Analytical solutions for Euler‐Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads

In this paper, the dynamic response of an infinite beam resting on a Pasternak foundation and subjected to arbitrary dynamic loads is developed in the form of analytical solution. The beam responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The mechanical resistance of the Pasternak foundation is modeled using two parameters, that is, one accounts for soil resistance due to compressive strains in the soil and the other accounts for the resistance due to shear strains. Because the Winkler model only represents the compressive resistance of soil, comparatively, the Pasternak model is more realistic to consider shear interactions between the soil springs. The governing equation of the beam is simplified into an algebraic equation by employing integration transforms, so that the analytical solution for the dynamic response of the beam can be obtained conveniently in the frequency domain. Both inverse Laplace and inverse Fourier transforms combined with convolution theorem are applied to convert the solution into the time domain. The solutions for several special cases, such as harmonic line loads, moving line loads, and travelling loads are also discussed and numerical examples are conducted to investigate the influence of the shear modulus of foundation on the beam responses. The proposed solutions can be an effective tool for practitioners. Copyright © 2017 John Wiley & Sons, Ltd.     

黏弹性地基上路面板在多轮荷载作用下的响应分析

将内配纵向钢筋的路面视为置于黏弹性Winkler地基上的正交各向异性薄板,假设板在荷载运动方向上为无限长,车辆轮载被简化成包含自振频率的矩形均布移动荷载。应用三角级数及Fourier变换方法,得到了单轮荷载作用下路面竖向位移的表达式。在Fourier变换域内,利用叠加原理分别得到了前后双轮荷载和平行双轮荷载作用下路面竖向位移的解析解答。在此基础上,推导出连续配筋混凝土路面上受4轮车辆荷载作用时轮载附近位移的稳态响应表达式。利用数值积分方法对表达式进行Fourier逆变换,得到了数值结果,并全面地分析了多轮荷载作用时荷载间距、相位差、荷载速度、频率、地基阻尼以及配筋率对路面竖向位移响应的影响。     

移动荷载下三维半空间动力安定性下限分析

针对交通移动荷载,基于第一动力安定性理论预测了弹塑性半无限空间动力安定性下限值。通过建立无限元边界的三维动力有限元模型,研究了移动荷载作用下弹塑性半无限空间的动应力分布,构造了稳态动应力下的残余应力场,基于残余应力场的构造提出了动力安定性下限值的预测方法,分析了交通动应力下的安定性下限值以及车辆移动速度对安定性的影响,同时分析了均布荷载和Hertz荷载分布对动力安定性下限值的影响。当荷载移动速度低于Rayleigh波速时,安定性极限值随着速度的增加而增加,而当速度超过Rayleigh波速时,随着速度的增加而减小。研究还发现,当荷载和总重相等时采用圆形均布荷载和Hertz荷载的安定性极限值较为相似。     

An explicit representation of steady state response of a beam on an elastic foundation to moving harmonic line loads

A closed‐form deflection response of a beam rest is presented in this paper using the integral transform method. The theory of linear partial differential equations is used to represent the deflection of beam subjected to a moving harmonic line load in integration form. The solution is finally carried out using the inverse Fourier transform. To evaluate the integration analytically, poles of the integrand are identified with the help of algebraic equation theory. Residue theorem is then utilized to represent the integration as a contour integral in the complex plane. Closed‐form deflections and numerical results are provided for different combinations of load frequency and velocity. Copyright © 2002 John Wiley & Sons, Ltd.     

Dynamic response of double beam rested on stochastic foundation under harmonic moving load

This paper deals with the dynamic response of infinite double Euler–Bernoulli beam supported by elastic foundation with stochastic stiffness subjected to an oscillating moving load, which is the first research in relevant literature review. In this matter, equations of motion for double beam are formulated in a moving frame of reference. Moreover, by employing the first order perturbation theory and calculating contour integration, the response of double beam is obtained analytically and validated by a stochastic finite element model. Sensitivity analyses on the various parameters of closed form solution such as velocity, load frequency, coefficient of variation of soil foundation and rail and slab bending stiffness show the significant effect of load frequency on the dynamic response of the doubled beam. From practical point of view, the obtained results of the present study can be utilized efficiently in analysis and design of slab track systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical analysis of soil vibrations due to trains moving at critical speed

High-speed train induced vibrations of track structure and underlying soils differ from that induced by low-speed train. Determining the critical speed of train operation remains difficult due to the complex properties of the track, embankment and ground. A dynamic analysis model comprising track, embankment and layered ground was presented based on the two-and-half-dimensional (2.5D) finite elements combining with thin-layer elements to predict vibrations generated by train moving loads. The track structure is modeled as an Euler–Bernoulli beam resting on embankment. The train is treated as a series of moving axle loads; the embankment and ground are modeled by the 2.5D finite elements. The dynamic responses of the track structure and the ground under constant and vibrating moving loads at various speeds are presented. The results show that the critical speed of a train moving on an embankment is higher than the Rayleigh wave velocity of the underlying soil, attributed to the presence of the track structure and the embankment. It is found that the dynamic response of ground induced by moving constant loads is mostly dominated by train speed. While for the moving load with vibration frequency, the ground response is mostly affected by the vibration frequency instead of train speed. Mach effect appears when the train speed exceeds the critical speed of the track–embankment–ground system.     

Effect of vehicle–pavement interaction on dynamic response of rigid pavements

In the present paper, dynamic response of rigid pavements subjected to moving vehicular loads is presented using an analytical procedure based on the finite element method. The vehicle–pavement interaction effects are taken into account while developing the solution algorithm. The concrete pavement is discretized by finite and infinite beam elements. Infinite elements are helpful in appropriate modeling of end conditions. The main purpose of the development of infinite elements is to model the unbounded domain. The underlying soil medium is modeled by Pasternak model which assumes the existence of shear interaction between the spring elements. The moving vehicle is represented by a mass supported by a spring-dashpot system. The vehicle -pavement interaction force is modeled with Dirac-delta function. Dynamic equilibrium equation is solved with Newmark-Beta integration scheme. It has been observed that the dynamic interaction between the moving load and the pavement has significant effect on pavement response. Parametric study is carried out to investigate the effect of vehicle–pavement interaction (VPI) and soil parameters on the response of pavement.     

交通荷载作用下Kelvin地基上不平整路面的动力响应分析

为了评价不平整路面在交通荷载作用下的动态响应,从路面几何不平整出发,假设路面不平整服从正弦函数曲线,并把1/4车身结构简化为2个自由度振动体系,得到了不平整路面上的动荷载表达式。又将此荷载模式看成是作用在Kelvin地基上无限大路面板表面的移动矩形荷载,采用双重Fourier变换方法得到了不平整无限大路面板竖向位移的积分形式解,此解析解可以退化为静荷载作用下的经典解答。并利用快速傅立叶变换(FFT)方法得到了数值计算结果。分析了荷载速度,路面不平整波长、不平整幅值以及地基阻尼和地基刚度对板的动力响应的影响。结果表明,板竖向位移的分布受荷载移动速度的影响;位移幅值随着路面不平整波长和荷载速度的变化出现2个峰值;路面不平整使得位移峰值出现滞后现象,且不平整波长越小或不平整幅值越大,滞后现象越明显。     

Propagation and localization of wave in multi‐span Timoshenko beams on elastic foundations under moving harmonic loads

Wave propagation and localization in ordered and disordered multi‐span beams on elastic foundations due to moving harmonic loads are investigated by using the transfer matrix methodology. The transfer matrix, as a function of the frequency and velocity of the moving harmonic load, of the periodic beam is formulated in a coordinate system moving with the load. The expressions of critical velocities, cut‐off frequency of an associated uniform beam without discrete spaced supports, are determined through the analysis of the wavenumbers, and the dynamic responses of the beam are also examined. For the ordered and disordered case, the propagation constants and localization factors are respectively employed to identify the velocity and frequency pass bands and stop bands in order to examine whether the perturbation can propagate along the structure or not. The effects of the periodicity, disorder level, excitation frequency, and moving velocity are studied in detail. The validity of the obtained results is confirmed by evaluating the transverse deformation of the beams through the finite element simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

列车荷载下冻土路基非平稳随机振动分析的时域显式法北大核心CSCD

随着我国冻土地区铁路运营里程的不断提升,冻土路基在随机列车荷载作用下的动力响应分析成为了急需解决的工程问题。本文以青藏铁路某路基横断面为例,采用时域显式蒙特卡罗模拟法,计算其在随机列车荷载作用下响应的统计特征。首先,提出列车驶过施加在路基顶面荷载的快速计算方法,并引入轨道不平顺及列车行驶速度两类随机参数,以生成随机分析所需的大量荷载样本。然后,采用数值积分方法将运动方程在时域上进行离散,建立任意离散时刻冻土路基动力响应关于列车荷载的显式表达式。基于该显式表达式,可以高效地实施蒙特卡罗模拟,得到在随机列车荷载作用下路基关键响应的均值、标准差和峰值等统计量。采用该方法,分析了路基在夏冬两个季节不同深度处的随机动力响应,发现在夏季路基浅层中的位移和速度响应最为剧烈。数值算例表明,时域显式蒙特卡罗模拟法在冻土路基随机振动分析中具有理想的计算精度和计算效率。     

列车引起的地基动应力特性

为分析列车引起的地基动应力特性,用移动荷载作用下的弹性地基Euler-Bernoulli梁模型计算得到地基表面与路堤之间的反力。将其视为作用于地基表面的移动面荷载,以单位移动荷载引起的半空间地基内部应力解为基础,对反力在作用空间上进行积分,得到了列车速度小于地基中Rayleigh波速时列车荷载在地基中引起的稳态应力响应解答。给出了应力随空间坐标和时间的变化,分析了应力的特性及分布规律。发现列车经过时在地基中产生的动应力是循环应力,具有明显的偏移应力。竖向动应力的影响范围随着列车速度的增大而不断增大。     

Analytical solutions for Euler–Bernoulli beam on visco‐elastic foundation subjected to moving load

Analytical solutions for the steady‐state response of an infinite beam resting on a visco‐elastic foundation and subjected to a concentrated load moving with a constant velocity are developed in this paper. The beam responses investigated are deflection, bending moment, shear force and contact pressure. The mechanical resistance of the foundation is modeled using two parameters k_s and t_s — k_s accounts for soil resistance due to compressive strains in the soil and t_s accounts for the resistance due to shear strains. Since this model represents the ground behavior more accurately than the Winkler spring model, the developed solutions produce beam responses that are closer to reality than those obtained using the existing solutions for Winkler model. The dynamic beam responses depend on the damping present in the system and on the velocity of the moving load. Based on the study, dynamic amplification curves are developed for beam deflection. Such amplification curves for deflection, bending moment, shear force and contact pressure can be developed for any beam‐foundation system and can be used in design. Copyright © 2012 John Wiley & Sons, Ltd.     

黏弹性地基板在矩形变速荷载作用下的振动分析

利用双重Fourier变换方法,首先推导得到了黏弹性地基板在多个矩形变速移动荷载作用下的动力响应一般解。然后以实际刚性路面为算例,采用快速傅立叶变换(FFT)方法对匀变速荷载作用下的板的动挠度进行了数值求解。与经典解进行了比较,显示出了较好的一致性。利用数值结果重点分析了矩形移动荷载的加速度、减速度以及初速度对板的动态挠度的影响,发现移动荷载加速度、减速度及初速度对黏弹性地基板动力响应的影响十分明显,随着加速度（减速度）和初速度的增大,板的动挠度在关键速度时明显减小;板的动挠度出现峰值的位置对应着不同的加速度（减速度）和初速度,但却对应着相同的关键速度。     

移动荷载下高铁路基段振动加速度频谱衰减特性

采用精细化和多尺度建模技术,建立了设计速度为350 km/h的双线高速铁路轨道-路基-地基非线性耦合系统的真三维动力分析模型,将地基土和路基填筑材料的非线性纳入分析模型中,采用三维黏弹性静-动力统一人工边界技术对地基无限域进行模拟,采用动接触算法模拟底座板底面和基床表层表面之间的相互作用,考虑移动荷载作用前路基-地基中客观的静应力状态对后续动力计算的影响,依托大规模并行计算技术和单元生死技术,模拟了地基的初始应力场生成、路基结构和轨道系统的施工过程,在此基础上模拟了与8辆编组的某型动车组轮对空间位置相对应的、作用于钢轨顶部的压力荷载的移动过程。基于分析结果,归纳了移动荷载作用下高速铁路轨道-路基-地基系统中振动加速度频谱的衰减特性。     

饱和土体中排桩对移动荷载的被动隔振效果分析

利用Muki和Sternberg的虚拟桩法,研究了饱和土体地基中排桩对移动荷载引起振动的被动隔振效果。隔振桩作为一维杆,饱和土体满足Biot理论。利用已有的移动荷载作用下的饱和土体的自由波场解及饱和土体内部受竖向圆形分布荷载作用下的基本解,建立了频域内土-桩的第2类Fredholm积分方程。通过Fourier逆变换得到时间域内评价隔振效果的振幅比。与已知文献结果相比较,验证了方法的正确性。数值结果表明,荷载速度对排桩的隔振效果有一定影响,即在相同隔振系统情况下,单排桩对低速荷载引起振动的隔振效果比高速移动荷载效果好。同时,较高速时的最佳桩长比低速时要短。     

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

The vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 John Wiley & Sons, Ltd.     

Moving load response of an axially loaded Timoshenko beam on a multilayered transversely isotropic half-space comprising different media

This paper investigates the dynamic response of an axially loaded Timoshenko beam coupled with a multilayered transversely isotropic (TI) half-space subjected to a moving load. An axial force induced by the thermal expansion is taken into account in the Timoshenko beam. The half-space considers the alternate distribution of an arbitrary number of TI elastic and poroelastic layers to model foundation soils with different properties and moisture conditions. To solve the governing equations, Fourier transform is adopted. The stratified foundation is formulated by extending an “adapted stiffness matrix method” to a more general scenario with an arbitrary number of layers. The beam is then coupled with the foundation to derive solutions to the system in the frequency-wavenumber domain. The final results in the time-spatial domain are recovered by the inverse Fourier transform. After confirming the accuracy of the method in this study, the influences of the pore water existence, the transverse isotropy of different parameters, and the axial force are investigated. It can be observed that the effect of pore water existence on the maximum beam deflection can reach 22% in this study. The transverse isotropy of the elastic and shear moduli influences the critical speed of the beam deflection by altering the phase velocity of the first wave propagation mode of the beam-foundation system. The vertical permeability coefficient is more important than the horizontal one in determining the excess pore pressure. The rise of the beam temperature (axial force) decreases the critical speed and magnifies the vibrations.     

移动荷载下黏弹性半空间体上双层板的动力响应

采用黏弹性半空间体上无限大双层板模型模拟路面结构,通过三重Fourier积分变换得到单位脉冲荷载作用下路面动力响应的Green函数。根据线性系统的叠加原理,利用广义Duhamel积分推导出移动荷载作用下路面动力响应的解析解。采用自适应Simpson法编制了计算奇异、振荡函数的广义积分计算程序,完成了路面动力响应从波数-频率域到时间-空间域的转换。结合算例,对移动荷载作用下路面的振动规律进行研究,讨论上、下层板厚度和板材料的弹性模量对路面动力响应的影响,明确了路面结构的振动特性,研究结果可为路面结构的设计、施工提供理论指导。     

Dynamic responses of unsaturated half‐space soil to a moving harmonic rectangular load

The problem of the dynamic responses of a semi‐infinite unsaturated poroelastic medium subjected to a moving rectangular load is investigated analytical/numerically. The dynamic governing equations are obtained with consideration of the compressibility of solid grain and pore fluid, inertial coupling, and viscous drag as well as capillary pressure in the unsaturated soil, and they can be easily degraded to the complete Biot's theory. Using the Fourier transform, the general solution for the equations is derived in the transformed domain, and then a corresponding boundary value problem is formulated. By introducing fast Fourier transform algorithm, the unsaturated soil vertical displacements, effective stresses, and pore pressures induced by moving load are computed, and some of the calculated results are compared with those for the degenerated solution of saturated soils and confirmed. The influences of the saturation, the load speed, and excitation frequency on the response of the unsaturated half‐space soil are investigated. The numerical results reveal that the effects of these parameters on the dynamic response of the unsaturated soil are significant.     

基于薄层法和移动坐标系法的地铁竖向振动在成层土层中传播

针对地铁竖向振动在成层地基中的传播,提出了移动荷载作用下层状地基的分析模型。基于该模型,利用狄拉克函数及三重傅里叶变换将时空域内单个移动简谐荷载转换为频率-波数域内的荷载,结合薄层法和移动坐标系法推导了单个移动简谐荷载和移动简谐线荷载作用下三维层状地基动力响应的解析解,并给出了移动简谐线荷载动力响应解析解中参数n的经验取值范围;分析了荷载的移动速度对层状地基动力响应的影响以及弹性模量、泊松比、阻尼比和荷载频率对土体临界速度的影响。结果表明：荷载的移动速度对不同频带的动力响应的影响范围不同;移动速度对低频响应的影响程度大于对高频响应的影响程度;相比于泊松比和阻尼比,弹性模量对土体临界速度的影响最大;频率越接近荷载振动频率的振动响应,其幅值越大,临界速度越小。