Investigation of correlation structures and weak stationarity using the CPT soil behavior classification index

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Investigation of correlation structures and weak stationarity using the CPT soil behavior classification index

M. Uzielli & G. Vannucchi

Department of Civil Engineering, University of Florence, Italy K.-K. Phoon

Department of Civil Engineering, National University of Singapore

Keywords: geotechnical reliability, spatial variability, random field, cone penetration testing, soil classifica- tion

ABSTRACT: The soil behavior classification index is a more direct indicator of soil type because it maps the Robertson chart onto a one-dimensional scale. Previous studies have alluded to the hypothesis that random field parameters depend on soil type; this study addresses this issue systematically using Ic. A set of 70 physically homogeneous soil profiles were first identified from 304 cone penetration testing (CPT) soundings taken at Turkish and North American sites. Only 33 Ic profiles were found to be weakly stationary and suit- able for estimation of random field parameters (vertical scale of fluctuation and coefficient of variation). Re- sults from this rigorous exercise indicate that the random field parameters of Icare both smaller in clayey soils in contrast to corresponding values obtained in sandy soils.

1 INTRODUCTION

The importance of spatial variability analysis in geotechnical engineering is increasingly recognized as reliability-based methods are adopted for the calibra- tion of new design codes. The recent introduction of a section on soil properties (Section 3.07) in the JCSS (Joint Committee on Structural Safety) Prob- abilistic Model Code is an example of this trend.

This publication can be downloaded from http://www.jcss.ethz.ch/. Models for geotechnical variability generally identify inherent soil variability, measurement uncertainty, and transformation uncertainty as the primary sources. Inherent soil variability results primarily from natural geologic processes that form and continuously modify the soil mass in situ. Measurement uncertainty arises from equipment, procedural-operator, and random testing effects. Transformation uncertainty is introduced when laboratory or field measurements are trans- formed into design parameters using theoretical, semi-empirical or empirical models. The relative contribution of these sources to the overall uncertainty in the design parameter depends on the site conditions, degree of equipment and procedural con- trol, and quality of the correlation model. Clearly, it is overly-simplified to cite a coefficient of variation for a design parameter without reference to the specific set of circumstances (site conditions, measurement techniques, correlation models) from which it was derived.

Uzielli et al. (2005) attempted to characterize inherent soil variability using stress-normalized CPT measurements; namely, the dimensionless cone penetration resistance (qc1N) proposed by Robertson (1990), calculated using the revised algorithm for the stress exponent (Robertson 1999), and the nor- malized friction ratio (FR) as defined by Wroth (1984). Using a large database, it was found that the vertical scale of fluctuation of qc1N increases with in- creasingqc1N, while the vertical scale of fluctuation of FR decreases with increasing FR. Because qc1N in- creases and FR decreases from clayey to sandy soils, this observation appears to indicate an underlying soil type effect. The same effect was observed for the coefficient of variation of inherent variability as well.

The objective of this paper is to study this soil type effect directly using the soil behavior classifica- tion index, Ic, which maps the CPT-based soil classification zones proposed by Robertson (1990) in the logarithmic FR-qc1N plane (shown in Fig. 1) onto a one-dimensional scale. It is more general - and physically more meaningful - to link inherent soil variability to soil type, rather than specific measurements. The soil behavior classification index is calculated from:

( ) ( )

[

3.47 log _c1_N ² 1.22 log _R ²

]

⁰^.⁵

c q F

I = − + + (1)

Robertson & Wride (1998) asserted that the bounda- ries between the soil behavior zones (hereinafter

ICOSSAR 2005, G. Augusti, G.I. Schuëller, M. Ciampoli (eds)

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SBZs) may be approximated in the log(FR)-log(qc1N) plane by a set of concentric circular arcs of radius Ic, and that the same chart may be used interchangeably with the soil behavior categories defined by Ic; hence, SBZs will be referred to on the basis of the corresponding interval of Ic values.

It is well known that vertical and horizontal correlation structures in soil properties are generally ani- sotropic, with greater variability in the vertical direc- tion. Here, inherent variability of Ic is addressed only in the vertical direction; thus, the results of the present study do not apply to the horizontal direction.

2 IDENTIFICATION OF HOMOGENEOUS SOIL UNITS

The identification of homogeneous soil units is an important prerequisite for variability analyses, as the correlation structure of soil properties has been shown to depend on soil type in terms of composition and behavior. Moreover, the assumption of statistical homogeneity – necessary for the application of statistical techniques – is not likely to apply unless the investigated volume of soil is fairly uni- form in composition or behavior (Fenton 1999).

For the present study, the database of 70 homogeneous soil units (HSUs) obtained by Uzielli et al.

(2005) from 304 soundings at five regional sites (greater Oakland area, California [OAK]; Mid- America Earthquake regions [MAE]; Texas A&M University site [TXS]; Adapazari, Turkey [ADP];

Treasure Island, San Francisco Bay area [TSI]) is used. Details of the identification procedure are

given in Uzielli et al. (2004).

To provide a concise overview of the 70 HSUs, the mean values of Ic and the corresponding standard deviations are plotted on the Robertson (1990) chart as points and error bars, respectively (Fig. 1). The degree of homogeneity in the identified HSUs was assessed by observing that the mean value of the co- efficients of variation of Ic in the HSUs was 0.02, and that values never exceeded 0.10. Fig. 1 shows that the identified HSUs cover a wide range of soil types (SBZs 3 to 6). It may also be observed that the standard deviation of Ic tends to increase from SBZ3 to SBZ6, i.e. from clay to sand, indicating that CPT profiles are more variable in cohesionless soils. This has been discussed in detail by Uzielli et al. (2005).

Given the presence of HSUs from different soil behavior types in most regional sites, it is possible to study trends based on geological context and soil type.

3 RANDOM FIELD MODELING: PROCEDURE In the present study, inherent soil variability is mod- eled as a zero-mean weakly stationary random field with finite-scale correlation structure (Vanmarcke 1983). This random field is added to a deterministic trend function. Reliable estimates of the parameters which describe a random field completely in the second-moment sense (i.e. scale of fluctuation and coefficient of variation) are obtained through a rigorous, integrated procedure summarized in the flow- chart in Fig. 2 and outlined in the following sections.

Details are given in Uzielli et al. (2005).

zone Ic range Soil behavior type 1 not appl. sensitive, fine grained 2 Ic>3.60 organic soils - peats 3 2.95<Ic<3.60 clay, silty clay 4 2.60<Ic<2.95 clayey silt to silty clay 5 2.05<Ic<2.60 silty sand to sandy silt 6 1.31<Ic<2.05 clean sand, silty sand 7 Ic<1.31 gravelly sand to sand 8 not appl. very stiff sand to clayey sand 9 not appl. very stiff, fine grained

Figure 1. Mean values and standard deviations of Icprofiles in the 70 HSUs as viewed within the context of Robertson’s (1990) FR-qc1Nsoil classification chart (adapted from Robertson & Wride 1998)

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3.1 Decomposition

In the present study, Ic profiles are decomposed into a linear trend function [t(z)] and a fluctuating com- ponent [w(z)], the latter representing inherent soil variability. As discussed by Uzielli et al. (2005), trend removal by linear least-squares regression adequately meets the following key requirements:

(a) existence of a physical cause; (b) compatibility with models for the estimation of random field parameters; and (c) compatibility with the adopted stationarity assessment criterion.

3.2 Scale of fluctuation

In finite-scale models, the scale of fluctuation (δ) is a concise indicator of the spatial extent of a signifi- cantly correlated domain. Here, δ is estimated through calculation of the sample autocorrelation function (ACF) using the method of moments, viz:

( )

_�^�

�

��

�

��

�

��

�

� ⋅

=

� �

⁻

=

−

= +

d j n

i i d j

n i

j i i

j w w w

R

1 2 1

ˆτ (2)

where nd is the number of data points in a profile.

Subsequently, plausible theoretical autocorrelation models (ACM) are fitted to the ACF derived from Eq. 2, and δ is evaluated through the characteristic model parameter in the best-fit ACM as shown in Table 1. Four ACMs are considered: a) single exponential (SNX); b) cosine exponential (CSX); c) second-order Markov (SMK); and d) squared exponential (SQX).

model parameter in the best-fit ACM as shown in Table 1. Four ACMs are considered: a) single exponential (SNX); b) cosine exponential (CSX); c) second-order Markov (SMK); and d) squared exponential (SQX).

Table 1.Autocorrelation models and relations between scale of fluctuation and characteristic model parameters

Autocorrelation model Autocorrelation

model EquationEquation Scale of fluc-

tuation Scale of fluctuation

SNX R( )τ = exp

(

−λτ

)

δ=2/λ

CSX R( )τ =exp −

( )

bτ cos( )bτ δ=1/b SMK R( )τ =

(

1+dτ

) (

exp−dτ

)

δ=4/d SQX ^R( )^τ ⁼^exp

[

⁻( )^a^τ²

]

^δ⁼ ^π^/^a

To increase the reliability of the estimated δ, the ACMs are fitted only to the initial part of the sample ACF exceeding Bartlett’s limits, viz:

B nd

r =1.96. (3)

linear trend t(z)

fluctuating com pone nt w (z) de co mp osition

calculate sample ACF

calcu late Bartlett's limit rB

at least 4 AC coeff. > r_B?

fit ACM

calculate scale o f fluctu atio n (δ)

is R²≥0.9 ? discard

assess weak stationarity byMBS

yes

no

yes no

is p rofile we akly stationary ?

accept δ

yes

discard no

HSU I_c profile

MBSR

calculate coeff.

of variation (η) rando m field

parame ters δ,η

if more th an one ACM is accep ted, ch oose on e with

maximu m R²

Figure 2. Procedure used to estimate random field parameters from homogeneous soil units

(repeat for each ACM)

discard

The coefficient of determination, R², is recorded for each ACM fit. Only ACMs producing R²>0.9, with at least 4 initial autocorrelation coefficients greater thanrBto ensure the significance of the fit, were ac- cepted. If more than one ACM satisfy the above conditions, the scale of fluctuation resulting from the ACM with the maximum R² is adopted. The procedure outlined above was applied to Ic profiles for all HSUs.

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3.3 Coefficient of variation of inherent variability 4 RANDOM FIELD MODELING: RESULTS 4 RANDOM FIELD MODELING: RESULTS For weakly stationary data sets, the dimensionless

coefficient of variation of inherent variability (η) [Phoon & Kulhawy 1999] is obtained by normaliz- ing the standard deviation with respect to the median value of the linear trend function (tM) in the HSU under investigation:

Only 33 Ic profiles out of the 70 HSUs could be analyzed further following the implementation of the MBSR procedure (i.e. weakly stationary and provid- ing a reliable estimate of the scale of fluctuation).

Out of these, 3 belonged to OAK sites, 18 to MAE, 8 to ADP, and 4 to TSI, while no profiles from TXS are amenable to further analyses.

Only 33 Ic profiles out of the 70 HSUs could be analyzed further following the implementation of the MBSR procedure (i.e. weakly stationary and provid- ing a reliable estimate of the scale of fluctuation).

Out of these, 3 belonged to OAK sites, 18 to MAE, 8 to ADP, and 4 to TSI, while no profiles from TXS are amenable to further analyses.

[ ]

M nd i d i

t

n

�

w

− =

= ¹

2

1 1

η (4) served that 12 out of 1

Fig. 3 shows the best-fit ACM for each HSUs that satisfy MBSR, as well as non-stationary (NST) and HSUs not amenable to MBSR (NAPP). It is ob-

6 non-stationary HSUs fall in SBZs 3 and 4, which correspond to clays, silty clays and clayey silts. Although CPT profiles in sands are more erratic than those in clays, the latter occasion- ally show large jumps embedded in predominantly smaller fluctuations, and some of these spikes bring very large – though isolated - peaks in Bartlett statistics (Uzielli et al. 2005). Non-applicability of the MBSR procedure resulted in most cases from insuf- ficient reliability of ACM fitting: out of the 21 NAPP HSUs, 16 were not amenable to MBSR due to insufficiently high R², while 5 did not meet the range of dimensionless factors set by Phoon et al.

(2003). NAPP HSUs are distributed equally among SBZs: 5 in SBZ3; 5 in SBZ4; 5 in SBZ5 and 6 in SBZ6. Hence, the MBSR procedure seems unbiased in terms of applicability to different soil types.

Fig. 3 shows the best-fit ACM for each HSUs that satisfy MBSR, as well as non-stationary (NST) and HSUs not amenable to MBSR (NAPP). It is observed that 12 out of 16 non-stationary HSUs fall in SBZs 3 and 4, which correspond to clays, silty clays and clayey silts. Although CPT profiles in sands are more erratic than those in clays, the latter occasion- ally show large jumps embedded in predominantly smaller fluctuations, and some of these spikes bring very large – though isolated - peaks in Bartlett statistics (Uzielli et al. 2005). Non-applicability of the MBSR procedure resulted in most cases from insuf- ficient reliability of ACM fitting: out of the 21 NAPP HSUs, 16 were not amenable to MBSR due to insufficiently high R², while 5 did not meet the range of dimensionless factors set by Phoon et al.

(2003). NAPP HSUs are distributed equally among SBZs: 5 in SBZ3; 5 in SBZ4; 5 in SBZ5 and 6 in SBZ6. Hence, the MBSR procedure seems unbiased in terms of applicability to different soil types.

3.4 Assessment of weak stationarity

Weak stationarity is an important requisite for random field modeling of soil properties. A profile is weakly stationary if: a) the mean is constant over a given spatial direction; and b) the autocovariance is only a function of the separation distance between observations. The modified Bartlett (MBS) procedure proposed by Phoon et al. (2003), which allows to avoid biased assessments in presence of correlated data, is employed here.

The integrated procedure proposed by Uzielli (2004), including: a) estimation of δ with restrictive conditions on ACM fitting; and b) weak stationarity assessment by MBS, is hereinafter referred to as MBSR. MBSR has been shown (Uzielli et al. 2004) to be more discriminating than other traditional clas- sical tests, as it: a) incorporates the correlation structure in the underlying data; and b) includes key as- sumptions in geostatistical analysis (stationarity, choice of trend function, and autocorrelation model).

conditions on ACM fitting; and b) weak stationarity assessment by MBS, is hereinafter referred to as MBSR. MBSR has been shown (Uzielli et al. 2004) to be more discriminating than other traditional clas- sical tests, as it: a) incorporates the correlation structure in the underlying data; and b) includes key as- sumptions in geostatistical analysis (stationarity, choice of trend function, and autocorrelation model).

A systematic analysis of the descriptive parameters of a random field in the second-moment sense (i.e. scale of fluctuation and coefficient of variation) is undertaken for 33 Ic profiles that satisfy MBSR.

First, it is verified that the statistics are not biased by the number of CPT measurement points within each HSU, the CPT measurement interval, and the best–

fit ACM. Second, the statistics computed from Ic

profiles are analyzed. Third, the effect of soil type is interpreted. Finally, if sufficient data is available, site-specific effects not explainable by mean Ic value and soil type are highlighted.

A systematic analysis of the descriptive parameters of a random field in the second-moment sense (i.e. scale of fluctuation and coefficient of variation) is undertaken for 33 Ic profiles that satisfy MBSR.

First, it is verified that the statistics are not biased by the number of CPT measurement points within each HSU, the CPT measurement interval, and the best–

fit ACM. Second, the statistics computed from Ic

profiles are analyzed. Third, the effect of soil type is interpreted. Finally, if sufficient data is available, site-specific effects not explainable by mean Ic value and soil type are highlighted.

Figure 3. Results of MBSR stationarity assessment for Ic profiles from the 70 identified HSUs

4.1 Checking for bias 4.1 Checking for bias

Possible bias produced by sample size (i.e. the number of data points in an HSU Ic profile) and CPT measurement interval is investigated by plotting the vertical scale of fluctuation against sample size (Fig.

4), with data points categorized by measurement interval, ranging from 12 to 50 mm in the source database. No bias is discernible from the lack of visible trend.

Possible bias produced by sample size (i.e. the number of data points in an HSU Ic profile) and CPT measurement interval is investigated by plotting the vertical scale of fluctuation against sample size (Fig.

4), with data points categorized by measurement interval, ranging from 12 to 50 mm in the source database. No bias is discernible from the lack of visible trend.

4.2 Vertical scale of fluctuation 4.2 Vertical scale of fluctuation

The vertical scales of fluctuation, [δ(Ic)], are plotted against mean Ic value according to the best-fit ACM (Fig. 5a). Estimates computed by one ACM do not rank consistently above or below the others. Values of [δ(Ic)] range between 0.13 and 1.18 m, with an The vertical scales of fluctuation, [δ(Ic)], are plotted against mean Ic value according to the best-fit ACM (Fig. 5a). Estimates computed by one ACM do not rank consistently above or below the others. Values of [δ(Ic)] range between 0.13 and 1.18 m, with an

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overall mean value of 0.52 m, and a dispersion around the mean given by a COV of 0.63. Fig. 5b shows the same plot with data points categorized by regional site. It may be seen in Fig. 5b that the lower and upper bounds of [δ(Ic)] decrease with in- creasing mean Ic when all regional sites are considered together. The same trend can also be discerned within the OAK, ADP and MAE regional sites. A possible physical basis for this observation is that Ic

partly depends upon normalized cone tip resistance, qc1N, which is influenced by a volume of soil around the cone tip that is larger than the sampling interval.

Hence, a few continuous values of Ic are affected by almost the same volume of soil as the cone pene- trates. The extent of the influence zone has been observed (e.g. Teh & Houlsby 1991) to increase with increasing soil stiffness; hence, it is tendentially larger in sand. These trends are more distinctive than those shown by Uzielli et al. (2005) (Fig. 8 and 9 in their paper) where qc1N and FR are considered sepa- rately. In addition, the hypothesis that random field parameters depend on soil type is established more firmly. Previous studies have alluded (Kay et al.

1991; Phoon & Kulhawy 1999) to this effect, but this study addresses this issue systematically using Ic.

Figure 4. Vertical scale of fluctuation [δ(Ic)] versus number of data points in each HSUIc profile for various measurement intervals.

The range of [δ(Ic)] (0.13-0.26 m) for the cluster of 4 TSI data points classified in SBZ3 - probably belonging to the Young Bay Mud layer - lies entirely below those of the other regional sites (Fig. 5b).

Hence, a site-specific effect appears to be present at TSI.

Figure 5. Scale of fluctuation vs. mean Ic in HSU for MBSR-valid HSUs, categorized by: (a) best-fit ACM; and (b) regional site

Figure 6. Coefficient of variation of inherent variability vs. mean Ic in HSU for MBSR-valid HSUs, categorized by: (a) best-fit ACM; and (b) regional site

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4.3 Coefficient of variation of inherent variability The coefficients of variation [η(Ic)] computed from each Ic profile are plotted against the respective mean Ic values according to the best-fit ACM (Fig.

6a) and by regional site (Fig. 6b).

In general, η(Ic) increases from SBZ3 to SBZ6 for all ACMs (Fig. 6a) and all sites (Fig. 6b). The overall mean of η(Ic) for SBZ3 (0.02) is lower than the corresponding mean for SBZ4 and SBZ6 (0.03 and 0.06, respectively) and the ranges between SBZ3 and the other SBZs are distinct, again con- firming that Ic profiles in cohesive soils are signifi- cantly less erratic. This trend is visible in Fig. 6a and 6b (decreasing because Ic increases from SBZ3 to SBZ6). Overall, The trends for [η(Ic)] are similar to those noted for δ(Ic).

5 CONCLUSIONS

This study addresses the spatial variability of the CPT-based soil behavior classification index (Ic), a more direct indicator of soil type, using a finite-scale weakly stationary random field model.

A set of 70 physically homogeneous CPT profiles were first identified from 304 soundings taken at Turkish and North American sites. These sites were grouped into five regional sites.

At the end of a rigorous selection procedure, only 33Ic profiles resulted adequate from both physical and statistical considerations to be analyzed further.

The majority of the acceptable profiles were found in sandy soils and cohesive-behavior fine-grained soils, with fewer HSUs located in intermediate soils.

The vertical scale of fluctuation and the coefficient of variation of inherent variability of Ic estimated in this paper support the existence of a marked soil type effect on the spatial correlation and variability of the mechanical response to cone penetration. The observations are compatible with the results of past research, which indicate that the extent of the failure zone increases with increasing shear strength and stiffness; hence, the spatial correlation is greater in sand.

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