variability on soil liquefaction, Griffiths and Fenton (1993), Dham and Ghanem (1995), Fen- ton and Griffiths (1996) on seepage through spatially random soils, ...
EFFECTS OF SPATIAL VARIABILITY ON SOIL LIQUEFACTION: SOME DESIGN RECOMMENDATIONS by: R. Popescu1, J.H. Prevost2 and G. Deodatis3 October, 1996
Accepted for publication (November 1996) in Geotechnique
Senior Research Engineer, C{CORE, Memorial University of Newfoundland, St. John's, NF, A1B 3X5, Canada 2 Professor, Department of Civil Engineering and Operations Research, Princeton University 3 Assistant Professor, Department of Civil Engineering and Operations Research, Princeton University 1
1
Abstract The e ects of spatial variability of soil properties on the behavior of saturated soil deposits subjected to seismic excitation are analyzed, and their implications on geotechnical design are discussed. A Monte Carlo simulation methodology, combining digital generation of non{Gaussian stochastic vector elds with dynamic, nonlinear nite element analyses is used for that purpose. The proposed procedure is applied to assess the soil liquefaction potential, which is found to be considerably a ected by the inherent spatial variability of soil properties. Parametric studies, involving the degree of soil liquefaction and the frequency content of the seismic input are performed. Finally, a characteristic percentile value of soil strength, which will predict a response equivalent to that provided by the more expensive Monte Carlo simulations, is proposed for use in deterministic design.
Keywords Statistical analysis, Liquefaction, Shear strength, Sands, Numerical modeling and analysis, In situ testing.
1 Introduction Many physical systems in general and soil materials in particular exhibit relatively large spatial variability in their properties, even within so called homogeneous zones. Deterministic descriptions of this spatial variability are not feasible due to prohibitive cost of sampling and to uncertainties induced by measurement errors (e.g. Vanmarcke, 1989). It is widely understood nowadays that probabilistic methods are therefore to be used, and a rst step in acknowledging the e ects of uncertainties in soil properties was made by introducing the concept of characteristic values as \cautious estimates" of those parameters a ecting the occurrence of limit state (e.g. Eurocode, 1994). At this stage, the assessment of characteristic values as percentiles of recorded soil properties is mainly based on the degree of con dence in the mean values, as estimated from soil test results. However, the natural variability of soil properties within geologically distinct and uniform layers is known to a ect the soil system behavior itself. The consequences of spatial variability are not well understood yet, and their exploration requires use of stochastic eld based techniques of data analysis. In this respect, we refer here to the work of Ohtomo and Shinozuka (1990), Fenton (1990), Ural (1995), Popescu (1995) on the e ects of spatial variability on soil liquefaction, Griths and Fenton (1993), Dham and Ghanem (1995), Fenton and Griths (1996) on seepage through spatially random soils, and Paice et al (1996) on settlements. All these approaches use the Monte Carlo simulation method which combines digital generation of stochastic elds representing the spatial variability of certain soil properties over the analysis domain, with nite element analyses using stochastic input parameters. 2
Soil liquefaction seems to be particularly sensitive to spatial soil variability. Previous studies (e.g. Popescu, 1995) have shown that both the extent and pattern of pore water pressure build{up in saturated soil deposits subjected to seismic excitation are di erent when computed by Monte Carlo simulations, which account for the random spatial variability of soil properties, as compared to numerical analyses, which are using average values. An explanation is that liquefaction is triggered by the presence of loose pockets in the soil mass, which can only be accounted for by considering the stochastic nature of spatial variability of soil properties (e.g. Popescu et al, 1996a). Within the framework of seismically induced soil liquefaction, this study rst presents the implications of spatial variability of soil properties on the behavior of saturated soil deposits. Monte Carlo simulations of soil liquefaction, combining digital generation of non{ Gaussian stochastic vector elds with dynamic, nonlinear nite element analyses, are used for that purpose. Since Monte Carlo simulations are too expensive for routine design, the study is aimed at estimating a characteristic percentile value of soil strength which, when used in a deterministic analysis results in an equivalent response (i.e. an equivalent amount of pore water pressure build{up which also leads to similar values of horizontal deformations). For that purpose, a range of deterministic computations, using some average values of soil properties, are performed with progressively lower soil strengths. The numerical results (displacements, excess pore{pressure build{up) predicted by the Monte Carlo simulations are compared to the deterministic analysis results. From these comparisons, and by performing parametric studies involving the seismic excitation duration and the loading rate (or frequency content of the seismic excitation), a characteristic percentile of soil strength is estimated. The theoretical developments are illustrated with numerical examples based on real in{ situ soil data. The soil geomechanical properties and the probabilistic characteristics of their spatial variability are estimated from the results of a statistically signi cant set of piezocone soil tests performed at an arti cial island in the Canadian Beaufort Sea (Figure 1, from Gulf, 1984).
2 Analysis of Randomly Variable Media
2.1 Representing the Variability of Soil Properties Soil properties in a homogeneous soil layer are a ected by a series of uncertainties: inherent spatial variability, random test errors, systematic test errors (or bias), etc. The natural heterogeneity in a supposedly homogeneous layer may be due to (Tang, 1984): small scale variations in mineral composition, environmental conditions during deposition, past stress history, variations in moisture content, etc. Since deterministic descriptions of this spatial variability are in general not feasible, the overall characteristics of the spatial variability and the uncertainties involved are mathematically modeled using stochastic (or random) elds. One of the important features of stochastic elds is the concept of statistical correlation 3
between eld values at di erent locations in space. Univariate stochastic elds describe the variability of one single soil property over the domain of interest, while multi{variate stochastic elds refer to the variability of several soil properties and to their spatial interdependence. Moreover, soil properties follow non{Gaussian marginal probability distributions (e.g. they do not assume negative values). 110m
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Figure 1: Caisson retained island: a. schematic cross section, showing the core lling procedure (after Je eries, 1985); b. plan view with locations of the piezocone pro les at Tarsiut P{45 (after Gulf 1984); the pro les selected for analysis are shown with black circles. With reference to the cone tip resistances recorded in a homogeneous soil deposit (Figure refcpta), the probabilistic characteristics of a stochastic eld can be described as follows:
{ Mean values, which sometimes may vary in space (with depth, for the case of soil
properties), as illustrated by the thick lines in Figure refcpta. For mathematical convenience, these systematic trends can be identi ed and separated from local deviations (or uctuations). { Variance, representing the degree of scatter of the uctuations about mean values. { Correlation structure, which describes the similarity between uctuations recorded at two points as a function of the distance between those two points. As illustrated in Figure 2a, some degree of coherence between the uctuations can be observed, 4
which becomes stronger as the measuring points are closer. Correlations between values of the same material property measured at di erent locations are described by the auto{correlation function. The main parameter of the auto{correlation function is called scale of uctuation (or correlation distance) and represents a length over which signi cant coherence is still manifested. The spatial dependence between two di erent properties measured at di erent locations is described by the cross{correlation function. The ensemble of auto{correlation and cross{correlation functions (which also include the variance) form the cross{correlation matrix. { Higher order moments. 9m
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Figure 2: Comparison between a. recorded in{situ cone tip resistances, and b. simulated values obtained at the same locations from one sample function. Unlike Gaussian elds where the rst two moments provide complete probabilistic information, non{Gaussian elds require knowledge of moments of all orders (to be precise, in the increasing order of completeness, the probability structure of a non{Gaussian stochastic eld is described by the probability density functions of all orders, e.g. Lin, 1967). Unfortunately it is practically impossible to estimate moments higher than order two from actual (non{Gaussian) data. Therefore, the simulation of soil properties, that are represented as non{Gaussian stochastic elds, is done in practice using the cross{correlation matrix and the marginal probability distribution functions. 5
Several methods to estimate the correlation structure of spatial variability from results of in{situ soil tests are described in the literature. A comprehensive review is presented by DeGroot and Baecher (1993). Once the probabilistic characteristics of the spatial variation of soil properties are known, sample functions of the underlying stochastic eld can be digitally simulated using the spectral representation method, ARMA models, Kriging, conditional probability, etc. (For a literature review the reader is referred to Soong and Grigoriu, 1993). Each sample function represents a possible realization of the relevant soil properties over the domain of interest. This is illustrated in Figure 2, where the in{situ recorded values of cone tip resistance (Figure 2a) are compared to the simulated values, obtained from one of the sample functions at the same locations (Figure 2b). Though the values of cone tip resistance shown in Figures 2a and 2b are di erent, their spatial distributions exhibit similar characteristics: for example, the pockets of loose or dense material have di erent locations but similar sizes and shapes.
2.2 Proposed Approach
Stochastic Finite Element Methods (SFEM) are a class of methodologies to analyze structures with uncertain material and/or geometric properties. The only universal SFEM that can be used for any structure (e.g. involving nonlinearity and large variations of the uncertain parameters, as exhibited by soil properties) appears to be the Monte Carlo simulation technique. According to this method, sample functions of the uncertain quantities are digitally generated to be compatible to their prescribed probabilistic characteristics, and the resulting deterministic problems are solved for a large number of sample functions. Statistics of the response are computed by ensemble averaging. The Monte Carlo simulation method presented in the following accounts for the e ects of stochastic spatial variation of material properties on the system performance under static and dynamic loads, for highly nonlinear materials. The ensemble of material properties over the domain of interest are modeled as a multi{variate, multi{dimensional (mV-nD), non{ Gaussian stochastic eld, each component of the vector eld representing one of the di erent properties. The proposed Monte Carlo procedure has basically three steps (Popescu, 1995): 1. Estimation of the probabilistic characteristics of spatial variability (e.g. probability distribution functions, cross{correlation structure) based on results of extensive measurement programs. 2. Digital generation of sample functions of an mV{nD, non{Gaussian stochastic vector eld, each sample function representing a possible realization of relevant measurement results over the analysis domain. 3. Finite element analyses using stochastic input parameters obtained from the generated sample vector elds of material properties. The method is applied in the following to saturated soil deposits subjected to seismic excitation. The simulation of the soil properties, that constitute a non{Gaussian vector 6
eld, is performed according to the limited probabilistic information that is available: the correlation structure and the marginal (non{Gaussian) probability distribution functions. For details on the simulation method, the reader is referred to Popescu et al, 1997.
3 Monte Carlo Simulations of Soil Liquefaction 3.1 Field Data Analysis
The eld test results selected for this study come from a set of piezocone soil tests (Gulf, 1984) performed for core veri cation at one of the arti cial islands (namely: Tarsiut P{45 { Figure 1a) constructed in the Canadian Beaufort Shelf and used as drilling platforms for oil exploration in shallow waters. The core was made of undensi ed hydraulically placed sand, using special provisions for limiting excess pore pressure build{up during placement and loading (median grain size D50 > 0:3mm and nes content f < 4%). Sand dredged from the Erksak borrow area was used for that purpose. For a detailed description of the construction and veri cation of the arti cial island at Tarsiut P{45 the reader is referred to Je eries et al (1985, 1988). A total of 32 piezocone tests, providing an almost uniform coverage of the core area (72m 72m) { Figure 1b { were carried out. Five borings were placed on a straight line, at 9 m distance intervals between centers (pro les MAC04, . . . , MAC08). Three other piezocone pro les (MAC31, . . . MAC33) were performed after a few months to assess the e ects of time on cone tip resistance. There are two pairs of pro les (MAC05{MAC32 and MAC08{MAC33) placed at 1m center{to{center distance. The piezocone test results obtained from those eight borings (which are represented by black dots in Figure 1b) were used in the present study. The results of piezocone tests are analyzed in terms of: (1) cone tip resistance, qc (Figures 2a and 3), which is mainly dependent on relative density and con ning stress, and (2) soil classi cation index, Ic, which is related to grain size and soil type. The expression of soil classi cation index proposed by Je eries and Davis (1993) is: q
Ic = [3 ; log Q(1 ; Bq )]2 + [1:5 + 1:3 log F ]2 (1) where Q is the normalized cone resistance, Bq is the piezocone pore pressure ratio, and F is the stress normalized friction ratio. The in{situ recorded cone tip resistances, averaged over 10cm intervals in elevation, are plotted with markers in Figure 3. The average values, assumed to vary linearly with depth, and estimated by linear regression, are shown with a thick continuous line in Figure 3. The main results of a stochastic analysis of eld data, leading to the probabilistic characteristics of the spatial variability of standardized (i.e. zero{mean, unit{variance) relevant soil properties, are listed in Table 1. The parameters of the theoretical models selected for probability distribution functions and cross{correlation structure are evaluated by nonlinear regression from the corresponding sample functions estimated from eld test results. For a 7
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(coeff. of determination: 2 R =53.4%) average 70-percentile 80-percentile 90-percentile MAC05 MAC06 MAC07 MAC08 MAC31 MAC32 MAC33
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Figure 3: In{situ measured values (markers), average values (continuous line), and percentiles (dashed lines) of cone tip resistance recorded in the sand core at Tarsiut P{45. detailed description of eld data analysis procedures (step 1 of the Monte Carlo simulation algorithm) the reader is referred to Popescu (1995). Table 1. Characteristics of spatial variability of eld data (after Popescu, 1995) Probabilistic characteristic Results of eld data analysis Probability distribution qc Beta distribution with b1 = 3:5, b2 = 18:4 functions Ic Beta distribution with b1 = 5:2, b2 = 120 Separable auto{correlation horiz. b1; : : :; b5 = 19:8; 0:60; 1:25; 0:43; 0:47 structure with parameters dir. correlation distance x = 12:1m b1; : : :; b5 (the same for vert. b1; : : :; b5 = 1:28; 4:80; 2:78; 1:98; 0:46 both qc and Ic) dir. correlation distance z = 0:95m Cross{correlation between qc and Ic with qI = ;0:58
3.2 Generation of Stochastic Fields
Stochastic analysis of eld data, in terms of cone tip resistance (qc) and soil classi cation index (Ic), produced the cross{correlation matrix and the probability distribution functions of these two soil properties, that were considered to be a bi{variate, two{dimensional (2V{ 2D) non{Gaussian stochastic vector eld. Twenty ve sample functions of this 2V{2D vector eld (representing 6 possible realizations of qc and Ic over the analysis domain) were then generated following the methodology described by Popescu (1995) and Popescu et al. (1997). This is the second step of the Monte Carlo simulation procedure described in x2.2. 8
3.3 Estimation of Constitutive Soil Parameters from Simulated Field Data Next, for each sample function at each spatial location, the simulated values of piezocone test results (qc and Ic) are used to estimate the constitutive parameters of the soil constitutive model to be employed for stochastic input nite element analyses (part of step 3 of the Monte Carlo simulation procedure).
3.3.1 Multi{Yield Plasticity Constitutive Soil Model The multi{yield constitutive soil model was selected to simulate the nonlinear behavior of soil materials. It is a kinematic hardening model based on a relatively simple plasticity theory (Prevost, 1985) and is applicable to both cohesive and cohesionless soils. A non{associative plastic ow rule is used for the dilatational component of the plastic deformation. The model has been tailored: (1) to retain the extreme versatility and accuracy of the simple multi{ surface J2{theory (Prevost, 1977) in describing observed shear nonlinear hysteretic behavior and shear stress{induced anisotropic e ects; and (2) to re ect the strong dependency of the shear dilatancy on the e ective stress ratio in both cohesionless and cohesive soils. Nested conical yield surfaces are used for that purpose. The required constitutive parameters of the multi{yield soil model are listed in Table 2. Table 2. Constitutive parameters of the multi{yield plasticity model Constitutive parameter Symbol Type Mass density - solid s State w Porosity n parameters Permeability k Low strain elastic moduli G0 ; B0 Low strain Reference e ective mean normal stress p00 elastic Power exponent n parameters Friction angle at failure Yield and Stress{strain curve coecient failure Maximum deviatoric strain "max parameters dev Coecient of lateral stress K0 Dilation parameter (cyclic) Xpp Dilation Dilation angle parameters Excepting for the dilation parameter, all the constitutive model parameters are expressed in terms of traditional soil properties, and can be derived from the results of conventional laboratory (\triaxial", \simple shear" { e.g. Popescu and Prevost, 1993) or in{situ (\cone penetration", \standard penetration", \wave velocity" { e.g. Popescu, 1995) soil tests. The dilation parameter Xpp, which controls the amount of plastic dilation, is evaluated based on the results of liquefaction strength analysis, as shown by Popescu and Prevost (1993) and Popescu (1995). 9
The multi{yield plasticity model, its implementation algorithm and calibration procedures have been veri ed in the past for soil liquefaction potential assessment using full scale data (e.g. Keane and Prevost, 1989) and centrifuge geotechnical models (e.g. Popescu and Prevost, 1993, 1995).
3.3.2 Parameter Estimation Method The index properties of Erksak sand are reported by Been et al. (1987, 1991). The correlation formulas used to evaluate the soil constitutive parameters as functions of cone tip resistance qc, e ective overburden stress v0 0, and soil classi cation index Ic are shown in Table 3. The speci c correlations are selected on the basis of (1) sand type: uniformly graded, medium to ne grained sand, with subrounded shaped particles and mainly quartz minerals; and (2) deposit history: normally consolidated hydraulic ll. The empirical correlations for relative density, Dr , and friction angle at failure, , are selected to accommodate the unusually large values of the coecient of lateral stress K0 (Je eries et al. 1985, 1988). For example, the range of average friction angle values and their variation with depth are estimated from 50{percentile values of the state parameter reported by Je eries et al. (1985), and also using a chart proposed by Marchetti (1985), relating friction angle, cone resistance and coecient of lateral stress. The resulted range and the tendency of friction angle to diminish with depth in the hydraulic ll core (as reported by Je eries et al. 1985, 1988) are best matched by the correlation formula proposed by Robertson and Campanella (1983) { Table 3. The soil classi cation index, Ic, does not appear explicitly in Table 3. However, the grain size (expressed in terms of D10 or Dmax), which in uences the values of permeability and stress{strain curve coecient, is expressed at each spatial location as a function of the soil classi cation index, according to Je eries and Davies (1993). The dilation parameter, Xpp, is evaluated by liquefaction strength analysis (e.g. Popescu and Prevost, 1993), using a correlation between observed eld performance and normalized cone tip resistance, qn , proposed by Shibata and Teparaksa (1988). An empirical formula relating dilation parameter, Xpp, in{situ recorded cone tip resistance, qc, and e ective overburden stress, v0 0, is obtained for Erksak 330/0.7 sand as (Popescu, 1995, Popescu et al. 1996b):
p 0 0:39 15 + q c pa =(v0 ) log Xpp = 7:07 ; 11:4 0:1 25 ; q pp =(0 )0:39 c a v0 "
#0:13
(2)
where pa is the atmospheric pressure. There are a series of soil parameters (mass density of solid particles { s , power exponent { n, maximum deviatoric strain { "max dev , coecient of lateral stress { K0 , and dilation angle { ) which are assumed invariant with respect to cone tip resistance and soil classi cation index, as shown in Table 3. All the other constitutive parameters of the multi{yield soil model are considered stochastic input parameters and are evaluated at each spatial location 10
( nite element centroid) from the corresponding values of cone tip resistance (qc), con ning stress (v0 0), and soil classi cation index (Ic) at the respective locations, for each of the 25 sample functions of the 2V{2D non{Gaussian stochastic vector eld simulated at step 2 of the Monte Carlo procedure. Consequently, the stochastic input parameters are spatially cross{correlated, according to the cross{correlation structure estimated from the stochastic analysis of the eld data (Table 1). Table 3. Estimation of geotechnical properties of Erksak sand Soil property Correlation / Estimated value Reference 3 Mass density s = 2660Kg/m Been et.al., 1991 8 0:17 0 > Overburden stress Shibata and > < v0 +0:07 ; v0 0:05MPa Teparaksa, 1988 correction factor Cq = > q > 0 : 0 pa=v0 v0 > 0:05MPa Kulhawy and Maine, (qn = Cq qc) 1990 i0:5 h q 1 n Ibid. Relative density Dr = 305 pa Void ratio e = emax ; Dr (emax ; emin ) Porosity nw = e=(1 + e) Permeability k = 1:2Cu0:735D100:89 1+e3e Das, 1994 0 : 5 2 Low strain shear G0 = 70 (2:171+;ee) ppa0 Hardin and Richard, modulus (MPa) 1963 Poisson's ratio = 0:35 estimated Power exponent n = 0:5 h estimated i q ; 1 c Robertson and Friction angle = tan 0:1 + 0:38 log v0 at failure Campanella, 1983 Stress{strain curve = 0:217 + 0:037Dmax Hayashi et.al., 1992 coecient ;0:027Cu = 0:26 Coe . of lat. stress K0 = 0:7 Je eries et al, 1988 Max. dev.strain "max = 7% compr., 4% ext. Been et.al., 1991 dev Dilation angle = 31 Been and Je eries, 1985 15 + q ( MPa ) Cyclic stress ratio CSR = 0:1 25 ; qn (MPa) Shibata and n Teparaksa, 1988 0 Dilation parameter Xpp = f(qc; v0) { Eqn. (2) Popescu, 1995 0
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3.4 Finite Element Analysis with Stochastic Parameter Input The numerical simulations are performed with the computer code DYNAFLOW (Prevost, 1995). It is a nite element program for nonlinear seismic site response analysis. The solid and uid coupled eld equations are based on an extension of Biot's formulation (Biot, 1962) in the nonlinear regime, and are applicable to multidimensional situations. The computations are performed in terms of e ective stresses. 11
3.4.1 Seismic Motion The base input motion used for the numerical applications is a horizontal seismic acceleration with peak value set equal to 0.15g. A procedure for digital generation of non{stationary time histories (Deodatis, 1996) capable of simulating seismic ground motion time histories that: (1) are compatible with prescribed response spectra, and (2) have a prescribed modulating function for amplitude variation, is used to generate the acceleration time histories. To assess the e ects of loading rate (frequency content of seismic excitation) on dynamic behavior, earthquake ground motions corresponding to di erent local soil conditions are simulated using di erent prescribed response spectra, as shown in Figure 4. Speci cally, the rst three types of response spectra plotted in Figure 4a correspond to the Uniform Building Code (Uniform, 1994), with type 1 for rock and sti soils, type 2 for deep cohesionless or sti clay soils, and type 3 for soft to medium clays and sands. Response spectrum type 4, with a range of maximum spectral values corresponding to frequencies that are lower than for types 1,2 and 3, is believed to be representative for locations close to the epicenter. Four acceleration time histories, corresponding to the four types of response spectra shown in Figure 4a, are plotted in Figure 4c. For demonstration purposes, these acceleration time histories are used as base input motions in the numerical applications presented hereafter. The e ectiveness of the algorithm is illustrated in Figure 4b, where the prescribed type 4 response spectrum (continuous line) is compared to the response spectrum (dotted line) computed from the corresponding simulated acceleration time history.
3.4.2 Stochastic vs. Deterministic Computational Results A loose to medium dense soil deposit, with geomechanical properties as well as spatial variability characteristics corresponding to the hydraulic ll at Tarsiut P{45 is selected for the numerical applications. The analysis domain extends 60m in horizontal direction (Fig. 6a). A 12m thick saturated sand deposit is overlaid by an 1.6m dry sand layer. An impervious layer is assumed at the base, where the input horizontal acceleration is applied. The liquefaction potential of the analyzed soil deposit is rst estimated using various empirical correlations between liquefaction resistance under level ground conditions and normalized cone tip resistance. The number of equivalent cycles at 0:65amax (e.g. Seed et al. 1975) is represented in Figure 5a as a function of time for the four acceleration time histories used in this study (and plotted in Figure 4c). The results of the simpli ed liquefaction risk assessment procedure (e.g. Das 1983) presented in Figure 5b indicate that initial liquefaction should occur after 10 equivalent cycles, or after about 8 sec of shaking. The analysis is performed using the average values of cone tip resistance recorded at Tarsiut P{45 (thick, continuous line in Figure 3). Finite element simulations using stochastic input parameters (step 3 of the Monte Carlo procedure described in x2.2) are performed next. The size of the nite elements for Monte Carlo simulations is selected small enough to capture the essential features of the underlying random eld. For the correlation structure exhibited by the soil properties recorded at 12
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Figure 4: Base input motions selected for the nite element simulations: a. prescribed response spectra; b. prescribed and resulted values for the type 4 response spectrum; c. response spectrum compatible acceleration time histories used in the study. a. Number of representative cycles at 0.65 amax
b. Liquefaction risk assessment 0
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Figure 5: Estimation of liquefaction potential using correlations between observed eld performance and normalized cone resistance. 13
Tarsiut P{45, a reasonable mesh size was found to be 3.0m 0.5m (Popescu, 1995). The Monte Carlo simulations are performed using soil parameters corresponding to the 25 sample functions of a 2V{2D stochastic eld with probabilistic characteristics derived from eld data analysis and listed in Table 1. The base input acceleration compatible with the type 2 response spectrum (Figure 4c) is used for this numerical application. Some representative results, in terms of excess pore pressure ratio with respect to the initial e ective vertical stress after 8 sec of shaking, are presented in Figure 6b (results obtained from six sample functions, out of 25, are only shown). For comparison, a deterministic analysis is performed using input soil parameters having the same values as the average values used in the Monte Carlo simulations. The results are presented in the upper left plot of Figure 6c. From the results shown in Figure 6 it can be concluded that (1) more pore pressure build{up is predicted by the stochastic input model than by the deterministic model (moreover, the stochastic input model predictions are in good agreement with the results of liquefaction assessment procedure based on observed eld performance, which are shown in Figure 5b); and (2) in the case of stochastic input, there are patches with large excess pore pressure, corresponding to the presence of loose pockets in the material (this predicted pattern explains better the phenomenon of sand boils observed in areas a ected by soil liquefaction). Finally, a set of deterministic input nite element analyses is performed using various percentile values of in{situ recorded cone tip resistance, some of which are plotted with dashed lines in Figure 3. From the results that are presented in Figure 6c it can be concluded that it is possible to de ne a characteristic percentile for the soil strength which, if used in deterministic design, will predict an amount of pore water pressure build{up equivalent to the one predicted by the more expensive Monte Carlo simulations. This is the object of the parametric study presented in the next section.
4 Characteristic Percentile of Soil Strength Pore pressure build{up and horizontal displacements predicted by Monte Carlo simulations are compared in the following to the corresponding results of deterministic nite element analyses for various percentiles of in{situ recorded cone tip resistance. The deterministic analyses are performed using the following values of cone tip resistance: (1) average values, shown with a thick continuous line in Figure 3, and (2) 70{, 80{ and 90{percentiles, shown with dashed lines in Figure 3. Parametric studies are conducted to investigate the e ects of load duration (or degree of liquefaction) and loading rate (or frequency content of the seismic input).
4.1 Comparison Criteria Characteristic for the behavior of saturated soil deposits subjected to dynamic excitation is the phenomenon of pore water pressure build{up. Increases in pore pressure lead to reduc14
a. Finite element analysis set−up free field boundary conditions
480 two−phase finite elements for porous media (4 dof/node) with dimensions: 3m x 0.5m
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Vertical scale is two times larger than horizontal scale
0.90...1.00 0.70...0.90 0.50...0.70 0.00...0.50
60.0m
b. Excess pore pressure ratio predicted using six sample functions of a stochastic vector field with cross−correlation structure and probability distribution functions estimated from piezocone test results
Sample function #1
Sample function #2
Sample function #3
Sample function #4
Sample function #5
Sample function #6
c. Excess pore pressure ratio predicted using deterministic input soil parameters (linear variation of qc and Ic with depth). The percentile values are computed for qc only.
(u /σv0)max = 0.44 Deterministic − average
50 − percentile
60 − percentile
70 − percentile
80 − percentile
90 − percentile
Figure 6: Set{up of nite element analysis (a.), and comparisons between results of Monte Carlo (b.) and deterministic (c.) computations in terms of excess pore pressure ratio with respect to the initial e ective vertical stress, at the onset of liquefaction (time T = 8sec, input acceleration compatible with the type 2 response spectrum).
15
tion of e ective normal stress and therefore of soil shear strength, with e ect on the overall behavior and stability of the soil deposit. Consequently, the following comparisons between results of Monte Carlo simulations of soil liquefaction and deterministic nite element analysis results are mainly presented in terms of predicted excess pore pressure u(x; t). This is expressed at each spatial location x and time instant t as a ratio with respect to the initial e ective vertical stress v0 0(x) { Figure 6b,c:
r(x; t) = u(0 x(;xt))
(3)
v0
Due to the large amount of information provided by the parametric studies, the predicted excess pore pressure ratio is further synthesized for comparison purposes. The following comparison indices are used: 1. Area of lique ed zone, or A80 index (Figure 7), where A80(t) represents the proportion of the saturated soil deposit for which the excess pore pressure ratio r(x; t) exceeds 80%, at a certain time instant, t. 2. Liquefaction index, computed by averaging the excess pore pressure ratio in horizontal direction: ZL Q(z; t) = L1 0 r(x; z; t)dx
(4)
where L = 60m is the length of the analysis domain. At each time instant t0 the liquefaction index characterizes the distribution of excess pore pressure in elevation [Q(z; t0)]. Comparisons in terms of predicted horizontal displacements are also provided. Di erential displacements between ground level and base of the analysis domain are considered:
(t) = ground level(t) ; base(t)
u σv0 > 0.80
(5)
Area
A80(t) =
u(t) σv0 > 0.80
Total Area
Figure 7: De nition of A80 index (area of lique ed zone). In order to simplify the comparison charts presented in Figures 8, 9, 11 and 12, the predictions of the Monte Carlo simulations are represented as ranges of results obtained from all the stochastic sample functions. 16
4.2 In uence of Loading Duration A comparison between Monte Carlo simulation results and deterministic analysis results using various percentiles of soil strength is performed to assess the in uence of the duration of seismic excitation (and therefore of the degree of liquefaction) on the characteristic percentile. The type 2 acceleration (Figure 4c) is used as base input motion. Predicted excess pore pressures are compared in Figure 8 in terms of A80 index and liquefaction index. As can be inferred from Figure 8, during the seismic excitation there is a phase of pore pressure build{up in the rst 10 sec of shaking, followed by a dissipation phase, between 10 and 20 sec. The values of liquefaction index are plotted at three representative time instants (as reported to the results of Monte Carlo simulations): after 8 sec of shaking at the onset of liquefaction, at 10 sec when maximum pore pressures are predicted to occur, and at 16 sec during the dissipation phase. It is postulated that a reasonably conservative prediction is provided by the upper limit of the Monte Carlo simulation response range, since in this manner the e ects of soil variability are accounted for and the worst possible situation is considered. (In case of limited eld information on soil strength, supplementary provisions would have to be made). In this situation, it is evident from the comparison presented in Figure 8 that deterministic analyses using average values of soil strength underestimate the pore pressure response, while those using a 90{percentile overestimate it, for all phases of the seismic analysis. The upper limits of Monte Carlo simulation response ranges are bounded by deterministic analyses using 80{percentile of soil strength. The same conclusions are obtained from the comparison in terms of horizontal displacements at the ground level, presented in Figure 9. The predicted peak displacements (in absolute values) are listed in Table 4. Table 4. In uence of loading rate on predicted horizontal displacements Response Peak horiz. displacements (absolute values) at the ground level (m) spectrum Monte Deterministic input Carlo average 70{percentile 80{percentile 90{percentile type 1 2.3 1.2 1.7 2.7 6.0 type 2 3.1 1.2 2.8 4.2 5.0 type 3 7.4 3.4 5.1 8.0 16.0 type 4 18.7 12.6 16.6 25.3 39.6 It can be concluded so far that: (1) the loading duration (or degree of liquefaction) has little or no in uence on the comparisons between results of Monte Carlo simulations and deterministic analyses, and (2) the 80{percentile is a suciently conservative value of soil strength to be used in deterministic design, for the type of soil and seismic input analyzed. 17
A80 - index
det. - average 70 - percentile 80 - percentile 90 - percentile range from Monte Carlo simulations
0.4 0
0.2
a. Area of liquefied zone - A80
5
10
15
20
b. Liquefaction index - Q
0
6
time = 16 sec
elevation - z (m)
12
0
6
time = 10 sec
elevation - z (m)
12
0
6
time = 8 sec
elevation - z (m)
12
time (sec)
0
0.25
0.5
0.75
1
liquefaction index - Q
Figure 8: Comparison between results of deterministic analyses and Monte Carlo simulations in terms of A80 index (a.), and liquefaction index (b.), for a soil deposit subjected to the type 2 input acceleration. The soil properties are estimated from the results of piezocone tests performed at Tarsiut P{45. 18
5 2.5 0 −2.5
det. − average 70 − percentile 80 − percentile 90 − percentile
−5
horizontal displacements (cm)
δh = δtop − δbase
0
4
8
range from Monte Carlo simulations time (sec) 12
16
20
Figure 9: Comparison between results of deterministic analyses and Monte Carlo simulations in terms of predicted displacements at the ground level, for a soil deposit subjected to the type 2 input acceleration.
4.3 In uence of Loading Rate
As a dynamical system, a soil deposit has its own characteristic frequency (or lower eigenfrequency value), which depends on soil properties (especially on shear modulus), soil deposit geometry, and degree of saturation. This characteristic frequency may decrease during dynamic excitation, due to the decrease in e ective shear moduli as a result of pore pressure build{up and/or large shear strains. Any mechanical system is more sensitive to dynamic loading as its characteristic frequency is closer to the frequency range corresponding to the maximum spectral values of the excitation. Consequently, the frequency content of the seismic excitation can make a signi cant di erence in the excess pore pressure response of saturated soil deposits. This is illustrated in Figure 10, where contours of excess pore pressure ratio predicted for the same soil deposit but using input motions with di erent frequency content are compared. It is mentioned at this point that the simpli ed liquefaction risk assessment procedure based on observed eld performance (Figure 5b) cannot account for the in uence of loading rate (frequency content of seismic excitation) on dynamic response. However, since the underlying empirical correlations are designed to envelop most cases of observed liquefaction in the eld, the simpli ed procedure is deemed as providing conservative results. A study on the in uence of frequency content of seismic excitation on the characteristic percentile of soil strength for deterministic design is presented hereafter. Monte Carlo simulations and deterministic analyses for the soil deposit described in x3.4.2 are performed using various types of base input motion (type 1, type 3 and type 4 acceleration time histories { Figure 4c). Predicted excess pore pressures are compared in Figure 11 in terms of A80 index and liquefaction index. The following conclusions result from Figure 11, regarding the in uence of loading rate on predicted pore pressure build{up, for the type of soil deposit and the range of loading rates employed in the study: 1. Larger pore pressures are predicted when using input motions with lower frequency 19
Base input acceleration compatible with: a. Type 2 response spectrum b. Type 4 response spectrum
T=8sec
T=10sec
T=16sec Grey scale range for the excess pore pressure ratio (u/σv0)
0.90...1.00 0.70...0.90 0.50...0.70 0.00...0.50
Figure 10: Excess pore pressure ratio predicted using base input accelerations with di erent frequency contents (the material properties are computed according to the stochastic eld #3 - soil data from Tarsiut P-45). content. For example, the predicted area of lique ed zone (Figure 11a) is about twice as large when using the type 4 input than for the type 1 input. 2. Peak values of predicted excess pore pressure ratio are situated at lower elevations as the loading rate decreases (Table 5). Table 5. In uence of loading rate on predicted pore pressure at time T = 8 sec Response Frequency Elevation of peak liquefaction index, Qmax, spectrum range of measured from the base (m) of seismic max. spectral stochastic deterministic excitation values input input (Fig. 4a) (Hz) (70. . . 90{perc.) type 1 2.6 | 6.7 9.25 8.75 type 2 1.8 | 6.7 9.00 8.00 type 3 1.1 | 6.7 8.50 7.25 type 4 0.7 | 2.0 8.00 6.75 Regarding the characteristic percentile of cone tip resistance to be used in deterministic analyses, it is apparent from the results presented in Figure 11 that an 00{percentile value is conservative enough The only case when the upper bound of the result ranges predicted 20
type 3 input
A80 - index
A80 - index
0.6
type 1 input
0
0.3
a.Area of liquefied zone (u/σv0 > 0.80) - A80 index
10
A80 - index
type 4 input
15
20
time (sec) det. - average 70 - percentile 80 - percentile 90 - percentile range from Monte Carlo simulations
0
0.3
0.6
5
5
10
15
20
b. Liquefaction index - Q
0
6
type 4 input
elevation - z (m)
12
0
6
type 3 input
elevation - z (m)
12
0
6
type 1 input
elevation - z (m)
12
time (sec)
0
0.25
0.5
0.75
1
liquefaction index - Q
Figure 11: Comparison between results of deterministic analyses and Monte Carlo simulations in terms of A80 index (a.), and liquefaction index at time T = 8sec (b.), for a soil deposit subjected to base input accelerations with various frequency contents. 21
10 -10
0
type 1 input
-10
0
10
type 3 input
det. - average 70 - percentile 80 - percentile 90 - percentile range from Monte Carlo simulations
0
10
20
30
type 4 input
time (sec)
-10
horizontal displacements (cm)
40
-20
horizontal displacements (cm)
20
horiz. displ. (cm)
by Monte Carlo simulations exceeded the deterministic analysis results obtained for the 80{ percentile of soil strength was the A80 index for type 4 input (Figure 11a). It is mentioned, however, that only two sample functions out of the twenty ve generated provided results in terms of A80 index higher than those of the deterministic analysis using 80{percentile of soil strength. Another important result is illustrated in the lower left plot of Figure 11b, where it is shown that use of overconservative parameters may lead to unrealistic prediction of pore water pressure distribution, with direct implications on the overall predicted behavior (peak pore pressure predicted using 90{percentile resulted lower than that predicted using 70{ and 80{percentiles). Similar conclusions result from the analysis of predicted horizontal displacements (Figure 12 and Table 4): the characteristic value of 80{percentile is conservative enough for all situations analyzed.
0
5
10
15
20
Figure 12: Comparison between results of deterministic analyses and Monte Carlo simulations in terms of predicted displacements at the ground level, for a soil deposit subjected to base input accelerations with various frequency contents. 22
5 Conclusions A Monte Carlo simulation procedure for evaluating the e ects of spatial variability of soil properties on soil behavior under dynamic loads is presented. These e ects are investigated by comparing results of Monte Carlo simulations and deterministic nite element analyses performed for a saturated soil deposit subjected to seismic excitation. It is concluded that both the pattern and the amount of dynamically induced pore water pressure build{up are strongly a ected by the spatial variability of soil properties. For the same average values of soil parameters, more pore water pressure build{up is predicted by the stochastic model than by the deterministic model. Parametric studies are performed to estimate a characteristic percentile of soil strength to be used in deterministic analyses, which would result in a response equivalent to that predicted by Monte Carlo simulations. Results of a sequence of deterministic analyses using various percentiles of soil strength are compared to the range of results provided by Monte Carlo simulations using 25 sample functions of a stochastic eld. (The probabilistic characteristics of the stochastic eld are evaluated by stochastic analysis of real in{situ soil data). The comparisons are performed for various degrees of liquefaction of the soil deposit and for seismic excitations with various ranges of maximum spectral response amplitudes. For the type of soil deposit analyzed (loose to medium hydraulically placed sand) and for the range of seismic motions considered the 80{percentile of soil strength is found to be a conservatively enough characteristic value to be used in deterministic dynamic analyses. Further, it is shown that, given the complex behavior of saturated soil materials subjected to dynamic loads, the use of overconservative parameters in deterministic analyses may yield unconservative results, due to the prediction of unrealistic distribution of pore water pressure (x4.3 and Figure 11b). It is mentioned that the e ects of stochastic variability of soil properties on seismically induced liquefaction are strongly dependent on the degree of variability and the probability distributions of di erent soil properties (for more details on the e ects of probability distribution functions, the reader is referred to Popescu, 1995, and Popescu et al, 1996a). Therefore caution should be used if extrapolations of the results of this study are to be used for other types of soil deposits.
Acknowledgements The work reported in this study was supported in part by a collaborative research agreement between Kajima Corporation, Japan and Princeton University. This support is gratefully acknowledged. The authors are also indebted to Dr. Michael G. Je eries for providing the eld data, reviewing the paper and o ering many valuable suggestions. This work was also supported by the National Science Foundation under Grant # BCS{9257900 with Dr. Cli ord J. Astill as Program Director. 23
References
Been, K. and Je eries, M.G., (1985). A state parameter for sands. Geotechnique, 35(2):99{ 112. Been, K., Je eries, M.G., and Hachey, J., (1991). The critical state of sands. Geotechnique, 41(3):365{381. Been, K., Lingnau, B.E., Crooks, L.H.A., and Leach, B., (1987). Cone penetration test calibration for Erksak (Beaufort Sea) sand. Canadian Geotech. J., 24:601{610. Biot, M.A., (1962). Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 33(4):1482{1498. Das, B.M., (1983). Fundamentals of Soil Dynamics. Elsevier. Das, B.M., (1994). Principles of Geotechnical Engineering. PWS Publishing Co., third edition. Deodatis, G., (1996). Nonstationary stochastic vector processes: Seismic ground motion applications. Journ. of Probab. Engnrg. Mech. DeGroot, D.J. and Baecher, G.B., (1993). Estimating autocovariance of in{situ soil properties. Journ. Geotechnical Engnrg., ASCE, 119(1):147{166. Dham, S. and Ghanem, R., (1995). Stochastic nite element analysis for multiphase ow in heterogeneous porous media. In P.D. Spanos, editor, Computational Stochastic Mechanics, pages 429{434, Balkema, Rotterdam. Eurocode 7: Geotechnical Design, (1994). European Committee for Standardization, Technical Report ENV 1997{1, October. Fenton, G.A., (1990). Simulation and Analysis of Random Fields. PhD thesis, Princeton Univ. Fenton, G.A. and Griths, D.V., (1996). Stochastics of free surface ow through stochastic earth dam. Journ. Geotechnical Engnrg., ASCE, 122(6):427{436. Griths, D.V. and Fenton, G.A., (1993). Seepage beneath water retaining structures founded on spatially random soil. Geotechnique, 43(4):577{587. Gulf Canada Resources Inc, (1984). Frontier development { Molikpaq, Tarsiut delineation { 1984/85 season. Technical Report 84F012, Gulf Canada Resources Inc. Hardin, B.O. and Richard, F.E., (1963). Elastic wave velocities in granular soils. J. Soil Mechanics and Found. Div.,ASCE, 89(SM1):33{65. Hayashi, H., Honda, M., and Yamada, T., (1992). Modeling of nonlinear stress strain relations of sands for dynamic response analysis. In Proc. 10th World Conf. on Earthquake Engrg., volume 11, pages 6819{6825, Madrid, Spain, Balkema, Rotterdam. Uniform Building Code, (1994). International Conference of Building Ocials. Je eries, M.G. and Davies, M.P., (1993). Use of CPTu to estimate equivalent SPT N60. Geotechnical Testing J., 16(4):458{468. 24
Je eries, M.G., Rogers, B.T., Stewart, H.R., Shinde, S., James, D. and Williams-Fitzpatrick, S., (1988). Island construction in the Canadian Beaufort Sea. In Proc. ASCE Spec. Conf. Hydr. Fills, pages 816{883, Fort Collins, CO. Je eries, M.G., Stewart, H.R., Thompson, R.A.A. and Rogers, B.T., (1985). Molikpaq development at Tarsiut P{45. In Proc. Conf. Arctic '85, ASCE, pages 1{27, San Francisco, CA. Je eries, M.G. and Wright, W.H., (1988) Dynamic response of Molikpaq to ice{structure interaction. In Proc. 7th Int. Conf. on O shore Mech. and Arctic Engrg., ASME, volume 4, pages 201{220, Houston, TX. Keane, C.M. and Prevost, J.H., (1989), An analysis of earthquake data observed at the Wildlife Liquefaction Array Site, Imperial County, California. Proc. 2n d U.S.{Japan Workshop on Liquef., Large Ground Deform, E . on Lifel., New York, September, pages 39{53. Kulhawy, F.H. and Mayne, P.W., (1990). Manual on estimating soil properties for foundation design. Final Report 1493{6, EL{6800, Electric Power Research Institute, Palo Alto, CA. Lin, Y.K., (1967). Probabilistic Theory of Structural Dynamics. Krieger. Marchetti, S., (1985). On the eld determination of k0 in sand. In Proc. 11th Int. Conf. Soil Mech. Found. Engrg., volume 5, pages 2667{2672, San Francisco, CA. Ohtomo, K. and Shinozuka, M., (1990). Two{dimensional spatial severity of liquefaction. In Proc. 8th Japan Earthquake Engnrg. Symp., Tokyo. Paice, G.M., Griths, D.V. and Fenton, G.A., (1996). Finite element modeling of settlements on spatially random soil. Journ. Geotechnical Engnrg., ASCE, 122(9):777{779. Popescu, R., (1995). Stochastic Variability of Soil Properties: Data Analysis, Digital Simulation, E ects on System Behavior. PhD thesis, Princeton University, Princeton, NJ. Popescu, R. and Prevost, J.H., (1993). Centrifuge validation of a numerical model for dynamic soil liquefaction. Soil Dynamics and Earthquake Engrg., 12(2):73{90. Popescu, R. and Prevost, J.H., (1995). Comparison between VELACS numerical \class A" predictions and centrifuge experimental soil test results. Soil Dynamics and Earthquake Eng., 14(2):79{92. Popescu, R., Prevost, J.H., and Deodatis, G., (1996a). In uence of spatial variability of soil properties on seismically induced soil liquefaction. In Proc. ASCE GED Spec. Conf. on Uncertainty in the Geologic Environ.: From Theory to Practice, Madison, Wisconsin. Popescu, R., Prevost, J.H., and Deodatis, G., (1996b). Stochastic input nite element simulation of the behavior of an arti cial island. In Proc. 1996 Can. Soc. Civil Engrs. Annual Conf., Edmonton, Alberta. Popescu, R., Deodatis, G., and Prevost, J.H., (1997). Simulation of non{Gaussian stochastic vector elds. Probabilistic Engnrg. Mechanics. 25
Prevost, J.H., (1977). Mathematical modeling of monotonic and cyclic undrained clay behavior. Int. J. Num. Meth. Geomech., 1(2):195{216. Prevost, J.H., (1985). A simple plasticity theory for frictional cohesionless soils. Soil Dynamics and Earthquake Engrg., 4:9{17. Prevost, J.H., (1995). DYNAFLOW: A nonlinear transient nite element analysis program. Technical Report, Dept. of Civil Engrg. and Oper. Research, Princeton University. Robertson, P.K. and Campanella, R.G., (1983). Interpretation of cone penetration tests { Part I: sand. Canadian Geotech. J., 20:718{733. Robertson, P.K. and Campanella, R.G., (1985). Liquefaction potential of sands using the CPT. J. Geotech. Engrg., ASCE, 111(3):384{403. Seed, H.B., Idriss, I.M., Makdisi, F. and Banerjee, N., (1975). Representation of irregular stress{time histories by equivalent uniform stress series in liquefaction analyses. Technical report, EERC 75{29 U.C. Berkeley, October. Shibata, T. and Teparaksa, W., (1988). Evaluation of liquefaction potential of soils using cone penetration tests. Soils and Foundations, 28(2):49{60. Soong, T.T. and Grigoriu, M., (1993). Random Vibrations of Mechanical and Structural Systems. Prentice Hall. Stark, T.D. and Olson, S.M., (1995). Liquefaction resistance using CPT and eld case histories. Journ. Geotechnical Engnrg., 121(12):856{869. Tang, W.H., (1984). Principles of probabilistic characterization of soil properties. InProbabilistic Characterization of Soil Properties: Bridge Between Theory and Practice, ASCE, Atlanta, Georgia, pages 74{89. Ural, D., (1995). The E ects of Spatial Variability of Soil Properties on Liquefaction. PhD thesis, Princeton Univ. Vanmarcke, E.H., (1989). Stochastic nite elements and experimental measurements. In Int. Conf. on Comp. Meth. Exp. Measur., Capri, Italy, pages 137{155. Springer{ Verlag.
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