One-dimensional modelling of the non-linear far field in soil–structure-interaction analysis

In a bounded domain elasto-plastic wave propagation can be modelled accurately using the finite-element method. As is even the case for an elastic analysis, an unbounded domain, e.g. a semi-infinite soil or fluid, can, however, not be represented in this manner, as any spatial discretization has to be avoided. For one-dimensional wave propagation with a bi-linear elasto-plastic material law involving one stress component an analytical solution exists. The latter is used in modelling the non-linear far field of an unbounded medium using a rigorous bookkeeping procedure of the generated elastic and plastic waves propagating in both directions. The need for a non-linear model of the far field arises, as in a two-dimensional representation of soil-structure interaction the surface waves do not decay.     

A hybrid modelling of soil–structure interaction problems for deeply embedded structures in A multilayered medium

Dynamic response of deeply embedded structures, such as underground tunnels and deep foundations, in a multilayered elastic half-space are analysed when the structure is excited by a plane P or SV wave propagating at some angle. The scattered field is represented by the sum of three Green's functions, corresponding to two oscillating forces and one oscillating moment at the centroid position of the buried structure. The amplitudes of these two forces and one moment are a priori unknown and are obtained by satisfying displacement and stress continuity conditions across the near-field/far-field boundary. The distinguishing feature of this technique from direct or indirect boundary integral techniques is that in these techniques a distribution of sources of unknown amplitude are considered at the near-field/far-field boundary, and a large number of sources are needed for different combinations of source-receiver arrangements. But in this technique the sources of unknown amplitude are placed at the location of the structure, not at the near-field/far-field boundary and, using the Saint Venant's principle, the scattered field is modelled. Thus, the number of sources required is reduced to only three. Two example problems are solved. The first one is for a deeply embedded footing in a three-layer soil mass and the second one is for a rectangular tunnel in a two-layer soil mass.     

Earthquake soil‐structure interaction of nuclear power plants,differences in response to 3‐D, 3 × 1‐D,and 1‐D excitations

In soil‐structure interaction modeling of systems subjected to earthquake motions, it is classically assumed that the incoming wave field, produced by an earthquake, is unidimensional and vertically propagating. This work explores the validity of this assumption by performing earthquake soil‐structure interaction modeling, including explicit modeling of sources, seismic wave propagation, site, and structure. The domain reduction method is used to couple seismic (near‐field) simulations with local soil‐structure interaction response. The response of a generic nuclear power plant model computed using full earthquake soil‐structure interaction simulations is compared with the current state‐of‐the‐art method of deconvolving in depth the (simulated) free‐field motions, recorded at the site of interest, and assuming that the earthquake wave field is spatially unidimensional. Results show that the 1‐D wave‐field assumption does not hold in general. It is shown that the way in which full 3‐D analysis results differ from those which assume a 1‐D wave field is dependent on fault‐to‐site geometry and motion frequency content. It is argued that this is especially important for certain classes of soil‐structure systems of which nuclear power plants subjected to near‐field earthquakes are an example.     

磁流体冲激波在非均匀磁场中的传播

本文应用Chisnell- no方法,求解了在理想介貭中,垂直磁流体冲激波在非均勻磁場中的传播問題。这种方法,把非均勻介貭分解成无限小的弱間断面,根据气体动力学中波与間断面相互作用的原理,算出激波通过弱間断面时的强度变化,然后用积分求得激波通过整个非均勻区时的强度变化。作者引入了激波的特征速度（它是激波在波前后介貭中传播速度的几何平均值）作为輔助参量,得到形式上比較簡单的激波传播方程。然后考虑了磁压力远大于气体压力的强磁介貭中的激波传播問題,并进行了数值积分。采用的介貭密度模型有三种:（1）阿尔芬波速为常数;（2）密度不变;（3）密度与磁場强度成正比。計算結果表明:当激波由弱磁場向强磁場传播时,激波的强度逐漸变弱。其中,在阿尔芬波速为常数的介貭中,激波强度的衰减最为緩慢;在密度不变的介貭中,激波强度的衰減最为迅速;而在密度与磁場成正比的介貭中,激波强度的衰減則介乎上述两种密度分布之間。作者联系磁流体冲激波在地球外层空間的传播問題进行了討論,密度的模型采取大气啃昔的观測結果（卽上述第三种密度分布）,并进行了适当的外推,估計了在十个地球半径处的磁流体冲激波传到地面时的强度,求出了激波在地面引起的磁場变化与激波初始速度之間的关系。根据上述簡化模型,計算結果表明,在十个地球半径处初始速度为10⁸厘米/秒的激波,传到地面引起的磁場变化大約为60伽（亻馬）,这个数值的量級恰好与中低緯度强磁暴的急始变幅相符。     

Short-and medium-term earthquake precursors as evidence of the sliding instability along faults

A theory of waves propagating along faults (trapped waves) is constructed. They have such a dispersion that their frequency decreases with approach to the catastrophic rupture threshold followed by a uniform slip on a fault (earthquake). At the same time, the group velocity of the trapped waves tends to infinity, which means the breaking of the wave pulse front. Together with the frequency decrease (spreading of the pulse), this phenomenon can serve as a short-term earthquake precursor. The sliding instability on faults dividing a real solid body into blocks leads to its unstable deformation, a significant decrease in the shear modulus, and development of plasticity. Because plasticity is accompanied by seismic anisotropy and precedes or is associated with earthquakes, seismic anisotropy can be regarded as a medium-term precursor of an earthquake.     

Synthetic seismograms ofP-waves propagating in solid wedges with free boundaries

A numerical method for the synthesis of seismograms for body wave propagation in solid wedges is presented. The method is based on the superposition of multiple reflections arising from the entrance of a plane primary wave. Therefore the method is restricted to those domains in time-space where no diffracted waves from the wedge axis occur. In spite of this restriction, the dispersion of body waves in wedges can well be studied by this method. Seismograms have been synthesized which show the dispersion of a primary,p-signal propagating in a solid 10°- and a 5°-wedge with free boundaries. For wedge anlges less than 10° the signal from (as distinguished from the wave front) suddently decreases its velocity from that in the infinite medium to about that of the plate wave as the signal approaches the wedge axis. A decrease of the dominant period of the interference signal occurs simultaneously in this transition zone. These observations are concordant with previous seismic model studies [1]. Particle motion diagrams disclose elliptical polarization of the interference signal in the neighboorhood of the wedge axis which changes its sense from prograde to retrograde on passing through the transition zone. This paper has been submitted for publication to Geophysics. It will appear, in Vol. 31 (1966).     

Elastic wave propagation in inhomogeneous anisotropic media

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New approach to analyzing soil-building systems

A new method of analyzing seismic response of soil-building systems is introduced. The method is based on the discrete-time formulation of wave propagation in layered media for vertically propagating plane shear waves. Buildings are modeled as an extension of the layered soil media by assuming that each story in the building is another layer. The seismic response is expressed in terms of wave travel times between the layers, and the wave reflection and transmission coefficients at layer interfaces. The calculation of the response is reduced to a pair of simple finite-difference equations for each layer, which are solved recursively starting from the bedrock. Compared with commonly used vibration formulation, the wave propagation formulation provides several advantages, including the ability to incorporate soil layers, simplicity of the calculations, improved accuracy in modeling the mass and damping, and better tools for system identification and damage detection.     

Propagation properties of guided wave in the anchorage structure of rock bolts

We investigated the properties of guided wave propagating in grouted rock bolts and the formation of the interface wave through theoretical analysis along with experimental and numerical simulations. Experimental and numerical simulations reveal that the wave propagating in anchorage structure is related to boundary conditions within the range of excitation wave frequencies. Waves with different frequencies have different propagation velocities and attenuation characteristics. The optimal excitation wave occurs in grouted rock bolts with minimized attenuation and maximized propagation distance, and the end reflection of grouted rock bolts can be observed clearly. Longitudinal wave propagating in rock bolts is very sensitive to anchorage strength. With the increase of anchorage strength, longitudinal wave gradually attenuates and eventually disappears. Subsequently, interface wave appears and the velocity of wave propagating in the grouted part becomes that of the interface wave. Based on these studies, ultrasonic guided wave was used to study the end reflection of embedded rock bolts with different anchorage strengths and bonding lengths. The relationships among anchorage strength, bonding length and attenuation coefficient K, as well as the means to inspect the bonding quality of the embedded rock bolts were also evaluated.     

Dynamic stress concentration in pre-stressed poroelastic media due to moving punch influenced by shear wave

During the occurrence of earthquake, the shear wave propagates in the rocks present inside/at the Earth’s crust. The propagation of shear wave may lead to the progression of punch present inside the rock medium. As a result of this, substantial stress accumulated at the vicinity of propagating punch inside rock medium which significantly affects the stability of various geological and human-made structure and, hence, may cause failure of structure. Therefore, the analysis of stress concentration at the vicinity of punch moving due to shear wave propagation has become prominent in the area of seismology. In the present paper, an analytical perspective has been employed to discuss the influence of velocity of moving punch associated with the propagation of shear wave on developed dynamic stress concentration (DSC) in three types of pre-stressed vertical transversely isotropic (VTI) poroelastic media viz. granite (an igneous rock); sandstone (a sedimentary rock); and marble (a metamorphic rock). The closed form expression of DSC for the force of constant intensity has been derived with the aid of Weiner-Hopf technique along with Galilean and two-sided Fourier integral transformations. The noticeable influence of different affecting parameters (viz. velocity of moving punch associated with the shear wave propagation, horizontal compressive/tensile initial stresses, vertical compressive/tensile initial stress, porosity, and anisotropy parameter) on dynamic stress concentration has also been reported. Numerical computation and graphical illustrations have been carried out for the aforementioned three different types of porous rocks to investigate the profound impact of affecting parameters on DSC. Moreover, some noteworthy peculiarities have also been derived from the obtained expression of dynamic stress concentration.     

High-resolution modelling of a large-scale river plume

Evolution of a large-scale river plume is studied numerically using the Massachusetts Institute of Technology general circulation model. The model parameters were set close to those observed in the area of the Columbia River mouth. The fine-resolution grid along with the non-hydrostatic dispersion included in the model allowed for the reproduction of detailed inner plume structure, as well as a system of internal waves radiated from the plume’s boundary. It was found that not only first-mode but also second- and third-mode internal waves are radiated from the plume at the latest stages of its relaxation when the velocity of the front propagation drops below an appropriate wave phase speed of internal baroclinic mode. The model output shows that the amplitude of these high-mode waves is of the same order as the leading first-mode waves, which in combination with the specific vertical structure (location of the maximum structure function beyond the pycnocline layer) creates favourable conditions for the generation of shear instabilities. High-resolution model output also reveals evidence of a fine internal structure of the plume characterized by the presence of secondary fronts inside the plume and secondary internal wave systems propagated radially from the lift-off area to the outer boundary. These structures intensify the mixing processes within the propagating plume with predominance of the entrainment mechanism developing on the lower boundary between the plume’s body and underlying waters. The scheme of horizontal circulation in the plume was reproduced by the methodology of Lagrange drifters released near the mouth at different depths.

     

High-resolution modelling of a large-scale river plume

Evolution of a large-scale river plume is studied numerically using the Massachusetts Institute of Technology general circulation model. The model parameters were set close to those observed in the area of the Columbia River mouth. The fine-resolution grid along with the non-hydrostatic dispersion included in the model allowed for the reproduction of detailed inner plume structure, as well as a system of internal waves radiated from the plume’s boundary. It was found that not only first-mode but also second- and third-mode internal waves are radiated from the plume at the latest stages of its relaxation when the velocity of the front propagation drops below an appropriate wave phase speed of internal baroclinic mode. The model output shows that the amplitude of these high-mode waves is of the same order as the leading first-mode waves, which in combination with the specific vertical structure (location of the maximum structure function beyond the pycnocline layer) creates favourable conditions for the generation of shear instabilities. High-resolution model output also reveals evidence of a fine internal structure of the plume characterized by the presence of secondary fronts inside the plume and secondary internal wave systems propagated radially from the lift-off area to the outer boundary. These structures intensify the mixing processes within the propagating plume with predominance of the entrainment mechanism developing on the lower boundary between the plume’s body and underlying waters. The scheme of horizontal circulation in the plume was reproduced by the methodology of Lagrange drifters released near the mouth at different depths.     

3-D shell analysis of cylindrical underground structures under seismic shear (S) wave action

The 3-D shell theory is employed in order to provide a new perspective to earthquake-induced strains in long cylindrical underground structures, when soil-structure interaction can be ignored. In this way, it is possible to derive analytical expressions for the distribution along the cross-section of axial, hoop and shear strains and also proceed to their consistent superposition in order to obtain the corresponding principal and von Mises strains. The resulting analytical solutions are verified against the results of 3-D dynamic FEM analyses. Seismic design strains are consequently established after optimization of the analytical solutions against the random angles which define the direction of wave propagation relative to the longitudinal structure axis, the direction of particle motion and the location on the structure cross-section. The basic approach is demonstrated herein for harmonic shear (S) waves with plane front, propagating in a homogeneous half-space or in a two layer profile, where soft soil overlays the bedrock.     

不滑移状态下深埋圆形隧道地震响应研究

地下结构的地震响应主要取决于由地震波传播产生的土体变形与土结相互作用。剪切波传播过程中将会引起隧道衬砌的椭圆化变形,进而降低衬砌有效承载力。剪切波作用下的深埋圆形隧道可认为处于均质的纯剪状态,基于相对刚度法的拟静力解析解可充分考虑土结相互作用对隧道结构内力的影响。基于此,本文将通过有限元数值分析获得的自由场地地基变形引入不滑移状态下深埋圆形隧道内力求解公式,并结合二维和三维数值模拟途径,将动力分析结果与解析解结果进行对比分析,以评价各种解析方法的适用性和数值途径的可靠性。     

Analytical solution of the solute transport equation for the binary homovalent ion exchange in groundwater

A binary homovalent ion exchange transport model governed by local chemical equilibrium is considered for a one-dimensional, steady flow in a homogeneous soil column. An analytical solution of the aqueous concentration distribution for the convex exchange is obtained by applying nonlinear shock wave theory. The main nonlinear feature is the breaking of fronts into shock waves. The corresponding mathematical theory is the method of characteristics with a special treatment of shock waves. The wave velocity and front thickness are also obtained to illustrate the front propagation and structure. The derivation of the solution presented may offer a wide range of application opportunities and may also provide a good approach for solving the binary heterovalent exchange transport model.     

1D harmonic response of layered inhomogeneous soil: Analytical investigation

The seismic response of inhomogeneous soil deposits is explored analytically by means of one-dimensional viscoelastic wave propagation theory. The problem under investigation comprises of a continuously inhomogeneous stratum over a homogeneous layer of higher stiffness, with the excitation defined in terms of vertically propagating harmonic S waves imposed at the base of the system. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous layer. The problem is treated analytically leading to an exact solution of the Bessel type for the natural frequencies, mode shapes and base-to-surface response transfer function. The model is validated using available theoretical solutions and finite-element analyses. Results are presented in the form of normalized graphs demonstrating the effect of salient model parameters such as layer thickness, impedance contrast between surface and base layer, rate of inhomogeneity and hysteretic damping ratio. Equivalent homogeneous soil approximations are examined. The effect of vanishing shear wave propagation velocity near soil surface on shear strains and displacements is explored by asymptotic analyses.     

Obliquely incident wave propagation across rock joints with virtual wave source method

Due to the presence of joints, waves are greatly attenuated when propagating across rock masses. Zhu et al. (2011) (Normally incident wave propagation across a joint set with virtual wave source method. J. Appl. Geophys.73, 283–288.) studied normally incident wave propagation across a joint set with the virtual wave source method (VWSM). The introduced VWSM has merits in some aspects, especially the capability of separating differently arriving transmitted waves. However, normal wave incidence is only the special case for wave incidence with arbitrary incident angles. Obliquely incident wave propagation across a joint set is more complicated than normally incident wave propagation due to wave transformation at the joints. As a continuation of the previous paper, this work is extended to analytically study obliquely incident wave propagation across joints with VWSM. Complete theoretical reflection and transmission coefficients across single joint described by displacement discontinuity model are derived through plane wave analysis. The superposition of P wave and S wave is for the first time mathematically expressed and studied. The VWSM is verified through comparison with the propagation matrix method. Through extensive parametric studies on wave transmission across single and multiple parallel joints, it is shown that transmitted wave energy is mainly constrained in the transmitted wave of the same type as the incident wave. And with increasing joint stiffness, the transmission coefficients across single joint increases except those whose wave type is different from the incident wave. The amplitude of superposed transmitted wave for P wave incidence increases with incident angle, which is coincident with field observations. Both joint spacing and number of joints have significant effects on transmission coefficients. We find that when joint spacing is sufficiently large, the transmission coefficient is no longer a constant as the normally incident wave propagation case (Zhu et al., 2011). And when joints are very closely spaced, wave attenuation depends little on the number of joints, which is different from the conclusions from equivalent medium method.     

Propagation of Kelvin waves along irregular coastlines in finite-difference models

In this paper, we examine the behavior of internal Kelvin waves on an f-plane in finite-difference models using the Arakawa C-grid. The dependence of Kelvin wave phase speed on offshore grid resolution and propagation direction relative to the numerical grid is illustrated by numerical experiments for three different geometries: (1) Kelvin wave propagating along a straight coastline; (2) Kelvin wave propagating at a 45° angle to the numerical grid along a stairstep coastline with stairstep size equal to the grid spacing; (3) Kelvin wave propagating at a 45° angle to the numerical grid along a coarse resolution stairstep coastline with stairstep size greater than the grid spacing. It can be shown theoretically that the phase speed of a Kelvin wave propagating along a straight coastline on an Arakawa C-grid is equal to the analytical inviscid wave speed and is not dependent on offshore grid resolution. However, we found that finite-difference models considerably underestimate the Kelvin wave phase speed when the wave is propagating at an angle to the grid and the grid spacing is comparable with the Rossby deformation radius. In this case, the phase speed converges toward the correct value only as grid spacing decreases well below the Rossby radius. A grid spacing of one-fifth the Rossby radius was required to produce results for the stairstep boundary case comparable with the straight coast case. This effect does not appear to depend on the resolution of the coastline, but rather on the direction of wave propagation relative to the grid. This behavior is important for modeling internal Kelvin waves in realistic geometries where the Rossby radius is often comparable with the grid spacing, and the waves propagate along irregular coastlines.©1998 Published by Elsevier Science Limited. All rights reserved     

Seismic response due to travelling shear wave including soil-structure interaction with base-mat uplift

The seismic response due to a travelling shear wave is investigated. The resulting input consists of a translational-and a torsional-acceleration time history, which depend on the ratio of the wavelength to the dimension of the footing. A nuclear reactor building is used for illustration. The combined result of the translational and torsional elastic response (the latter arises even in an axisymmetric structure) will not, in general, be larger than that encountered in the case of a spatially uniform earthquake. If the footing slips or becomes partially separated from the soil, a non-linear dynamic analysis has to be performed to determine the response. Substantial motions in all three directions will take place. The peak structural responses and the floor-response spectra are found to be highly non-linear for high acceleration input values.     

Soil macroporosity and infiltration characteristics of a forest podzol

Spatial distribution of soil macroporosity was determined for a forest podzol from tension infiltrometer measurements at the soil surface. Surface‐derived macroporosity values were compared with point infiltration characteristics obtained from soil water content and soil water chemistry measurements during an experimental irrigation, and with parameters of a kinematic wave model applied to soil water content data. Macroporosity estimated by the tension infiltrometer ranged from 0·00087 to 0·0219% of soil volume, and infiltration at these two sites was dominated by propagation of a well‐defined wetting front through the soil profile and bypass flow via soil macropores, respectively. Infiltration at sites with intermediate macroporosities reflected a combination of these two processes, although results were inconclusive at one site owing to lateral flow at the base of the soil profile. There was no agreement between macroporosities estimated by the tension infiltrometer and the kinematic wave model. The maximum soil conductance parameter within the profile at a site, however, was related directly to the surface‐derived macroporosity. The partial agreement between surface‐derived macroporosity estimates and point infiltration characteristics shown here supports the use of tension infiltrometry as a rapid, non‐destructive method of assessing spatial variations in the relative contribution of macropore flow to the infiltration process. Copyright © 2000 John Wiley & Sons, Ltd.