Modelling Propagation of Stress Waves through Soil Medium for Ground Response Analysis

During past earthquakes, damages occurred to buildings located at soil sites are more compared to damages observed on buildings located at rock sites. Modelling wave propagation through soil medium helps to derive the primary and secondary wave velocities. Most of the time soil mediums are heterogeneous, layered and undergoes nonlinear strains even under weak excitation. The equivalent linear approximation with one dimensional wave propagation is widely adopted for modeling earthquake excitation for layered soil. In this paper, importance of local soil effects, the process of wave propagation through three dimensional elastic medium, layered medium situated on rigid rock, attenuation of stress waves due to material damping, equivalent linear approximation, the concept of one dimensional wave propagation, and a case study of one dimensional wave propagation as a part of site-specific ground response analyses for Delhi region are included. The case study brings out the importance of carrying out site-specific ground response analyses of buildings considering the scenario earthquakes and actual soil conditions for Delhi region.


Introduction
During many of the past earthquakes (Kachh, 1819, Mexico City, 1985, Loma Prieta, 1989, Chi Chi, 1999, Kobe, 1995) it has been observed that damages occurred to buildings located at soil sites are more compared to damages observed on buildings located at rock sites as reported in literature [1][2][3][4][5][6].Figure 1 shows the ground motions recorded at two adjacent sites viz., a rock site (UNAM) and a soil site (SCT) located 350 km away from epicenter during 1985 Mexico city earthquake.The response spectra of UNAM site and SCT site are shown in Figure 2. The earthquake has caused only moderate damage near the epicenter and caused severe damage in the lake zone underlain by 38 to 50 m of soft soil (site period was 1.9 to 2.8 sec).Most of the buildings in the 5 -20 storey range got severely damaged and the buildings of less than five stories or more than 30 stories suffered lesser damage.This pattern of damage was partly attributed to the resonance effect of time period of soil deposit and the time period of the building.Initially, it was felt that soil amplification can be observed for week motions only and for strong shaking there may not be considerable amplification due to damping of soil. Figure 3 shows the nonlinear relation of peak ground acceleration (PGA) for weak and strong motion as observed in Mexico City, Loma Prieta earthquake and numerical calculations.Adapting to the response spectra (Figure 4) of Seed and Idriss [3] which depicted the variation of response of different site conditions, building codes viz.,    Uniform Building Code [7], Indian seismic code [8] have introduced three types of response spectra for hard, medium and soft soil deposits (Figure 5).Later the classification of soil sites based on average shear wave velocity of top 30 m has been introduced and the modification to response spectra after implementing soil amplification factors has been brought into International Building codes [9] as shown in Figure 6.Since it was felt that, maximum amplification can occur only due to soil layers present in the top 30 m, soil classification has been proposed based on average shear wave velocity of top 30 m  soil [9].However, studies are being carried out [10,11] on response of deeper deposits which can result in longer time periods capable of imposing higher demands on tall buildings.
The necessity of design ground motions for carrying out time history analysis is felt essential for analysis of important structures; hence the methodologies to arrive at the modified ground motion including the effect of change in amplitude, frequency content and duration due to the presence of soil layer are developed.Site effects include the modification of ground motion due to basin and topography effects also.For geotechnical problems viz., checking the stability of slopes, construction of dams and reservoirs it may be required to include the effects of basin and topography.
The wave front of shock waves created during the occurrence of earthquake, consists of all four types of waves viz., primary (P) waves, secondary or shear (S) waves, Rayleigh (R) waves and Love (L) waves.Out of these, shear waves can cause maximum damage to buildings.Hence modeling the shear wave propagation through rock and soil layers is being felt essential for engineering purposes.
Ground motions felt at the surface where no structure is present are known as free field motions.Ground motions observed or simulated at the top of exposed rock esponse analysis will include the pr are known as outcrop motions and the ground motions buildings and structures.In this paper, brief introduction about wave propagation through soil medium and a case study of site-specific ground response analysis for Delhi region are included.
felt below the soil layer are denoted as bedrock motions.
There are different definitions for bedrock, seismic bedrock (shear wave velocity in the range of 3.2 km/sec) or engineering bedrock (shear wave velocity more than 400 m/sec).When the foundation is proposed to be located below the ground level for specific applications it is required to arrive at the ground motion at the base of the soil layer using the surface level ground motions (simulated or recorded design ground motion for a specified risk level on surface).This process of obtaining the bedrock motion from free field motion is known as de-convolution.Knowledge of propagation of horizontal component of shear wave through soil medium located on rigid or elastic rock is essential for carrying out soil amplification studies.A complete ground r

Wave Propagation through Soil Medium
For carrying out dynamic analysis, structural elements made of materials viz., Reinforced Concrete (RC) and steel can be idealized as discrete elements with quantifiable stiffness and mass.When the dynamic load is applied to continuous medium like soil, the deformation that takes place in the soil medium causes stress waves.Propagation of stress waves through soil can be modelled by either of these three methods 1) stress waves in an elastic unbounded medium 2) stress waves in a longitudinal bar 3) stress waves in elastic half space.
ocess of modeling the rupture mechanism at the source and the path attenuation and wave propagation through soil medium.The response spectrum at the soil surface is significantly different from that of bedrock response spectrum due to the modification of ground motion as it passes through the soil layers overlying the bedrock.Building codes are simplified tools and do not adequately represent any single earthquake event from a probable source for the site under consideration.Recently, it has been recommended [11][12][13][14] that in addition to the use of seismic codes, site-specific analysis which includes generation of strong ground motion at bedrock level and propagating it through soil layers and arriving at the design ground motions and response spectra at surface should also be carried out in the design of important

Three Dimensional Modeling of Wave Propagation through Soil Medium Situated on Rigid Rock
To understand the propagation of stress waves in infinite elastic medium and bounded elastic medium equations of motion can be written in terms of stresses.Let the normal and shear stresses acting on a soil element with sides dx, dy and dz are  x ,  y ,  z and  xy ,  yx ,  yz ,  zy ,  zx , and  xz respectively as shown in Figure 7.When u, v, and w are the displacement components in x, y and z directions the equation of equilibrium along x, y and z directions can be written as given in Equations ( 1)-(3) [16,17] wherein the unbalanced external forces are balanced by an inertial force, where  is the mass per unit volume or the density.
Referring to theory of elasticity [18] and writing the eq o uations for normal, shearing strains in terms of partial derivatives of displacements and linking the stresses and strains by Hook's law with material constants viz., young's modulus (E), shear modulus (G), bulk modulus () and poisson's ratio () and substituting  xy =  yx ;  yz =  zy and  xz =  zx the equation of motion for x component is btained as given in Equation ( 4).Similarly by getting the equation of motion in other components and differentiating with respect to x, y and z and by adding Equation ( 5) which relates volumetric strain (ε) and primary wave velocity (v p ) of soil medium is obtained. where, The propagation of stress waves in a bounded elastic medium is similar to Equation ( 5) and can be expressed as For shear waves or S waves the equation of motion in x direction reduces to the following form where ot cause any rotation and S-wave does volume change.The ratio of P-wave velo P waves do n not cause any city and S-wave velocity is given by where 0.3v 1.87 The body waves t hemispherical wa front and Rayleigh waves travel with cylindrical wave front.Th ravel with ve e amplitude of body waves is proportional to 1/r and amplitude of Rayleigh wave is proportional to 1 r .To illustrate the wave propagation at the interface of neous elastic layers, the problem of harmonic stress wave travelling along a constrained rod in the positive x-direction and approaching an interface between two different materials is often chosen (Figure 8).

Wave Propagation in a Layered
Satisfying the compatibility conditions of displacements and continuity of str resses of incident, reflected and transmitted waves are related by the following equations Impedance ratio of zero means free boundary conditions (surface), amplitude of displacement at boundary is twice as that of displacement of incident wave and the stresses are equal with opposite sign.Impedance ratio of infinity means (rigid rock) displacement is zero, amplitude of incident and reflected waves are equal but with opposite signs.Stress at this boundary is twice as that of incident wave.Impedance ration of unity means, all the incident waves are getting transmitted and no component is reflected back.Response of a dynamically loaded system can be solved by making use of Fourier transforms and transfer functions in frequency domain.This approach is widely used for ground response analysis, wherein the applied time history at rock level is converted to Fourier transforms and multiplied with transfer functions of the soil layer and converted back to time domain by inverse Fourier transforms and the ground motion at surface is obtained.
The problem is now to get the transfer functions of the soil layers, which is the ratio of maximum displacement of the topmost and bottommost point of the soil layer.The modulus of transfer function gives the amplification function.The soil layer is seldom homogeneous and the heterogeneity of soil layers can be modeled by inclusion of more number of soil layers.The response of layered soil on elastic rock can be determined using the procedure described in the following sections.
During earthquake shaking fault ruptures below the earth surface and body waves travel away from the source when met with boundaries between different geological materials they get reflected and refracted.Due to the lesser velocity of materials present at shallower depths inclined rays that strike horizontal layer boundaries are usually reflected to a more vertical direction.Assumption of one dimensional ground response analysis is that soil boundaries are horizontal and the response of a soil deposit is predominantly caused by SH-waves propagating vertically from the underlying bedrock.

Attenuation of Stress Waves Due to Material
The )-( 7) represent wave propagation with-

Damping
Equations (1 out change in amplitude, which cannot be practical.During the propagation of wave through soil medium, dissipation of energy takes place which results in decrease in amplitude.If the soil medium is idealized as visco elastic material with spring stiffness G and viscous damping constant  the as shown in Figure 9, and the shear stress ()-strain () relationship is given by One dimensional equation of moti on for vertically pro- pagating SH waves can be written as, Substituting Equation (10) in Equation ( 11) Considering soil as a visco-elastic m N horizontal layers where N th layer is bedrock ( 10 expressed as where A and B are the amplitudes of waves travelling in the upward and downward direction, k * is the complex wave number extending the results of Equations ( 8) and ( 9), (14) the complex impedance ratio   m   between layers m and m+1 can be given by At ground surface, the shear stress must be equal to zero and A 1 = B 1 the functions relating amplitudes at layer m and layer 1 are given by The transfer function relating the displacement amplitude at lay layer is known, the ground motion for the any other layer can be calculated using the transfer functions.

Equivalent Linear Approximation
Soil undergoes inelastic strains even under very small level of ground shaking, hence nonlinear behavior of soil needs to be accounted.The hysteresis loop of soil under symmetric cyclic loading is given in .The width of the hysteretic loop increase wi n cyclic shear strain hence the damping ratio increases with increase in shear strain.Both modulus reduction ratio and damping ratio are influenced by plasticity characteristics, and the variation of modulus reduction ratio and damping ratio curves for different plasticity indices as developed by Vucetic and Dobry [19] are reproduced from Kramer in Figures 14 and 15.

Two and Three Dimensional Analyses
ces, heavy structures, stiff or embedded structures, e two dime l strong ground motions including source path effects using stochastic finite fault model [20, One dimensional analysis may not be adequate for the structures located on sloping and irregular ground surfa walls and tunnels and henc nsional or possibly three dimensional analysis may need to be carried out.

Site-Specific Ground Response Analysis for Delhi Region-A Case Study
In order to bring out the importance of site-specific analysis, three soil sites (viz., site 1, site 2 and site 3) have been chosen at Delhi, capital city of India as shown in Figure 16.Artificia   compared with similar simulation from literature [22].One dimensional equivalent linear vertical wave propagation analysis is the widely used numerical procedure for modeling soil amplification problem [2,23] as discussed in the earlier section.In one dimensional wave propagation analysis, soil deposit is assumed to be having number of horizontal layers with different shear modulus (G), damping () and unit weight () as shown in Figure 10.Equivalent linear analysis program SHAKE [24,25] is used in the present study.Equivalent linear modulus reduction (G/G max ) and damping ratio () curves generated from laboratory test results are adopted from Vucetic and Dobry [19] depending on the plasticity index of different soil layers.Three actual soil sites designated as site 1, site 2 and site 3 located in Delhi as shown in Figure 16 are chosen in the present study.The layer wise soil characteristics (medium type) and the depth to the base of the layer from the surface is given elsewhere [10,26] The variation of shear wave velocity along the depth in the present study is obtained by using the correlations suggested for Delhi region by Rao and Ramana [27] as given in Equation ( 19).

(19a) (19b)
From the ground response analyses results, it has been observed that the PGA amplifications and the response spectra of the three sites are quite different for the earthquakes considered.
Using the site-specific response spectra, storey shear of three storey and fifteen storey building (Figure 18) are estimated using response sp m method.The com-hree sites considered.As per IBC [9] guidelines site-specific a ommended for soil type F only for which a wave velocity of top 30 m is less than 180 m/s.T sites considered in the present study are moderate sites an damping, equivalent linear approximation, the concept of one dimensional wave propagation analysis, and a case study of site-specific ground response analyses for Delhi region are presented.
In the case study, rock outcrop motions have been generated for Delhi for the scenario earthquakes of magnitude, M w = 7.5, M w = 8.0 and M w = 8.5.Three actual soil sites have been modeled and the free field surface motions and the response spectra have been obtained through one dimensional wave propagation analyses.Further, the response of a three storey building and a fifteen storey building are studied and it is observed that, for the three sites considered the response of the building varies significantly.The studies made, brings out the importance of carrying out site-specific ground response analyses of buildings considering the scenario earth-varies significantly from the storey shear obtained using Indian seismic code BIS 1893-2002 Part 1 [8] code.The linear displacements for the two buildings are obtained by linear static analyses program and the comparison has been made for the three sites as shown in Figure 20.It is seen that displacement response also varies significantly for the t nalysis is recverage shear he soil d do not come under the category of F type.The studies made, bring out the importance of carrying out site-specific ground response analyses of buildings considering the scenario earthquakes and actual soil conditions for Delhi region.

Summary
In this paper, importance of local soil effects and procedure for modeling wave propagation through three dimensional elastic medium, layered medium situated on rigid rock, attenuation of stress waves due to material

Figure 7 .
Figure 7. Stresses in an elastic solid.

Figure 8 .
Figure 8.One dimensional wave propagation at materia interface.l

) 2 . 4 .
Transfer Function aterial consisting of Figure ), the solution to the wave equation can be

Figure 19 .
Figure 18.Plan of a three storey and a fifteen storey RC framed building.

Figure 20 .
Figure 20.Comparison of linea .0;M w = 8.5;(b) Fifteen storey building M w = 7.5; M quakes and actual soil conditions for Delhi region.