ELSEVIER 0267-7261(94)00038-7

Soil Dynamics and Earthquake Engineering 14 (1995) 79-92 © 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0267-7261/95/$09.50

Comparison between VELACS numerical 'class A' predictions and centrifuge

,experimental soil test results

Radu Popescu & Jean H. Prevost Department of Civil Engineering and Operations Research, Princeton, NJ 08544, USA

(Received for publication 22 August 1994)

The numerical 'class A' predictions performed within the framework of the VELACS Project are compared to the experimental results recorded in the centrifuge experiments. The comparisons are made in terms of: (1) the root mean square error of the predictions with respect to the mean of the experimental results; and (2) the size of a confidence interval centered at the predicted value which contains the estimated true value of the experimental results with a 75% probability. An assessment of the capability of various groups of constitutive soil models to predict excess pore pressures induced by dynamic loading is also presented.

Key words: 'class A' predictions, centrifuge experiments, soil liquefaction, confidence intervals.

1 INTRODUCTION

Numerous constitutive laws have been proposed for the numerical analysis of dynamic induced liquefaction of soil materials. In order ~:o be useful to the engineering practice, a numerical technique requires verification and validation by comparison of its predictions with observed full-scale in-situ field performance. However, much data are unlikely to be obtained soon, because of the scale of the structures involved, the cost of testing and the low probability of having a particular instrumented geotechnic, al system subjected to design load. The situation is a common problem in geo- technical engineering and is exacerbated in earthquake engineering. Even if a large number of sites were instrumented to monitor liquefaction, there would be only a few cases of liquefaction recorded during actual earthquakes. Consequently, some form of model study is desirable to enable alternative analysis procedures to be checked and validated. In this context, the VELACS (VErification of Liquefaction Analysis by Centrifuge Studies) project offers a good opportunity to verify the accuracy of various analytical procedures. This NSF sponsored study on the effects of earthquake-like loading on a variety of soil models was aimed at better understanding the mechanisms of soil liquefaction and at acquiring data for the verification of various analysis

79

procedures. The numerical predictions were intended to be 'class A' predictions, and thus were made before the relevant experiments were performed. The verification and validation of the various analysis procedures were to be carried out by comparing their predictions with the measurements recorded in the centrifuge experiments (in terms of excess pore water pressure, acceleration and displacement time histories). However, it should be pointed out that to this date no firm experimental verification and validation of centrifuge test results with the corresponding prototype situations have been made.

A centrifuge 1-4 is used to simulate gravity-induced stresses in soil deposits at a reduced geometrical scale through centrifugal loading. Conceptually, the technique consists of increasing the confining environ- ment in the model soil, so that the confining stress is identical in both model and prototype at homologous points. The technique allows soil liquefaction tests to be performed at a conveniently reduced scale and provides data applicable to full-scale problems. A program of dynamic centrifuge model tests consisting of nine geotechnical models constructed from or founded on liquefiable soil deposits have been performed within the framework of the VELACS Project, 5 as listed in Table 1.

The 'class A' predictions were based on the results of conventional laboratory soil tests 6 performed on the soil materials used in the centrifuge models, and on the

80 R. Popescu , J. H . P r e v o s t

Table 1. Centrifuge experiments for VELACS Project

Centrifuge (University) Experiments

Primary Duplicating

Refered as

Rensselaer Polytechnic Inst. California Inst. of Technology University of California, Davis Colorado University at Boulder Cambridge University (UK) Princeton University

Models 1 and 2 Model 3 Models 4a, 4b, 6 Model 7 Model 11 Model 12

Models 3, 4a, 4b, 7, 12 RPI Models 2 and 4a Caltech Models 1, 2, 3 (NA), 7, 12 UC Davis Models 1 and 4b CU Boulder

Cambridge Princeton

(NA) __ results not available for this study.

anticipated base input motion (viz. 'target motion'). Overview presentations of 'class A' predictions and comparisons with the experimental results have already been made within the framework of the VELACS Project. 7 The present study provides a unitary and concise presentation of the prediction performance relative to the experimental measurements, in terms of the excess pore water pressures induced by dynamic loading. The results of the comparison study are dependent on the assumption that the systematic errors or bias in centrifuge experimental results can be neglected. As discussed in a previous study, 8 this assumption is imposed by the current lack of knowledge on the bias in centrifugal experimental results, as compared to corresponding full-scale prototype behaviour.

A summary of the reliability of the experimental results obtained in the centrifuge experiments performed within the framework of the VELACS Project 8 is first presented in Section 2. The methodology for comparing predictions to experimental results, and the introduction of two comparison indices are presented in Section 3. In Section 4, the comparison results for individual predictors are presented. For each centrifuge model and pore pressure transducer location, the predicted time histories are compared with the mean of the experiment recordings in terms of a root mean square error. The variability in the experimental recordings and, hence the reliability of the centrifuge experiment results is accounted for by constructing a confidence interval for each prediction, with respect to the estimated 'true value' of the mean of the experiments. A comparative assessment of the performance of various constitutive models in predicting dynamically induced excess pore pressure is presented in Section 5.

2 CENTRIFUGE EXPERIMENTAL TEST RESULTS RELIABILITY

The centrifuge modeling technique offers the ability to create relatively realistic full-scale stress states, together with uniform and measurable soil properties and, hence, it can provide useful data for verification of analytical procedures. However, when compared to full-scale measurements, the centrifuge experimental results can

be affected by a number of errors, due to: boundary conditions, details of sample preparation, recording equipment functionality and placement, failure to reproduce the desired input motion, etc. For seven of the nine centrifuge models of the VELACS Project, some of the experimental errors were minimized by duplicating the centrifuge experiments, as shown in Table 1. The duplicating experiments were intended to be performed under conditions identical to the primary experiment, using the same type of equipment and following the same specifications for sample prepara- tion. The primary and duplicating tests for each model were carried out at different laboratory facilities, to minimize the bias in results induced by human factors.

A previous study 8 was conducted to assess the reliability of the pore water pressure recordings obtained in the centrifuge experiments which had been duplicated. The study sought to address the reliability of the mean of a series of test results as an estimate of the (unknown) true value, regarded as the true mean as would result from very many nominally identical experiments. Under the reserve of some simplifying assumptions (viz. random sampling, gaussianity of the test outcomes) and of the low number of experiments (typically 2 or 3) imposed by the circumstances, some of the conclusions are summarized in Fig. 1 (b) in terms of the likelihood 7 that the true value of the test outcome lies within a +10% interval of the mean of the experiment measurements. The reliability analysis was performed in terms of the integral of the recorded excess pore pressure time histories over a significant time interval (Fig. l(a)). The results of the study, evaluated for all the transducer locations where more than one pore pressure recording was available, are shown with black dots. For some locations, where one of the records was found to be clearly different from the others, the experimental result reliability was improved by not accounting for that record. The results are shown in Fig. l(b) for such cases with open dots: using all the records, and with black dots: for improved reliability. A subjective acceptability limit was set in Ref. 8 by accounting for the degree of complexity of the centrifuge experiments and by comparing the reliability of the centrifuge results with those obtained in conventional laboratory soil tests. 6 According to that limit, four out

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H - number of experimemid results avdabb 5x each tratooduoef kx:/iofl

Fig. 1. Results of a reliability analysis of the VELACS Project centrifuge experiments, in te:nns of the recorded excess pore pressures: (a) quantification of the recorded time histories; (b) analysis results: centrifuge medels 2, 4a, 4b and 12 are found to provide results which are most appropriate for comparisons

with 'class A' predictions (after Ref. 8).

of the seven centrifuge experiments which had been duplicated provide reliable excess pore water pressure results, which can be used for meaningful comparisons with 'class A' predictions.

The results of similar studies performed to assess the reliability of the recorded displacements and accelera- tions, are shown in Figs 2 and 3, respectively. The same methodology as presented in Ref. 8 was used. The

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; 100% L V D T i S S S 4 S ~ 1 2 S 4 5 ~ 1 2 S 4 6 6 1 V 3 1V3 ~ V = 1

M M i I M V V v VV 141414 VV 141414 Model #1 12 83 #4 - #4b #7 t1:1

N - number of ~perirnerdad re. its Mtldddo for eech I I l ~ t J c o r Ioc~on H - horizontid displacement; V - vertk:al displscement

Fig. 2. Realibility assessment of the VELACS Project centrifuge experimental measurements, in terms of the

displacements recorded at the end of shaking.

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l • • • • 0 •

• • • O O •

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2 0 %

40%

6O%

8O%

1 ( X ) %

Fig. 3. Realibility assessment of the VELACS Project centrifuge experimental measurements, in terms of the

recorded acceleration magnitude.

displacement results are compared in terms of the values recorded at the end of shaking. The recorded accelera- tion time histories are compared by using a measure of their amplitudes, computed as the root mean square of the acceleration time histories:

o = (at, - ct) 2 (1) i = l

where at, is the value recorded at time ti; tl = 0 and tn is the time at the end of shaking; n is the number of recordings between tz and t,; and ~ is the average value (approximately zero).

The results presented in Figs 2 and 3 show that, due to the large variability of the recorded values, the displacement and acceleration records have consider- ably lower reliability than the recorded excess pore pressures (see Fig. 1). Consequently, in the following the comparisons 'class A' predictions vs centrifuge experi- ment results are only performed in terms of excess pore pressures.

R e m a r k s

(1) The reliability studies for recorded excess pore pressures and accelerations are both performed in terms of quantities obtained by integrating the recorded values over a certain time interval. On the other hand, the reliability of recorded displacements is evaluated in terms of a single value, which inherently is more variable than the integral values used for pore pressures and accelerations. Therefore, a lower acceptability limit might be appropriate when assessing the reliability of recorded displacements, than used for pore pressures and accelerations. Such a limit might be taken as: 7 = 33%, corresponding to the reliability of a series of experimental results which are uniformly distributed in a

82 R. Popescu, J. H. Prevost

+100% interval with respect to the true value (see Ref. 8 for more details), and is shown in Fig. 2 by a dashed line. Even with respect to this lower limit, most of the centrifuge experiments provide relatively low reliable results for displacements.

(2) As pointed out in Ref. 8, the centrifuge experiment reliability results presented in Figs 1, 2 and 3 are strongly dependent on the number of experiments made available for this study. A larger number of centrifuge experiments for each model might have resulted in higher reliability of the results, as in the case of centrifuge model # 12.

3 M E T H O D F O R COMPARING PREDICTIONS VS EXPERIMENTAL RESULTS

A time domain representation, based on the time histories of the predicted and recorded excess pore pressures, is used for the comparison in the following.

3.1 Data smoothing

In general, both recorded and predicted excess pore pressure time histories display some high frequency oscillations, with a frequency related to the dominant frequency of the base input motion. To compute more realistic values for the comparison indices presented in the following, the records are first smoothed by filtering out the high frequency component as shown in Fig. 4. The filtering method uses a nonrecursive time domain digital filter and Kaiser windowing. 9 The parameters of the low pass filter (high cut-off frequency vrt and transition band 6) are selected by minimizing the errors induced by the filtering procedure (Fig. 4(c)):

[ r~.a, - (t)] [p°rig (t) pf i l t e red d t 4Tinta e (2) ef

]Tn.,, p°rig'(t) dt 4 Tinitial

while acquiring a satisfactory smoothing, i.e. high cut- off frequency less than the frequency corresponding to the second significant peak in the Fourier transform of the original time history (Fig. 4(b)). The resulting maximum error induced by the filtering procedure was less than 1% for all cases.

3.2 Time shifting

Some of the base input accelerograms used in the centrifuge experiments do not have the same time origin as the target motion used for the 'class A' predictions. Such an example is presented in Fig. 5(a). By shifting one record relative to the other and by computing the resulting correlation coefficient between the two time series (see e.g. Ref. 10), one can obtain the correlation

w' t .

O

d~ ~ o

i

a. Excess pore pressure recorded at Cambridge Model 11 - tnmsdueer PPT3

! ~. , , : ,~ . fdtered

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time (see)

b. Normalised Fourier Transform

filtered reemt . . . . . . . . o~,inal record

I I I I I I 0 0.5 l 1.5 2 2.5 3

frequency (Hz)

c. Filter parameter evaluation Selected: v n = 1.0 Hz, 8 = 0.02 Hz

**q~r

O

vH--0.sm - . . . . . . . v~ l .0I -Iz

. . . . v~l.2X-Xz / . / - - - " . . . .

I I I I 0 0 .0 l 0 .02 0.03 0 .04

wansition band 8(Hz)

Fig. 4. Data smoothing: original and filtered excess pore pressure time histories: (a) in the time domain; (b) in the

frequency domain; (c) selection of the filter parameters.

coefficient between the two time histories as a function of the time shift (Fig. 5(b)). The difference in the time origin can then be conveniently evaluated as the time shift corresponding to the maximum value of the correlation coefficient. Consequently, to obtain mean- ingful comparisons with 'class A' predictions, some experimental recordings are shifted along the time axis by the corresponding amount.

3.3 Root mean square error

In order to assess the closeness of two time histories (e.g. recorded vs predicted), one may look at the extreme values only, or one may plot and compare the whole time histories. The first method provides very scarce and incomplete information in the case of dynamic events. The second one becomes tedious when a large amount

V E L A CS numerical 'class A ' predictions 83

it. Base input horizontal acceleration (Model 3) cenlrifuge experiment - Caltech

O

tr~

~o

I I I I I 0 5 10 15 20 25 30

° a l _JIt dl . d , ~ J , ~ t u ~ . x . . . .

J°t o " r'rl l' ,rr , . " I I I I I

0 5 10 15 20 25 time (sec)

b. Correlation experiment vs. target motion - -1 (Model 3 - primary experiment. Caltech)

"T

. _ AAAAAAA ,

v v V V V V V v

I 30

I I I I I -4 -3 -2 -1 0

time ~ t (see)

Fig. 5. Time shifting: (a) base input acceleration used in one of the centrifuge experiments and target motion for the same experiment; (b) correlation coefficient between the base input motion and the target motion, as a function of the relative

time shift.

of information is availab]e.ll In the following the pore pressure time histories ar,~ compared in terms of a root mean square error computed as follows: given the two pore pressure time h:istories pp(t) and pr(t), over a certain time interval Tinitia 1 < t < Tnnal, as shown in Fig. 6, the overall root mean square error of pp (predicted)

with respect to Pr (recorded) is computed as:

= , oTi~i~at [pp(t) -- Pr(t)] 2 d t} 1/2

e r m s Gvl0x/Tfinal- Tinitial (3)

where the denominator is a normalizing factor, expressed in terms of the initial effective vertical stress

/

tr,0 and the duration (Tnnal- Zinitial). For each transducer, the errors are evaluated by using

the average of the recorded values, i.e. Pr in eqn (3) is computed as:

N~xp 1 ~ recorded, t"

pr(t) = Nexp ~ Pi ~ ) (4) i=1

where Nexp is the number of centrifuge experiments recorded reported for that particular model and Pi (t) are the

values recorded in experiment 'i ' . The root mean square error as defined by eqn (3) can also be used to assess the quality of each centrifuge experiment, by comparing each experiment with the mean of the experiments. Thus, whenever more than one experimental result is available for a specific transducer, a range of experi- mental error is computed with respect to the mean of the recordings and the prediction errors are reported to that range. For the models for which only one centrifuge experiment is reported (models 6 and 11) the recorded values are used as the basis for the comparison.

Remark The root mean square error provides information about how close the predicted values are to the average of the experimental recordings, during the time interval Tinitia I ~ t < Tfinal. By computing the root mean square error of each individual experiment, one can also assess the scattering of the measurements. However, a comparison based on the root mean square error takes for granted the validity of the average of the experi- mental recordings and does not account for the influence that the reliability of the experimental results (function of the scattering of the experimental results and the number of tests 8) should exercise over the quantitative reliability assessment of the predictions. That issue is addressed by another comparison index proposed hereafter.

! t i m e

Tinttia I t Tflnal

Fig. 6. Computation of the root mean square error index, ~ s .

3.4 Size of confidence interval centered at the prediction, containing the 'true value'

A second comparison index is proposed to account for the influence that the quality of the experimental results has on the reliability to be assessed to the 'class A' predictions. The index is based on the measure of the half length 'b' of the confidence interval centered at the predicted value 'p' which contains the estimated true mean value 'mR' of the experiment with the probability /~. In other words, ~ is the likelihood that the true value

84 R. Popescu, J. 1t. Prevost

mR lies within the confidence interval of length 2b, centered at the predicted value p:

/3 = P i p - b < mR < p + b] (5)

This 'b' index can only be evaluated when more than one experimental result is available for comparisons with 'class A' predictions. The methodology is briefly described in the following.

Before a series of N experiments are performed, their results R1,R2, . . . ,R u are random variables with expected values E[Ri] and standard deviations aR,, i = 1 ,2 , . . . , N. The following assumptions are made:

(1) the experiments are performed under identical conditions;

(2) the experiment outcomes are mutually independ- ent (random sampling), i.e, bias can be neglected;

(3) the results are normally distributed.

The validity of these assumptions is discussed in Ref. 8 for laboratory soil tests, and for geotechnical centrifuge experiments in particular. The 'random sampling' assumption is imposed by the lack of knowledge in quantifying the amount of correlation between experi- mental results and the bias in results given by centrifugal testing as compared to full-scale situations.

Under the above assumptions, the results R; are identically distributed random variables, i.e. E[Ri] = E[R], O'Ri = f i R , for all i = 1, . . . , N (assumption 1), the expected outcome equals the true value, E[R] = mR (assumption 2), and the average outcome /~= ( l / N ) ~ x Ri is a normally distributed random vari- able, even for small N (assumption 3). The mean and standard deviation o f /~ are: 12 E [/~] = E [R] = mR and crk/x/'N, respectively. It follows that the random variable: ( ( R - mle)x/-N)/a R is standard normal. Since the real standard deviation of the test results is not known, it is replaced by its estimate - - the sample standard deviation #R, evaluated after the experiments have been performed. The new random variable:

T = (/~ - mR)x/-N (6) #R

can be shown to follow a Student's t distribution with N - 1 degrees of freedom, x2 Denoting by Tu-i the respective cumulative distribution, and replacing mR in eqn (5) using eqn (6), one gets:

/3 : P[x I < T ( X2] = TN_I(X2) - - TN_I(X1) (7)

with:

[/~ - p + ( - 1)kb]v/-N xk = , k = 1,2 (8)

#R

After the experiments have been performed and the results r~ - - realizations of R; - - are known at every time instant t, the random variable /~ in eqn (8) is replaced by its realization ~ = ( l / N ) ~ 1 ri, and, given

a. i f " ( ~ ) - - ' - •

- ~ l u e ? experiments b. ~f~i(mR) [ ~=sam le eanof | ~experimen~

interval Fig. 7. Evaluation of the size b of/3 confidence interval: (a) probability density function of the average outcome/~ of the experiments; (b) location of the true value m R with respect to the sample mean ~ of the experimental outcomes, under the

assumption of negligible bias or systematic errors.

a value for /3, eqn (7) can be numerically solved for b -- b(/3, t).

The meaning of the size b of the/3 confidence interval is illustrated in Fig. 7. Before the experiments are performed, the random variable k has the probability distribution function fR, shown in Fig. 7(a). If the standard deviation of the test results aR is known, f,~ is derived from the normal distribution, and it comes from the Student's t distribution if cr R is approximated by the sample standard deviation ~R. After the experiments, is replaced by its realization ~, but the true value m R remains unknown. Under the 'random sampling' assumption, the uncertainty in locating the true value mR relative to ~ can be expressed by the same probability distribution function, as shown in Fig. 7(b). Finally, given the predicted value p, the/3 confidence interval has a size 2b, corresponding to an area which equals the likelihood/3 (hatched area in Fig. 7(b)).

An overall measure of how close the predicted time history is to the experimental measurements is obtained by averaging the value b(/3, t), computed at every time instant t, over the analysis time interval Tinitia I < t < Tfinal. Finally, to get a nondimensional index, the average value is normalized with respect to

I , the initial effective vertical stress a~0.

[r.~, 6(/3, t) dt bnaV(/3) = ,r,~,~ (9)

a'0(r~.al- Ti.itial) The quantity b~V(/3), evaluated for a certain/3 value, will be referred to as 'size of/3 confidence interval'.

In order to illustrate how the size b,~V(/3) of the /3 confidence interval index accounts for the reliability of the experimental results, as compared to the root mean square error index erms, defined previously, an example is presented in Fig. 8, for/3 = 75%. Let ri, i = 1,2, 3, be the experimental recordings at a certain time instant,

VELACS numerical 'class A' predictions 85

p,.

d

o

,.~ r3 . r or .,g 0 p r ed i c t i on

I I 0 0.2 0.4 01.6 01.8 l

8 - measure o f scat ter in exper iments

Fig. 8. Influence of the experhnental result scatter on the size of 75% confidence interval.

assume that they are such that r2 = (rl + r3)/2 = f (the sample mean) and denote by 6 = ( r 3 - r t ) / f = a measure of the scatter in the experimental results.

I Also, for simplicity, assume that av0 = r. If one only looks at one time instant, then the root mean square error is erms = [P - ~1/~, and the size of 75% confidence interval is b~V(75%) = b(75%)/f shown in Fig. 8 as a function of t5 and erms.

Under the assumptions stated at the beginning of this section, one can see from Fig. 8 that, even for constant root mean square error e,ans, the confidence interval of the predicted value, baY(75%), increases with /5, a measure of the scatter in the experimental data (i.e. with decreasing reliability of the experimental results). The 'perfect prediction' in terms of the root mean square error (E~m~ = 0 - - continuous line in Fig. 8) is shown to be affected by uncertainty when the experimental results are scattered (b(75%)/~ > 0 for i5 > 0).

3.5 Remark

A drawback of the two comparison indices presented in this section, root mean square error e~ms and size b~V(/3) of/3 confidence interval, is that they only offer a measure of the overall closeness of the predicted time histories with the experimental recordings. No information is provided as to the sign of the errors (i.e. lower or higher predicted than recorded values), nor as to the time instants at which the ,differences in predictions vs recordings are concentretted. However, the fact that these two indices convert the information contained in a whole time history into single numbers makes them convenient for comparison studies involving a large amount of data.

4 ASSESSMENT OF 'CLASS A' PREDICTION PERFORMANCE

The comparison indices previously presented - - normalized root mean square error, erms and normal- ized size bav of 75% confidence interval - - are used for a concise presentation of the 'class A' predictions performance. According to the reliability study reported in Section 2, the relatively low reliability of the recorded experimental displacements and accelera- tions makes the comparisons with numerical predictions pointless and is not addressed here. Consequently, the present comparison study is restricted to predicted and recorded excess pore pressures.

As presented in Table 2, about 75% of the 'class A' predictions submitted for the VELACS Project were available for this study. Since no previous experience exists in terms of the indices proposed in Section 3, the comparison study presented hereafter cannot provide information on the absolute quality of the 'class A' predictions. However, the large amount of predictions involved (56 'class A' predictions, for nine centrifuge models with a total of 41 pore pressure transducers) allows a meaningful comparative evaluation of these predictions.

4.1 Comparison indices

Accounting for the reliability of the experimental results (Fig. 1 and Ref. 8), the comparison indices computed at each transducer location for each experiment are presented as follows:

- - in Fig. 9: for centrifuge models 2, 4a, 4b and 12, deemed as providing reliable results in terms of recorded excess pore pressures and, therefore, allowing meaningful comparisons with numerical predictions;

- - in Fig. 10: for centrifuge models 1, 3 and 7, with apparently low reliable results, according to the study in Ref. 8;

- - i n Fig. 11: for centrifuge models 6 and 11, which were not duplicated and, hence, whose reliability cannot be assessed.

Therefore, it must be emphasized that the results presented in Figs 10 and 11 should be carefully evaluated and by no means used for arriving at definite conclusions.

With reference to Figs 9, 10 and 11, some further explanations and details are provided hereafter:

- - The comparisons between predicted and measured excess pore pressure time histories are presented in terms of (1) the root mean square error index normalized with respect to the initial effective vertical stress at each particular transducer location and computed over a time interval 'T', which is specified in each figure, and (2) the size of the 75%

86 R. Popescu, J. H. Prevost

Table 2. Class 'A' predictions for V E L A C S Project 5

First predictor Referred as Predicted excess pore pressures for model # (in this study)

1 2 3 4a 4b 6 7 11 12

Constitutive model, as in t4

Bardet - - Univ. of Southern California Been - - Golder Associates (UK) Chan - - Glasgow University (UK) Hamada - - Tokai University (Japan) Iai - - Port and Harbour Research

Institute (Japan) Ishihara - - University of Tokyo

(Japan) Kimura - - Tokyo Inst. of Techn.

(Japan) Lacy - - Tufts University Li - - University of California, Davis Manzari - - Univ. of California, Davis Muraleetharan - - The Earth

Technology Corporation Prevost - - Princeton University Rollins - - Brigham Young

University Roth - - Dames and Moore Shiomi - - Takenaka Corp. (Japan) Siddharthan - - University of Nevada,

Reno Sture - - Colorado

University at Boulder Towhata - - University of Tokyo

(Japan) Yogachandran - - Leighton &

Associates Zienkiewicz - - University

College of Swansea (UK)

USC A

Glasgow Univ. A

NA

U. Tokyo A (Ishihara)

NA

Tufts Univ. A UC Davis A

NA ETC A

Princeton Univ. A Brigham A

Young U. Dames & Moore NA Takenaka Corp. A U. Nevada Reno A

A NA bounding surface NA classical plasticity A A A A A A generalized pl.

NA NA direct (empirical) NA NA NA multi-mechanism

bounding surface

NA NA NA multi-mechanism

A A A A nested surfaces bounding surface bounding surface

A bounding surface

A A A A A A A A nested surfaces NA A total stresses

NA A direct (emp.) A A A classical plasticity A A A direct (empirical)

CU Boulder NA A

U. Tokyo (Towhata) Leighton & Assoc. A A A A

U. Coll. Swansea A A A A A A A A

NA A classical plasticity

A direct (empirical) A

bounding surface

generalized plasticity

A: predicted excess pore pressures - - available for this study; NA: predicted excess pore pressures - - not available for this study.

confidence interval index, normalized with respect to the initial effective vertical stress and averaged over the same time interval as the first index. The second index is only used for the centrifuge models for which more than one experiment was reported.

- - The 'significant time interval ' , T in Figs 9-11 , was selected to include both pore pressure build-up and dissipation phases (see Fig. 1).

- - I n general different initial effective vertical stress values were reported by different experimenters for the same transducer location. 5 Further, not all predictors reported their computed initial stress values. Therefore, a single value was selected for each location, as computed in the 'class A ' pre- dictions submitted by Princeton University and reported in Refs. 11 and 13, for the results presented in Figs 9-11.

- - A specific marker is assigned to each predictor and used th rough all the centrifuge models. The same type o f markers (e.g. stars, circles, etc.) is assigned to predictors using similar constitutive models, as classified in Ref. 14 and listed in Table 2.

- - F o r the centrifuge models with more than one experiment reported, the range o f roo t mean square

errors and 75% confidence intervals o f the experi- mental records themselves are shown with shaded areas. It should be obvious that the ranges o f the compar ison indices computed for the experimental results should, in general, be smaller (i.e. better) than the ones computed for the predictions, since they are computed with respect to the recordings themselves.

- - As shown in Fig. 1 and explained in Ref. 8, at the transducer locations where one o f the experimental recordings is clearly different f rom the others, that record was no t used for comput ing the indices shown in Figs 9 and 10. At these locations, an addit ional lighter shaded area is shown for that part icular experimental record in Figs 9 and 10.

- - In centrifuge models 4a and 4b, the pore pressure t ransducer PPT A, located in the silt layer, is placed at a locat ion with too low a confining stress. Further, in some centrifuge experiments, the t ransducer was placed deeper than its target location. These led to very scattered and uncertain experimental results 15 and, consequently, that location (PPT A) is not considered in the present compar ison study, shown in Fig. 9(b) and (c).

- - Only two experiments have been made available for

V E L A C S numerical 'class A ' predictions 87

e. model #2

;i;ii;i;i;i;i+;~ )i))+i)+++++)+i)))~ ))++)i)+)++i)+)i+~

++i+))+)++++i+:++~i

v,~_m LVOT1

~::::::iii!iiiiii~i:~i!!ii~i ................ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

~+++++++++++++NN++++++++~++++++++++++++++++++++++++++++++++ +~+iiiiiE5+iiE5~i+++~!i+i~E+E+E)E~i+++~+++i~i+++E+E+++EE~+~++!~+i+i+iiE~++~+i5~iii~E+iE~i+~:!++~!i5Eii55iEii++5:~+~+::+~+E+iiiEii

~iii))~ii~))?iiiii)iiiii~?~iiiiiiiiiiiii)ii~i)?~i~i)?i)i)iiiiiiii)i)ii~i?~)iii?i)i)i!)ii)ii~?~)~iii!)iiiiiii)~)ii

- - " 22.86 m

• LVDTI

LYDT4

I LVDTS

LVDTS

I PPT I N°rmalized root.mean-square error ~ (T = 2 5 s 10.0 ~+ or2 014 016 0.8 1.0

I ~i~ii ~ N ( ~ ' E~ ii I I I

m " ~:++++I+++++++++++++++++++++++++++P++++,+++++++++++++:++)+++++++++++++++++I ~ ' m _ l::.+i:,+.+, ~ , • x , , ,

• ':::::::~:.': :~:::1 ~ ~ • ~ t n

J Normalized size bn B of 75% confidence interval

r ::++ t~ ° ' ' ' . + ~ + , , , 7 [~+)++++11++~:+)+1::++++:+ :::u;;.r:: ~:+~:;::::;+:+:+:~:::::::::::::::::::::::: I I

li++i~+4+::+:+::+::+ i.+.++.+.l++.+L......+)ii+.)i++.+i+++++++!++ ' '

I++++t p , , * ' , , 8 :~.~.,..'~<~ I l l A I i ::::::::::::::::::::::::::: , i

d. model #12 ~W,~l~_~cce

~i+iiiiiiiiMiiiiii!+iiiiiiiiiiiiiiiiiii!iiiiii] ~ ~i~)!~ii~)i~iiii!iiiiiMi~ii!~~~iiii~i~i~i~ii~!~!~!~ii~ii~i~i~iM~iiii~i~i~i~i~i~iiiiii~ ~, ,~ ~ . : °,'-. ~ii~i~i~i!~!!ii!i!!i!M!)!!iiil!Mi)!iiiiiilMi!i!i!)ii!i]~

12,50 ~ _ I • I ~ 12.50

PPT Normalized rcol-mean-square error Ermm (T-15S) 0.0 Or2 014 016 Ore 1 . 0

1 +++1+i+++!+ = , ¢ . , , x ; : : :

2 i + i ! ~ . *~ + l * 1 3 ~i~i) ~ ' ~ ' X l I I

l Normalized size bn my of 75% confidence Interval PPT]0 2~/, 4~/o 69% 8({%0 100%

I I I X 1 l)+i)])iP ," k + ,, ,

a I++))+++,~. ; <" , × , , 3 l++++)+)~++)+++l ',+ × ,' ,' ,'

E+IE$~+I~I:.': +i++~++:. )!~++++~++ I I I + liii+ii+i++jii+iti+~ii' , , × ,

b. model #4a

I ~ ~ i

PPT

B

C

~lormalizad root-mean-square error E~m (T-30S) 0.0 01.2 014 016 OrS 1.0 +) o~v, , • , , ~i o~, i i ° i +,,~ ~ , , , )~.~ • • , • C3) i

PPT ] Normalized size bn my of 75% confidence interval - - - I0 2~o 4~/o 69%o 8(~Yo 100~

ii))I))i I I i I I B I+1.~*, I , • I ° I

I c I N i + l ' ~ l : - ~ , : c. model #4b

• ~ A~p ~ T , . ~ . A ~ = I

~- m ~"~72::'+++'+'+'+' ............. +'++++'=+~+~+~++++++~++++.. +++++++++++I=. ~, PPT Normalized root-mean-square error Ern~ (Tm30S)

0.0 C 2 014 016 Or8 1 .0

a i)ii)i!~i)+~) , • o , i$~$~"'%:$"::: ; ; c +!ii~ii!)~1; , • ~ ,

PPT Normalized size bn av of 75% confidence interval 0.. .......................... ~ ......... 4~%0 6q%* 8(~ 100~

S i)!iiii~iiii)ii!i!i)iiii/iiii~i!)i!!iil ' , l 13 c iiiiiii!i+!i+iii~!!( ,' ', • o,'

"Class A" p red ic t ions Glasgow Univ. (3 U. Nevada Reno • Tufts Univ. O CU Boulder X ETC ¢- U. Tokyo (Towhata) z~ Princeton Univ. • Leighton & Assoc. Brigham Young U, HI" U. Coll. Swansea • Takenaka Corp. X I expected value of the bn av index

experiment ~ recordings used for comparison with "class A* predictions record

range [ ~ i i : i I all recordings included

Fig. 9. Comparison between 'class A' predictions and experimental recordings, in terms of E~s and bav indices, for the centrifuge models providing reliable experimental results.

88 R. Popescu, J. H. Prevost

a . m o d e l # 1 - - LVDT2 LVDTI ~7

AX2a+-J I ~ ~ , . ~ AID

l:liiii++iii+iiiii+i!iiiiii~+i~+iiiiiiiiiiiii!!N~iiii++i i~!iii+iii++iiii+iiiiiiii+iiiiiii++ii+iiii+!iiiiiii!iii o+. I] ++ ++ + +

• !{iii~ii~iiiil N~iN~,',iii',',i~,',~/~!i~i ~ N~4iiii{i~i{!~i~i!i~i~iii{i~i!~!~:i~i~i!~!i~i~iiiii~E~i!i~i~ii~i~i~i~iii~ • ~,~,~:]i~,iii',;}+}i',}i i{}}!i!~i} ', )i',iii]i',; ~, i};}}}i}i}; ii~i}i] i',i{i};}iiii;}ilili~;i ;i;}}}ili}: ',i',;}i{i}i)!~,i}ii ii}iil Nevldl Sand- O~llO% l}i 1 }vi-,, l I ...................................................... [ ........................... i .................... ! ........................................................................... +::!:

~ LV91'S

~ L.V~T4

~LVm 1

~ LYDTI :

b. m o d e l # 3

22.88 m

P P T Normalized root-mean-square error Erm, (T=25s) 0.0 C .2 014 016 01.8 1.0

++++ Fo , 5 +:,+++++:+++ = O . , . .

i v i,,~PI

17.80 .~_!

P P T +++++++++':+++++"+"++++++++ +~ o * : * + ' i~)'J-~.~'+!I i+.0 I , t 3.0 0r2 014 016 01B

7 ..... ~ ~" " * I ~ O ' ~ onee~edmental ::.~+:::~,:~:: ~ 1 :3 , , , , 4" I I 2 O 11" ~,~,.d I • I ::: ::':.::::+.:::: ::': ~. :: $:

iii+ilii~iiii ! ':'+':':':~'+'+::: .................. P ~ ~ ~ . . ~ i~ i~ 3 i!i!i~!!!+ i].. @ ~ i+i:!:':':+:i!i!~++i! i i i • II II

8 iiiiiii+li~iii I ' ~ - I I I I * ~ " ' ' 4 :::~""'~: .................. ::!+.:. .::- 0 • • :!: +::.::::..'-::~:!:i~, , , l 5 :~ • t 0 • ° n • expel+mental

Normalized size bn "v of 7 5 0 confidence interval +++++++++~f~l~lO I • I ; P P T ) 2<:1% 4(p°/. 6q% 8q% 100~ 6 ...... + ......

:t=~i;=,~ i - , , , }]+i++}}~+)iii+{++ +I+l++~",,o. , , ~ - . , _ 7 ++...+.+~ o " , , ,

5 ::' ~.++Ms.,'.+'i<z.++..<q +.'..++a::+++++LmJl~ v .,b "A", x ##++]| I • • ~ I I I $::~$:: $~:::~.:~'~:!:. <::~ : :~:.:,: :::~:.>.: ~,~ . e.:.'. .............. .,. ................... :.'.:::::.+::: ........ ::-~-:.I:.:.'..:.:..':'~:FvuJ 8 ~}!!!] t:::l 0 U / I +++,++~ ~ ......... + ',

6 +++. . . . .+~ ~ , , 9 ++++ +++ .+ o o ~ I 7 , , 10 ).(({+)+:++~+li.{I o ~ m ; ; + . +>...++pro ~]' , ~ ,

++~..:~++ ..... ..,+++++ ++.:,'..'.;~ +..':'~,:m:+:::.... ~+..4 P i i

+.++:+ .+~++.+"~ _ o ~ k * ~ , " 8 +++:+'.+++ ++.+,..:+.+.+.~+.>:+.++ i :+++++ ++.+$~-+] + ~ 1 ! , r . * !

c. model #7

~pj. Nevadii Sand - Dr - 6(3% Bonny ~It " ACCi ":J- " ~ , ~ 1 7 . 7 5 , ~ - ~ I ~ 1 7 . 0 0 v ~ l ~ 17.75 ~--I

INormalized root-mean-square error Erms (T-25s) PPT 10.0 Or 2 ot 4 0~6 Or8 1.0

iii!!iiiiiiiii@il • • O I [++++:+...:....,+,++ . . ~ , ..

, I+++++++++))+++++I ~ , , ;,, o , I+++++++++~t t~+ 4 ' , ,

. ~ i ~ ~iii:;":'!::ii . . . . . . . . . I I I 4 l+a++:.+++~+ , , ,

Normalized root-mean-square error £rms (T130S) 1.0

PPT I Normalized size bn~V of 75% confidence Interval - - - 1.0 2q% 4?% 6(p°]o 81:1% 100°~ :::::::::::::::::::::::::::::::

• $"::~:..~:.$~ ~:.::.+:~'.'~i:" $!:'~!:: •

4 [}++~+f++++++}+!++:.~++~ll • ============================================================ ~ / I I • .:. • .,,.:+.,...:* ,'. • :.+:$..... ,'~., I I I -:-...:~m.~!+.:.+.:,...>:~..~+

I [:~:.:++~ .................... ,+.+.,.+~, ~ ,

- ¢;":'+~!:+:'+::+'+'+:;P'W, i i ::::: :+< :~',~ >+::::.~:, s b,+. .++.++l ~

:::::::.++.:+:..: &.::::::

+ ' ] l ~ I ~ ~ ~ l l /

"C lass A" pred ic t ions

USC Brigham Young U.

Normalized size bn =v of 75% confidence Interval P P T o 2 ~ . 4(p°/. 6_(p% s(p+o 100°4 i ............. ~ ........................... +~ ...................

I .+!i+~+!+++++l+i+~+~I~i+l ll.. ,.o,. o

~+~+;+ . . . i ~ i~ I ' ' 3 l++'...++#+~+++++...:++--..n | , ,

LVD'I'4

LVDTI

LVDT2

LVDT1

O ¢. Glasgow Univ. I~ Dames & Moore • U. Tokyo (Ishihara) ,k Takenaka Corp. I I Tufts Univ. O U. Nevada Reno • UC Davis -A- Leighton & Assoc. * ETC ~, U. Coll. Swansea • Princeton Univ. • I expected value of the bn av index

recordings used for comparison experiment with "class A" predictions record

range | ~ i ! i t all recordings included

--->- out of plot range

Fig. 10. Compar ison between 'class A' predictions and experimental recordings, in terms of erms and b~ v indices, for the centrifuge models providing experimental results with apparently low reliability.

V E L A C S numerical 'class A ' predictions 89

a. model #6

I~IAO~ ~CCS - ~ - Nevada San¢l - D r = 60% L... ~.OOr. ..._I

PPT

A B C D

Normalized root-mean-square error P-~r, (T-30s) o.o 0,4 0,6 o,a i.o

I • I O I ~ X I I ~ I , I I ~ I xm I I I I I I

• ,,m , i X I I X l l m I , • m m l l r I

b. model #11 Lead sheet LVm= ACC~2

~ ,'I 'L .I. A ACC l l /

. ~ H I ~ I LVDT1

]

i#~.~!~i~i~iL~.~.~1~i~i~`..~ii~i~i~.~.~I~i~.~i~i~ii~i~i~i~i~ii~.~.~ ~ ~ .:-:-¢.: :¢.:.:.:.:::.:,~'.: .x~.:!~..:::$..<,.:<-:. . . : :..:~.1.:.:.:+:.:.:.:.:.:...:.:<.:.:<..:.:.-:~..:... ..A<.: • :~. . : :~:.~:~ ~ . : : : : :~ : :~ '~ ~. : :~ .~ . .~ ::::::: :::::..:: .:..'~.¢:~ ~:~:.',~::::.~::::.~:::::~.:.~::: ~: ":~.:.~ ::: 4.::.<:: .~.?.~.. ' . . ~ : .~.: ~ ~.:.¢ .~ . . . , . .~ : : >. . . . . . . . . . . . ~. ::~ .P.:::::. ::::. :::::.:.~:~.:..: .> .... +:+:. ~ ...'.:,, ., :::... ..::., ~ :::~:: A ~ ' ~ / .............. .~.,.:~.,.,..,.,,.~.~ ......... :~.,..,,,.,,. ............................ . ~ ) : . . , ~ ) ............. ~ > Retaln,r¢l wall ~:~:~::,:~:~:.;~ ~ i~ i~ ~.:...:~:~- ,~.~:?~ ~-L.,~E.,..~,~ L~ ~ . ; : ~ i : ~ : g ~ - ~ . ~ < . ~ i ~ :~:`.<...!~:::.~:::..:::.?.~::::.~.!..`:!~:.~.~...~:::~::~.::.::.::.:..V:~:.::.:.~:::::..;.(~:~:~:~:~:: . . . . . . . . . ;:.~!¢-:.~" ' .:.,.':..':: ' ":,Y.~ • .~.:~ ;~ ~.~ ~ ~:.:::..-~::p.~: ,..::::.:.:::. ::: :.:::.: :::.: + ~::<, :~ ~:.:~:::..~ ..+:..~:.:.:....'.::: . . .~ . . ~:. ....:.:.: . : .~ : . . .> : AC~I

:~!:,.' ~ i ' ~ : ~ : : : . : ~ : . ~ : ~ ~:~:~: :::~ ::::: ::~.~-'.~ ~::.::~ :::~::~::::~ ~.I.'.:,'.:.~ ~ i~ : . . . . : ' ~ - : : :~ .: x . : : : : : ~ ' ~ p . ~ ::::.~: :~-:.:.....:~.: ..: :~.: ..--.:~.:¢ ~.~2~:.....: .+:-:::-:-:-:-.-:-:-:-..: ..~ +..:.:~.: ~..:.:.~.:.:.~:~ < ~:.:..~:...<.:..~:...:-:...:.:...:...:~ ~ ...,-.:.:<....:.: ~.:.:.~:.....:...<.

::::.:'~::::'~::::::.'::::~:"::,.':<'.'~ :?.~:'::::::~":'~:~::~:'):.::~:::::::::::::.::.~ ~'E?~$::::.':::::::':~:'::: ":::'"':':~:::~.~.::~::::::::::::~'~::: ::~:::¢'::::':::::.:"$::':::'-::::::::: "::::".':::: ~:~:]:'~i

1 7 . O r n ~ I ~ 5 .Ore . _ 1 1 2 . s m L. ( 2 0 . O r e I ~ ~ ~ ~ 0 . 0

l o

P P T

2 3 4 5 6 7

8

Normalized root-mean-square error grm s (T-I 4s) ).0 0,2 0# 0,6 0re 1.0

,I om, ,

l i l I oom i x i n A

e o m i 6e i A i I r - I I

• 01 m',w , i i i i i i , • , Lx , $

I I I I m A

i I I ~ i I

"Class A" prediction: Glasgow Univ. O U. Tokyo (Towhata) z~ Princeton Univ. • Leighton & Assoc. Takenaka Corp. It U. Coll. Swansea • CU Boulder X

Fig. 11. Comparison between 'class A' predictions and experimental recordings, in terms of the Crms index, for the centrifuge models which had not been duplicated.

centrifuge model #3 --- the primary test performed at Caltech 5 and the second duplicating test per- formed at RPI. The first duplicating test had been performed by RPI 5 under conditions which are different from those 'stated by the primary experi- menter. Therefore, OnlLy the results of the second test, reported later 16 and performed under the standard conditions, are considered for this comparison study.

4.2 Expected average value of the bav index

The vertical bars in the figures containing the size bav of the 75% confidence intervals represent the expected average value 6~, v, computed at each transducer location function of the experimental result reliability, as explained hereafter.

As shown in Section 3.4 and Fig. 8, the values of the bav index are dependent on the experimental result reliability. This becomes obvious if one compares the values of the b~ v indices computed for the predictions obtained for centrifuge models 2, 4a, 4b and 12 (having acceptable experimental Iesult reliability (Fig. 9)), to the bn av indices computed for models 1, 3 and 7 (providing

apparently unreliable experimental results (Fig. 10)): the overall average value of the b, ~v index at the transducers located in one of the models 1, 3 and 7 is about 45% higher than in models 2, 4a, 4b and 12. This dependence is illustrated in Fig. 12, by plotting the values of the bav index, averaged for each transducer vs the measure "7 of the reliability of the experimental results. As discussed in Section 2, the "7 index represents the likelihood that the mean of the experimental recordings lies within a +10% interval of the true value. This index is described in details in Ref. 8 and its values are shown in Fig. 1 for all the centrifuge models and pore pressure transducers for which more than one experimental record was made available.

The overall sample mean, or the expected value 6 av, of the bav indices is evaluated for each particular centrifuge model and transducer location as a function of the reliability index "7 of the experimental recordings at that location, i.e.:

/~av =f("7) (10)

which is represented by the regression curve passing through the points plotted in Fig. 12.

90 R. Popescu, J. H. Prevost

.+-

>.

¢3

ra averages ofb." from models 2, 4a, 4b and 12 o av~N~ of b." fsxn raodcls 1.3 ~ad 7

k ~," = ~(~- m,. o2); ̂ =+.m, B.0.~8, c.0.~6

[ ]

o []

o ~ - - - : 9 ~- 0 [ ]

1"I

I I 0 0~2 0.4 0.6 01.8 ;

y- experimental result quality index

Fig. 12. Evaluation of the expected value/~av as a function of the experimental result reliability index ~, (continuous line), using the averages of the b~ av indices for each transducer

location (markers).

The functional relation (10) is evaluated based on the numerical example presented in Section 3.4 and Fig. 8; for e=~ = 0 and with reference to Ref. 8, eqn (10) becomes:

D /~v (,,/) = (11)

where D is a constant which depends on the /3 value (e.g. D ~ 1 . 6 for / 3=75%) , N = 3 (the n u m b e r of experiments), and T is the Student's t cumulative distribution function• For the real case, with erms ~ 0 and various number of experimental results reported at each transducer location, the following expression is suggested:

C bav(7) = A +/3")' + (12) )

where A, B and C are constants to be evaluated by curve fitting. The resulted best fit curve, obtained by using the least square error method, is shown in Fig. 12. It is to be noticed that by using a linear or a quadratic regression curve, the error sum of squares is about 56% or 21%, respectively, higher than computed by using eqn (12).

A comparative assessment of the quality of each 'class A' prediction, in terms of the bn av index, is provided in Figs 9 and 10 by locating where the value of the bav index of that prediction (marker) is situated with respect to the expected average/~av (vertical bar). Since at each transducer location, the expected value/~av is estimated using the values of the bav indices computed for the 'class A' predictions for all the centrifuge models, this value offers a fair comparison, in the sense that it is not much influenced by the quality of the predictions submitted for a particular centrifuge model.

~.entrlfoge[ Nommllzed root-mean-squara error ~ml mo~l Io•o ~2 o# 0 # ors ~•(

, , , I = i i i + • ~:i I I I I

] | ,. I+]+ . .%- , • + , I ~ 4b |~+++.~I I • I I"I I

" " ,, :i< . 1 [++++++~ 0 0 . ~ + '

• = ~ |~;i'i;~;.~:~:~':::::11 l ~ W ~ I I • ~ l ~ + + t . , % o , , ,

' l + + : m , ,

! ~ J • I l

Cen rifu0e I Normilzed slze bn w ot 75% conlldeace Interval

= ' " F+@++,.~+t i • o ' ' '

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• ::::: ~ : : : : : : :~¢ , : ' : : ' : !~ . : ] ~ 0 " ~ I I

• + 1+++%~++I R i • , , 7 I~:,.:.+:.~:•.+++~+.:.:,~:, ,~

PREDICTOR COOE SYMBOL MODEL Princeton Univ. DYNAFLOW • nesled Tufts Univ. DYNAFLOW O sudacas U. CoIL Swansea SWANDYNE4 • generalized Glasgow U n i v • DIANA-SWAND• [3 plaslldly leighton & Aseoc• DYSAC2 . ETC DYSAC2 ~ bounding USC ENS 0 surface UC Davis SUMDES -A- U• Tokyo (Ishiham) *, CU Boulder UQCA x daseical Takenoka Corp• II p lu l ld ly U. Nevada Rono TARA-2 • direct Dames & Moore FLAC • (empirical) U• Tokyo (Towhala) dose form sdn• L~ Brigham Young U• QUAD4 4. total ,"rose

experimental record range

Fig. 13. Summary of the results presented in Figs 9-11: averages of the erms and bav indices over all the transducers of

each centrifuge model.

5 ASSESSMENT OF VARIOUS CONSTITUTIVE MO D ELS PERFORMANCE

An attempt to summarize the performance of various constitutive models 14 in predicting excess pore pressures, on the basis of the 'class A' predictions which were available for this study, is presented in Fig. 13. For each centrifuge model, the root mean square errors and 75% confidence intervals computed for each predictor were averaged over all pore pressure transducers and are shown in Fig. 13. Three types of elasto-plastic con- stitutive models, adopted by the predictors, are selected for discussion: multi-yield or nested surfaces plasticity represented with circles, generalized plasticity repre- sented with squares, and bounding surface plasticity

V E L A C S numerical 'class A ' predictions 91

represented with stars in Fig. 13. Under the reserve that the experimental results for models 1, 3 and 7 were deemed as having low reliability, 8 and that no information exists on the reliability of models 6 and 11, which have not been duplicated, the following conclusions can be drawn from Fig. 13:

- - For homogeneous layered soil models (models 1 and 2), close predictions are provided by all three models; the closest predictions in model 2 are provided by the generalized plasticity constitutive models, while the nested surface and bounding surface constitutive models predict pore pressure build-up too early (see Ref. 5 for more details).

- - F o r the nonhomogeneous sand centrifuge model (#3), close predictions are provided by both general- ized plasticity and nested surface constitutive models.

- - For nonhomogeneous layered soils (models 4a and 4b), very close predictions in the sand are provided by the nested surface and bounding surface con- stitutive models, while the generalized plasticity models predict much lower excess pore pressures than recorded in the experiments. 15

- - For embankments (models 6 and 7) and structural centrifuge models (11 and 12), close predictions are provided by the nested surface constitutive model; the predictions provided by the generalized plasticity constitutive models are within the average of the other predictions; the number of available 'class A' predictions using bounding surface constitutive models is too small (only one prediction for one model) for drawing any conclusion.

6 CONCLUSIONS

(1) The VELACS Project offers a good opportunity to verify and validate various analytical procedures for numerical simulation of soil liquefaction, by comparing the 'class A' predictions with the centrifuge experimental results.

(2) Results of reliability studies, involving the experi- mental recordings of excess pore pressures, displace- ments and accelerations;, are discussed. Due to the relatively low reliability of the displacement and acceleration experimental measurements, the com- parisons with 'class A' predictions are only performed in terms of excess pore pressures.

(3) A methodology to assess the goodness of fit of predicted pore pressure time histories with respect to the values recorded in the experiments is presented. The proposed comparison indices are: (1) normalized root mean square error (Crms index) of the predicted values with respect with the average of experimental record- ings, and (2) normalized size bav of a confidence interval centered at the predicted value, which contains the

estimated true value of the experimental results with 75% probability.

(4) The results of a comparative evaluation of about 75% of the 'class A' predictions submitted for the VELACS Project are presented in an informative and concise manner, in terms of the C~ms and bn ~v indices. The expected value of the bn av index is evaluated at each transducer location, as a function of the experimental result reliability. This allows an assessment of the quality of each prediction, relative to all the predictions available for this study.

(5) A comparative assessment of the capability of various groups of soil constitutive models to predict excess pore pressures induced by dynamic loading is presented. The performance of multi-yield plasticity, generalized plasticity and bounding surface plasticity constitutive models is discussed in detail.

(6) Once more, it is important to emphasize that the comparison study was performed under the assumption that the centrifuge experimental results have negligible bias or systematic errors. Further studies, involving comparisons between centrifuge experiment results and corresponding full scale prototype behaviour, are needed to verify this assumption.

ACKNOWLEDGEMENTS

The work reported in this study was supported in part by a grant from Kajima Corporation, Japan to Princeton University. This support is most gratefully acknowledged. The authors are also indebted to the VELACS Project Steering Committee and experi- menters, for providing the results of the 'class A' predictions and centrifuge experiments discussed in this study. And last but not least, the authors would like to thank Professor Erik H. Vanmarcke for his valuable suggestions.

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