MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

瑞雷波斜入射下层状半空间中沉积谷地周围的三维散射研究

针对层状半空间中沉积谷地对斜入射瑞雷波的三维散射问题, 采用直接刚度法计算自由场波场, 以层状半空间中移动斜线均布荷载动力格林影响函数求解三维散射波场, 建立了求解该问题的间接边界元方法. 通过与已有结果的比较, 验证了该方法的正确性, 并以均匀半空间以及弹性基岩上单一土层场地中沉积谷地为例进行了计算分析. 研究结果表明: 层状半空间与均匀半空间中沉积谷地对瑞雷波的散射存在显著差别; 层状半空间中瑞雷波的振动模态对沉积附近位移幅值有着重要影响; 土层刚度和厚度等参数也对沉积附近位移幅值大小及空间分布有着显著的影响.      

Soil characterization of Tınaztepe region (İzmir/Turkey) using surface wave methods and nakamura (HVSR) technique

To determine the shear wave velocity structure and predominant period features of T?naztepe in ?zmir, Turkey, where new building sites have been planned, active–passive surface wave methods and single-station microtremor measurements are used, as well as surface acquisition techniques, including the multichannel analysis of surface waves (MASW), refraction microtremor (ReMi), and the spatial autocorrelation method (SPAC), to pinpoint shallow and deep shear wave velocity. For engineering bedrock (V _s > 760 m/s) conditions at a depth of 30 m, an average seismic shear wave velocity in the upper 30 m of soil (AV_s30) is not only accepted as an important parameter for defining ground behavior during earthquakes, but a primary parameter in the geotechnical analysis for areas to be classified by V _s30 according to the National Earthquake Hazards Reduction Program (NEHRP). It is also determined that Z1.0, which represents a depth to V _s = 1000 m/s, is used for ground motion prediction and changed from 0 to 54 m. The sediment–engineering bedrock structure for T?naztepe that was obtained shows engineering bedrock no deeper than 30 m. When compared, the depth of engineering bedrock and dominant period map and geology are generally compatible.     

面向地下结构的最优地震动峰值指标随埋深变化规律

选取了50条实际地震动,采用一维场地等效线性化方法分别对均匀半空间场地和成层半空间场地进行地震响应分析,同时选择效益性作为判别标准来探究最优地震动峰值指标(峰值加速度PGA,峰值速度PGV,峰值位移PGD)随埋深变化的规律.研究结果表明:对于选取的两类场地,最优地震动峰值指标均随埋深的改变而变化,埋深浅时PGA效益性最...     

Two-dimensional scattering of SH waves in a soil layer underlain with a sloping bedrock

A two-dimensional analysis is applied to examine the effect that a sloping bedrock half-space has on the amplification of an anti-plane shear wave. The direct boundary integral equation method is used for the two-dimensional analysis. The particular soil–rock configuration investigated includes a homogeneous soil layer underlain by a sloping rock half-space. The rock half-space dips for a horizontal distance L and then becomes horizontal so that the overlying soil layer has a thickness H that remains constant from this point to infinity. The materials in the soil–rock configuration are considered viscoelastic except in the rock half-space below soil layer thickness H, which is considered elastic. This limitation in damping is due to the correction used for the truncation of the half-space boundary. Four cases are used to study the relationship between rock slope and surface displacement, vertical, 1:2, 1:4, 1:8. Surface displacements are determined for each of these cases for half-space incidence angles of 90, 75, and 60°. To allow for applicability to a wide range of problems, results are determined as a function of dimensionless parameters. In addition, solutions from a one-dimensional analysis are compared with the results of the two-dimensional analysis to establish limits outside of which a one-dimensional analysis suffices.     

Surface and downhole shear wave seismic methods for thick soil site investigations

Shear wave velocity–depth information is required for predicting the ground motion response to earthquakes in areas where significant soil cover exists over firm bedrock. Rather than estimating this critical parameter, it can be reliably measured using a suite of surface (non-invasive) and downhole (invasive) seismic methods. Shear wave velocities from surface measurements can be obtained using SH refraction techniques. Array lengths as large as 1000 m and depth of penetration to 250 m have been achieved in some areas. High resolution shear wave reflection techniques utilizing the common midpoint method can delineate the overburden-bedrock surface as well as reflecting boundaries within the overburden. Reflection data can also be used to obtain direct estimates of fundamental site periods from shear wave reflections without the requirement of measuring average shear wave velocity and total thickness of unconsolidated overburden above the bedrock surface. Accurate measurements of vertical shear wave velocities can be obtained using a seismic cone penetrometer in soft sediments, or with a well-locked geophone array in a borehole. Examples from thick soil sites in Canada demonstrate the type of shear wave velocity information that can be obtained with these geophysical techniques, and show how these data can be used to provide a first look at predicted ground motion response for thick soil sites.     

Two-dimensional scattering of in-plane body waves due to a discontinuity in bedrock

The direct boundary integral equation technique is used to study in-plane surface amplification of in-plane seismic body waves for the case of an inhomogeneity in a bedrock half-space. In the studied soil configuration, a soil layer rests on a rock half-space which includes a rock inclusion. The rock inclusion considered is a semi-infinite horizontal rock layer in which its upper boundary borders the soil layer. Materials in the soil–rock configuration are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to in-plane body waves. The study investigates varying the stiffness and the thickness of the rock inclusion for a range of frequencies and wave incidence angles. Anti-plane waves for this type of soil-rock configuration have been addressed in a previous article by Heymsfield (Earthquake Engng. Struct. Dyn. 28 : 841–855 (1999)). Copyright © 1999 John Wiley & Sons, Ltd.     

Two-dimensional scattering of SH waves due to a discontinuity in bedrock

The direct boundary integral equation technique is applied to determine the impact on surface amplification caused by an inhomogeneity in a bedrock half-space. The particular soil-rock configuration studied is one in which a soil layer rests on a rock half-space which includes a rock inclusion. The particular rock inclusion considered for this study is a semi-infinite rock layer with its upper boundary bordering the soil layer. Materials are considered viscoelastic except for the section of the rock half-space below the level of the rock inclusion which is considered elastic. A parametric study is performed to determine controlling factors for surface displacement due to a vertically incident shear wave. The study includes varying the stiffness and the thickness of the inclusion for a range of frequencies. Solutions from a one-dimensional analysis are compared with the results of a two-dimensional analysis in order to establish limits inside of which a two-dimensional analysis is required. Copyright © 1999 John Wiley & Sons, Ltd.     

层状半空间中洞室对平面SH波的放大作用

利用间接边界元法,求解了弹性层状半空间中洞室对入射平面SH波的放大作用问题,并以基岩上单一土层中洞室对入射平面SH波的放大作用为例进行了数值计算分析。本文模型的特点之一是考虑了层状场地的动力特性,因而更接近于实际工程;特点之二是计算精度非常高。研究表明,层状半空间中洞室对波的放大作用与均匀半空间中情况有着本质的差别;层状半空间中洞室附近地表动力响应由土层动力特性和洞室对波的散射二者共同决定。土层动力特性不仅影响洞室附近地表位移的幅值,还会影响地表位移的频谱。在土层的前几阶共振频率附近,随着基岩与土层剪切波速比的增大,土层的影响随之增大,而随着土层厚度的增加,土层的影响随之减小,并逐渐趋于均匀半空间情况。     

ESTIMATING MODAL PARAMETERS FOR A SIMPLE SOFT-SOIL SITE HAVING A LINEAR DISTRIBUTION OF SHEAR WAVE VELOCITY WITH DEPTH

Simple approximate expressions are derived for estimating equivalent modal parameters of a single soft-soil layer that has a straight line distribution of shear wave velocities and is on a flexible half space which is more rigid than the overlying soil layer. It is shown that radiation damping from the flexibility of the half space can be converted approximately into an equivalent material damping and that a site of a single soft-soil layer on a flexible half space can be replaced by an equivalent single layer, with the same shear wave velocity distribution, on a rigid bedrock. The implementation for modal analyses and non-linear analyses of the equivalent site under earthquake excitation is discussed.     

The estimation of shear-wave statics using in situ measurements in marine near-surface sediments

Shear‐wave statics in marine seismic exploration data are routinely too large to be estimated using conventional techniques. Near‐surface unconsolidated sediments are often characterized by low values of V_s and steep velocity gradients. Minor variations in sediment properties at these depths correspond to variations in the shear‐wave velocity and will produce significant static shifts. It is suggested that a significant proportion of the shear‐wave statics solution can be estimated by performing a separate high‐resolution survey to target near‐surface unconsolidated sediments. Love‐wave, shear‐wave refraction and geotechnical measurements were individually used to form high‐resolution near‐surface shear‐wave velocity models to estimate the shear‐wave statics for a designated survey line. Comparisons with predicted statics revealed that shear‐wave statics could not be estimated using a velocity model predicted by substituting geotechnical measurements into empirical relationships. Empirical relationships represent a vast simplification of the factors that control V_s and are therefore not sufficiently sensitive to estimate shear‐wave statics. Refraction measurements are potentially sensitive to short‐wavelength variations in sediment properties when combined with accurate navigational data. Statics estimated from Love‐wave data are less sensitive, and sometimes smoothed in appearance, since interpreted velocity values represent an average both laterally and vertically over the receiver array and the frequency–depth sensitivity range, respectively. For the survey site, statics estimated from near‐surface irregularities using shear‐wave refraction measurements represent almost half the total statics solution. More often, this proportion will be greater when bedrock relief is less.     

SH Wave Number Green’s Function for a Layered,Elastic Half-Space. Part I: Theory and Dynamic Canyon Response by the Discrete Wave Number Boundary Element Method

We present a closed-form frequency-wave number (ω – k) Green’s function for a layered, elastic half-space under SH wave propagation. It is shown that for every (ω – k) pair, the fundamental solution exhibits two distinctive features: (1) the original layered system can be reduced to a system composed by the uppermost superficial layer over an equivalent half-space; (2) the fundamental solution can be partitioned into three different fundamental solutions, each one carrying out a different physical interpretation, i.e., an equivalent half-space, source image impact, and dispersive wave effect, respectively. Such an interpretation allows the proper use of analytical and numerical integration schemes, and ensures the correct assessment of Cauchy principal value integrals. Our method is based upon a stiffness-matrix scheme, and as a first approach we assume that observation points and the impulsive SH line-source are spatially located within the uppermost superficial layer. We use a discrete wave number boundary element strategy to test the benefits of our fundamental solution. We benchmark our results against reported solutions for an infinitely long circular canyon subjected to oblique incident SH waves within a homogeneous half-space. Our results show an almost exact agreement with previous studies. We further shed light on the impact of horizontal strata by examining the dynamic response of the circular canyon to oblique incident SH waves under different layered half-space configurations and incident angles. Our results show that modifications in the layering structure manifest by larger peak ground responses, and stronger spatial variability due to interactions of the canyon geometry with trapped Love waves in combination with impedance contrast effects.     

REFLECTION OF ELASTIC WAVES FROM PERIODICALLY STRATIFIED MEDIA WITH INTERFACIAL SLIP*

A periodically stratified elastic medium can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding is a special instance of such a medium. Slowness surfaces are derived for all plane wave modes through the equivalent medium and reflection coefficients for a half-space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for the reflection of SH-waves from a half-space with parallel slip interfaces is found following the matrix method of K. Gilbert applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half-space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror. The method for the reflection of P- and SV-waves is fully outlined, and reflection coefficients are shown for a particular example. The solution requires finding the eigenvalues of a 4 × 4 transfer matrix, each eigenvalue being associated with a particular wave. At higher frequencies, unexpected eigenvalues are found corresponding to refracted waves for which shear and compressional parameters are completely coupled. The two eigenvalues corresponding to the transmitted wavefield give amplitude decay perpendicular to the stratification along with up- and downgoing phase propagation in some other direction. Much of this work was performed while the author was at the Department of Geophysics and Planetary Sciences, Tel-Aviv University, Ramat-Aviv, Israel. The author is grateful for illuminating discussions with K. Helbig and K. Gilbert.     

地震动输入界面的选取对地震动参数的影响

地震反应分析中的输入界面是一种规定的假想基岩面,其选取具有很大的不确定性,对确定地震动参数的影响很大.本文以北京地区存在剪切波速较高的卵石硬夹层场地为研究对象,探讨实际工程场地中能否将硬夹层选作地震动输入界面的问题.利用一维等效线性化波动方法,对390个含有剪切波速高于500m/s的卵石层场地模型进行地震反应分析计算,讨论卵石层的上覆土层、下伏土层以及卵石层自身特性对地震反应分析结果的影响.计算结果显示,厚度5m左右、剪切波速小于600m/s的卵石层作为地震动输入界面是否合理,主要取决于上覆土层和下伏土层,而卵石层厚度和剪切波速的影响很小.对于一般性工程,当上覆土层厚度大于15m,下伏土层厚度在10m以内时,选取硬夹层顶面与选取钻探揭示的基岩或剪切波速不小于500m/s的土体顶面为输入界面,计算结果差别不大;当上覆土层厚度达到60m,下伏土层的变化对计算结果影响很小.研究表明,在钻探没有揭示出基岩或坚硬土体时,埋深大于60m的硬夹层可以选为地震动输入界面,这一结论对实际的地震安全性评价工作具有一定参考价值.     

Effect of preloading on the amplification characteristics of soil profiles

Preloading is a temporary loading, usually an embankment, applied to improve subsurface soils by densification. This paper studies the effect of preloading on the amplification characteristics of soft sites with an elaborate parametric analysis. The soil type, the depth of the bedrock, the water table depth, the level of preloading, the applied earthquake, the shear wave velocity of the bedrock and the shear modulus and damping versus shear strain relations were varied in a systematic manner. The analysis was performed by the commonly used one-dimensional equivalent-linear dynamic method. The shear wave velocity versus depth and the effect of preloading on shear velocity are computed with well-established soil mechanics equations. The results illustrated that the seismic response at the top of the profile generally decreases as a result of preloading. A more detailed analysis of results shows that the effect of preloading on the seismic response depends on the soil type and the depth of the bedrock. Based on these results, a method is proposed by which a practicing engineer involved with improvement of soft ground can simulate the effect of preloading on the seismic motion.     

Side effect of using response spectral amplification ratios for soil sites—variability and earthquake-magnitude and source-distance dependent amplification ratios for soil sites

We investigate a special type of variability in response spectral amplification ratios computed from numerical “engineering” models for a soft soil site. The engineering models are defined by shallow soil layers over “engineering” bedrock with a shear-wave velocity over 600–700 m/s and the model is subjected to vertical propagating shear waves. The variability, perhaps unique in earthquake engineering, is a result of the “perfectly accurate” computational procedure. For example, an engineering soil site model, subjected to two rock site records or the two horizontal components of a rock site record, produces different response spectral amplification ratios. We use a large number of strong-motion records from “engineering” rock sites, with a reasonably balanced distribution with respect to magnitude and source distance, generated by subduction earthquakes in Japan, to investigate the nature of the variability. In order to avoid any approximation in removing the effect of soil nonlinear response, we use a simple model, a single horizontal soil layer over a bedrock, modelled as elastic. We then demonstrate that a similar type of variability observed in the one- or two-dimensional nonlinear soil models is caused by the nature of response spectral amplification ratios, not a direct result of soil nonlinear response. Examination of variability reveals that the average of response spectral amplification ratios systematically depends on both earthquake magnitude and source distance. We find that, at periods much longer than the site natural periods of the soil sites, the scatter of the amplification ratios decreases with increasing magnitude and source distance. These findings may have a potential impact in establishing design spectra for soft soil sites using strong-motion attenuation models or dynamic numerical modelling.     

处于海洋环境的桩柱在瑞利波作用下的响应

本文以一海洋深水环境桩柱为例,考虑到海洋地基为两相饱和土介质,基于Biot两相饱和多孔介质理论,通过数值算例分析了饱和介质中瑞利波加速度、位移分布规律并与单相弹性介质地基条件下进行了对比,结果表明,瑞利波主要作用于地表附近且振动幅度随深度衰减很快,饱和地基下的瑞利波影响深度要大于单相弹性地基,在瑞利波地震作用下,考虑两相饱和介质下的桩柱位移响应大于单相弹性介质地基。并且由于桩柱处海洋深水环境,本文对环境水体对桩柱振动响应的影响进行了研究,结果表明,流固耦合效应可明显降低桩柱运动响应,在海洋深水环境地震响应分析时,有必要考虑饱和地基以及流固耦合效应对桩柱运动响应的影响。     

Vibration isolation by trenches in continuously nonhomogeneous soil by the BEM

Vibration isolation of structures from ground-transmitted waves by open trenches in isotropic, linearly elastic or viscoelastic soil with a shear modulus varying continuously with depth is numerically studied. Both an exponential and a linear shear modulus variation with depth are used in this work. Waves produced by the harmonic motion of a rigid surface machine foundation are considered. The problem is solved by the frequency domain boundary element method employing the Green's function of Kausel-Peek-Hull for a thin layered half-space. Thus only the trench perimeter and the soil-foundation interface need essentially to be discretized. The proposed methodology is first tested for accuracy by solving two Rayleigh wave propagation problems in nonhomogeneous soil with known analytical solutions and/or for which experimental results are available. Then the method is applied to vibration isolation problems and the effect of the inhomogeneity on the wave screening effectiveness of trenches is studied.     

饱水介质波动分析的谐波输入初始间断的处理方法

本文以饱水两相介质的土力学模型为研究对象,在假定两相介质为弹性介质条件下,采用了显式有限元法和透射边界进行了饱和弹性半空间动力响应问题的研究。为避免谐波输入初始间断的影响,文中提出了一个处理函数,并以弹性半空间为算例,对饱水介质和单相介质分别进行了在底边界P波垂直入射时的动力响应分析,验证了该处理函数的有效性和实用性。     

APPLICATION AND STATISTICAL ANALYSIS OF RELATIONSHIP BETWEEN SHEAR WAVE VELOCITY AND DEPTH OF SOIL-LAYERS

The shear wave velocity is one of the important parameters in seismic engineering.The common mathematical models of relationship between shear wave velocity and depth of soil-layers are linear function model,quadratic function model,power function model,cubic function model,and quartic function model.It is generally believed that the regression formulae based on aforementioned mathematical models are mainly used for preliminary estimation of the local shear wave velocity.In order to increase the value of test data of wave speed in boreholes,the calculation formulae for the thickness of ground cover layer are derived based on the aforementioned mathematical models and their fitting parameters.The calculation formulae for the mean shear wave velocity of soil-layers are derived by integral mean value theorem.Accordingly,the calculation formulae for the equivalent shear wave velocity of soil-layers are derived.The calculation formulae for the depth of reflective waves in time-depth conversion of the reflection seismic exploration are derived.Through the statistical analysis of test data of shear wave velocity of soil layers in Changyuan County,Henan Province,regression formulae and their fitting parameters of aforementioned mathematical models are obtained.The results show that in the determination of the quality of these regression formulae and their fitting parameters,the adjusted R-square,root mean square error and residual error,the matching on the statistical range between the geometry of function of mathematical models used and the scattergram of the measured data,the application purpose and the simplicity of the regression formulae should be considered.With the aforementioned new formulae,the results show that the calculated values of equivalent shear wave velocity of soil-layers and thickness of ground cover layer meet the engineering needs.The steps for statistics and applications of the relationship between shear wave velocity and depth of soil-layers for a new area are as follows:(1) Analyze the relevant data about the site such as the drilling and wave speed test data,etc.and divide the site into seismic engineering geological units;(2) In a single seismic engineering geological unit,make statistical analysis of the data of borehole wave speed test,comprehensively identify and select mathematical models and their fitting parameters of the relationship between shear wave velocity and depth of soil-layers;(3) Substitute the selected fitting parameters into the formulae,based on their mathematical models for the thickness of ground cover layer,or the equivalent shear wave velocity of soil-layers,or the depth of reflective wave,then the thickness of ground covering layer,equivalent shear wave velocity,and depth of reflective wave are obtained.