Abstract
A phenomenological constitutive model, `BWGG', is developed for the non-linear one-dimensional ground response analysis of layered sites. The model reproduces the nonlinear hysteretic behaviour of a variety of soils, and possesses considerable flexibility to represent complex patterns of cyclic behaviour such as stiffness decay and loss of strength due to build-up of pore-water pressure, cyclic mobility, and load induced anisotropy. It also has the ability of simultaneously generating realistic modulus and damping versus strain curves, by a simple calibration of only three of its parameters. The model is implemented through an explicit finite-difference algorithm into a computer code which perform integration of the wave equations to obtain the nonlinear response of the soil. The code, NL-DYAS', is then applied to study the seismic response of a soft marine normally-consolidated clay. The results are compared with those of widely used codes. Finally, the records of the Port Island array during the Kobe 1995 earthquake, are utilized, and the model is shown to "predict" the observed response with sufficient accuracy.
INTRODUCTION
Site response analysis is a very important issue in earthquake engineering. It is usually the first step in the calculation of structural response in soil-structure interaction problems. Thus, a substantial effort has been devoted in the last three decades in developing analytical techniques and numerical methods for evaluating the response of soil deposits to strong earthquake motions. The majority of practical methods used for describing nonlinear soil amplification are classified into two categories: (a) frequency domain equivalent linear and (b) time domain nonlinear methods.
Although equivalent linear type of analyses are the most popular, they have certain well-known limitations under strong seismic shaking.
On the other hand, many of the commercially available nonlinear models: (a) are incapable of simultaneously fitting the observed shear modulus degradation and damping curves, usually overestimating hysteretic damping at large strains (if the Masing rule for unloadingreloading is used), and (b) are not versatile in properly modeling the shape of various experimental stress-strain loops for various types of soil behaviour.
Clearly, the Masing criterion is handly appropriate for cyclic soil behaviour, as demonstrated in the examples of Fig. 1 . A phenomenological 1-D constitutive model, designated as BWGG model, is developed in this paper for the static and dynamic response of soils. The model avoids the aforementioned disadvantages and is quite versatile, capable of reproducing even some of the most complex nonlinear characteristics of cyclic behaviour , such as the cyclic mobility and liquefaction response depicted in Fig. 2 . The model predictions can simultaneously match any experimental modulus decline and damping growth versus shear strain curve by properly adjusting only three of its parameters. The other parameters do not affect the match and can be set equal to their default values , or simply eliminated from the model. Identification and/or complete calibration of the model parameters is beyond the scope of this paper. Nevertheless, a methodology for relating the model parameters to the physical and mechanical soil properties (from in-situ and/or laboratory tests) is briefly discussed. An outline of the model is given below.
THE MODEL: EQUATIONS AND PARAMETERS
The constitutive model proposed here in for modeling the soil simple shear stress-strain relation of a soil element is given by; (1) where T and y are the shear stress and strain respectively , Gmax is the initial tangent shear modulus, cx is a parameter that controls the post yielding shear stiffness , Ty is the value of shear stress at initiation of yielding in the soil . The parameter C= (t) is hysteretic dimensionless quantity that controls the nonlinear response of soil, and is governed by the following differential equation; (2) in which A, b, g and n are dimensionless quantities which control the shape of the stress-strain loop; yy is the value of shear strain at "initiation of yielding" in the soil. The model of (1) and (2) was originally proposed by Bouc (1971) and subsequently extended by Wen (1976) and used in random vibration studies of inelastic systems. The expressions given by (1) and (2) have been previously used from Pires (1989) and Loh et al. (1995) for probabilistic analysis of the seismic response of multilayered soil deposits.
By differentiating Eq. (1) with respect to the shear strain y, one obtains;
(3) where (4) In monotonic loading conditions Eq. (2) collapses to; (5) where (6) is the strain ductility that the soil element experiences during loading. It can be easily shown that when y tends to 0, Ea. (6) reduces to:
Substituting Eq. (7) to Eq. (3) and setting a= 0 yields (8) When A= 1, Gmax (= Ty 1 yy) becomes the small-amplitude shear modulus, while a becomes the ratio of the post-to pre-yielding stiffness. From Eq. (1) it is obvious that the maximum value of the shear stress, Tmax, is reached when y and take their maximum value. For monotonic loading the maximum value of is obtained by setting c1C1 dy = 0 , and by virtue of Eq. (3) this maximum takes the value; (9) Substituting Eq. (9) to Eq. (1) and setting T = z max, OC = 0, and C= Cmax one obtains; (10) Note that the maximum shear stress becomes equal to T y when (11) In undrained loading conditions , Ty is equal to the undrained shear strength Su. (Ishihara, 1996) Monotonic Loading Curve
The parameter n governs the sharpness of the transition from the linear to nonlinear range during initial virgin loading. Its range of values is between 0 and (x), with the T -y curve approaching bilinear behaviour as n approaches (x). However elastic-perfectly plastic behaviour is practically achieved when n takes values greater than 10. The effect of parameter n on the monotonic loading curve is portrayed in Fig. 3 . As the values of n decrease plastic straining appears even at low loading levels. Monotonic loading curves for different values of the post yielding parameter a and for constant value of n are also presented in Fig. 4 .
Loading-Unloading-Reloading Rule
Parameters b and g control the shape of the unloadingreloading curve. As is shown in Fig. 5 there are four basic hysteretic shapes, which depend on the relation between b and g. As b tends to 1 and by virtue of b+g= 1, the reversal stiffness tends to 0. For the special case b= 1 and g= 0, the stress-strain loop collapses to the monotonic loading curve (nonlinear elastic behaviour). On the contrary as g reaches 1 under the condition of b+ g= 1, the reversal stiffness becomes greater than the initial stiffness (at virgin loading). Finally when b = g = 0.5 the reversal stiffness equals the initial stiffness and the Masing criterion for loading-unloading-reloading arises.
Stiffness and Strength Degradation
with Cyclic Loading The model is capable of reproducing stiffness and strength degrading behaviour. Stiffness decay is achieved by introducing the parameter n in Eq . (2), giving; (12) Prescribing q to be an increasing function of time will induce stiffness decay. ri can be expressed as a function of the dissipated hysteretic energy and/or the cumulative strain ductility. Note that decreasing ri is equivalent to reducing A, b and g in proportion. The flexibility of damping curves cover a very broad range, all the shear modulus curves fall within a very narrow zone, revealing the independence between the two group of curves, primarily thanks to the key parameter s1. Note that only when s1 tends to infinity, the damping tends to (the unrealistically high) 2/it at large strains. The proposed model is also versatile in simulating strength degradation with cyclic loading. This can be achieved in two different ways: (a) By making parameters b and g increasing functions of hysteretic energy and/or of cumulative strain ductility, while fulfilling the condition of equal reduction of b and g; (b) both strength and stiffness degradation can be affected by incorporating the parameter r in Eq. (1) giving; (15) data. By fitting these data, r can be expressed as: (16) where A u is the incremental pore pressure build up, o-c, is the initial vertical effective stress, WN is a dimensionless energy term (Law et al., 1990 ) and a and ,6 are determined from laboratory tests. Parameter r can be adapted to incorporate any pore pressure model (e.g. Ishihara and Towhata, 1980; Finn et al., 1977) . Figure 7 depicts stressstrain loops for soil elements experiencing stiffness and strength degradation with cyclic loading, computed with the proposed model for soils. The loop of Fig. 7 (a) corresponds to values of b and g equal to 0.5, whereas that of Fig. 7 (b) to b=-0.6 and g= 0.4. The stress-strain loops of Figs. 7(a) and 7(b) are characteristic of a sand behaviour experiencing pore-water pressure build up, just a little before the initiation and during the cyclic mobility, respectively. To simulate the transition between these two different phases of soil behaviour would require calibration against laboratory data of parameter r, or alternatively of parameters b and g, in time domain. For example r should be an increasing function of time before the initiation of cyclic mobility, experiencing fluctuations with time (for simulating the strain hardening response). Since identification and complete calibration of the model parameters is beyond the scope of this paper, the stress-strain loops in Fig. 7 shall be considered purely as a demonstration of the potential capabilities of the proposed model for soil.
To further extend the capabilities of the model in simulating the cyclic response of soil elements undergoing cyclic mobility, Eq. (12) is modified by introducing the term; (17) leading to the following differential equation (18) Ls and (I are dimensionless parameters which control the narrowing of the hysteretic loop around the center of the stress-strain axis. C2 is a constant that controls the shift of the backbone curve on the C axis. The influence of parameter Ls on the shape of stress-strain loop is illustrated in Fig. 8 . Notice that the larger the Ls, the more pronounced the narrowing of the hysteretic loop around the center of stress-strain axis. The mechanical analog of Eq. (18) is that of two springs placed in parallel. The first spring "constant" represents the tangent shear stiffness of the soil and is given by; (19) with C* governed by Eq. (12) . The modulus of the second spring, PG), stands for improving the modelling of dilative soil behaviour.
To this end, Eq. (18) can be rewritten in the equivalent form;
(20) Figure 9 depicts a typical stress-strain loop computed with the use of Eq. (18), corresponding to the cyclic behaviour of a dense sand undergoing cyclic mobility.
Asymmetric Response with Loading Direction
Finally, the non-symmetric cyclic response of soil with where co is a constant that controls the shift of the backbone curve on axis. Figure 10 illustrates the stressstrain monotonic loading curves computed for selected values of Co.
SHEAR MODULUS REDUCTION AND DAMPING CURVES
One of the major advantage of the proposed model for soils is its ability to independently match the observed shear modulus degradation and damping for any given soil, by simply calibrating only three of its parameters , n, Si and s2 (defined in the previous section). An optimization procedure has also been developed to evaluate the three parameters to fit the observed behaviour , the presentation of which however is beyond the scope of this article. As can be seen from the graphs in Fig . 11 To this end, nonlinear optimization techniques could be used (such as time domain least square procedure, and dynamic neural networks). For details about a comprehensive method for the identification of the Bouc-Wen model parameters, the reader is referred to the work of Sues, Mau, and Wen (1988) . The development of a complete methodology for the identification of the model parameters is not in the scope of this paper. Nevertheless, it is very important to mention that the model has a powerful advantage: the number of the necessary parameters for analysis purposes can be adapted to the amount of the available geotechnical data. The other parameters can be set equal to their default values or simply eliminated from the model. The lesser the in situ and/or laboratory data, the fewer the number of the necessary-for the analysis parameters, but of course, the larger the uncertainty in the model. For example, supposing we have to conduct a site response analysis and the only available information about the soil are the shear wave velocity profile Vs,the soil classification, and the Atterberg limits, the following steps shall be carried out for calibrating the model parameters: (i) Calculate the small-amplitude shear modulus Gmax. (ii) Correlate the small-amplitude shear modulus with the undrained shear strength (for clay) and/or the relative density with the internal friction angle (for sand). (iii) Set the yield shear stress T y equal to the shear strength of the soil. (iv) Calculate the yield shear strain yy=ry /Gmax.(v) Choose the appropriate shear modulus reduction and damping curves from the literature, according to soil type, plasticity index, and effective confining pressure. (vi) Calibrate the parameters n, s1, and s2 for the calculated shear modulus and damping curves to match the experiment. The other parameters of the model are not needed since the geotechnical data are not adequate for calibrating them. Therefore, they must be set equal to their default values or even be eliminated from the model. That is: set b= 0.5, and eliminate a, r, Ls,Ci, C2, and Co.
On the other hand, supposing that (except from the aforementioned data) we have also in our disposal results from cyclic undrained triaxial tests, then the proposed methodology is further extended to incorporate additional parameters to account for the effect of the pore-water pressure build up on the stress-strain loop. As we have already mentioned in a previous section, this can be achieved in two different ways: (i) by expressing parameters b and g as a function of hysteretic energy and/or of cumulative strain without affecting r, and (ii) by prescribing r as a function of the effective confining pressure, without affecting b and g. Reevaluation of the strength parameter Ty could also be done, by using the results of the triaxial tests which are more accurate than those from the empirical correlations.
A question then arises about the role of the 'pinching'
parameters Ls, and C2, and the parameter for asymmetric response with loading direction Co, in simulating the shape of a stress-strain loop. These parameters aim at further improving the capability of the model in matching the measured soil behaviour. They are simply curve-fitting parameters, not associated with the soil properties. Besides, it is well-known that the larger the number of parameters in a soil model, the more flexible it becomes in reproducing the observed soil behaviour.
NUMERICAL MODELING OF ONE-DIMENSIONAL WAVE PROPAGATION
The problem studied herein is that of a layered soil profile subjected to seismic excitation. The one-dimensional vertical shear wave propagation through a continuum is described by the differential equation;
where z is depth from surface, t is time, u is soil displacement, T is soil shear stress, p is soil density, and c is viscoelastic constant. The boundary conditions at the base (rock outcrop motion) and at the top of the soil profile are, respectively; where pr and Vr the density and shear wave velocity respectively of the rock, H is the thickness of the soil deposit, and ug= ug(t) the displacement history of the input "rock outcrop" motion.
The proposed model is schematically illustrated in Fig. 12 . An explicit finite difference technique is used for the solution of the field Eq. (22) which is coupled with the constitutive Eqs. (15) and (18) , and the boundary Eqs. (23) and (24). We have incorporated the BWGG model into a computer algorithm named NL-DYAS for the nonlinear one-dimensional ground response analysis of layered sites (Gerolymos, 2002) .
NONLINEAR SEISMIC RESPONSE ANALYSIS OF A LAYERED SITE-COMPARISON WITH OTHER METHODS
For a first evaluation of the model, a soil amplification analysis is performed for a comparison with codes widely used in geotechnical engineering. More specifically, the seismic response of a representative soil profile of the St. Stefanos bay area in Halkida (Greece), is studied . The soil profile and available geotechnical data are given in Fig. 14. The calibration of the model parameters was based on the optimum fit of the published experimental shear modulus reduction and damping curves, using the methodology developed in a previous section. Specifically, the values of the parameters used in the analysis are: n= 0.6, b = 0.5, s1=1.2, and s2= 0.25. The other parameters are eliminated from the model. The profile was subjected to a seismic accelerogram (Sepolia record) from the Athens 1999 Earthquake (Ms = 5.9, A = 10 km), which had a peak ground acceleration of 0.42 g, in order to examine the modeling ability at large strains. The results of the analyses for the three soil models are compared in Figs. 15-17 . The following remarks are worthy of note:
(a) Whereas the equivalent-linear SHAKE analysis predicts pga = 0.58 g, the nonlinear inelastic analyses (NL-DYAS and DESRA) predict 0.32 g and 0.19 g, Fig. 13 . The soil profile and available geotechnical for our example:
The shear modulus is taken as proportional to Su and hence proportional to depth Fig. 15 . The seismic excitation (Sepolia Record, Athens 1999 earthquake) and acceleration time histories at the ground surface computed with three models for the soil profile of Fig. 14 respectively. This reveals that significant errors could emerge when ignoring the true nature of soil nonlinearity.
(b) The hyperbolic model in conjunction
with the Masing rule for unloading-reloading imposes a systematic cutting-off the acceleration peaks. The value of the limiting acceleration, 0.19 g , is a function of both the minimum value of Su at the surface, Su (0) and the rate of increase of Su with depth, dSu /dz. Of course, no such cutting-off exists in the equivalent linear analysis. On the other hand, the BWGG model restrains the acceleration peaks but without completely cutting them off at a constant value. This intermediate behaviour stems partially from the hardering parameter a. It is believed to be in closer agreement with reality.
(c) A substantial difference exists between the shear strain amplitudes computed with the equivalent linear and the two nonlinear methods. At a depth of about 4 m we compute max y 1.2% with the SHAKE analysis, while max y :,--, 0.25% according to both DESRA and NL-DYAS. It is precisely this large strain value which in SHAKE leads to the doubling of acceleration: from the value max A --, 0.30 g at z= 4 m to the value max A , 0.58 g at z = 0 ma phenomenon reminiscent of the "whip -lash" effect in flexible systems vibrating in a higher mode (Travasarou and Gazetas, 2003) .
(d) The frequency content of the surface motions, as seen in the acceleration histories, and their response spectra, is rather sensitive to the soil model used in the analysis. Indeed, the equivalent linear analysis filters substantially the high-frequency components of the excitation, such as those in the time internal 4.5-6.0 sec. This happens for two reasons: (i) throughout shaking the material damping ratio in the equivalent-linear analysis remains constantly 18% near the ground surface, and (ii) also throughout shaking, the effective stiffness remains constantly low at about 20% of the original Gmax (at Yeff''''' (2/3) max y -,-, 0.8%). Apparently spuriously high damping and low stiffness are the result of the preceding highest amplitude of acceleration at r--,-4 sec, which controls the equivalent linear response. The two effects combined give the observed artificial filtering of the high frequency components associated with small amplitude oscillations. In contrast, the hyperbolic-Masing model filters (perhaps excessively) the low-frequency highamplitude components, due to the aforementioned cutting-off process. BWGG captures more accurately than the above two models both the high and low amplitude periods of shaking, as will be further proven in the following case history.
SEISMIC RESPONSE OF PORT ISLAND IN THE 1995 KOBE EARTHQUAKE: ANALYSIS AND RECORDS
The BWGG model for soils is further used to analyse the seismic response in Kobe's Port Island at the in-depth seismographic array where surface and downhole instruments (at 16 m, -32 m, and -82 m) have recorded the shaking during the Great Hanshin (Kobe) Earthquake (1995) . These records are the benchmark against which any new method must be tested.
Site Conditions and Seismic Records
A downhole accelerometer array was situated at the north-west corner of Port Island (Fig. 18 ). The array consisted of triaxial accelerometers located at the three geotechnical investigation of the array site, performed before and after the earthquake, the soil profile is known with sufficient accuracy (as is summarized in Fig. 19 ) . The subsurface layer down to a depth of 19 m consists of the granular fill layer, underlain by an alluvial clay layer between 19 m and 27 m depth, followed by sandygravelly strata interlayered with clay down to 61 m depth. Below is a diluvial clay layer from 61 m to 82 m depth , and a gravelly sand layer starting at 82 m depth . The water table is located at 4 m depth approximately . Figure 19 (after Iwasaki, 1995) shows the soil profile together with the Standard Penetration Test (SPT) N values and the shear wave velocity distributions. The low SPT values in the granular fill justify the high excess pore pressures /liquefaction that took place in this layer. The N44W components of the recorded downhole accelerations are presented in Fig. 20 (after Iwasaki, 1995) .
Previous Numerical Simulations of the Seismic Response of the Site
According to Yamazaki et al. (1995) who computed the seismic response of the site, liquefaction may have had occurred in more than one layer at different times. First at 27 m to 33 m depth, and then in the loose layer at 10 m to 16 m depth. Elgamal et al. (1995) who also carried out soil amplification analysis of the site, showed that below 32 m depth the soil response was essentially linear with no appreciable reduction in stiffness. On the other hand , at shallow depths a reduction in soil stiffness with a slight shear strain hardening at elevation 24 m, and an abrupt sharp loss of stiffness accompanied by reduction of yield strength at 8 m depth, was predicted.
Site Response Analysis
The profile is analysed using: (a) the BWGG model implemented in NL-DYAS, and (b) the hyperbolic model in conjunction with Masing rule for unloading and reloading (DESRA). Although the proposed model is capable of reproducing cyclic liquefaction response, the calibration of the model parameters was based exclusively on the best fit to the experimental shear modulus and damping curves (Kokusho et al., 1996) . Stiffness and strength degradation due to pore-pressure rise was not taken into account.
Specifically, the values of the parameters used in the analysis are: For the sand and gravel layers, n= 1.2, b = 0.5, si = 1.1, 52= 0.15, and a = 0.035. For the clay layers, n= 0.6, b = 0.5, si = 1 .2, 52= 0.25, and a = 0.035. The other parameters are simply eliminated from the model. The N44W earthquake record at 83 m depth ( Fig. 20) was used as input excitation of the soil profile. Figures 21 and 22 show comparison between recorded and calculated motions at the ground surface for both the BWGG model (NL-DYAS) and the hyperbolic plus Masing (DESRA). Stress-strain loops computed by the two models at depths of 20 m and 32 m respectively are also compared in Fig. 23 .
The agreement for the proposed model with the record appears to be excellent in the region of the first 6 sec (i.e ., before liquefaction has apparently occurred). This is also reflected in the comparison of Fig. 24 in which the predicted spectrum values corresponding to the first 6 sec of the seismic motion, match the recorded spectrum with very good accuracy. As expected, the post-liquefaction response acceleration (after the 6 sec) is overestimated by the BWGG model. This is because no pore water pressure build-up was considered in the calibration of the model parameters.
Comparison between the spectral accelerations of the complete record at the ground surface and those computed by the BWGG model for the whole duration of motion is shown in Fig. 25 . The agreement is satisfactory except for periods around 2 sec for which the response is overpredicted for the aforementioned reason. Figure 26 compares the distribution of maximum accelerations computed by the BWGG model and the hyperbolic-plus-Masing model, together with the recorded values. The agreement for the BWGG model is indeed good, and far better than that of the hyperbolic-plus-Masing model. The latter leads to a systematic cutting-off the acceleration time history at ground surface, even before the initiation of liquefaction (Fig. 22 ). This is mainly for the two aforementioned reasons: (a) the overestimation of the material damping at large strains, and (b) the zero post-yield hardening characterizing the model. These weaknesses of the hyperbolic-plus-Masing model in simulating cyclic behaviour, are shown in the stress-strain loops, at depths of 20 m and 32 m respectively, plotted in Fig. 23 . The inability of the hyperbolic-plus-Masing model to capture the actual response is also evident in Figs. 24 and 25 where the computed spectral accelerations for periods between 1 and 3 seconds, are about two times lower than the recorded.
CONCLUSIONS
A phenomenological constitutive model, BWGG, is developed in this article for static and dynamic response of soil elements. The model is incorporated into a novel algorithm named NL-DYAS for the nonlinear inelastic one-dimensional ground response analysis of layered sites. It is shown that the model is very versatile in representing complex nonlinear characteristics of the cyclic behaviour of soil elements, such as stiffness decay with strain amplitude, loss of strength due to pore-water pressure development, cyclic mobility, and non-symmetric behaviour with loading direction. The proposed model has the ability to independently match experimental shear modulus and damping curves, by properly adjusting only three of its parameters. The NL-DYAS code was utilized to simulate the cyclic behaviour of a layered site subjected to seismic excitation. The results are compared with predictions of widely available and extensively used methods of analysis, such as the equivalent-linear and the hyperbolic nonlinear model. The comparison reveals the weakness of the aforementioned methods of analysis to simulate realistically the nonlinear cyclic behaviour of a soil deposit, at large strain levels. To provide some verification of the model, the records of the Port Island array during the Kobe 1995 earthquake
were utilized. The model was shown to "predict" the observed response with sufficient accuracy .
The hyperbolic model in conjunction
with the Masing rules for unloading-reloading was also used for soil amplification analysis but was shown to be less successful in reproducing the reality.

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