Composite Euler Bernoulli Beams Research Articles

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.

Nonlinear dynamic study of non-uniform microscale CNTR composite beams based on a modified couple stress theory

This study aims at investigating the nonlinear dynamic behavior of microscale carbon nanotube reinforced (CNTR) composite Euler–Bernoulli beams with a non-uniform cross-section, based on a modified couple stress theory (MCST). The nonlinear partial differential equations (PDEs) of motion are established based on the Von-Karman nonlinear strain–displacement relationship and Hamiltonian principle. The coupled PDEs are reduced to a single PDE, by neglecting the effects of the axial inertia and considering two different types of boundary conditions (i.e. clamped–clamped and clamped–free). At the same time, the single PDE is reverted to a nonlinear ordinary differential equation (ODE) by means of the Galerkin approach, and it is solved by using a semi-inverse method and the method of multiple time scales (MTS) for a free and forced vibration analysis, respectively. A large systematic numerical analysis is here performed to check for the sensitivity of the nonlinear response of CNTR composite beams to different boundary conditions and reinforcement parameters, with useful scientific insights for further computational investigations on the topic.

Nonlinear analysis of the ballast influence on the train-bridge resonance of a simply supported railway bridge

In the present work, the influence of the ballast-deck interface on the dynamic response of single ballasted track, simply supported high speed railway bridges is investigated. The ballasted track railway bridge is modelled by a composite Euler-Bernoulli beam with two layers connected between them through longitudinal springs and dampers representing the nonlinear friction behavior occurring at their interface. A constant moving load model termed (HSLM-A) is utilized to simulate the moving trains. The governing nonlinear transverse vibration equation of the system has been derived by using the Hamilton’s principle. By applying the Galerkin method and using the Differential Quadrature Method (DQM) and the Integral Quadrature Method (IQM), the time history of the studied beam is numerically solved by the Runge-Kutta scheme. Considering two railway bridges having different lengths, the obtained results have shown that the dynamics of the system is governed essentially by a Duffing like oscillator, where the ballast superstructure contributes through both additional nonlinear stiffness and damping. The ballast-bridge interface was found to affect largely the dynamic response at resonance of short-span bridge than that of lengthier structures. In this last case the ballast influence can be reduced to just an additional mass.

Adhesive toughness and instability in bonded heterogeneous films

Peeling of bonded films has applications to many engineering as well as biological systems. While significant prior research describes the mechanics of peeling of bonded homogeneous films, the studies on bonded heterogeneous films were relatively rare. Recently, bonded heterogeneous films have gained attention due to their potential for increasing the load required to propagate a debond without altering the adhesion characteristics of the interface. Although, it is known that the bending rigidity of a bonded heterogeneous film has significant influence on the adhesive toughness, the potential instability during the peeling process does not appear to be fully studied in the literature. In this paper, the heterogeneous film is simplified as a composite Euler-Bernoulli beam in a finite deformation framework. A semi-analytical model based on cohesive zone fracture description is developed, where both displacement-controlled and force-controlled loading conditions are considered. The semi-analytical solution is validated against experimental data in literature as well as finite element simulations. The study also addresses the transfer and distribution of energy in the film structure, and the variation of fracture process zone size with changing bending rigidity of the film. Additionally, comparisons are made between thickness enhancement and material enhancement to improve the peel performance of a heterogeneous film. The developed model provides insight into the mechanics of peeling, and it faithfully captures the instability often observed during debonding of heterogeneous films. The unstable load response to peeling can be leveraged to design systems where fracture toughness and overall reliability is improved without modifying the interface.

An analytical formulation for postbuckling and buckling vibration of micro-scale laminated composite beams considering hygrothermal effect

The current paper is concerned with the micro-scale laminated composite Euler-Bernoulli beams under the hygrothermal environment. The aim is mainly focused on the static and dynamical responses of micro-scale beams under consideration in both prebuckling and postbuckling cases. Meanwhile, the mechanism how temperature and hygrothermal concentration affect the critical buckling load is discussed. The material properties are considered to be temperature- and hygroscopic concentration-dependent. And the hygrothermal deformation caused by both variations in temperature and hygroscopic concentration is taken into account. The modified couple stress Euler-Bernoulli beam model involving geometric nonlinearity due to the mid-plane stretching is adopted. Then, an analytical formulation of postbuckling and buckling vibration is derived as a function of the external compressive axial load. Exact solutions for the critical buckling load, the postbuckling configuration and the natural frequency of vibration around the buckling configuration of the beam with various boundary conditions are investigated.

Nonlinear free vibration of laminated composite Euler-Bernoulli beams based on finite strain using generalized differential quadrature method

ABSTRACTIn this research, free vibration of laminated composite Euler-Bernoulli beams has been studied using finite strain assumptions. Governing equations of motion and boundary conditions have been obtained using the Green-Lagrange strain tensor and employing Hamilton's principle. The generalized differential quadrature method has been engaged to solve the equation of motion. Frequencies of the beam have been computed for carbon/epoxy material with various lay-ups subjected to different boundary conditions. The effects of fiber orientation change and lay-ups sequences of angle-ply composite beams on the frequency of the beam have been investigated. Also, dimensionless frequencies for different ratios of length to thickness of the beam subjected to different boundary conditions have been obtained.

Modified couple stress theory and finite strain assumption for nonlinear free vibration and bending of micro/nanolaminated composite Euler–Bernoulli beam under thermal loading

In this paper, the effect of finite strain on the nonlinear free vibration and bending of the symmetrically micro/nanolaminated composite beam under thermal environment within the framework of the Euler–Bernoulli and modified couple stress theory is studied. The governing equation of motion and boundary conditions are obtained using Hamilton’s principle, and then they are solved by generalized differential quadrature method. The bending and free vibration of the beam are investigated for both carbon/epoxy and glass/epoxy materials based on the finite strain and von Karman assumptions subjected to different boundary conditions. Also, two different fiber orientations including unidirectional and cross-ply are considered in this research. Comparison of the bending results show that there is a significant difference between the finite strain and von Karman particularly for [Formula: see text]. Furthermore, it is found that the natural frequencies predicted by the finite strain are more than the von Karman. Also, when the microbeam is inserted under thermal loading, the natural frequencies increase.

Corotational Formulation for Bonded Joint Finite Elements

Enhanced finite elements are elements with an embedded analytical solution that can capture detailed local fields, enabling more efficient mesh independent finite element analysis. In earlier research, this method was applied to adhesively bonded joints. The adherends were modeled as composite Euler–Bernoulli beams, and the adhesive layer was modeled as a bed of linear shear and normal springs. The field equations were derived using the principle of minimum potential energy, and the resulting solutions for the displacement fields were used to generate shape functions and a stiffness matrix for a single bonded joint finite element. In this study, the capability to model large rotations and nonlinear adhesive constitutive behavior is developed, and progressive failure of the adhesive is modeled by remeshing the joint as the adhesive fails. The results obtained using this enhanced joint element are compared with experimental results.

The effect of a piezoelectric layer on the mechanical behavior of an electrostatic actuatedmicrobeam

In this paper, the deflection and natural frequency of a microbeam under combinedelectrostatic and piezoelectric actuations is obtained for the first time. The microbeam hasbeen assumed as a composite Euler–Bernoulli beam with a piezoelectric layerdeposited on it. The nonlinear equation of motion has been derived using theHamilton principle, and solved using the Galerkin method. It is assumed that theneutral axis of bending is stretched when the beam is deflected. The axial load orpre-stress has been considered in derivations of the motion equations, and theboundary conditions in transverse vibration are assumed as clamped–clamped. Ithas been shown that the pull-in voltage, natural frequency and deflection of thesystem depend on the value of electrostatic actuation and the location, thicknessand applied voltage of the piezoelectric layer. Thus the deflection and naturalfrequency of an electrostatic actuated microbeam may be tuned to a suitablevalue by altering the applied voltage to the piezoelectric layer, which is moreconvenient than altering the axial load, as used in previous methods. Also, this showsthat a new sensor–actuator system may be constructed which is actuated byapplying the voltage to the piezoelectric layer, and this actuation is sensed bythe value of the output electric current induced from movement of the polarizedmicrobeam.

Exact static analysis of partially composite beams and beam-columns

The ordinary differential equations and general solutions for the deflection and internal actions and, especially, the pertaining consistent boundary conditions for partially composite Euler–Bernoulli beams and beam-columns are presented. Static loading conditions, including transverse and axial loading and first- and second-order analyses are considered. The theoretical procedure is applicable to general loading and boundary conditions for uniform composite beams and beam-columns with interlayer slip. Further, the exact closed form characteristic equations and their associated exact buckling length coefficients for composite columns with interlayer slip are derived for the four Euler boundary conditions. It is shown that these coefficients are the same as those for ordinary fully composite (solid) columns, except for the Euler clamped-pinned case. For the clamped-pinned case, the difference between the exact buckling length coefficient and the corresponding value for solid columns is less than 1.8% depending on the so-called composite action parameter and relative bending stiffness parameter. Correspondingly, the maximum deviation between the exact and approximate buckling load is at most 2.5%. These small differences can in most practical cases be neglected. Also, the maximum theoretical range for the relative bending stiffness for partially composite beams and beam-columns is derived. An effective bending stiffness, valuable in the determination of the critical buckling load for partially composite members, is derived. This effective bending stiffness is also suitable for analysing approximate deflections and internal actions or stresses in composite beams with flexible shear connection. The beam-column analysis is applied to a specific case. The difference in the approaches to the first- and second-order analysis is illustrated and the results clearly show the magnification in the actions and displacements due to the second-order effect. The magnification of the internal axial forces is different from magnifications obtained for the other internal actions, since only that portion of an internal axial force that is induced by bending is magnified by the second-order effect.

Elastic flexural buckling analysis of composite beams of variable cross-section by BEM

In this paper a boundary element method is developed for the elastic flexural buckling analysis of composite Euler–Bernoulli beams of arbitrary variable cross-section. The composite beam consists of materials in contact. Each of these materials can surround a finite number of inclusions or openings. All of the cross-section’s materials are firmly bonded together. Since the cross-sectional properties of the beam vary along its axis, the coefficients of the governing differential equation are variable. The beam is subjected to a compressive centrally applied load together with arbitrarily axial and transverse distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problems are solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The influence of the boundary conditions on the buckling load is demonstrated through examples with great practical interest. The flexural buckling analysis of a homogeneous beam is treated as a special case.

Global and Local Wave Vibration Characteristics of Materially Coupled Composite Beam Structures

A unique feature of fiber-reinforced composite materials is that it allows structural tailoring for favorable dynamic performance, due to the directional nature of composite materials. The directional nature causes material coupling, which results in coupled vibrational modes and complicates dynamic analysis. Most of the up-to-date composite structure related dynamic studies focus on free vibration analysis. In this paper, the local wave transmission and reflection characteristics at various discontinuities are studied first. Such discontinuities include general point supports, boundaries and change in sections. The matrix relations between the injected waves and externally applied forces and moments are also derived. By assembling these matrices, both free and forced vibration responses of materially coupled composite Euler–Bernoulli beams are obtained. The wave-based vibration analysis approach is found concise and systematic. Numerical examples are given.

Effect of material coupling on wave vibration of composite Euler–Bernoulli beam structures

Fiber-reinforced composite materials are an important development in the continuing search for lightweight materials for great strength and stiffness. Lighter structures are usually more prone to vibrations. Furthermore, due to material coupling, composite beams normally vibrate with coupled vibration modes, which complicates vibration analysis. This paper presents the effect of coupling between bending and torsional deformations on vibrations of composite Euler–Bernoulli beams from a wave vibration standpoint. In the study, it is found that the torsional mode is only unaffected by the material coupling at low frequencies. The flexural modes are found affected over the entire frequency band. Unlike their uncoupled metallic counterparts, the wavenumbers for propagating and decaying wave components are no longer the same due to material coupling. Free vibration analysis is performed from a wave standpoint. Free vibration of composite beam with the coupling effect taken into account is of great importance. This is because the free vibration characteristics can be designed favorably by suitable choice of ply orientations and stacking sequence—one of the unique features of composite structures. Numerical examples for which comparative results are available in the literature are presented.