Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

横向载荷作用下轴向运动梁的屈曲失稳以及非线性振动特性

梁的轴向运动会诱发其产生横向振动并可能导致屈曲失稳,对结构的安全性和可靠性产生重大的影响。本文重点研究了横向载荷作用下轴向运动梁的屈曲失稳及横向非线性振动特性。基于Hamilton变分原理,建立了横向载荷作用下轴向运动梁的动力学方程,获得了梁的后屈曲构型。使用截断Galerkin法,将控制方程改写成Duffing方程的形式。用同伦分析方法确定载荷作用下轴向运动梁的非线性受迫振动的封闭形式的表达式。结果表明,后屈曲构型对轴向速度和初始轴向应力有明显的依赖性。通过同伦分析法得出非线性基频的显式表达式,获得了初始轴向力会影响非线性频率随初始振幅和轴向速度的线性关系。另外,轴向外激励的方向也会改变系统固有频率。     

Analytical dynamic displacement response of rigid pavements to moving concentrated and line loads

This paper presents the formulation of a plate of infinite dimensions (without boundary conditions) on an elastic foundation, subjected to a moving concentrated and line load of constant amplitude and speed, using a triple Fourier transform. The solution is carried out integration by residues. A closed-form solution of displacement field has been obtained for a moving load with subsonic, transonic and supersonic speeds. It is found that the maximum response of the slab occurs beneath the moving load and travels with the load at the same speed. It is also shown that a critical speed exists. If the moving load travels at critical speed, slab displacement becomes infinite in amplitude.     

正交各向异性地基平面问题动力响应研究

以位移分量为基本未知量,在直角坐标系下建立正交各向异性地基的平面应变问题动力偏微分方程．采用Laplace-Fourier变换和逆变换方法,引入初始条件和边界条件,推导了任意形式表面动荷载作用下正交各向异性地基平面问题在时域内动力反应的积分形式解．基于理论解,编制了相应的计算程序,并对正交各向异性土
体表面作用线性移动谐振荷载进行了算例分析,研究了土体参数、荷载移动速度、荷载频率不同而导致的土体表面各点竖向位移幅值的变化规律,以及荷载速度对竖向应力分量的影响规律．数值分析结果表明：土体的各向异性、荷载频率和移动速度对表面位移幅值有较大影响,土体阻尼比对于荷载中心点附近的位移幅值影响较小;荷载移动速度对于竖向应力分量有较大影响,这对工程实践具有重要指导意义．     

不确定性移动载荷激励下弹性简支梁的动态响应边界分析及应用

不确定性移动载荷激励下的弹性梁振动是土木、机械和航空航天等工程领域普遍存在的一类重要问题。在许多实际工程中,不确定移动载荷的样本测试数据有限或测试成本较高,本文引入区间过程模型对此类动态不确定性参数进行描述,提出了一种求解不确定移动载荷激励下弹性梁振动响应边界的非随机振动分析方法。首先,介绍了确定性移动载荷激励下弹性梁的振动微分方程及其解析求解方法;其次,引入区间过程模型,以上下边界函数的形式对不确定性移动载荷进行度量,进而基于模态叠加法发展出弹性梁振动响应边界求解的非随机振动分析方法;最后,将上述非随机振动分析方法应用于车桥耦合振动问题。     

Reddy高阶组合梁弯曲的一般解析解法

基于Reddy高阶梁的轴向位移模式,考虑组合梁界面滑移变形,利用最小势能原理建立了Reddy组合梁弯曲问题的控制微分方程和边界条件,,并将控制方程转化为含12个基本未知量的一阶常微分方程组,给出一般求解方法和解表达式。其次,研究了横向均布荷载作用下Reddy简支组合梁的弯曲,所得结果与有限元解吻合良好,说明本文解析解的有效性和可靠性。最后,数值分析了组合梁界面滑移剪切刚度kcs、弹性模量-剪切模量比E/G、梁长-高比L/h和子梁厚度比hs/hc等参数对Reddy简支组合梁弯曲的影响。分析表明：滑移刚度显著影响横截面应力的分布;组合梁长-高比越小、弹性模量-剪切模量比越大或界面滑移刚度越大,组合梁的剪切效应对其挠度影响越显著,此时不宜忽略其剪切变形。     

DYNAMIC RESPONSE OF A DOUBLE-LAYERED SUBGRADE WITH ROCK SUBSTRATUM TO A MOVING POINT LOAD

In this paper, an analytical solution for the dynamic response of a double-layered subgrade with rock substratum to a moving point load is derived. The subgrade profile is divided into two layers. The upper layer is modeled by an elastic medium and the lower layer by a fully saturated poroelastic medium governed by Biot’s theory. In the meanwhile, the subgrade is resting on the rock substratum. The analytical solutions for stress, displacement and pore pressure are derived by using the Fourier transform. Numerical results obtained by using the inverse fast Fourier transform (IFFT) are used to analyze the influence of the moving load velocity, the thickness of an elastic medium layer and a fully saturated poroelastic medium layer on the dynamic response.     

弹性地基双层梁理论下的混凝土路面力学分析

为建立混凝土路面结构受力分析计算模型,以Winkler弹性地基梁模型为基础,推导出了弹性地基双层梁理论的表达式;给定边界条件,利用MATLAB软件获得了无限长弹性地基梁在集中力作用下的挠度表达式。将混凝土路面结构简化为弹性地基上的双层梁,当车辆荷载作用于混凝土路面时,在集中载荷的作用下,建立了面层与基层的微分平衡方程。应用广义“初参数”法,得到了双层梁位移和应力的解析解。通过算例,对面层及基层的变形和应力进行了分析,结果表明:增大面层、基层的轴惯性矩和地基的弹性常数,可以有效地减少面层和基层的变形量,降低最大应力数值,但抗弯刚度对基层和面层的弯矩受力影响不大。最后将结果与ANSYS分析结果进行了比较,佐证了解的可靠性,研究结果可为混凝土路面结构设计提供依据。     

爆炸荷载作用下饱和半空间中衬砌隧道弹性动力响应IBIEM模拟

基于Biot两相介质理论,采用一种高精度间接边界积分方程法（IBIEM）研究了饱和半空间中浅埋衬砌隧道在内部爆炸荷载作用下的瞬态弹性动力反应。通过典型算例,给出了爆炸荷载作用下隧道附近地表位移、衬砌动应力、围岩径向位移和衬砌表面孔隙水压的时程响应,并对比分析了饱和半空间和全空间中隧道动力响应的区别。研究表明：覆土层厚度对浅埋隧道-围岩整体动力响应特征具有明显影响;衬砌表面透水状态对爆炸荷载的时程响应的影响不显著;随半空间饱和介质孔隙率增加,围岩受隧道内部爆炸影响程度降低,衬砌承担的爆炸作用增大;当和直达波、衬砌内部反射波的峰值叠加作用时,半空间表面反射波对衬砌隧道拱顶附近响应影响显著,使得衬砌动应力幅值、径向位移相比深埋情况大幅度增加。     

非饱和土半空间Lamb问题及能量传输特性

地球表面绝大多数土层处于非饱和状态, 故采用传统饱和两相介质理论进行动力学分析时, 结果往往与实际情况不符. 针对这一问题, 本文以非饱和半空间作为研究对象, 基于连续介质力学和多孔介质理论, 考虑非饱和多孔介质中各相的质量守恒方程、动量守恒方程、本构方程以及有效应力原理等基本方程, 建立了以骨架位移、孔隙水压力和孔隙气压力为基本未知量的动力学控制方程. 针对非饱和半空间表面在竖向集中简谐荷载作用下的动力学响应及能量传输问题, 建立了频域内经典Lamb问题的轴对称计算模型, 采用Helmholtz分解法, 通过引入势函数Φ和Ψ表示骨架的位移分量, 结合本构方程获得了不同边界条件下半空间表面位移场和能量场等物理量的解析解答, 并通过数值算例对荷载参数(激振频率)、材料参数(饱和度、渗透系数)等影响因素进行了分析与讨论. 结果表明: (1)饱和度的升高或者激振频率下降, 都会提高非饱和半空间的表面位移幅值; (2)当渗透系数下降至一临界值时, 地表位移幅值会趋于一极限值, 并且透水(气)边界与不透水(气)边界条件下渗透系数的影响表现出明显的差异性.      

Effect of debonding in active constrained layer damping patches on hybrid control of smart beams

A new model for a smart beam with a partially debonded active constrained layer damping (ACLD) patch is presented, and the effects of the debonding of the ACLD patch on both passive and hybrid control are investigated. In this model, both shear and compressional vibrations of the viscoelastic material (VEM) layer are considered. The moment inertia and the transverse shear effect are also taken into account based on the Timoshenko’s beam theory. The adhesive layer between the host beam and the piezoelectric sensor patch is modeled as an elastic load transferring media. The debonding of the ACLD patch is approximated by removing the VEM between the constraining layer and the host beam in the debonding region, and the continuity conditions are imposed based on displacement continuity and force balance. A modal velocity observer-based modal control scheme is also given to perform the active modal control of the beam. In order to examine the effects of part debonding of the ACLD patch, the characteristic equation of the beam treated with an ACLD patch is derived. The simulation example results show that an edge debonding of the ACLD patch can significantly affect both passive and hybrid control. It is also found that the additional mode induced by the debonding has unique effects on the modal damping ratios and frequencies of both open-loop and closed-loop system.     

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half-space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of (sub-) hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of (sub-) soft beam and hard half-space but are similar to each other for the system of (sub-) hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity (the shear wave velocity of the beam for the system of soft beam and hard half-space). The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave radiation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.     

Stresses of orthotropic laminated beams subjected to high temperature and mechanical load

Thermo-elastic analysis of simply-supported orthotropic laminated beams subjected to high temperature and mechanical load is presented on the basis of the exact two-dimensional thermoelasticity theory. The beam is composed of several orthotropic layers, each with temperaturedependent material properties. The governing equation for each layer is analytically solved using the state space method. The displacement and stress solutions of the beam are obtained using the transfer-matrix method. A numerical example is included to study the effects of temperature on the mechanical responses of a sandwich beam. The results reveal two main effects of temperature:(i) inducing deformations and stresses by itself;(ii) affecting the deformations and stresses induced by the mechanical load.     

等效刚度对非线性压电俘能系统性能影响研究

基于Hamilton原理,考虑几何非线性和梁的不可伸长条件,建立了五层压电双晶片叠合梁俘能器在直接和参数激励作用下的运动微分方程。利用Galerkin法和谐波平衡法获得了俘能器的位移、输出电压和输出功率的解析解。引入随时间变化的扰动,提出了非线性方程解的稳定性条件。为了对压电俘能器的结构-性能关系进行综合分析,研究了被动层的配置形式、被动层与主动层的厚度比和弹性模量对压电俘能系统性能的影响。结果表明,在叠合梁厚度不变的情况下,采用五层的压电双晶片叠合结构,选择合理的被动层与主动层厚度比、被动层弹性模量、被动层厚度比和负载电阻,可以有效提高能量俘获的效率。     

硬夹心矩形夹层板的整体稳定性分析

摘要：本文在Reissner型理论给出的位移模式基础上,修正其软夹心假设,考虑夹心层面内刚度,给出了硬夹心夹层板的几何方程、物理方程,建立了硬夹心夹层板结构在面内纵向载荷作用下的平衡微分方程,并对方程进行了简化,通过理论计算得到了四边简支条件下硬夹心矩形夹层板整体失稳临界载荷的解析解,并分别计算了夹心层材料的弹性模量 、厚度 、泊松比 对硬夹心夹层板临界载荷的影响,结果证明,对于硬夹心夹层结构,夹心层面内刚度对硬夹心夹层板整体失稳临界载荷的影响较大,考虑其面内刚度是必要的。     

高速荷载下多孔饱和地基的动力响应

研究高速荷载作用下梁与多孔饱和半空间的动力响应。由Fourier变换求解多孔饱和固体的动力基本方程，根据梁与半空间的接触条件得出多孔饱和半空间上梁的垂直位移的表达式。文中的数值算例考虑了荷载移动速度对梁的动力位移的影响，并与相应的弹性半空间问题作了对比。从算例中可以发现荷载移动速度对动力位移有很大的影响，当移动速度与半空间的表面波速相近时，地面会当产生很大的振动，同时还发现当速度大于介质的剪切波速时，多孔饱和半空间上梁的动力响应与弹性半空间上梁的动力响应有很大的差别。     

Dynamic displacements of an infinite beam on a poroelastic half space due to a moving oscillating load

This investigation is concerned with the dynamic displacements of a beam on a poroelastic half space under a periodic oscillating load of constant velocity. The governing equations for the proposed analysis are solved using Fourier transform. The expression for the vertical displacement is obtained according to the contact condition between a beam and a half space. The effects of the moving velocity and vibration frequency of the load on the dynamic displacement are considered in the numerical examples. The results show that the load velocity has significant influence on dynamic displacement. It is also noted that large differences exist between the dynamic responses for a beam on a poroelastic half space and on an elastic half space when the load velocity is larger than the shear wave speed of the medium. The reported work is supported by the National Natural Science Foundation of China (Project No. 10372073).     

A numerical model for the isolation of moving-load induced vibrations by pile rows embedded in layered porous media

A numerical model is developed to analyse the isolation of moving-load induced vibrations using pile rows embedded in a layered poroelastic half-space. Based on Biot’s theory and the transmission and reflection matrices (TRM) method, the free wave field solution for a moving load applied on the surface of a layered poroelastic half-space and the fundamental solution for an harmonic circular patch load are determined. Using Muki and Sternberg’s method, the second kind of frequency domain Fredholm integral equations for the dynamic interaction between pile rows and the layered poroelastic half-space are derived. The time domain solution is recovered via inverse Fourier transform in order to obtain the amplitude reduction ratio and thus assess the vibration isolation efficiency of pile rows. A special case of the present model shows good agreement with an existing solution. Numerical results of this study show that the speed of moving loads has an important influence on the isolation of vibrations by pile rows: the same pile rows can achieve better isolation efficiencies for higher speed loads than for lower speed loads. Pile rows embedded in a two-layered poroelastic half-space with a softer overlying layer usually generate better vibration isolation effects than those with a stiffer overlying layer. Finally, better isolation vibration may be realized by increasing the pile length and decreasing the net spacing between neighboring piles in a pile row.     

压电激励圆形曲梁的静态响应及位移控制

研究压电激励圆形曲梁的静态位移响应及位移控制的参数特性。将压电夹层圆形曲梁等效为单层结构,基于一维小曲率曲梁理论,建立其控制方程。在集中弯矩和径向集中力以及电载荷作用下,分析了带压电激励器的圆形悬臂曲梁的静态响应。与有限元结果比较表明:本文的理论模型能够模拟压电激励的小曲率圆形曲梁的静态响应。压电夹层圆形曲梁在任意位置的径向集中力载荷作用下,控制其自由端径向位移响应为零,求得控制电压的解析表达,数值分析表明:随着集中力载荷的位置变化和梁长的变化,最优控制电压将出现峰值和反号。     

Dynamic analysis to infinite beam under a moving line load with uniform velocity

I.IntroductionAninfinitebeamonelasticfoundationnotonlycanbelookedonasdynamicmodelforaSuspensionbridgeoratensiondiagonalbridgel']butalsocanbeusedindynamicsanalysistoarailtrack,therefore,dynamicresponseofinfinitebeamundermovillgloadhasbecomethefocusofdiscussioninpastseveraldecades.TheproblemalsohasbeeninvestigatedbyTimoshenko12j,Frybal'],Steelel'],Lee15],etal.Wefind,however,thatallofthestudiesonthissubjectonlydiscussedmovingpointloadproblem.Actually,eitherautomobileloadoratrainloadisalinedist…