Euler Bernoulli Research Articles

Optimizing Coupled Fluid-Structure Simulations for Nuclear-Relevant Geometries

Abstract The numerical simulation of fluid-structure interactions (FSI) has gained interest to study flow-induced vibrations. Nevertheless, the high computational resources required by such simulations can represent a significant limitation for their application to industrial configurations. Therefore, simplified modeling approaches, when physically applicable, can represent an interesting compromise. This can be the case of slender structures (tubes, rods) often encountered in nuclear power plants. In this paper, an Euler–Bernoulli beam finite element model is implemented inside the computational fluid dynamics (CFD) code code_Saturne. With the goal of finding CFD methods less expensive than large eddy simulations (LES), unsteady Reynolds Navier–Stokes (URANS) and hybrid URANS/LES approaches are considered. The resulting fluid-structure model is able to calculate the vibration response of cantilever beams under a fluid flow, avoiding the necessity of CFD-finite element method (FEM) code coupling. The first part of the paper describes the model and its implementation: it allows to perform 2-way explicit fluid-structure coupling, using the Arbitrary Lagrangian-Eulerian approach to account for the structure deformations. Validation test cases are presented in the second part: first, the model is validated in terms of frequency, added mass, and damping for a cylinder vibrating in static air and water; then, the model is validated toward the vortex-induced resonance and lock-in mechanisms for a cylinder subjected to water cross-flow. The model is then applied to a real experimental configuration of two in-line cylinders in water cross-flow: the calculated vibrations are found to be in good agreement with the experimental measurements.

Dynamic analysis and stability of variable length vertical pipe containing fluid using Timoshenko’s beam theory

In this article, the dynamics of pipes containing fluid is investigated according to Timoshenko’s beam theory. The problem’s equations were extracted with the help of Hamilton’s principle, and then their responses were obtained using Galerkin’s numerical approach. The modeling of the vertical pipe containing fluid was done using Timoshenko’s beam theory. It is prominent to mention that the considered vertical pipe containing fluid has variable length. The results of this analysis will be compared with previous researches, which were according to Euler–Bernoulli theory. Also, the gravity effect on the pipe stability will be analyzed. Finally, by examining the impact of rotational inertia and shear deformation separately, it was determined that the effect of shear deformation on the pipes’ dynamic behavior containing fluid will be more than the effect of rotational inertia.

Study on coupled mode flutter parameters of large wind turbine blades

As the size of wind turbine blades increases, the flexibility of the blades increases. In actual operation, airflow flow can cause aerodynamic elastic instability of the blade structure. Long blades may experience coupled mode flutter due to the bending torsion coupling effect, leading to blade failure. Based on Euler Bernoulli beam theory combined with Theodorsen non directional aerodynamic loads, a blade flutter characteristic equation is established through finite element method. Taking NREL 5 MW wind turbine blades as an example, analyze the influence of parameter changes in different regions of the blades on flutter characteristics. Research has found that paramter changes in the tip region of blade have the greatest impact on flutter characteristics. The vibration frequency shows an overall upward trend with the increase of waving stiffness and torsional stiffness. The flutter velocity of the three regions tends to stabilize as the bending stiffness decreases. The blade flutter speed increases with the increase of torsional stiffness. The radius of gyration is inversely proportional to the flutter frequency and flutter velocity. The impact of centroid offset on blade structure flutter frequency is minimal, but the centroid offset in the tip region has a greater impact on flutter velocity. Increasing the torsional frequency can prevent coupled mode flutter and provide a theoretical basis for blade flutter prevention design.

A Numerical Analysis of the Non-Uniform Layered Ground Vibration Caused by a Moving Railway Load Using an Efficient Frequency–Wave-Number Method

The ground vibration caused by the operation of high-speed trains has become a key challenge in the development of high-speed railways. In order to study the train-induced ground vibration affected by geotechnical heterogeneity, an efficient frequency–wave-number method coupled with the random variable theory model is proposed to quickly obtain the numerical results without losing accuracy. The track is regarded as a composite Euler–Bernoulli beam resting on the layered ground, and the spatial heterogeneity of the ground soil is considered. The ground dynamic characteristics of an elastic, layered, non-uniform foundation are investigated, and numerical results at three typical train speeds are reported based on the developed Fortran computer programs. The results show that as the soil homogeneity coefficient increases, the peak acceleration continuously decreases in the transonic case, while it gradually increases in the supersonic case, and the ground acceleration spectrum at a far distance obviously decreases; the maximum acceleration occurs at the track edge, and a local rebound in vibration attenuation occurs in the supersonic case.

Analytical solution for the interfacial stress and energy release rate at failure initiation of the three-point bending test (ISO 14679:1997)

Determining adhesive failure in adhesively bonded joints is still challenging in the field. While the three-point bending test (3PBT) (ISO 14679:1997) has been helpful in identifying critical forces and displacements related to bond strength, it only allows for a qualitative assessment of the bond line. Recently, a new quantitative methodology has been developed to determine critical stress and fracture toughness using the coupled stress–energy criterion and the 3PBT. However, these assessments require a significant effort using semi-analytical or finite element (FE) analysis. Therefore, the present study proposes a reliable set of analytical equations to determine peel and shear stress distributions using a weak interface formulation and 1D Euler–Bernoulli approach. More precisely, a particular numerical method computes the integration constants from these equations. A new method has been proposed for calculating the interfacial stiffness in peel and shear mode, based on the material and geometrical parameters of the geometry. This method differs from the previous approach, where the interfacial stiffness was calibrated from the experimental behavior of the test. The whole approach has been validated through experimental and numerical analysis, including the costly 3D FE analysis. An analytical expression for interfacial energy release is suggested, developed following the works of Fraisse and Schmit on J-integral assessment of sandwich-type overlaps and depending on the applied force and a rotation, which could be experimentally measured. Therefore, this work is significant progress in determining bond strength using a simple mechanical test and equations applicable to various industries.

Periodic Solutions of the Euler–Bernoulli Quasilinear Vibration Equation for a Beam with an Elastically Fixed End

Periodic Solutions of the Euler–Bernoulli Quasilinear Vibration Equation for a Beam with an Elastically Fixed End

A numerical investigation on the vibration of a two-deck Euler–Bernoulli beam flooded by a potential flow

Abstract This work concerns the structural vibration of a bladeless wind turbine, modelled by a two-deck Euler–Bernoulli beam, due to a surrounding potential flow. The deflection is governed by the Euler–Bernoulli equation which is studied first by a linear theory and then computed numerically by a finite difference method in space with a collocation method over the arc length, and an implicit Euler method in time. The fluid motion in the presence of gravity is governed by the full Euler equations and solved by the time-dependent conformal mapping technique together with a pseudo-spectral method. Numerical experiments of excitation by a moving disturbance on the fluid surface with/without a stochastic noise are carried out. The random process involved in generating the noise on the water surface is driven by a Wiener Process. A Monte Carlo method is used for stochastic computations. The generated surface waves impinge on the beam causing structural vibration which is presented and discussed in detail. By elementary statistical analysis, the structural response subject to the stochastic hydrodynamic disturbance caused by white noise is found to be Gaussian.

Energy harvesting from vibrations of a beam under moving mass using auxetic cantilever beams: Theoretical and experimental investigations

Recent research aims to enhance piezoelectric energy harvesting for low-power electronic circuits like sensors and wireless systems using cantilever energy harvesters with piezoelectric substrates featuring auxetic structures such as reentrant, missing rib, and double arrowhead patterns. Employing the Wentzel–Kramers–Brillouin (WKB) method is a novel method to analyze these structures based on Euler–Bernoulli beam theory. The study comprises theoretical analysis, finite element simulations, and experimental tests. The highest power output is achieved when consecutive masses excite the bridge using the double arrowhead auxetic unit cell substrate, yielding power outputs of 1.152 μ W / N . The effect of increasing unit cells has also been investigated. Highlights Creating unique analytical methods for auxetic substrate use in energy harvesters. Analyzing three auxetic harvester substrates, emphasizing their unique performance. Auxetic harvesters were studied under various bridge simulator base excitations. The study examines three auxetic structures' effect on harvested power. Study of moving masses' influence on bridge simulator, weights, and velocities.

Improved lumped parameter model for shear wall–frame structures with geometric and material nonlinearities

Lumped parameter macroscopic structural models, such as those consisting of springs and rigid sticks, are useful for parametric studies and probabilistic analyses of individual building structures as well as for regional-level assessments on building stock. The existing lumped parameter models (LPMs) for shear wall–frame dual systems exhibit some drawbacks. There is a discrepancy in their arrangement of springs. They are calibrated to match only two modal periods of the original multi-story or even high-rise building structure. The existing models also have limitations when confronted with the P-Δ effect. After a careful investigation, this paper developed new schemes of spring-stick assemblies that accurately reproduce the elastic stiffness matrix of a two-dimensional Euler–Bernoulli or Timoshenko beam–column element and cope with geometric and material nonlinearities. The developed assemblies, which are essentially lumped parameter substitutes for beam–column elements, can be used to build LPMs for moment frames and shear walls. For moment frames, the traditional fishbone model is converted into a LPM that not only reflects the beam-to-column stiffness ratio and strength ratio but also captures the P-Δ effect. Combining the LPM for shear wall with the lumped parameter fishbone model results in an improved LPM for wall–frame dual systems. Together with lumped floor masses, the improved LPM is able to accurately predict all the global vibration modes and nonlinear seismic responses of a dual system. The proposed LPM possesses distinct advantages when simplified macroscopic models are preferred for building structures.

Verification of Euler–Bernoulli beam theory model for wind blade structure analysis

This paper presents a computational tool for analyzing the structural response of wind turbine blades using the advanced Euler–Bernoulli Beam theory. The objective is to achieve efficient computation of blade structural response under aerodynamic loading without sacrificing computational accuracy. This study involves the development of a Python-based computer program as a numerical tool, which utilizes inputs such as blade geometry and aerodynamic loading from Blade Element Momentum theory. The program outputs axially varying moments of inertia, spatial deflections of the blade, and stress distributions within the blade. The numerical tool is verified by comparing the results with both analytical solutions and far more time-consuming commercial finite element solvers. One of the key contribution of this research is the application of the polygon algorithm, which efficiently calculates the moments of inertia of web-stiffened turbine blades along the blade length. It is shown herein that this algorithm significantly reduces computational time while maintaining numerical accuracy as compared to the results obtained from commercially available finite element codes. The study confirms that the advanced Euler–Bernoulli beam theory, when combined with the polygon algorithm, is capable of accurately predicting the structural response of cambered and twisted wind turbine blades during the initial design phase.

Analysis of bending vibrations of a three-layered pre-twisted sandwich beam with an exact dynamic stiffness matrix

This paper presents the novel development of the dynamic stiffness matrix for a three-layered symmetric pre-twisted sandwich beam (PTSB), aiming to investigate its free vibration characteristics. The outer layers of the beam are modeled using the Euler–Bernoulli theory, while the core is assumed to deform solely in shear. The boundary conditions and governing partial differential equations of motion are derived based on Hamilton's principle. By applying harmonic variations of displacements, the governing equations of motion are expressed as a tenth-order equation, which is solved to obtain the desired dynamic stiffness matrix. To compute the natural frequencies of in-plane and out-of-plane free vibration for both uniform and PTSBs, the Wittrick–Williams algorithm is employed. The computed frequencies are compared with the results obtained by other authors as well as those obtained from ABAQUS simulations. Various vibration modes of uniform and twisted sandwich beams are plotted and thoroughly discussed. Interestingly, contrary to straight symmetric sandwich beams, the results indicate that flexural displacements in pre-twisted symmetric sandwich beams exhibit coupling in two planes. Additionally, although there are minor changes in vibration frequencies, the mode shapes undergo significant transformations as the pre-twist angle is altered.

The Shear Effect of Large-Diameter Piles under Different Lateral Loading Levels: The Transfer Matrix Method

In various analytical models, modeling the behavior of large-diameter monopiles and piles can be challenging due to these foundations with huge body sizes carrying mechanisms of lateral loads to the surrounding soils. In this paper, the transfer matrix method with the Timoshenko beam theory was used to modify the shear rotation of pile sections under different loading stages, including serviceability limit stages and the ultimate loading stage. In this transfer matrix method, a large-diameter pile is considered according to the Timoshenko beam theory, and the recurring variables in the matrix equation are replaced with constants to simplify the calculation steps. Two model test cases were used to verify the accuracy of the method. Then, a series of comparisons between the Timoshenko beam and the Euler–Bernoulli beam theories, with the relative pile–soil stiffness being equal to 0.15, 0.45, and 0.75, was conducted to investigate the differences in pile response after considering the shear deformation. The results show that the effect of shear deformation of large-diameter piles changes with different loading levels. The values of the pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory were in the range of 1.0 to 1.10, and they increased slightly with increasing loads, reaching their maximum value, and then rapidly decreased to 1.0 when close to the ultimate lateral load; the maximum value was influenced by the relative pile–soil stiffness. Furthermore, the ratio of the shear rotation of the pile section to the slope of the deflection curve was in the range of 1.0 to 1.10; these also showed similar but more moderate trends compared with the values of pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory.

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.

Elastic solids under frictionless rigid contact and configurational force

A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a two-dimensional formulation. With a strong analogy to fracture mechanics, it is shown that (i.) a path-independent J–integral can be defined for frictionless contact problems, (ii.) which is equal to the energy release rate G associated with an infinitesimal growth in the size of the frictionless constraint, and thus gives the value of the configurational force component along the sliding direction. Furthermore, it is found that (iii.) such a configurational sliding force is the Newtonian force component exerted by the elastic solid on the constraint at the frictionless contact. Assuming the kinematics of an Euler–Bernoulli rod for an elastic body of rectangular shape, the results (i.)–(iii.) lead to a new interpretation from a nonlinear solid mechanics perspective of the configurational forces recently disclosed for one-dimensional structures of variable length. Finally, approximate but closed-form solutions (validated with finite element simulations) are exploited to provide further insight into the effect of configurational forces. In particular, two applications are presented which show that a transverse compression can lead to Eulerian buckling or to longitudinal dynamic motion, both realizing novel examples of soft actuation mechanisms. As an application to biology, our results may provide a mechanical explanation for the observed phenomenon of negative durotaxis, where cells migrate from stiffer to softer environments.

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.

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.

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.

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.