to cumulative forces acting on the system. This amplification is a consequence of the cumulative forces acting on the system. Hence, this research provides valuable insights for designers and engineers to optimize foundation design, thereby reducing structure-borne vibrations transmitted to the ground.

Nonlinear oscillation of microscale fiber-reinforced composite laminated beams under a thermal loading

The work analyzes the nonlinear dynamic response of fiber-reinforced composite laminated microbeams in thermal environment, accounting for the dependence of the material properties on temperature. The governing equations of the nonlinear partial differential equations are based on the Euler-Bernoulli beam theory, and the modified couple stress theory involving the von Kármán geometrical nonlinearity. The governing nonlinear equations are reduced to a single equation by neglecting the axial inertia, which is discretized in the form of nonlinear ordinary differential equation, according to the Galerkin method, and it is solved analytically using the multiple time scales (MTS) method. A systematic investigation checks for the effect of the material length scale parameter, temperature increase, and damping parameter of the microscale beam on its time history and phase plane trajectory. Different stacking sequences are also considered within the parametric investigation. Based on results, it is noticed that the system is in a damped period-doubling bifurcation for four cycle oscillation states. It is also noticed that the stacking sequence affects significantly the nonlinear dynamic behavior of the system. It is found that the microscale composite laminated beam with [0/0/0] lay-ups has the highest dynamical deflection and the highest nonlinear frequency, followed by [0/90/0], [30/-30/30] (angle ply), [90/0/90], [45/-45/45] (angle ply) and [60/-60/60] (angle ply). However, the microscale composite laminated beam with [90/90/90] lay-ups has the lowest dynamical deflection and the lowest nonlinear frequency.

Homogenization Model for Multistable Honeycomb Metastructures with Beam-like Behavior

Reduced order models facilitate initial design space investigations and enable assessing the benefits of compliant structures utilized for shape adaptability. This work presents a simple model to determine the flexural rigidity of a beam-like, multistable metastructure used as a spar in a hybrid spanwise morphing wing. The model considers the more complex metabeam as a homogeneous beam described by Euler-Bernoulli beam theory with an equivalent flexural rigidity. The analytical model's validity is established by comparing the obtained static and dynamic responses to finite element simulations. A closed-form expression of the flexural rigidity is then given, drawing from the multistable honeycomb's material properties and the metabeam’s geometry. The model's limitations are addressed by examining several specific cases of the metabeam’s morphed configurations and a more complex metabeam structure.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.

A study on vortex-induced vibration of a long flexible catenary cable in perpendicular flow

In this paper, a numerical model based on Euler-Bernoulli beam theory and a single van der Pol wake oscillator model is proposed to study the vortex-induced vibration of a long flexible catenary cable with a low mass ratio. The wake model uses only one oscillator to describe the in-line and cross-flow vibration with consideration of their nonlinear coupling. An experiment on a flexible cable model in perpendicular flow is carried out to validate the proposed numerical model. Numerical results and experimental measurements are compared on the vortex-induced vibration responses including the maximum amplitude, axial tension, and vibration frequency. A sensitivity analysis of the parameters used in the numerical model has been carried out. Numerical simulations by the proposed numerical model with properly selected parameters have demonstrated that the vibration frequency and the axial tension can be well predicted, and the variation of the vibration amplitudes can also be properly captured. The lock-in phenomena and the axial tension variation of the long flexible catenary cable in perpendicular flow have been carefully studied both numerically and experimentally.

Longitudinal analysis model for segment lining uplift during shield tunnelling considering shearing dislocation of circumferential joints

During shield tunnelling, longitudinal deformation occurs in the shield tunnel structure owing to the buoyancy generated by the synchronous slurry, which has adverse effects on the assembly quality and even the structural integrity of the segmental ring structure. However, the current longitudinal beam-like model for analysing the uplift response during shield tunnel construction adopts the Euler–Bernoulli beam to describe the shield tunnel. This model does not consider the shearing dislocation effect, causing errors in the structural longitudinal deformation analysis. In this context, this study proposes a shield tunnelling uplift model based on the Timoshenko beam theory, which can effectively consider the shearing dislocation effect, aiming to evaluate the longitudinal response of shield tunnels more accurately. A Pasternak two-parameter foundation with springs reflecting the resistance of the overburden with different thicknesses was adopted to simulate the supporting effects of the synchronous grouting layer and surrounding soil layer composite on the tunnel, considering the influences of both the shearing effect of the foundation and tunnel buried depth on the shield tunnel uplift. The dynamic buoyancy coefficient, determined through an inverse analysis of the field measurement data, was introduced to accurately calculate the magnitude of the dynamic buoyancy generated by the flow of synchronous slurry. The proposed model was validated by comparing its results with actual measurements. The uplift characteristics of the shield tunnel during construction were obtained by using the proposed model in an actual project. It was found that the longitudinal analysis model for shield tunnel uplift based on the Euler–Bernoulli beam theory underestimates the uplift and overestimates the internal forces generated during shield tunnelling.

Free vibration analysis of soil-beam-fluid interaction with variable Winkler backfill using DTM

This study aims to analyze the free vibrations of the Euler-Bernoulli beam vibrating in the vicinity of an inviscid, irrotational, and incompressible fluid, as well as Winkler soil, using the differential transformation method. To show the robustness of the proposed formulation, it is used in different numerical tests in which the free vibration of different soil-beam-fluid systems is assessed. The differences between these systems are rooted in their supports and the soil characteristics. Furthermore, the results achieved from the proposed method are compared with those of the finite element strategy. It is emphasized that the differential transformation method employed in this study showcases sufficient precision and reliability throughout the analysis process.

Pseudo-static Winkler springs for longitudinal underground structures subjected to shear waves

ABSTRACT Shear waves in underground longitudinal structures impose deformations that need to be accounted for in structural design. Simplified approaches for such structures typically consider the Winkler foundation model, assuming Euler-Bernoulli beam theory, which neglects shearing-induced distortions, especially significant when shear waves are a dominant component of the seismic motion. In order to overcome these limitations, Timoshenko beam models have been proposed in the literature. These approaches however depend on an appropriate determination of the ground springs. Existing analytical formulations often assume plane-strain conditions, inadequate for representing low frequencies and thus are not directly applicable for pseudo-static interaction analyses. The present paper develops analytical solutions to determine transverse and rotation Winkler springs for structures subjected shear waves. The proposed springs overcome the drawbacks of plane-strain models and can be construed as a generalisation of them. The springs are obtained as a function of the seismic wavelength and the ground-structure stiffness contrast. Results obtained are validated against solutions from the literature and numerical results from a full 3D finite-element model. A non-dimensional parametric study is also presented, that allow an expedited evaluation of ground springs for practical applications.

Non-smooth dynamics of impacting viscoelastic pipes conveying pulsatile fluid

This paper aims to investigate the stability and nonlinear dynamics of an initially curved viscoelastic clamped–clamped pipe conveying pulsatile fluid, with special attention to the non-smooth dynamics of this gyroscopic system under the influence of initial curvature and gravity. In this study, the impacting contact is modeled as a trilinear repulsive force. Based on the Euler–Bernoulli beam theory, the nonlinear governing equations, which take into account geometric, curvature, and damping nonlinearities, are developed using Hamilton’s principle. The continuous model is discretized using the Galerkin method and then solved using the arc-length technique and a time integration method. A linear analysis of the natural frequency and complex mode shape shows the unexpected frequency intersection and modal asymmetry. The qualitative changes in the nonlinear dynamics of the impacting pipe in both sub- and super-critical regions are examined and presented in the form of bifurcation diagrams, frequency–amplitude curves, force–amplitude curves, time histories, and phase portraits. Numerical results show that some complex dynamical behaviors can occur in different impact constraint cases, including a grazing bifurcation, sticking, period doubling, and chaos at the impact location, may occur. The initial curvature and gravity result in different behaviors, such as the asymmetry change of the equilibrium configuration and the occurrence of the flutter instability. The current analysis is significant to understand the non-smooth dynamic mechanisms and also provide design and technology guidance for the pipe system with motion-limiting constraints.

Modeling and experiments on flexible side-electrodes electrostatic actuators: Influence of the contact interface on the available force

This paper presents an in-depth study of electrostatic actuators based on flexible side-electrodes focusing on the available electrostatic force and the derived useful force. A design tool is proposed considering technological aspects as aspect ratio, insulating layer material and voltage level. A space discretized mechanical approach based on Euler-Bernoulli beam theory is developed to model the bending deflection of the flexible movable electrode and “zipping” contact. Analytical developments and simplifications are achieved, rapidly and efficiently implemented in a proposed algorithm. ANSYS® simulations validate the procedure for updating the contact zone through the updating of the electrostatic pressure distribution. Optimum movable electrode width and appropriate “opposable” stiffness for guiding system are discussed. A “residual air thickness” parameter is introduced to address imperfect contact between the electrodes. Experimentalcharacterization studies various prototypes with different structural parameters as thickness (50 µm and 200 µm), gaps (5 µm and 15 µm) and material for insulating layers (silicon oxide and parylene C). The metrology campaign highlights the impact of aspect ratio and oxidation/deoxidation on electrode sidewall quality and hence mechanical performance.A comparison with well-known comb-drive shows higher electrostatic force for flexible side-electrode actuators. As an example, for a 10 µm stroke, the available electrostatic force for flexible electrodes with irregular sidewalls (residual air thickness 0.6 µm) is 2.2 times greater and rises to 7.3 times greater for perfect sidewalls (no residual air).