纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

Modeling of planar nonshallow prestressed beams towards asymptotic solutions

Employing the geometrically exact approach, the governing equations of nonlinear planar motions around nonshallow prestressed equilibrium states of slender beams are derived. Internal kinematic constraints and approximations are introduced considering unshearable extensible and inextensible beams. The obtained approximate models, incorporating quadratic and cubic nonlinearities, are amenable to a perturbation treatment in view of asymptotic solutions. The different perturbation schemes for the two mechanical beam models are discussed.     

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.     

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.     

Thermal post-buckling of an elastic beams subjected to a transversely non-uniform temperature rising

IntroductionAxiallycompressedstresseswilloccurinaconstrainedelasticbeamsubjectedtoatemperaturerising .Ifthemagnitudeofthecompressedstressesexceedacertainlimit,thermalbucklingoftheheatedbeam ,whichisoutofitsinitialconfiguration ,willtakeplace .So ,investigationsonthermalbucklingofrodsandbeamsareverynecessaryandimportantforthedesignofstructuresworkinginhightemperatureenvironmentsandofsomethermalsensitiveelasticelements.Becausethermalelasticpost_bucklingofbeamsandrodsareinducedbythethermallyaxial…     

挠性根部梁的动力学建模

挠性根部梁具有整体平动和转动自由度，其传统模型只适宜根部挠性很小的梁．采用柔性多体系统的建模方法建立了挠性根部Euler—Bernoulli梁的非线性动力学模型及线性耦合模型，所建模型不受根部挠性大小的限制；既可描述挠性根部梁的耦合振动，也可分别退化为固支梁或刚性梁的动力学模型；且线性耦合模型可线性变换为挠性根部梁传统模型．作为算例，采用假设模态法分析了两类线性模型的振动特性，表明线性耦合模型优于挠性根部梁传统模型．     

Moment-curvature models under reverse cyclic straining

The paper describes the development ofM-? models from stress-strain curves resulting from tests on axially loaded mild-steel specimens subjected to cyclic alternating plastic strains. These “push-pull” tests lead to the formulation of a power law which can be used to predict the nonlinear response of sections in bending under similar ambient conditions. These predictions are compared with experiments on beams subjected to pure bending involving strain control. The resulting load-deflection curves, derived directly from experimentalM-? models and indirectly from theoreticalM-? models arising from stress-strain data, are themselves compared with similar experimental data in the case of cantilever beams under tip loads.     

Bending and vibration analyses of coupled axially functionally graded tapered beams

The nonlinear bending and vibrations of tapered beams made of axially functionally graded (AFG) material are analysed numerically. For a clamped–clamped boundary conditions, Hamilton’s principle is employed so as to balance the potential and kinetic energies, the virtual work done by the damping, and that done by external distributed load. The nonlinear strain–displacement relations are employed to address the geometric nonlinearities originating from large deflections and induced nonlinear tension. Exponential distributions along the length are assumed for the mass density, moduli of elasticity, Poisson’s ratio, and cross-sectional area of the AFG tapered beam; the non-uniform mechanical properties and geometry of the beam along the length make the system asymmetric with respect to the axial coordinate. This non-uniform continuous system is discretised via the Galerkin modal decomposition approach, taking into account a large number of symmetric and asymmetric modes. The linear results are compared and validated with the published results in the literature. The nonlinear results are computed for both static and dynamic cases. The effect of different tapered ratios as well as the gradient index is investigated; the numerical results highlight the importance of employing a high-dimensional discretised model in the analysis of AFG tapered beams.     

Free vibrations of a beam with elastic end restraints subject to a constant axial load

A general model for vibration of beams restrained with two transversal and two rotational elastic springs subject to a constant axially load is presented. The frequency equations and the shape functions are derived analytically. The proposed model can be employed for simulating the dynamic responses of elastically supported beams in tension or compression for most classical boundary conditions. Some simplifications in the degenerate cases are deduced to evaluate the effectiveness of the model. Numeric examples are given for engineering applications. This model unifies most of the previous vibration models and provides a convenient tool for the analyses of various beam vibrations in tension and compression conditions.     

THE STABILITY OF AN AXIALLY ACCELERATING BEAM ON SIMPLE SUPPORTS WITH TORSION SPRINGS

The axially moving beams on simple supports with torsion springs are studied. The general modal functions of the axially moving beam with constant speed have been obtained from the supporting conditions. The contribution of the spring stiffness to the natural frequencies has been numerically investigated. Transverse stability is also studied for axially moving beams on simple supports with torsion springs. The method of multiple scales is applied to the partialdifferential equation governing the transverse parametric vibration. The stability boundary is derived from the solvability condition. Instability occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies or is two fold natural frequency of the unperturbed system. It can be concluded that the spring stiffness makes both the natural frequencies and the instability regions smaller in the axial speed fluctuation frequency-amplitude plane for given mean axial speed and bending stiffness of the beam.     

Experimental Validation of Reduction Methods for Nonlinear Vibrations of Distributed-Parameter Systems: Analysis of a Buckled Beam

An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integro-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach.     

Numerical analysis of large elastic-plastic deformation of beams due to dynamic loading

Numerical calculations were performed for two examples of the response of elastic-plastic beams subjected to dynamic loads. These were a simply supported, axially restrained beam under suddenly applied uniform pressure, and an axially restrained, clamped beam with a central mass that is impacted by a projectile. Large elastic-plastic deflections were considered, and the method of finite differences was used. Two different constitutive equations were assumed: the elástic-perfectly plastic relation, and a special elastic-viscoplastic, strain hardening model. Analysis of the results included examining the interaction between the bending moment and the axial force, the variation of the axial force, bending moment and deflection with time, and the propagation velocities of the various phenomena during motion. Experiments were carried out in which a rifle projectile hit a central mass which had been fastened to a clamped beam. Comparison between the theoretical and experimental dynamic deflections shows good agreement for relatively short response times.     

Exact solution of supercritical axially moving beams: symmetric and anti-symmetric configurations

We present an exact solution for supercritical configurations of axially moving beams with arbitrary boundary conditions. We take into account the geometric nonlinearity of the traveling beams in supercritical regime, and the nonlinear buckling problem is analytically solved. A closed-form solution for the supercritical configuration in terms of the axial speed is obtained. Some typical boundary conditions, such as fixed-fixed, fixed-pinned and pinned-pinned, are discussed. More importantly, based on the exact solution, we found a new anti-symmetric configuration for the fixed-fixed axially moving beams. The traveling beam may vibrate around the new anti-symmetric configuration at sufficiently high traveling speeds. A good accuracy of the solution is confirmed by a comparison with the data available in the literature, and with our own numerical results.     

On the cross-section distortion of thin-walled beams with multi-cell cross-sections subjected to bending

This paper deals with distortion of the cross-section contour of thin-walled beams with simple multi-cell closed rectangular cross-sections. The cross-section distortion is considered in the limit case. It is assumed that beam plates are hinged together along their longitudinal edges. Double symmetric three and two-cell closed cross-sections are considered. The stresses and displacements are obtained in the closed analytical form. The additional stresses and displacements due to distortion with respect to the stresses and displacements of the ordinary theory of bending are obtained. The boundary conditions are given in the general form. Some illustrative examples are given.     

Refined theories may be needed for vibration analysis of structures with overhang

The effect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated. A recently proposed modified Timoshenko-type equations of motion were used to analyze the vibration of the structure. Two different sets of boundary conditions, with either a fixed or hinged end support, were studied. The results were compared with those obtained for the classical Bernoulli–Euler beam theory. The comparison shows that for a hinged end beam with very long overhang, where the span between the supports is less than one tenth of the overall beam length, the classical theory significantly overestimates the values of the fundamental natural frequencies, even for isotropic shear rigidity. On the other hand, the span effect is reversed for the clamped end beam, for which a relatively significant difference between the classical theory and shear theory results may occur only for a long span. For transversely isotropic beams, the refined theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory, especially for short span hinged end beams and long span clamped end beams.     

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.     

Shear deformable bending solutions for nonuniform beams and plates with elastic end restraints from classical solutions

Summary The relationships of bending solutions between Timoshenko beams and Euler-Bernoulli beams are derived for uniform and non-uniform beams with elastic rotationally restrained ends. Extensions of these relationships for the cylindrical bending of Mindlin and Kirchhoff plates and for the bending of symmetrically laminated beams are also discussed. The new set of general relationships is useful because the more complex Timoshenko beam and Mindlin plate solutions may be readily obtained from their simpler Euler-Bernoulli beam and Kirchhoff plate solutions respectively, without much tedious mathematics. Received 16 March 1997; accepted for publication 26 November 1997     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

Vibration and stability of an axially moving beam immersed in fluid

Vibration and stability are investigated for an axially moving beam in fluid and constrained by simple supports with torsion springs. The equations of motion of the beam with uniform circular cross-section, moving axially in a horizontal plane at a known rate while immersed in an incompressible fluid are derived first. An “axial added mass coefficient” and an initial tension are implemented in these equations. Based on the Differential Quadrature Method (DQM), a solution for natural frequency is obtained and numerical results are presented. The effects of axially moving speed, axial added mass coefficient, and several other system parameters on the dynamics and instability of the beam are discussed. Particularly, natural frequency in terms of the moving speed is presented for fixed–fixed, hinged–hinged and hybrid supports with torsion spring. It is shown that when the moving speed exceeds a certain value, the beam becomes subject to buckling-type instability. The variations of the lowest critical moving speed with several key parameters are also investigated.