Euler Beam Theory Research Articles

Size-dependent mechanics of viscoelastic carbon nanotubes: Modeling, theoretical and numerical analysis

This research presents the complex mechanics of the size-dependent viscoelastic carbon nanotube (CNT) subjected to the electric load. The mechanical model describing the motions of CNT is formulated by the Euler beam theory, Kelvin-Voigt viscoelastic model, and nonlocal continuum theory. The nonlinear governing equation and end conditions are deduced by the generalized Hamilton principle. The homoclinic structure of the CNT system is constructed using the reductive perturbation approach and the infinite-dimensional global perturbation method. Finally, using the extensive parametric analysis, the impact of electric excitation, nano-scale effect, and viscoelastic effect on the complex mechanics of CNT is simultaneously highlighted. Additionally, the comparison is also conducted with published researches to test the model and theoretical analysis.

Energy absorption characteristics of three-layered sandwich panels with graded re-entrant hierarchical honeycombs cores

Two innovative re-entrant hierarchical sandwich panels constructed by substituting the cell walls of re-entrant honeycombs with isotropic regular hexagon substructure (RHH) and equilateral triangle substructure (RHT) are proposed in this paper, and their crashworthiness performance has been investigated systematically. Based on the Euler beam theory, the Young's moduli of units of re-entrant hierarchical sandwich panels are derived. A comparison between the re-entrant hierarchical sandwich panels and re-entrant sandwich panel has been conducted to study their energy absorption ability. Furthermore, the parametric analysis based on numerical analysis has been carried out to discuss the effects of the gradient, arranged orientation of hierarchical sandwich cores and the impact velocities. Results show that both the two proposed sandwich panels can greatly improve the in-plane stiffness and the energy absorption capacity, and the energy absorption mechanisms are discussed. In addition, the graded sandwich panels can effectively reduce initial peak force under quasi-static compression, and the configuration of large- medium -small (LMS) performs the best under high impact velocity. The cross-arranged cores can significantly improve the impact resistance ability of RHT sandwich panels. This work provides a new solution for designing lightweight sandwich structures.

On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity

In the present paper, the effects of material properties, nonlocal parameter, Lorentz and electric forces on maximum static deflections and natural frequencies of actuated hybrid carbon/boron-nitride nanotubes (CBNNT) subjected to thermal loads are studied for the first time. The displacement field of the nanotube satisfies assumptions of the Bernoulli–Euler beam theory. The Green-Lagrange small strains and moderate rotations for geometric nonlinearity of the nanotube are taken into consideration. Two nonlinear governing equations of motion for carbon and boron-nitride parts of the clamped nanotube are derived using the d'Alembert principle and the stress-driven nonlocal integral elasticity theory. Solution to the formulated nonlinear boundary value problem was obtained using the Galerkin modal expansion method. Validation of obtained results and parametric study are comprehensively presented. Obtained results show effect of variation of temperature, nonlocal parameter, ratio of length of carbon and boron-nitride components, direct current (DC) and alternating current (AC) voltages on shifting of the fundamental frequency and maximum deflection of hybrid nanotube.

Analytical solution for a jointed shield tunnel lining reinforced by secondary linings

Based on the curved Euler beam theory, an analytical solution is proposed for a jointed shield tunnel lining reinforced by secondary linings, which can be idealized as a partial-interaction composite curved beam. The effect of joints and the interaction between linings and surrounding soils are simulated by a series of linear mechanical springs and Winkler type springs, respectively. The solutions for the internal forces and deformations can be derived by the state space method for conditions involving arbitrary loadings and variable joint distributions. The capability of the proposed analytical solution is verified by the finite element method and other existing analytical solutions. Furthermore, the effect of bending-direction-dependent joint stiffness is discussed, and the variation trends of internal forces and deformations, varying with soil reaction stiffness, are examined under different joint stiffness conditions.

Flexural wave propagation in periodic tunnels with elastic foundations

In order to meet the needs of vibration reduction in complex vibration environments, one-dimensional phononic crystals constituted with periodic tunnels on elastic foundations are proposed. The governing equations of the flexural wave in a periodic structure are established based on Euler beam theory and its band structure is calculated by the plane wave expansion method. The theoretical and numerical results show that there are distinct forbidden bands and pass bands when the flexural wave propagates in the periodic tunnel structure. The elastic foundation stiffness, elastic modulus ratio, and axial prestress have significant effects on the location and width of the band gap. The results of the paper could be used for the vibration mitigation of the tunnel structure.

Active tuning of flexural wave in periodic steel-concrete composite beam with shunted cement-based piezoelectric patches

In this paper, flexural wave in a steel-concrete composite beam with periodically surface-bonded cement-based piezoelectric patches (CPP) is studied. The composite beam is designed using the idea of the phononic crystals. The external capacitor shunt circuit is connected to the CPP for regulating the band gap of the composite beam. The flexural wave equations are established by Euler beam theory. Using plane wave expansion method and transfer matrix method, the complex band structures of the flexural wave are calculated to investigate the frequency range and the vibration reduction in band gap. From the numerical results, it is shown that by tuning the external capacitor shunt circuit, the band gap of the periodic structure can be regulated conveniently. The regulation capability of the negative capacitance shunt circuit is stronger than that of the positive capacitance. We can obtain different starting low-frequency and wide band gaps by using negative capacitance. The proposed approach gives a promising way to design intelligent beams whose band gaps can be tuned without modifying the structures.

Blast resistance evaluation of urban utility tunnel reinforced with BFRP bars

To improve corrosion-resistance of shallow-buried concrete urban utility tunnels (UUTs), basalt fiber reinforced polymer (BFRP) bars are applied to reinforce UUTs. As the UUT must have excellent survival capability under accidental explosions, a shallow-buried BFRP bars reinforced UUT (BBRU) was designed and constructed. Repetitive blast experiments were carried out on this BBRU. Dynamic responses, damage evolutions and failure styles of the BBRU under repetitive explosions were revealed. The tunnel roof is the most vulnerable component and longitudinal cracks develop along the tunnel. When the scaled distance is larger than 1.10 m/kg1/3, no cracks are observed in the experiments. When the BBRU is severely damaged, there are five cracks forming and developing along the roof. The roof is simplified as a clamped-supported one-way slab, proved by the observation that the maximum strain of the transverse bar is much larger than that of the longitudinal bar. Dynamic responses of the roof slab are predicted by dynamic Euler beam theory, which can consistently predict the roof displacement under large-scaled-distance explosion. Compared with the UUT reinforced with steel bars, the BBRU has advantages in blast resistance with smaller deflections and more evenly-distributed cracks when the scaled distance is smaller than 1.260 m/kg1/3 and the steel bars enter plastic state. Longer elastic defamation of the BFRP bars endows the UUT more excellent blast resistance under small-scaled-distance explosions.

Dynamic stability and bifurcation phenomena of an axially loaded flexible shaft-disk system supported by flexible bearing

The present research studies the vibration and bifurcation analysis of a spinning rotor-disk-bearing system to carefully scrutinize the dynamic stability under extrinsic mass unbalance and pulsating axial load. The shaft is flexible and taken into account the geometrical nonlinearities due to large elastic deformation in bending. The rotor is supported by flexible bearings, which are modeled as an equivalent spring-damper system having linear and nonlinear stiffness elements. Equation of motion of the rotating system, which includes flexible shaft, rigid disk, and flexible bearing, is derived using extended Hamilton principle with the assumption of the Euler's beam theory. We studied initially the modal analysis to determine the modal parameters, i.e. natural frequency and mode shapes prior to the investigation of the dynamics of the system. Further, we developed the bifurcation diagram for steady-state solutions and to study the subsequent dynamic stability and verify with the findings solved numerically. The interactive behavior among the nonlinear shaft-bearing, axial load and an unbalance has been analyzed. Numerical simulation tools, i.e. frequency–response characteristics, time history, phase trajectories, and Poincaré map have been used to highlight the presence of nonlinear phenomena and its important role towards evaluating the system dynamics and subsequent stability. This current research showed that flexible bearings stabilize the system as a result of increasing the restoring force. Analyzing the bifurcation diagrams of pulsating axial load, we found that the system exhibits complex phenomena as multiple but stable periodic orbits leading to period doubling. The present system is highly vulnerable to catastrophic failure due to the S–N and Pitchfork bifurcation. The present research enables the notability of axial load and mechanical unbalance on the overall system behavior and stability in real working conditions.

A modified beam theory for bending of eccentrically supported beams

According to the conventional beam theories, the bending analysis of a beam under transverse loading depends on support conditions in addition to the other parameters. The conventional beam theories assume that supports are solely placed at the mid-plane of the beam. However, in practice, the beams often are supported at the point different from their centers. In this study, a modified beam theory is presented for the static bending analysis of an eccentrically simply-supported beam under transverse uniform loading by the application of the MacLaurin series. This study presents a theoretical approach to the analysis of eccentrically supported beams. To overcome the limitation of the Euler beam theory mainly applicable to concentrically supported beams both the transverse deflection and the axial deflection are considered in order to deal with eccentrically supported conditions. The effects of eccentric supports on the flexural rigidity of the beam have been investigated. Analytic equations derived were used to investigate the effect of varying support positions through the thickness on bending analysis of beams under transverse loading. The findings revealed that the flexural rigidity of beams is significantly influenced by eccentric pin–pin support. The accuracy of the equations was verified by comparing the results obtained with the finite element solutions. The results show that the derived equations are not only accurate but also simple in predicting the bending of beams. Communicated by Francesco Tornabene

Size-dependent buckling analysis of nanobeams resting on two-parameter elastic foundation through stress-driven nonlocal elasticity model

The instability of nanobeams rested on two-parameter elastic foundations is studied through the Bernoulli–Euler beam theory and the stress-driven nonlocal elasticity model. The size-dependency is incorporated into the formulation by defining the strain at each point as an integral convolution in terms of the stresses in all the points and a kernel. The nonlocal elasticity problem in a bounded domain is well-posed and inconsistencies within the Eringen nonlocal theory are overcome. Excellent agreement is found with the results in the literature, and new insightful results are presented for the buckling loads of nanobeams rested on the Winkler and Pasternak foundations.

Free and steady forced vibration characteristics of elastic metamaterial beam

The propagation of elastic waves in infinite elastic metamaterials (EMs) is studied by using the effective medium theory (EMT). However, when EMs are applied in engineering practice, finite EM structures should obviously be the most cases, in which the propagation of elastic waves corresponds to the problem of vibration. Therefore, it is necessary to establish a reasonable vibration analysis method for typical EM waveguides and investigate their unique modal characteristics and related vibration characteristics. At present, the relevant exploration is relatively limited, especially analytical analysis. As one typical EM finite structure, an EM beam was chosen to be investigated in this study. Based on the EMT, the Euler beam theory is extended to the analysis of EM finite beams, and the analysis process of free and steady forced vibration is established, the particular characteristics are revealed, including the gathering of natural frequencies in the vicinity of band edges, the absence of natural frequencies within bandgap (BG), and the particular modal correspondence before and after BG. Then, the formation mechanisms of the characteristics are explained from the perspective of standing waves. The obtained results may give insights into the vibration analysis of other finite EM structures, such as EM rods, shafts, plates, and shells.

Analytic springback prediction in cylindrical tube bending for helical tube steam generator

This paper newly proposes an efficient analytic springback prediction method to predict the final dimensions of bent cylindrical tubes for a helical tube steam generator in a small modular reactor. Three-dimensional bending procedure is treated as a two-dimensional in-plane bending procedure by integrating the Euler beam theory. To enhance the accuracy of the springback prediction, mathematical representations of flow stress and elastic modulus for unloading are systematically integrated into the analytic prediction model. This technique not only precisely predicts the final dimensions of the bent helical tube after a springback, but also effectively predicts the various target radii. Numerical validations were performed for five different radii of helical tube bending by comparing the final radius after a springback.

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a “spring-mass” resonator periodically attached to a piezoelectric/elastic phononic crystal (PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion (PWE) method. In order to reveal the unique wave propagation characteristics of such a model, band structures of locally resonant (LR) elastic PC Euler nanobeams with and without resonators, band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. Results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.

Analytical approach for circular‐jointed shield tunnel lining based on the state space method

SummaryThis paper presents a new analytical solution to the circular segmental tunnel lining subjected to overburden and surrounding earth pressures. The governing equations are derived by adopting the curved Euler beam theory and the principle of minimum potential energy. Based on the state space method, the displacements and the relevant energy‐conjugated internal forces are treated as the fundamental unknown state vector and can be obtained by solving the state function. The inter‐segment joints on the lining are modeled by a set of linear springs, including shear, compression, and rotation. The presented method allows for the arbitrary distribution of the segmented joints and loads along the circumferential direction. The most striking advantages of the proposed method include the rigid body displacement treatment, lining‐displacement‐dependent soil reaction, and internal force direction dependency of the joint stiffness. Using this method, the displacements and internal forces of the entire lining can be obtained conveniently and simultaneously under the arbitrary loading and joint distribution conditions. The verification of the analytical solutions is provided by several examples.

Improved Study on the Fluctuation Velocity of High-Speed Railway Catenary Considering the Influence of Accessory Parts

To improve the prediction accuracy of the wave propagation velocity of high-speed railway catenary, a method for identifying the fluctuation velocity of catenary (FVC) considering the influence of the accessory parts of catenary (APC) is proposed. Based on the classical Bernoulli-Euler beam vibration model, the single Euler beam model under ideal conditions is improved considering the influence of the inertial forces exerted by accessory parts on the catenary. The calculation expression of FVC considering the influence of APC is deduced according to the theory of elastomer vibration mechanics. The key parameters $\alpha $ and $\beta $ that affect the FVC are obtained, and the relation curves between the fluctuation velocity and the key parameters $\alpha $ and $\beta $ are analyzed respectively. Through the fluctuation velocity test for the actual catenary, the theoretical fluctuation velocity of contact wire considering the influence of APC are compared and verified with the measured fluctuation velocity. Research results show that the relative error between the theoretical fluctuation velocity obtained from the modified fluctuation velocity expression of contact wire considering the influence of APC and the measured fluctuation velocity is less than 10%. Compared with the fluctuation velocity theoretical formula based on the string/cable and Euler beam theory, the modified expression of the fluctuation velocity of contact wire (FVCW) is more accurate in identifying the FVCW, and the recognition accuracy is increased by 3.75%, which verifies the accuracy of the recognition method and provides a more accurate theoretical basis of fluctuation velocity for designing the structural parameters of catenary.

Highly tunable anisotropic co-deformation of black phosphorene superlattices.

Controlling mechanical deformation is one of the state-of-the-art approaches to tune the electronic properties of 2D materials. We report a new mechanism for tuning a phosphorene superlattice with intercalated amphiphiles by its strong anisotropic co-deformation. Anisotropic co-deformation of a phosphorene superlattice is found to follow tunable sinusoidal and Gaussian functions, which exhibit adjustable mechanical actuation, curvature and layer separations. We analysed the controlling mechanism and tuning strategy of co-deformation as a function of amphiphile assembly topology, van der Waals interactions, interlayer separation and global deformation based on Euler-beam theory. Our first-principles calculations demonstrate that the co-deformation mechanism can be used to achieve a theoretical bandgap tunability of 0.7 eV and a transition between direct and indirect bandgaps. The reported tuning mechanisms pave new ways for designing a wide range of tunable functional electronics, sensors and actuators.

Numerical modelling and field experimental validation of the axial load transfer on the drill-strings in deviated wells

This paper studies the axial load transfer along the drill-strings in deviated wells by developing a finite element model based on the Euler beam theory and the augmentation Lagrangian contact algorithms. The model can simulate the entire drill-strings showing nonlinear contact model between drill-strings and casing. Special attention is given to the axial load loss, the pipe-casing contact force distribution and the slender pipe deformation. The efficacy of the proposed model is validated experimentally using a packer releasing procedure. Various drill-string factors, such as deviation angle, dogleg severity, hook load magnitude and buckling configurations, are considered for evaluating the efficiency of axial load transfer. Our analysis shows that the dogleg severity has a significant influence on the transfer, and the helical buckling of the drill-strings due to excessive loading could make it worse. This study provides a theoretical understanding of the variation of the contact force and the axial load transfer for the drill-strings in deviated wells. It can be used to better understand the working condition of downhole and guide field drilling.

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain–displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress–strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli–Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

Nonlinear vibration of buckled nanowires on a compliant substrate

Buckling of thin nanowires on a pre-strained compliant substrate has been widely used to make nanowire-based stretchable electronics. On nanometer scale, surface effect plays an important role on a buckled nanowire structure. In addition, as the amplitude of the deflection of the buckled nanowire is larger than its thickness, geometrical nonlinearity should be taken into account. Taking the kinetic energy caused by the out-of-plane motion into account, and on the basis of Euler beam theory, a theoretical model for a nanowire-substrate structure is established, combined with the influences of the nano-scale surface effect and geometrical nonlinearity. By means of Lagrange's equation, the equation of motion is derived and then solved by the Symplectic (Partitioned) Runge–Kutta method (PRK). Several numerical examples are analysed to study the nonlinear vibration of the structure. The analytical expressions of stable and unstable equilibrium points, and the relationship between the vibration amplitude and the natural frequency are obtained. The influences of surface effect and pre-strain on the dynamic behaviour are analysed. Through these numerical results, one can find that when the surface elastic modulus and surface residual stress are considered, the number of unstable equilibrium points would increase to three. The frequency obtained with positive surface elastic modulus is greater than that obtained with negative surface elastic modulus, implying that the positive surface elastic modulus can make the nanowire-substrate structure stiffer. Furthermore, when the pre-strain increases, the locations of stable and unstable equilibrium points move further away from the initial displacement, and the homoclinic orbits become expanded. The results presented in this paper should be useful to guide the design of nanowire-based stretchable electronics.

Dynamic stiffness approach to vibration transmission within a beam structure carrying spring–mass systems

The dynamic stiffness method is developed for the dynamics of a beam structure carrying multiple spring−mass systems. Based on classical Bernoulli–Euler beam theory, three types of vibrations, namely, bending, longitudinal and torsional motions, are formulated in terms of dynamic stiffness matrix. Similar to finite element technique, the local dynamic stiffness matrices for individual beams and spring−mass systems are assembled into global dynamic stiffness matrices so as to address vibration transmission from machines to flexible beamlike foundations. Using our proposed method, the vibration transmission within a beam frame, carrying multiple spring−mass systems is addressed. Through numerical analysis, the calculated vibration responses agree well with those from finite element method, which demonstrates that dynamic stiffness formulation has great potential in modeling the dynamics of built-up beam structures that supports machinery, especially in characterizing vibration isolation due to wave reflections, wave conversions, and other underlying mechanisms.