2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves

The problem of harmonic wave diffraction by tunnels in an infinite poroelastic saturated soil obeying Biot's theory is studied numerically under conditions of plane strain and the effect of poroelasticity on the response is assessed through some parametric studies. The method is based on the theory of Mei and Foda, which considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer at the free boundaries. Both problems are solved numerically by the boundary element method in the frequency domain. Results dealing with the response of a circular tunnel with and without an elastic concrete liner in an infinite poroelastic medium to incident harmonic P and SV plane waves are provided and compared against analytical ones as well as to those corresponding to linear elastic soil behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

Dynamic interaction of twin vertically overlapping lined tunnels in an elastic half space subjected to incident plane waves

The scattering of plane harmonic P and SV waves by a pair of vertically overlapping lined tunnels buried in an elastic half space is solved using a semi-analytic indirect boundary integration equation method. Then the effect of the distance between the two tunnels, the stiffness and density of the lining material, and the incident frequency on the seismic response of the tunnels is investigated. Numerical results demonstrate that the dynamic interaction between the twin tunnels cannot be ignored and the lower tunnel has a significant shielding effect on the upper tunnel for high-frequency incident waves, resulting in great decrease of the dynamic hoop stress in the upper tunnel; for the low-frequency incident waves, in contrast, the lower tunnel can lead to amplification effect on the upper tunnel. It also reveals that the frequency-spectrum characteristics of dynamic stress of the lower tunnel are significantly different from those of the upper tunnel. In addition, for incident P waves in low-frequency region, the soft lining tunnels have significant amplification effect on the surface displacement amplitude, which is slightly larger than that of the corresponding single tunnel.     

圆弧形凹陷饱和土场地对平面P波散射问题的解析解

以Biot 饱和多孔介质动力学理论为基础，利用Fourier Bessel级数展开法, 得到饱和多孔介质半空间中圆弧形凹陷地形对平面P波的散射问题的解析解. 数值计算给出地表位移幅值，分析了入射波波长、入射角、圆弧高宽比对地表位移幅值的影响，并与现存的在单相介质情况下得到的结论进行对比.     

半球形凹陷饱和土半空间对入射平面SV波三维散射问题的解析解

以Biot饱和多孔介质动力学理论为基础,利用Fourier Bessel级展开法,给出了饱和多孔介质半空间中半球形凹陷地形对入射平面SV波散射问题的解析解.利用这一解析解数值计算地表位移幅值,分析入射角、入射波频率对地表位移幅值的影响,并与现存的单相介质情况下三维半球形凹陷问题解析解进行对比得出如下结论：(1)本文两相介质模型与单相介质模型有很大区别;(2)入射角、入射频率对场地表面位移分量有重要影响;(3)随着入射角、入射频率的增大,地表位移分布变得更复杂,且放大效应比单相介质模拟结果更加显著.     

Dynamic response of a piecewise circular tunnel embedded in a poroelastic medium

Recently, considerable efforts have been devoted to evaluation of seismic dynamic response of a circular tunnel. Conventional approaches have considered integral liners embedded in an elastic medium. In this study, we re-examine the problem with piecewise liners embedded in a porous medium. Surrounding saturated porous medium of tunnels is described by Biot's poroelastic theory, while the liner pieces and the connecting joints are treated as curved beams and characterized by curved beam theories. The scattered wave field in the porous medium is obtained by the wave function expansion method. The differential equations governing the vibration of a curved beam is discretized by the General Differential Quadrature (GDQ) method. The domain decomposition method is used to establish the global discrete dynamic equations for the piecewise tunnel. The surrounding soil and the tunnel are coupled together via the stress and the displacement continuation conditions which are implemented by the boundary collocation method. Numerical results demonstrate that the stiffness difference between the liner piece and the connecting joints has a considerable influence on the internal forces of the liner piece.     

3-D seismic response analysis of long lined tunnels in half-space

The dynamic response of infinitely long lined tunnels with a uniform cross-section buried into an elastic or viscoelastic half-space to body and surface harmonic seismic waves is numerically determined by a special direct boundary element method in the frequency domain. The waves have an arbitrary direction of propagation with respect to the axis of the tunnel and this renders the problem three-dimensional. However, this problem is effectively reduced to a two-dimensional one by a coordinate transformation and appropriate integration of the full-space dynamic fundamental solution along the direction of the tunnel axis. Quadratic isoparametric boundary line elements and advanced numerical integration techniques for the treatment of singular integrals produce results of high accuracy. Numerical results are presented for the case of an infinitely long lined tunnel of circular cross-section and compared against those of a full three-dimensional boundary element analysis, as well as those of other methods. Thus the proposed method is illustrated and its merits demonstrated.     

具有饱和土沉积层的充水河谷对平面波的散射

把波函数展开方法用于地震波散射问题的研究中,首次在频域内给出了具有饱和土沉积层的圆弧形充水河谷对平面P波和SV散射问题的解析解答. 其中半空间场地用单相介质弹性动力理论模拟,河谷中的饱和土沉积层用饱和多孔介质的Biot动力学理论模拟,河谷中的水用无黏性流体(理想流体)介质模拟. 文中还给出算例,计算了不同高宽比的河谷谷底的位移幅值,分析表明河谷地形的存在使得半空间介质表面的位移幅值随着观察点位置的变化变化较大.     

Transient Scattering of Elastic Waves by Dipping Layers of Arbitrary Shape Part 1: Antiplane Strain Model

Scattering of elastic waves by two dimensional multilayered dipping sediments of arbitrary shape embedded in an elastic half-sapce is investigated by using a bondary method. The displancement field is evaluated throughout the elastic media for both steady state and transient incident SH waves. The unknown scattered field is expressed in terms of wave functions which satisfy the equation of motion, traction-free boundary condition and appropariate radiation conditions. The transient response is constructed from the steady state solution by using the fast Fourier transform technique. The numerical results presented demonstrate that scattering of waves by subsurface irregularities may cause locally very large amplification of surface ground motion. The motion can be affected greatly by the scattered surface waves in the sediments. The results clearly indicate that the surface ground motion depends upon a number of parameters present in the problem, such as frequency and the angle of incidence of the incoming wave, impedance contrast between the layers and location of the observation point.     

基于粘性-滑移界面接触模型半空间隧道衬砌对平面P波的散射

基于粘性-滑移模型模拟隧道衬砌和围岩之间的接触状态,采用间接边界积分方程法求解弹性半空间中隧道衬砌对平面P波的散射,着重考察了地表位移放大和衬砌动应力集中效应。结果表明:接触滑移边界的刚度系数和粘度系数对隧道衬砌的动应力分布影响较大,滑移刚度较小时,共振频率段动应力集中效果更加显著;随着粘度系数的增大,衬砌内部环向应力幅值逐渐减小。另外,界面刚度系数和粘度系数对隧道衬砌动力反应的影响程度也受控于入射频率和角度。     

Scattering of plane harmonic SH,SV, P and rayleigh waves by non-axisymmetric three-dimensional canyons: A wave function expansion approach

Scattering of elastic waves by three-dimensional canyons embedded within an elastic half-space is investigated by using a wave function expansion technique. The geometry of the canyon is assumed to be non-axisymmetric. The canyon is subjected to incident plane Rayleigh waves and oblique incident SH, SV and P waves. The unknown scattered wavefield is expressed in terms of spherical wave functions which satisfy the equations of motion and radiation conditions at infinity, but they do not satisfy stress-free boundary conditions at the half-space surface. The boundary conditions are imposed locally in the least-squares sense at several points on the surface of the canyon and the half-space. Through a comparative study the validity and limitations of two-dimensional approximations (antiplane strain and plane strain models) have been examined. It is shown that scattering of waves by three-dimensional canyons may cause substantial change in the surface displacement patterns in comparison to the two-dimensional models. These results emphasize the need for three-dimensional modelling of realistic problems of interest in strong ground motion seismology and earthquake engineering.     

饱和土沉积谷场地对平面SV波的散射问题的解析解

把波函数展开方法用于饱和多孔介质中波的传播的研究中,给出了不同土层界面条件（透水条件和不透水条件）下具有饱和土沉积层的圆弧形沉积河谷场地对平面SV波散射问题的解析解. 其中沉积谷软土场地用饱和多孔介质的Biot动力学理论模拟,半空间场地用单相介质弹性动力理论模拟. 对于入射角大于临界入射角时,产生的面波的波函数用有限Fourier级数展开,这种方法适用于较大的入射波频率范围,这是现存的数值方法所不能比拟的一大优点. 文中算例分析了入射波频率和入射角对地震地面运动的影响.     

Spatial dispersion of surface waves at the free surface of a general anisotropic elastic medium

A new technique relates the wave velocity of the surface waves in anisotropic elastic medium to its elastic constants. Anisotropic propagation of surface waves is studied in a half-space occupied by a general anisotropic elastic solid. The phase velocity expressions of quasi-waves, in three-dimensional space, are used to derive the secular equation of surface waves. The complex secular equation is resolved, analytically, into real and imaginary parts and is then solved, numerically, for phase velocity along a given phase direction on the surface. The complete procedure is thus analogous to the one used for conventional Rayleigh waves in isotropic medium. A non-linear equation relates the ray direction of the surface waves to its phase direction on the (plane) surface of the medium. The analytical differentiation of secular equation yields the directional derivative of phase velocity. This derivative is used to calculate the wave velocity of surface waves. Spatial variations of phase velocity, wave velocity and ray direction over the free plane surface are plotted for the numerical models of crustal rocks with orthorhombic, monoclinic and triclinic anisotropies.     

Dynamic soil-tunnel interaction in layered half-space for incident plane SH waves

The dynamic soil-tunnel interaction is studied by indirect boundary element method (IBEM), using the model of a rigid tunnel in layered half-space, which is simplified to a single soil layer on elastic bedrock, subjected to incident plane SH waves. The accuracy of the results is verified through comparison with the analytical solution. It is shown that soil-tunnel interaction in layered half-space is larger than that in homogeneous half-space and this interaction mechanism is essentially different from that of soil-foundation-superstructure interaction.     

Diffraction of plane sh waves in a half-space

Scattering of antiplane shear waves (SH) in two dimensions by surface and near-surface defects in a homogeneous, isotropic elastic semi-infinite medium has been studied. Attention has been focused here in the range of medium to long wavelengths. A combined finite element and analytical technique has been used to study the problems of scattering by semi-circular and triangular canyons. The results for the former case are compared with the known exact solution and those for the latter case are compared with some available approximate solutions. Finally a problem of multiple scattering by a triangular canyon and a nearby circular tunnel is studied. Numerical results are presented showing the effects of multiple scattering and different angles of incidence. These results are of interest in earthquake engineering.     

Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation

An exact analysis for two-dimensional dynamic interaction of monochromatic progressive plane compressional and shear seismic waves with a permeable circular tunnel lining of circumferentially varying wall thickness buried in a boundless porous elastic fluid-saturated formation is presented. The novel features of Biot dynamic theory of poroelasticity in conjunction with the translational addition theorems for cylindrical wave functions, along with the appropriate wave field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. The analytical results are illustrated with numerical examples in which an air-filled and water-saturated permeable tunnel lining of variable wall thickness, embedded within water-saturated surrounding formations of distinct frame properties (soft, stiff, and stiff viscoelastic soil), is insonified by fast compressional or shear waves at selected angles of incidence. The effects of liner eccentricity, interface permeability, formation material type, incident wave frequency, and angle of incidence on the hoop stress amplitude are evaluated and discussed for representative values of the parameters characterizing the system. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

黏弹介质中勒夫波频散问题的统一解及其动态特征分析

目前完全弹性介质中面波频散特征的研究已较为完善,多道面波分析技术(MASW)在近地表勘探领域也取得了较好的效果,但黏弹介质中面波的频散特征研究依然较少.本文基于解析函数零点求解技术,给出了完全弹性、常Q黏弹和Kelvin-Voigt黏弹层状介质中勒夫波频散特征方程的统一求解方法.对于每个待计算频率,首先根据传递矩阵理论得到勒夫波复频散函数及其偏导的解析递推式,然后在复相速度平面上利用矩形围道积分和牛顿恒等式将勒夫波频散特征复数方程的求根问题转化为等价的连带多项式求解问题,最后通过求解该连带多项式的零点得到多模式勒夫波频散曲线与衰减系数曲线.总结了地层速度随深度递增和夹低速层条件下勒夫波频散特征根在复相速度平面上的运动规律和差异.证明了频散曲线交叉现象在复相速度平面上表现为:随频率增加,某个模式特征根的移动轨迹跨越了另一个模式特征根所在的圆,并给出了这个圆的解析表达式.研究还表明,常Q黏弹地层中的基阶模式勒夫波衰减程度随频率近似线性增加,而Kelvin-Voigt黏弹地层中的基阶模式勒夫波衰减程度随频率近似指数增加,且所有模式总体衰减程度强于常Q黏弹地层中的情况.     

Seismic ground amplification by unlined tunnels subjected to vertically propagating SV and P waves using BEM

Recent researches have revealed that the seismic ground response above tunnels can be different from the free-field motion during earthquakes. Nevertheless, to the best of the authors׳ knowledge, neither building codes nor seismic microzonation guidelines have yet considered this matter. In the present study, the seismic response of a linear elastic medium including a buried unlined tunnel subjected to vertically propagating incident SV and P waves are addressed. For analysis purposes, a numerical time-domain analysis is performed by utilizing a robust numerical algorithm working based on the boundary element method. It is observed that the amplification of the ground surface underlain by a tunnel is increased in long periods. The variation of the amplification factor and characteristic period of the medium versus the buried depth of the tunnel are depicted as the major results of this study. Some simple and useful relations are proposed for estimating the seismic microzonation of the areas underlain by tunnels. These relations can also be used for the preliminary seismic design of structures located on underground structures.     

Transient scattering of elastic waves by dipping layers of arbitrary shape. Part 2: Plane strain model

Scattering of elastic waves by dipping layers of arbitrary shape embedded within an elastic half-space is investigated for a plane strain model by using a boundary method. Unknown scattered waves are expressed in the frequency domain in terms of wave functions which satisfy the equations of motion and appropriate radiation conditions at infinity. The steady state displacement field is evaluated throughout the elastic medium for different incident waves so that the continuity conditions along the interfaces between the layers and the traction-free conditions along the surface of the half-space are satisfied in the least-squares sense. Transient response is constructed from the steady state one through the Fourier synthesis. The results presented show that scattering of waves by dipping layers may cause locally very large amplification of surface ground motion. This amplification depends upon the type and frequency of the incident wave, impedance contrast between the layers, component of displacement which is being observed, location of the observation station and the geometry of the subsurface irregularity. These results are in agreement with recent experimental observations.     

平面SH波作用下衬砌隧道对地下地震动的影响

以地下隧道对附近场地动力特性的影响为研究目标,基于弹性波动理论,利用波函数展开法和镜像法,分析了弹性半空间中圆形衬砌隧道对平面SH波入射产生的散射问题,得到了地下圆形衬砌隧道附近场地位移的级数解答。通过数值算例分析了地下圆形衬砌隧道对场地动力响应的影响,重点考察了SH波入射角度、入射频率和隧道埋深、衬砌刚度对隧道周围土体动力响应随深度变化的影响规律。结果表明,地下隧道对沿线场地的地下地震动影响显著。     

Scattering of plane SH waves by a circular- arc hill with a circular tunnel

IntroductionEarthquakedamageinvestigationsandtheoreticalanalysesshowthatahillamplifiesincidentwavestremendouslyduetomulti-reflectionofthewaveswithinthehill;notabledynamicstressconcentrationisobservedonwallofatunnel(Pao,Mow,1973;Lee,Trifunac,1979).Therefore,itmaybeassumedthattheexistenceofatunnelinahillmighthavegreateramplificationeffectonthegroundmotionnearby,andthedynamicstressconcentrationofthetunnelinthehillmightbemorenotable.Itiscommonthattherailwaysorhighwaysarethroughahilloramountainwi…