Plain strain soil–structure interaction model for a building supported by a circular foundation embedded in a poroelastic half-space

A simple theoretical model for soil–structure interaction in water saturated poroelastic soils is presented, developed to explore if the apparent building–foundation–soil system frequency changes due to water saturation. The model consists of a shear wall supported by a rigid circular foundation embedded in a homogenous, isotropic poroelastic half-space, fully saturated by a compressible and inviscid fluid, and excited by in-plane wave motion. The motion in the soil is governed by Biot's theory of wave propagation in fluid saturated porous media. Helmholtz decomposition and wave function expansion of the two P-wave and the S-wave potentials is used to represent the motion in the soil. The boundary conditions along the contact surface between the soil and the foundation are perfect bond (i.e. welded contact) for the skeleton, and either drained or undrained hydraulic condition for the fluid (i.e. pervious or impervious foundation). For the purpose of this exploratory analysis, the zero stress condition at the free surface is relaxed in the derivation of the foundation stiffness matrix, which enables a closed form solution. The implications of this assumption are discussed, based on published comparisons for the elastic case. Also, a closed form representation is derived for the foundation driving forces for incident plane (fast) P-wave or SV wave. Numerical results and comparison with the full-scale measurements are presented in the companion paper, published in this issue.     

The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid

This paper discusses surface displacements, surface strain, rocking, and energy partitioning during reflection-of-plane waves in a fluid-saturated poroelastic half-space. The medium is modeled by Biot's theory, and is assumed to be saturated with inviscid fluid. A linear porosity-modulus relation based on experimental data on sandstones is used to determine the material parameters for Biot's model. Numerical results in terms of angle of incident waves and Poisson's ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the amount of solid frame stiffness controls the response of a fluid-saturated porous system. A poroelastic medium with essentially dry-frame stiffness behaves like an elastic medium, and the influence of pore fluid increases as dry-frame stiffness is reduced. The effects of a second P-wave become noticeable in poroelastic media with low dry-frame stiffness.     

Dynamic response of a group of flexible foundations to incident seismic waves

The dynamic response of a finite number of flexible surface foundations subjected to harmonic incident Rayleigh or SH waves is presented. The foundations are assumed to be resting on an elastic half-space. The results show that the foundation stiffness has a marked effect on the vertical response, while there is only a minor effect on the horizontal displacements. In general, the dynamic response decreases with increasing foundation stiffness. In cases of Rayleigh wave incidence, the existence of an adjacent foundation generates a certain amount of horizontal response in the direction perpendicular to the incident wave and subsequently causes the system to undergo a torsional motion; while in cases of horizontally incident SH waves, a vertical response has been observed and its magnitude is comparable to the response in the direction of the incident wave.     

Transient foundation solution of saturated soil to impulsive concentrated loading

The transient dynamic response of saturated soil under suddenly applied normal and horizontal concentrated loading is studied in this paper. The behavior of saturated soil is governed by Biot's consolidation theory. The general solutions for Biot equations of equilibrium are derived in terms of displacements and variations of fluid volume, using Laplace–Hankel integral transforms. The solutions in the time domain can be evaluated by numerical inverse Laplace–Hankel transforms. Selected numerical results for displacements, stresses, and pore pressures are presented. Comparisons with existing closed-form solutions for the elastic half-space are made to confirm the accuracy of the present solutions. The solutions can be used to study a variety of transient wave propagation problems and dynamical interactions between saturated soil and structures.     

Study of pore pressure variation during liquefaction using two constitutive models for sand

Numerical analyses of liquefiable sand are presented in this paper. Liquefaction phenomenon is an undrained response of saturated sandy soils when they are subjected to static or dynamic loads. A fully coupled dynamic computer code is developed to predict the liquefaction potential of a saturated sandy layer. Coupled dynamic field equations of extended Biot's theory with u–P formulation are used to determine the responses of pore fluid and soil skeleton. Generalized Newmark method is employed for integration in time. The soil behavior is modelled by two constitutive models; a critical state two-surface plasticity model, and a densification model. A class ‘B’ analysis of a centrifuge experiment is performed to simulate the dynamic response of level ground sites. The results of the numerical analyses demonstrate the capability of the critical sate two-surface plasticity model in producing pore pressures that are consistent with observations of the behavior of liquefiable sand in the centrifuge test.     

New artificial boundary condition for saturated soil foundations

A new artificial boundary model based on multi-directional transmitting and viscous-spring artificial boundary theories is proposed to absorb stress waves in a saturated soil foundation in dynamic anal...     

Scattering of plane P waves by circular-arc alluvial valleys with saturated soil deposits

This paper presents an analytical solution for two-dimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow saturated soil deposits. The solution is based on Biot's dynamic theory for saturated porous media, and derived by employing Fourier–Bessel series expansion technique. In this analysis, soil deposits in the circular-arc valley are modeled as saturated porous media based on Biot's dynamic theory, and the circular-arc valley is assumed to be imbedded in an infinite half-space, filled with elastic single-phase media. Numerical results from this solution show that the amplitudes of displacement at the surface of an alluvial valley are mainly relative to the angle of incidence, the dimensionless frequency of incident P wave, the degree of saturation and porosity of soil deposits, and the stiffness and Poisson's ratio of the solid skeleton of the soil deposits. Furthermore, the proposed solution is compared with the previous solution, in which the soil deposit was modeled as an elastic single-phase solid.     

基础刚度对砖石古塔地震响应影响研究

以西安万寿寺塔纠偏工程为研究背景,分别建立增加圈梁、圈梁-地梁组合以及圈梁-地梁-托盘组合模型下的塔体与基础协同工作模型。采用时程分析法,对不同基础刚度条件下塔体的地震响应进行分析及评估,结果表明:(1)增大基础刚度的同时结构的地震位移响应将会被放大,但应力时程曲线的波动范围随之缩小、幅值减小;(2)增设圈梁时由于塔底三向受力,塔底应力将被放大;(3)进行基础完全托换能够大幅改善塔底在地震作用下的受力状态。在塔体加固维修时应综合考虑地震作用下塔体的位移放大效应与塔体应力分布的影响,选取合适的基础刚度。     

横观各向同性饱和地基中埋置荷载的非轴对称瞬态响应

基于孔隙介质的Biot理论,首先利用Laplace变换,给出圆柱坐标系下横观各向同性饱和弹性多孔介质在变换域上的波动方程;将波动方程解耦后,根据方位角的Fourier展开和径向Hankel变换,求解了Biot波动方程,得到以土骨架位移、孔隙水压力和土介质总应力分量的积分形式的一般解;借助一般解,建立了有限厚度饱和土层和饱和半空间的精确动力刚度矩阵,并由土层的层间界面连续条件建立三维非轴对称层状饱和地基的总刚度方程;在此基础上,系统研究了横观各向同性饱和半空间体在内部集中荷载激励下的动力响应,并给出了问题的瞬态解答.该研究为运用边界元法求解饱和地基动力响应奠定了理论基础.     

Correction factors for SSI effects predicted by simplified models: 2D versus 3D rectangular embedded foundations

The effects of soil‐structure interaction (SSI) are often studied using two‐dimensional (2D) or axisymmetric three‐dimensional (3D) models to avoid the high cost of the more realistic, fully 3D models, which require 2 to 3 orders of magnitude more computer time and storage. This paper analyzes the error and presents correction factors for system frequency, system damping, and peak amplitude of structural response computed using impedances for linear in‐plane 2D models with rectangular foundations, embedded in uniform or layered half‐space. They are computed by comparison with results for 3D rectangular foundations with the same vertical cross‐section and different aspect ratios. The structure is represented by a single degree‐of‐freedom oscillator. Correction factors are presented for a range of the model parameters. The results show that in‐plane 2D approximations overestimate the SSI effects, exaggerating the frequency shift, the radiation damping, and the reduction of the peak amplitude. The errors are larger for stiffer, taller, and heavier structures, deeper foundations, and deeper soil layer. For example, for a stiff structure like Millikan library (NS response; length‐to‐width ratio ≈ 1), the error is 6.5% in system frequency, 44% in system damping, and 140% in peak amplitude. The antiplane 2D approximation has an opposite effect on system frequency and the same effect on system damping and peak relative response. Linear response analysis of a case study shows that the NEHRP‐2015 provisions for reduction of base shear force due to SSI may be unsafe for some structures. The presented correction factor diagrams can be used in practical design and other applications.     

Dynamic stiffness matrices for two circular foundations

A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed.     

An experimental investigation of vertical vibration of model footings on sand

Many free and forced vertical vibrations tests were conducted on surface and embedded models for footings on dry and moist poorly graded sand that has the following properties: D₁₀ = 0·21 mm, D₃₀ = 0·305 mm, D₆₀ = 0·423 mm, C_u = 2, C_c = 1·05, G_s = 2·66, γ_max = 1·74 cm⁻³ and γ_min = 1·4 g cm⁻³. The tests were conducted at relative density Dr = 82%. The effect of mass, area, geometry, embedment, saturation, load amplitude and frequency were studied. For this purpose square, rectangular and circular models of concrete footings were chosen. Swieleh sand was chosen as the foundation soil. Results have been obtained for models having different mass, same base shape and area; models of different base area and equal base shape and mass; and models of different base shape geometry and about equal masses and base areas.Forced vertical vibration tests results showed an increase in natural frequency and a reduction in amplitude with the increase in embedment depth, degree of saturation and footing base area. Increasing the mass of model footing resulted in a decrease in the natural frequency while the dynamic response increased. Also, results showed that the circular model footing gives low values of dynamic response in comparison to other models.Free vertical vibration test results showed an increase in damping ratio with increase in the base area of the model footing, depth of embedment and saturation of sand. On the other hand, the results showed a decrease in damping ratio with increase in the footing mass. Circular footing gives the highest value of damping ratio among other footings.Results showed that the best method for evaluating the dynamic stiffness for vertical vibration is using Dobry and Gazetas¹ (J. Geotech. Engng, ASCE, 1986, 112, 109–133) equations together with the formula proposed by Ronald and Bojan² (J. Geotech. Engng, ASCE, 1995, 121, 274–286) for evaluating the equivalent shear modulus.     

In-plane soil–structure interaction in layered,fluid-saturated,poroelastic half-space I: Structural response

Linear in-plane soil–structure interaction in two dimensions (2D) is studied in fluid-saturated, poroelastic, layered half-space using the Indirect Boundary Element Method (IBEM). The structure is a shear wall supported by a rigid embedded foundation. Exact stiffness matrices for the soil layer and half-space, and Green׳s functions of uniformly distributed loads and pore pressure on an inclined line are derived. Results of the system response in the frequency domain are presented for the special case of single soil layer over bedrock, semi-circular foundation and zero seepage force. The effects of water saturation, soil porosity, depth of soil layer, rigidity contrast between layer and bedrock are investigated in the frequency domain for incident plane P- and SV waves. The results suggest that water saturation may cause increase of the system frequency by more than 10%.     

Response of L-shaped rigid foundations embedded in a uniform half-space to traveling seismic waves

A simplified indirect boundary element method is applied to compute the impedance functions for L-shaped rigid foundations embedded in a homogeneous viscoelastic half-space. In this method, the waves generated by the 3D vibrating foundation are constructed from radiating sources located on the actual boundary of the foundation. The impedance functions together with the free-field displacements and tractions generated along the soil–foundation interface are used to calculate the foundation input motion for incident P, S and Rayleigh waves. This is accomplished by application of Iguchi's averaging method which, in turn, is verified by comparison with results obtained rigorously using the relation between the solutions of the basic radiation (impedance functions) and scattering (input motions) problems. Numerical results are presented for both surface-supported and embedded foundations. It is shown how the seismic response of L-shaped foundations with symmetrical wings differs from that of enveloping square foundations. The effects of inclination and azimuth of the earthquake excitation are examined as well. These results should be of use in analyses of soil–structure interaction to account for the traveling wave effects usually overlooked in practice.     

An approximate method for soil-structure interaction for sh-waves—the born approximation

An approximate method is proposed for the scattering of SH-waves by foundations of irregular shape and the resulting soil-structure interaction problems. The scattering of elastic waves by the rigid foundation embedded in half-space is solved approximately by using integral representation of the wave equation. The procedure is the Born approximation which has been widely used in quantum mechanics for collision and scattering theory though not well-known in elastodynamics. This paper extends the previous work of the authors on the scattering of waves to account for soil-structure interaction. The motion of the foundation is evaluated by the balance of momentum under stresses due to the incident waves as well as the waves generated by its own motion and the forces coming from the superstructure. The model investigated consists of an infinitely long elastic shear wall of height H and thickness h erected on a rigid infinitely long foundation. Results are presented for the cases with circular, elliptical and rectangular foundations. For a circular foundation, excellent agreement is found with the exact solutions for the foundation displacement and the relative displacement between the top and bottom of the structure for the entire range of wave numbers. For an elliptical foundation, accuracy decreases with increasing wave numbers. Foundation displacements are compared for foundation shapes that are shallow elliptical, deep elliptical, rectangular and circular. It is observed that foundation displacements are dependent on the angle of incidence except for a semi-circle. The results on the details of the scattered field are, however, not as accurate.     

Analytical moment–rotation curves for rigid foundations based on a Winkler model

Analytical equations for the moment–rotation response of a rigid foundation on a Winkler soil model are presented. An equation is derived for the uplift-yield condition and is combined with equations for uplift- and yield-only conditions to enable the definition of the entire static moment–rotation response. The results obtained from the developed model show that the inverse of the factor of safety, χ, has a significant effect on the moment–rotation curve. The value of χ=0.5 not only determines whether uplift or yield occurs first but also defines the condition of the maximum moment–rotation response of the footing. A Winkler model is developed based on the derived equations and is used to analyze the TRISEE experiments. The computed moment–rotation response agrees well with the experimental results when the subgrade modulus is estimated using the unload–reload stiffness from static plate load–deformation tests. A comparison with the recommended NEHRP guidelines based on the FEMA 273/274 documents shows that the choice of value of the effective shear modulus significantly affected the comparison.     

Vertical and rocking impedances for rigid rectangular foundations on soils with bounded non-homogeneity

The vertical and rocking response of rigid rectangular foundations resting on a linear-elastic, compressible, non-homogeneous half-space soil model is studied. The non-homogeneity is described by a continuous yet bounded increase of shear modulus with depth. The mixed boundary value problem is solved by means of the semi-analytical method of the subdivision of the foundation/soil contact area whereby the influence functions for the sub-regions are determined by integration of the corresponding surface-to-surface Green's functions for the particular soil model. Impedance functions are given for representative values of the non-homogeneity parameters, the Poisson's ratio and the foundation geometry over a wide range of frequencies. Significant features associated with the soil non-homogeneity are pointed out. Copyright © 1999 John Wiley & Sons Ltd.     

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.     

Discrete model for dynamic through-the-soil coupling of 3-D foundations and structures

An efficient discrete model for predicting the dynamic through-the-soil interaction between adjacent rigid, surface foundations supported by a homogeneous, isotropic and linear elastic half-space is presented. The model utilizes frequency-independent springs and dashpots, and the foundation mass, for the consideration of soil–foundation interaction. The through-the-soil coupling of the foundations is attained by frequency-independent stiffness and damping functions, developed in this work, that interconnect the degrees of freedom of the entire system of foundations. The dynamic analysis of the resulting coupled system is performed in the time domain and includes the time lagging effects of coupled dynamic input due to wave propagation using an appropriate modification of the Wilson-θ method. The basic foundation interaction model is also extended to the evaluation of coupled building-foundation systems. © 1998 John Wiley & Sons, Ltd.     

Antiplane response of a dike on flexible embedded foundation to incident SH-waves

The response of an elastic circular wedge on a flexible foundation embedded into a half-space is investigated in the frequency domain for incident pane SH-waves. The problem is solved by expansion of the motion in all three media (wedge, foundation and half-space) in cylindrical wave functions (Fourier-Bessel series). The structural model is simple, but accounts for both differential motions of the base and for the effects of soil-structure interaction. Usually, structural models in earthquake engineering consider either differential ground motion, but ignore soil-structure interaction, or consider soil-structure interaction, but for a rigid foundation, thus ignoring differential ground motion. The purpose of the study is to find how stiff the foundation should be relative to the soil so that the rigid foundation assumption in soil-structure interaction models is valid. The shortest wavelength of the incident waves considered in this study is one equal to the width of the base of the wedge. It is concluded that, for this model, a foundation with same mass density as the soil but 50 times larger shear modulus behaves as ‘rigid’. For ratio of shear moduli less than 16, the rigid foundation assumption is not valid. Considering differential motions is important because of additional stresses in structures that are not predicted by fixed-base and rigid foundation models.