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Reliability assessment of centrifuge soil test results
Soil Dynamics and Earthquake Engineering 14 (1995) 93-101 © 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0267-7261/95/$09.50
0267-7261(94)00037-9
ELSEVIER
Reliability assessment of centrifuge soil test results Radu Popescu & Jean H. Prevost Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA (Received for publication 22 August 1994) A methodology to assess the reliability of the mean value obtained from a series of soil test results is employed to assess the quality of the results obtained from laboratory soil tests and centrifuge experiments performed within the framework of the VELACS Project.
Key words: soil tests, centrifugeexperiments,result reliability,confidenceintervals.
1 INTRODUCTION
recorded in the centrifuge experiments (in terms of excess pore water pressure, acceleration and displacement time histories). However, the recordings obtained in the experiments themselves may be biased and/or imperfect due to: (1) imperfect model scalings; and (2) experimental errors due to the details of soil preparation, input motion, instrumentation placement and functionality, etc. Therefore, a direct comparison of analysis results with all recorded experimental data may prove misleading. In the context of multiple testing (i.e. repetitions of what is nominally the same experiment), the experimental errors can be detected and quantified, as explained hereafter. The results of the experiments can then be used to provide a sound and scientific basis to benchmark various analysis procedures, and to allow an objective and critical assessment of their merits and shortcomings.
The results of laboratory/in-situ tests are used to estimate parameters needed for analyses. When dealing with soil test data, one often has to accommodate: (1) a small number of tests; and (2) a relatively large variability in the typical test results. Under these circumstances, adopting the average of the test outcomes as an estimate of the true value may lead to large and perhaps unacceptable errors. A method to evaluate such errors and to asses,,; the degree of reliability of the estimates is presented in the following. Field data on the performance of full scale systems are much needed to verify and validate various numerical analysis procedures. 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 geotechnical system subjected to design load. The situation is a common problem in geotechnical engineering and is exacerbated in earthquake engineering. Consequently, some form of model study seems inevitable to enable alternative design or analysis methods to be checked. Centrifuge soil experiment results have been used to validate numerical models. 1-4 An example is the VELACS (VErification of Liquefaction Analysis by Centrifuge Studies) project. 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 procedures. The numerical predictions were intended to be 'class A' predictions, and thus were made before the centrifuge tests were performed. The verification and validation of the various analysis procedures were to be carried out by comparing the predictions with the measurements
2 METHOD FOR RELIABILITY ASSESSMENT 2.1 Errors in soil tests Consider a series of N soil tests to be performed under nominally identical conditions. Before the experiments, the outcomes R1,R2,...,RN can be modeled as identically distributed random variables with (common) expected value E[R] and standard deviation an. When interpreting the results of such tests, one focuses, quite logically, on the sample average/~, itself a random variable (before the experiments), defined as: N
1 E Ri
(1)
i=1
with N - - the number of tests to be performed. Since k 93
R. Popescu, J. H. Prevost
94
is function of the random variables Ri, its statistics are also related to the statistics of the underlying random variables. In particular, it can be easily shown (see e.g. Ref. 5) that: (1) the expected value of k is equal to the expected value of the underlying random variables (E[R] = E[R]); and (2) the standard deviation of the sample mean crk is related to the standard deviation of the individual experiment outcomes ~R via:
PR,Rj
(2)
i=1 j=i+l
where PR;Rj is the correlation coefficient between the outcomes of the experiments i and j. Let ~ denote the realization of/~ after N experiments have been performed (~ is the sample mean). If ~ is used as an estimate of the expected value E [R], a measure of the random error associated with this estimator is the standard deviation (root-mean-square error): 6 e = {E[(/~ - E[R])2]} 1/2 = crk
(3)
However, the error measure in eqn (3) only accounts for scattering of the test outcomes and is computed using the expected value of the test results E [R], which can be different from the true value mR if the test results are biased. The scatter in test results does not provide any information about the bias b = E [R] - m R. The bias or systematic errors, and the scatter in test results are due to numerous sources. A non-exhaustive enumeration is presented below: (1) Errors related to soil sampling, due to: (a) sample disturbance; (b) inherent spatial variability of the soil properties within the same layer. (2) Errors induced by the testing method related to: (a) type of test/boundary conditions; (b) type of equipment; (c) other factors (rate of testing, confining pressure, water content, etc.). (3) Errors induced by human factors, which may be related to: (a) details of sample preparation; (b) recording equipment placement and reading; (c) test result interpretation. The effects of these errors are: (1) variability in the test results - - denoted by crR; and (2) systematic errors, or bias between the expected value of the results E [R] and the true value of the test outcome mR. For commonly used testing methods, the bias induced is well studied and quantified, based on the results of large numbers of soil tests (see e.g. Refs 7 and 8 for soil strength parameters). However, for centrifuge soil tests, no such exhaustive studies are available at this stage. Therefore, the bias in centrifuge experiments must be deemed to be negligible and is neglected in the following. Assuming that the statistical dependence between test
results is minimized, and that the bias induced by the testing method can be inferred from previous experience, the expected value of the test results E [R] becomes almost as good as the true value m R and the correlation between test outcomes can be neglected (PR;Rj ~ 0 for all i ~ j ) , i.e. the individual observations R1, R2,... RN are assumed to be mutually independent random variables - - the random sampling assumption. It should be pointed out that, in practice, it is virtually impossible to ascertain that the true random sampling is being achieved, and usually it is not. The reasons for this assumption are: mathematical simplicity, 5 on the one hand, and lack of knowledge in quantifying the amount of correlation (PR,Rj), on the other hand.
2.2 Expected error magnitude Under the random sampling assumption, to compare the reliability of various types of experiments, a nondimensional (normalized) root-mean-square error is proposed, for the case of tests with strictly positive results. It is obtained by normalizing ~ in eqn (3) with respect to the sample mean ~ and using eqn (2) with PR;Rj = O:
1 Cn = - e -
OR
(4)
~v~
Since, in general, the value crR is not known a priori, it can be approximated by an unbiased estimator, the sample standard deviation: ~R=
N NI--~_IE ( R / - / ~ )
] 1/2 2
(5)
i=1
Consequently, after the experiments have been performed, the normalized root-mean-square error in eqn (4) can be estimated as:
~R in = ?x/~
(6)
The estimate gn will be referred as expected error magnitude. It is obvious from eqn (6) that, under the random sampling assumption, in decreases with increasing number of tests. 2.3 Acceptability limit To have a measure of the scattering of some particular series of experimental results, one can compare the expected error magnitude in eqn (6) with a similarly derived quantity, evaluated for a completely disordered set of numbers generated within a certain interval. Uniformly distributed random numbers generated within a 4-100% error interval about the true value mR are suggested for this purpose (Fig. 1). The corresponding normalized root-mean-square error is,
95
Reliability assessment o f centrifuge soil RI 2,0
( ( R - m R ) V ~ ) / ~ R , which follows a Student's t distribution with N - 1 degrees of freedom. 5 Denoting by TN-1 the respective cumulative distribution and taking advantage of its symmetry, the probability 7 in eqn (10) can be evaluated as:
+ 100% error Ri is U (0.5; 2)
1.25,
sample mean true w~lue
•:
1.0'
1.25
0.5'
- 100% error
(~R = ~/3 / 4
"~ = 2TN_~(q) -- 1
0.0
with:
q=
arv
O~
~
En
(13) Fig. 1. Upper bound for the expected error magnitude - - the results R~ are assumed to be uniformly distributed. from eqn (4): e~d_
V~/4 1.25~
0"35 ~ ~
(7)
An acceptability criterion for a series of test results can be set by using the 'upper bound' given by eqn (7): < c . d ad
(8)
with 0 < e < 1, a function of the importance of the parameter evaluated from the test results and of the sensitivity of the numerical model to its variation. 2.4 Confidence intervals for the m e a n
It is important to observe that, if the bias is ignored or negligible, the reliability of experimental results very much depends on the number of tests performed: for a certain series of experiments, with given 5R and for a given interval size a, one can increase the level of confidence 7 (or decrease the expected error gn) as much as needed, at the expense of additional tests. This is illustrated in Fig. 2(a) for a constant coefficient of variation: VR = # R / f = 0"4. Conversely, for a fixed % the interval [mR - a f ; mR + a~] can be made as small as one wishes by increasing the number of tests. Due to the particularities of the Student's t distribution, the reliability of the experiment results increases with N even for constant expected error magnitude ~n, as shown in Fig. 2(b). In spite of an increasing coefficient of
Another way to express how close the realization f of the experimental-result average /~ is to the expected value E JR] is through confidence intervals for the mean. Let 7 denote the probability that the outcome of/~ lies within a certain interval centered at the true value: "7 = P[rnR -- a <_ [~ <_ ,n R + a];
a > 0
a. Constant variation coefficient:VR---~R/ F = 0.4 degree of ¢onfklenee¥
........
(9)
After the experiments ihave been performed, the value a can be replaced by a fraction of ~, the realization of/~: 7 = P[rnR - a~ < k < mR + a~];
for example a = 0-1
(10) I
Assuming the sample m e a n / ~ is normally distributed, eqn (10) can be written as: k - mR ] 7 = P - q <- - - e r r -< q
3
with:
4
af _ afx/~ q -- ~r~ aR
I
I
8 12 Number of experiments N
I
I
16
20
b. Constant e x i s t e d error magnitude to = 0.1
(11) t4D O
which leads to: "~ = 2~(q) - 1
S
(12)
where • is the cumulative standard normal distribution. D a t a indicate that various soil properties are, with reasonable accuracy, normally distributed, 9 so that the assumption o f normality for their sample average is a fortiori justified. The solution (12) i:~ only valid if the standard deviation aR of the test results is known, which in general is not the case. By replacing a R by its estimate - - the sample standard deviation 5R - - the expression (R--mR)/a[~ in eqn (11) is approximated by
O
o "~ ~ d-
dcgreeofce~klcnc,o¥
j
0
o
;2
Io
Number of experiments N Fig. 2. Variation of the non-dimensional quality assessment indices with the number of experiments, for ct = 0.1.
R. Popescu, J. H. Prevost
96
model 2
model 1
LVOrs LVCIT4 LVDTS ~VOTO
model 3 LVDTS
L1/DTS
model 4b I
"~"
,~,
model ,6
ldln?l.,.'~c~
I
LVOT1
•
~ ~ I|:::::~:::::::::::::::::::::::::::::::::::~t~::::::::::::::::::::::::::::::~::::~:~!ii:::~:i::..,.i:.~i~!:.~~i~i:i~ii~:iiii:iii}i~}~ii~3i~ii:i~ii~ii!ii}~i1~!!~i~?~!~!i!:!~!i~i::!~i:iii::: i:i:i:i;:~:::~:::::::::~:::~i!~::~:ii!i~:i~::i~i~i~ii:. ilii?: ~
model 7
(1
12.50
~l r
[eadsheet
1_....
17.0m
~1~
tyro . ' ~
5.0m ~
,,.1,.,,
12.5m1_...
~ ~ I
~
1~0 .
.
.
.
model 11
,20.0m.,.
Fig. 3. Centrifuge experiments performed for the VELACS project.
....!
i
Reliability assessment of centrifuge soil n o <> +
effective confining stress effective confi~ag stress effective confining stress effective confin~agstress
= = = =
(1) Conventional laboratory soil tests performed by The Earth Technology Corporation and several other laboratories, II providing information about the geo-mechanical properties of the soil materials to be used in the centrifuge experiments. This information was given to the predictors, to calibrate their numerical models. (2) Centrifuge experiments, which comprised nine soil models (Fig. 3). Except for models # 6 and # 11, a primary and two duplicating experiments were performed by different laboratories for each centrifuge model, l°
320 KPa 160 KPa 80 KPa 40 KPa
[] 0
8.
O O
O
o o
oo ++
0
I
30
40
!
97
I'
I
5O 6O Relative density D r (%)
70
Fig. 4. Low strain shear moduli (G at 3' = 10-4) for Nevada sand, evaluated from the results of resonant column tests.l~ variation VR, the experiment reliability increases with the number of tests, especially for small N. A lower bound for the confidence level 3' can be set as in Section 2.3, replacing ~ in eqn (13) by c. e~nd - - eqns (7) and (8). This is only valid if the mean of the uniform random variables Ri, considered in Section 2.3, is normally distributed, which is true for large N (central limit theorem) and satisfactorily close for values of N as low as 3 or 4. 5
- - RELIABILITY O F S O M E LABORATORY EXPERIMENTS PERFORMED FOR T H E VELACS P R O J E C T 3 EXAMPLES
To illustrate how the proposed method works and to illustrate the meaning of the expected error magnitude ~, and degree of confidence 7, several numerical applications are presented in tile following. The examples are taken from the results of laboratory soil tests and centrifuge experiments performed for the VELACS project] ° The project had two main experimental components:
3.1 L a b o r a t o r y soil tests
The low strain shear modulus is provided by the results of resonant column tests (Fig. 4). Two tests were performed for each value of the confining stress ,p,.ll Each pair of tests is considered as a test series, with outcome G i (i = 1 , 2 ) . However, all the test results (for all confining stresses) are considered to evaluate the power exponent 'n' in the relation:
(p)n
G = Go P0
(14)
expressing the variation of the low strain modulus with the confining stress. The results are presented in Table 1, for two different relative densities of the sand used in the experiments. Most of the experiments provide accurate estimates for the low strain shear moduli (the error magnitudes ~, are 5-10 times smaller than the upper bound enrnd). Higher degrees of confidence can be assessed to the estimates of the power exponent values n, which are evaluated from a larger number of tests. It is to be noticed that here 'random sampling' can be considered a quite strong assumption, since all the results are evaluated from the same type of test, and all the experiments were performed by the same soil laboratory. The expected accuracy of the friction angle at failure
Table 1. Reliability of the results of the resonant column tests performed by ETC ll for Nevada sand
Relative density Dr
40%
60%
Soil property
Confining stress p (KPa)
Nr. of tests N
Error magnitude in
low 40"0 strain 80"0 shear 160.0 modulus 320.0 power exponent n
2 2 2 2 6
0'0425 0'0526 0"0265 0-0342 0"0104
low str. 80"0 shear 160-0 modulus 320"0 power exponent n
2 2 2 4
0"0760 0"0540 0"0318 0"0441
Upper limit enr n d
0-245 0'122 0'245 0' 131
Degree of confidence 7 (%) a = 5%
a = 10%
55"1 48"4 69-0 61 "8 99"5
74"4 69"2 83"5 79"0 >99'9
37"0 47"5 63'9 66"0
58-6 68"5 80"4 89"2
98
R. Popescu, J. H. Prevost
Table 2. Refiability o f the results o f the isotropieaHy-consolidated monotonic triaxial tests performed by E T C and other laboratories I 1 for Nevada sand
Relative density
Test typea
Dr
40%
50% 60%
70%
Nr. of tests N
Error magnitude ~,,
Upper limit enrnd
Degree of confidence -y (%) a = 5%
a = 10%
CIUC CIDC CIUE CIDE
16 3 4 5
0.0054 0.0236 0.0850 0.0824
0.087 0.200 0-173 0.155
>99.9 83.2 40.3 42.3
>99-9 94.9 67-6 70.8
CIUC
3
0.0020
0.200
99-8
>99.9
CIUC CIDC CIUE CIDE
13 4 4 3
0.0069 0.0094 0.0320 0.1800
0.096 0.173 0.173 0.200
>99-9 98.7 78.6 19-2
>99.9 99.8 94.9 36.5
CIUC
3
0.0015
0.200
92-2
97.9
aCIUC - - isotropically-consolidated undrained compression tests; CIDC - - isotropically-consolidated drained compression tests; CIUE - - isotropically-consolidated undrained extension tests; CIDE - - isotropically-consolidated drained extension tests. ~blim, as determined from the results of isotropically consolidated, monotonic triaxial tests performed for the VELACS project, 11 is presented in Table 2. ~blim is evaluated as a function of the stress ratio r/ at failure, as: q~lim =
sin-1 3lnl
(15)
6+n with the stress ratio for triaxial conditions: r / = (o"1 -- o'3)/p , and p = ½(al + 2o'3) is the effective confining stress. The reliability analysis is performed for each relative density of the sand and, to minimize the type 2 errors discussed in Section 2.1, for each type of experiment separately. The computed values of the friction angle at failure resulting from each of the tests are plotted vs relative density in Fig. 5. Excepting for one test series ( C I D E at D r : 6 0 % ) all the tests are found to provide relatively accurate estimates of the friction angle at failure and half of them can be characterized as offering a relatively high degree of confidence.t It is worthwhile to compare the scatter of the test results presented in Figs 4 and 5, on the one hand, and the results of the reliability analysis performed for the two types of experiments (Tables 1 and 2), on the other hand: although the estimated friction angles are more scattered than the shear moduli values (larger coefficient of variation), the friction angles obtained from the triaxial test results have more reliability and less expected error magnitude than the shear moduli tHere, each predictor should also consider the results of a sensitivity analysis involving the respective soil parameters. For example, an error of 5% in estimating the friction angle can have much larger consequences on the numerical predictions than the same error in estimating the low strain moduli.
obtained from the resonant column tests. This is due to the larger number of triaxial tests performed, a s w a s already illustrated in Fig. 2. 3.2 Centrifuge
experiments
The object of this section is to assess the reliability of the outcome of the centrifuge experiments which were duplicated, in order to point out to those models which offer meaningful results for comparisons with 'class A' predictions. The centrifuge models (Fig. 3) which are discussed are listed in Table 3, along with the laboratories performing the experiments and the number of tests performed for each model available for this study. The assumptions stated in Section 2 are now considered. With reference to the type of errors listed in Section 2.1, it is noticeable that, in the case of centrifuge experiments, type 1 errors do not occur. Moreover, for the particular situation of the VELACS
÷
o floo
I"1
da
t~ta
o
o
1:3 o 0 +
CIUC CIDC CIUE CIDE
(undrained - compression test) (drained - compression test) (undrained - extension test) (drained - extension test)
3o
;o
;o
Relative density D (%)
Fig. 5. Friction angles at failure for Nevada sand, evaluated from the results of the isotropically-consolidated triaxial tests.~I
Reliability assessment of centrifuge soil Table 3. Duplicated centrifuge
Centr. model
4a
4b
12
99
experiments 1°
Test type
Experimenter~
primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating
RPI UCD CU Boulder RPI UCD Caltech Caltech RPI UCD b UCD RPI Caltech UCD RPI CU Boulder CU Boulder RPI UCD PU (4 tests) RPI UCD (2 tests)
51~
Avail. tests
,n,..~
:
3 b. ~=
i
0 all vslues Included • final results I "
I
3 --
•
•1 •
2
3
;
i
PPT
I
!
i
I
i
SSTSiSSTaI134sTn9101ABelABeI12341123 4
I
I
I
3
OI I
•-t • •
I.
o
le I oOO I
I
I
I
I
i e I IO I
I• I I I
I ~ I I0 0 I O0
i I I I
acceptsblllty limit: "~ = 45%
I
•1
7
20%
n
40%
60%
-
project, with each centrifuge experiment duplicated by different soil laboratories, the bias induced by type 3 errors is minimized. However, there are some uncertainties related to the validity of the proposed assumptions, a. Model
a'ansdueerPPT-C
4a -
,:'
L'
[
,,
~
UC Davis
:.--:_.
; ~ ~
xoo,, i I0
0
i 20
I 30
40
dine (scc) b. Model 7 - transducer PPT-2 '#t
!
~"l'-,~",'tt" ~,'~; ~"~"~1" ''"'~""'"
[
0.0
Mode I 1 I 2 3 I 4a I 4 b l 7 , 12 (3) I [ (3)(3)(3)'(4)(4)(5X4) 92 332132231222222221333133~1~2 2213 34 3 0
aRPI Rensselaer Polytechnic Institute; UCD - - University of California, Davis; CU Boulder - - Colorado University at Boulder; Caltech - - California Institute of Technology; PU - Princeton University. bResults not made available. -
0.3
I ee e I le Oi • 0.2 I I I I I acceptability limit: c ~ m d = 0 , 1 4 • I 03 ! N=3; c=0,7 (upper bound) O o, ,
,
3
I I
O
I i:
:
~
I•
,
/
e~'%% .
""
UC Davis
. . . .
~?,~ ',\:.~,~..,~,~#~'.',,':,';'¢" ,'~ ......................... , jlr
. . . . .
,
,
I
I
I
I0
20
30
40
~ne (sec) • Fig. 6. Excess pore pressures recorded during repeated centrifuge experiments: (a) reliable experimental results; (b) scattered excess pore pressure measurements.
80%
100% N - number of experimental results available for each transducer location
Fig. 7. Reliability analysis for the VELACS project centrifuge experiments: (a) quantification of experimental measurements; (b) reliability analysis results evaluated for the centrifuge experiments which were made available: centrifuge models 2, 4a, 4b and 12 are found to provide results which are most appropriate for comparisons with 'class A' predictions. due to the low number of test results available for each model (typically 2 or 3) and to the possibility of a scaling generated bias in the experimental results. As discussed previously, the bias or systematic errors in centrifuge experiments have not been studied and therefore must be deemed to be negligible and are neglected in this study. The study is restricted to recorded excess pore pressures, since they best represent the liquefaction phenomenon. The dynamically induced excess pore pressures, recorded during the experiments, are provided as time histories, as illustrated in Fig. 6. F r o m Fig. 6, it is clear that the mean of the experimental results obtained at the same location by different laboratories is relevant if the records are relatively close to each other (e.g. Fig. 6(a)), but leads to questionable conclusions if the records are scattered (e.g. Fig. 6(b)). The reliability analysis is performed in terms of the integral of the recorded excess pore pressure time histories over a significant time interval (Fig. 7(a)); this is made possible by the fact that all the recorded excess pore pressures are positive. The results are presented in Fig. 7(b), in terms of expected error magnitude - - ~, in
100
R. Popescu, J. H. Prevost
eqn (6) - - and likelihood that the mean of measurements lies within a + 10% interval of the true result - - 7 in eqn (13), with a = 0 " l . The various transducer locations are identified in Fig. 3 for each model. Remarks --The numbers N in Fig. 7(b) represent the actual number of test results used to assess the reliability at every transducer location. This number can be less than the total number of tests reported (Table 3), due to missing or malfunctioning transducers. - - A t locations where one of the records is clearly different from the others, the reliability can be improved by eliminating that record. Results are shown in Fig. 7(b) for such cases: with open dots - using all the records; and with black dots - - for improved reliability. - - F o r reference, a comparison with the expected reliability of conventional laboratory soil tests is provided, using the average values resulting from the previous study (Section 3.1). - - Acceptability limits - - upper bound for the expected error magnitude in and lower bound for the confidence level 7 - - are proposed; they are evaluated for experiments with N = 3 tests. Accounting for the complexity of the centrifuge experiments and comparing the results (in, 7) with those obtained from conventional laboratory soil tests, a relatively high value: c = 0.7 is set in the acceptability relationship (8).
The reliability results shown in Fig. 7(b) lead to the following conclusions: • From the viewpoint of result reliability, there are three groups of models: (1) Models 4a (sand layer only) and 12, with overall highly reliable excess pore pressure results, having a degree of confidence of the same order as some conventional laboratory soil tests. (2) Models 4b (sand layer only) and 2, with acceptably reliable results. (3) Models 1, 3 and 7, deemed as providing results with low reliability, in the sense that the comparisons between numerical predictions and the average values obtained from the recordings should be carefully evaluated and not used for arriving at definite conclusions. • All the pore pressure transducers placed in the silt material provided unreliable results. For models 4a and 4b, one of the causes was found to be the low confining stress environment at the transducer locations. 12 • The importance of repeating tests is again emphasized: - - t h e recordings at PPT7 (model 2) and PPT3
(model 12) have the same expected error magnitude 4, ~ 0.05; however, larger confidence in the mean value is offered by model 12 (7 = 87%, N = 4) than by model 2 (7 = 68%, N = 2); - - t h e average expected error magnitude is almost similar for both triaxial and resonant column tests. However, the reliability of the results obtained from the triaxial tests (which have a larger number of tests performed) is clearly superior to those provided by the resonant column tests. Remarks (1) Two out of the three centrifuge experiments labeled as providing low reliable results (models 3 and 7) have special problems related to the complexity of the soil model involved, e.g. vertical (model 3) or steep sloped (model 7) interfaces between different soil materials, which required special methods for test preparation and resulted in likely deviations from the original specifications. 1° (2) The results of the present study only refer to those centrifuge experiment recordings which have been made available to the authors. Since both reliability indices used in the study (in index and 7 index) are dependent on the number of experiments, a higher reliability of the experimental results and, hence, a more sound basis for verification and validation of analytical procedures (see Ref. 13) might have been possible from a larger number of reported centrifuge experiment results. As shown in Table 3, for most of the centrifuge models, three series of experimental results were made available for this study. However, for model #3 only two experimental results were available, while for model # 12 a total of seven experiments were available. This fact must have influenced the reliability study results shown in Fig. 7(b).
CONCLUSIONS A methodology to assess the reliability of a series of soil test results is presented. Under the random sampling assumption, expected error magnitudes with respect to the 'true value' are derived. Within the context of normally distributed experimental results, confidence intervals for the mean are evaluated, and resulting acceptability limits are proposed. The importance of repeating tests is emphasized through both theoretical considerations and numerical examples. The method is employed to assess the comparative reliability of some results of the centrifuge experiments performed for the VELACS project. The analysis results quite clearly point out to those centrifuge models which offer a meaningful data base for comparisons with 'class A' predictions.
Reliability assessment of centrifuge soil 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 Professor Erik H. Vanmarcke for reviewing the paper and offering valuable suggestions. The help of the VELACS Project Steering Committee in providing the experimental results is al,;o acknowledged.
REFERENCES 1. Bucky, P. B. Use of models for the study of mining problems. In AIMM£ Technical Publications, Amer. Inst. Mining, Metall. and Petroleum Engng, 1931. No. 425. 2. Pokrovskii, G. I. & Fiodorov, I. S. Centrifugal Modelling in Civil Engineering, Gosstroyizdat Publishers, Moscow, 1968, p. 247. 3. Pokrovskii, G. I. & Fiodorov, I. S. Centrifugal Modelling in the Mining Industry, Niedra Publishing House, Moscow, 1969, p. 270. 4. Roscoe, K. H. Soils and model tests. J. Strain Analysis, 1968, 3(1) 57-64.
101
5. Benjamin, J. R. & ComeU, C. A. Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, 1970. 6. Bendat, J. S. & Piersol, A. G. Random Data, Analysis and Measurement Procedures, John Wiley & Sons, 1986. 7. Kulhawy, F. H. & Maine, P. W. Manual on estimating soil properties for foundation design. Final Report 1493-6, EL-6800, Electric Power Research Institute, Palo Alto, CA, 1990. 8. Mitchell, J. K. Fundamentals of Soil Behaviour. John Wiley & Sons, 1993. 9. Schultze, E. Frequency distributions and correlations of soil properties. In 1 st Int. Conf. Appl. Stat. Prob. Soil Struct. Engng, 1971, pp. 371-87. 10. Arulanandan, K. & Scott, R. F. (eds). Proc. Int. Conf. on Verif. Numerical Proc. for the Analysis of Soil Liq. Problems, Vol. 1, Balkema, Rotterdam, 1993. 11. The Earth Technology Corporation. VELACS laboratory testing program soil data report. Technical report, Earth Technology Project No. 90-0562, 1992. 12. Prevost, J. H. & Popescu, R. Overview presentation of predictions for Models No. 4a and 4b. In Proc. lnt. Conf. on Verif. Numerical Proc. for the Analysis of Soil Liq. Problems, Vol. 2, Balkema, Rotterdam, 1994, pp. 14691514. 13. Popescu, R. & Prevost, J. H. Comparison between VELACS numerical 'class A' predictions and centrifuge experimental soil test results. Soil Dyn. & Earthq. Engng, (1995) 14, 79-92.
0267-7261(94)00037-9
ELSEVIER
Reliability assessment of centrifuge soil test results Radu Popescu & Jean H. Prevost Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA (Received for publication 22 August 1994) A methodology to assess the reliability of the mean value obtained from a series of soil test results is employed to assess the quality of the results obtained from laboratory soil tests and centrifuge experiments performed within the framework of the VELACS Project.
Key words: soil tests, centrifugeexperiments,result reliability,confidenceintervals.
1 INTRODUCTION
recorded in the centrifuge experiments (in terms of excess pore water pressure, acceleration and displacement time histories). However, the recordings obtained in the experiments themselves may be biased and/or imperfect due to: (1) imperfect model scalings; and (2) experimental errors due to the details of soil preparation, input motion, instrumentation placement and functionality, etc. Therefore, a direct comparison of analysis results with all recorded experimental data may prove misleading. In the context of multiple testing (i.e. repetitions of what is nominally the same experiment), the experimental errors can be detected and quantified, as explained hereafter. The results of the experiments can then be used to provide a sound and scientific basis to benchmark various analysis procedures, and to allow an objective and critical assessment of their merits and shortcomings.
The results of laboratory/in-situ tests are used to estimate parameters needed for analyses. When dealing with soil test data, one often has to accommodate: (1) a small number of tests; and (2) a relatively large variability in the typical test results. Under these circumstances, adopting the average of the test outcomes as an estimate of the true value may lead to large and perhaps unacceptable errors. A method to evaluate such errors and to asses,,; the degree of reliability of the estimates is presented in the following. Field data on the performance of full scale systems are much needed to verify and validate various numerical analysis procedures. 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 geotechnical system subjected to design load. The situation is a common problem in geotechnical engineering and is exacerbated in earthquake engineering. Consequently, some form of model study seems inevitable to enable alternative design or analysis methods to be checked. Centrifuge soil experiment results have been used to validate numerical models. 1-4 An example is the VELACS (VErification of Liquefaction Analysis by Centrifuge Studies) project. 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 procedures. The numerical predictions were intended to be 'class A' predictions, and thus were made before the centrifuge tests were performed. The verification and validation of the various analysis procedures were to be carried out by comparing the predictions with the measurements
2 METHOD FOR RELIABILITY ASSESSMENT 2.1 Errors in soil tests Consider a series of N soil tests to be performed under nominally identical conditions. Before the experiments, the outcomes R1,R2,...,RN can be modeled as identically distributed random variables with (common) expected value E[R] and standard deviation an. When interpreting the results of such tests, one focuses, quite logically, on the sample average/~, itself a random variable (before the experiments), defined as: N
1 E Ri
(1)
i=1
with N - - the number of tests to be performed. Since k 93
R. Popescu, J. H. Prevost
94
is function of the random variables Ri, its statistics are also related to the statistics of the underlying random variables. In particular, it can be easily shown (see e.g. Ref. 5) that: (1) the expected value of k is equal to the expected value of the underlying random variables (E[R] = E[R]); and (2) the standard deviation of the sample mean crk is related to the standard deviation of the individual experiment outcomes ~R via:
PR,Rj
(2)
i=1 j=i+l
where PR;Rj is the correlation coefficient between the outcomes of the experiments i and j. Let ~ denote the realization of/~ after N experiments have been performed (~ is the sample mean). If ~ is used as an estimate of the expected value E [R], a measure of the random error associated with this estimator is the standard deviation (root-mean-square error): 6 e = {E[(/~ - E[R])2]} 1/2 = crk
(3)
However, the error measure in eqn (3) only accounts for scattering of the test outcomes and is computed using the expected value of the test results E [R], which can be different from the true value mR if the test results are biased. The scatter in test results does not provide any information about the bias b = E [R] - m R. The bias or systematic errors, and the scatter in test results are due to numerous sources. A non-exhaustive enumeration is presented below: (1) Errors related to soil sampling, due to: (a) sample disturbance; (b) inherent spatial variability of the soil properties within the same layer. (2) Errors induced by the testing method related to: (a) type of test/boundary conditions; (b) type of equipment; (c) other factors (rate of testing, confining pressure, water content, etc.). (3) Errors induced by human factors, which may be related to: (a) details of sample preparation; (b) recording equipment placement and reading; (c) test result interpretation. The effects of these errors are: (1) variability in the test results - - denoted by crR; and (2) systematic errors, or bias between the expected value of the results E [R] and the true value of the test outcome mR. For commonly used testing methods, the bias induced is well studied and quantified, based on the results of large numbers of soil tests (see e.g. Refs 7 and 8 for soil strength parameters). However, for centrifuge soil tests, no such exhaustive studies are available at this stage. Therefore, the bias in centrifuge experiments must be deemed to be negligible and is neglected in the following. Assuming that the statistical dependence between test
results is minimized, and that the bias induced by the testing method can be inferred from previous experience, the expected value of the test results E [R] becomes almost as good as the true value m R and the correlation between test outcomes can be neglected (PR;Rj ~ 0 for all i ~ j ) , i.e. the individual observations R1, R2,... RN are assumed to be mutually independent random variables - - the random sampling assumption. It should be pointed out that, in practice, it is virtually impossible to ascertain that the true random sampling is being achieved, and usually it is not. The reasons for this assumption are: mathematical simplicity, 5 on the one hand, and lack of knowledge in quantifying the amount of correlation (PR,Rj), on the other hand.
2.2 Expected error magnitude Under the random sampling assumption, to compare the reliability of various types of experiments, a nondimensional (normalized) root-mean-square error is proposed, for the case of tests with strictly positive results. It is obtained by normalizing ~ in eqn (3) with respect to the sample mean ~ and using eqn (2) with PR;Rj = O:
1 Cn = - e -
OR
(4)
~v~
Since, in general, the value crR is not known a priori, it can be approximated by an unbiased estimator, the sample standard deviation: ~R=
N NI--~_IE ( R / - / ~ )
] 1/2 2
(5)
i=1
Consequently, after the experiments have been performed, the normalized root-mean-square error in eqn (4) can be estimated as:
~R in = ?x/~
(6)
The estimate gn will be referred as expected error magnitude. It is obvious from eqn (6) that, under the random sampling assumption, in decreases with increasing number of tests. 2.3 Acceptability limit To have a measure of the scattering of some particular series of experimental results, one can compare the expected error magnitude in eqn (6) with a similarly derived quantity, evaluated for a completely disordered set of numbers generated within a certain interval. Uniformly distributed random numbers generated within a 4-100% error interval about the true value mR are suggested for this purpose (Fig. 1). The corresponding normalized root-mean-square error is,
95
Reliability assessment o f centrifuge soil RI 2,0
( ( R - m R ) V ~ ) / ~ R , which follows a Student's t distribution with N - 1 degrees of freedom. 5 Denoting by TN-1 the respective cumulative distribution and taking advantage of its symmetry, the probability 7 in eqn (10) can be evaluated as:
+ 100% error Ri is U (0.5; 2)
1.25,
sample mean true w~lue
•:
1.0'
1.25
0.5'
- 100% error
(~R = ~/3 / 4
"~ = 2TN_~(q) -- 1
0.0
with:
q=
arv
O~
~
En
(13) Fig. 1. Upper bound for the expected error magnitude - - the results R~ are assumed to be uniformly distributed. from eqn (4): e~d_
V~/4 1.25~
0"35 ~ ~
(7)
An acceptability criterion for a series of test results can be set by using the 'upper bound' given by eqn (7): < c . d ad
(8)
with 0 < e < 1, a function of the importance of the parameter evaluated from the test results and of the sensitivity of the numerical model to its variation. 2.4 Confidence intervals for the m e a n
It is important to observe that, if the bias is ignored or negligible, the reliability of experimental results very much depends on the number of tests performed: for a certain series of experiments, with given 5R and for a given interval size a, one can increase the level of confidence 7 (or decrease the expected error gn) as much as needed, at the expense of additional tests. This is illustrated in Fig. 2(a) for a constant coefficient of variation: VR = # R / f = 0"4. Conversely, for a fixed % the interval [mR - a f ; mR + a~] can be made as small as one wishes by increasing the number of tests. Due to the particularities of the Student's t distribution, the reliability of the experiment results increases with N even for constant expected error magnitude ~n, as shown in Fig. 2(b). In spite of an increasing coefficient of
Another way to express how close the realization f of the experimental-result average /~ is to the expected value E JR] is through confidence intervals for the mean. Let 7 denote the probability that the outcome of/~ lies within a certain interval centered at the true value: "7 = P[rnR -- a <_ [~ <_ ,n R + a];
a > 0
a. Constant variation coefficient:VR---~R/ F = 0.4 degree of ¢onfklenee¥
........
(9)
After the experiments ihave been performed, the value a can be replaced by a fraction of ~, the realization of/~: 7 = P[rnR - a~ < k < mR + a~];
for example a = 0-1
(10) I
Assuming the sample m e a n / ~ is normally distributed, eqn (10) can be written as: k - mR ] 7 = P - q <- - - e r r -< q
3
with:
4
af _ afx/~ q -- ~r~ aR
I
I
8 12 Number of experiments N
I
I
16
20
b. Constant e x i s t e d error magnitude to = 0.1
(11) t4D O
which leads to: "~ = 2~(q) - 1
S
(12)
where • is the cumulative standard normal distribution. D a t a indicate that various soil properties are, with reasonable accuracy, normally distributed, 9 so that the assumption o f normality for their sample average is a fortiori justified. The solution (12) i:~ only valid if the standard deviation aR of the test results is known, which in general is not the case. By replacing a R by its estimate - - the sample standard deviation 5R - - the expression (R--mR)/a[~ in eqn (11) is approximated by
O
o "~ ~ d-
dcgreeofce~klcnc,o¥
j
0
o
;2
Io
Number of experiments N Fig. 2. Variation of the non-dimensional quality assessment indices with the number of experiments, for ct = 0.1.
R. Popescu, J. H. Prevost
96
model 2
model 1
LVOrs LVCIT4 LVDTS ~VOTO
model 3 LVDTS
L1/DTS
model 4b I
"~"
,~,
model ,6
ldln?l.,.'~c~
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~ ~ I|:::::~:::::::::::::::::::::::::::::::::::~t~::::::::::::::::::::::::::::::~::::~:~!ii:::~:i::..,.i:.~i~!:.~~i~i:i~ii~:iiii:iii}i~}~ii~3i~ii:i~ii~ii!ii}~i1~!!~i~?~!~!i!:!~!i~i::!~i:iii::: i:i:i:i;:~:::~:::::::::~:::~i!~::~:ii!i~:i~::i~i~i~ii:. ilii?: ~
model 7
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,,.1,.,,
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~
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.
.
.
model 11
,20.0m.,.
Fig. 3. Centrifuge experiments performed for the VELACS project.
....!
i
Reliability assessment of centrifuge soil n o <> +
effective confining stress effective confi~ag stress effective confining stress effective confin~agstress
= = = =
(1) Conventional laboratory soil tests performed by The Earth Technology Corporation and several other laboratories, II providing information about the geo-mechanical properties of the soil materials to be used in the centrifuge experiments. This information was given to the predictors, to calibrate their numerical models. (2) Centrifuge experiments, which comprised nine soil models (Fig. 3). Except for models # 6 and # 11, a primary and two duplicating experiments were performed by different laboratories for each centrifuge model, l°
320 KPa 160 KPa 80 KPa 40 KPa
[] 0
8.
O O
O
o o
oo ++
0
I
30
40
!
97
I'
I
5O 6O Relative density D r (%)
70
Fig. 4. Low strain shear moduli (G at 3' = 10-4) for Nevada sand, evaluated from the results of resonant column tests.l~ variation VR, the experiment reliability increases with the number of tests, especially for small N. A lower bound for the confidence level 3' can be set as in Section 2.3, replacing ~ in eqn (13) by c. e~nd - - eqns (7) and (8). This is only valid if the mean of the uniform random variables Ri, considered in Section 2.3, is normally distributed, which is true for large N (central limit theorem) and satisfactorily close for values of N as low as 3 or 4. 5
- - RELIABILITY O F S O M E LABORATORY EXPERIMENTS PERFORMED FOR T H E VELACS P R O J E C T 3 EXAMPLES
To illustrate how the proposed method works and to illustrate the meaning of the expected error magnitude ~, and degree of confidence 7, several numerical applications are presented in tile following. The examples are taken from the results of laboratory soil tests and centrifuge experiments performed for the VELACS project] ° The project had two main experimental components:
3.1 L a b o r a t o r y soil tests
The low strain shear modulus is provided by the results of resonant column tests (Fig. 4). Two tests were performed for each value of the confining stress ,p,.ll Each pair of tests is considered as a test series, with outcome G i (i = 1 , 2 ) . However, all the test results (for all confining stresses) are considered to evaluate the power exponent 'n' in the relation:
(p)n
G = Go P0
(14)
expressing the variation of the low strain modulus with the confining stress. The results are presented in Table 1, for two different relative densities of the sand used in the experiments. Most of the experiments provide accurate estimates for the low strain shear moduli (the error magnitudes ~, are 5-10 times smaller than the upper bound enrnd). Higher degrees of confidence can be assessed to the estimates of the power exponent values n, which are evaluated from a larger number of tests. It is to be noticed that here 'random sampling' can be considered a quite strong assumption, since all the results are evaluated from the same type of test, and all the experiments were performed by the same soil laboratory. The expected accuracy of the friction angle at failure
Table 1. Reliability of the results of the resonant column tests performed by ETC ll for Nevada sand
Relative density Dr
40%
60%
Soil property
Confining stress p (KPa)
Nr. of tests N
Error magnitude in
low 40"0 strain 80"0 shear 160.0 modulus 320.0 power exponent n
2 2 2 2 6
0'0425 0'0526 0"0265 0-0342 0"0104
low str. 80"0 shear 160-0 modulus 320"0 power exponent n
2 2 2 4
0"0760 0"0540 0"0318 0"0441
Upper limit enr n d
0-245 0'122 0'245 0' 131
Degree of confidence 7 (%) a = 5%
a = 10%
55"1 48"4 69-0 61 "8 99"5
74"4 69"2 83"5 79"0 >99'9
37"0 47"5 63'9 66"0
58-6 68"5 80"4 89"2
98
R. Popescu, J. H. Prevost
Table 2. Refiability o f the results o f the isotropieaHy-consolidated monotonic triaxial tests performed by E T C and other laboratories I 1 for Nevada sand
Relative density
Test typea
Dr
40%
50% 60%
70%
Nr. of tests N
Error magnitude ~,,
Upper limit enrnd
Degree of confidence -y (%) a = 5%
a = 10%
CIUC CIDC CIUE CIDE
16 3 4 5
0.0054 0.0236 0.0850 0.0824
0.087 0.200 0-173 0.155
>99.9 83.2 40.3 42.3
>99-9 94.9 67-6 70.8
CIUC
3
0.0020
0.200
99-8
>99.9
CIUC CIDC CIUE CIDE
13 4 4 3
0.0069 0.0094 0.0320 0.1800
0.096 0.173 0.173 0.200
>99-9 98.7 78.6 19-2
>99.9 99.8 94.9 36.5
CIUC
3
0.0015
0.200
92-2
97.9
aCIUC - - isotropically-consolidated undrained compression tests; CIDC - - isotropically-consolidated drained compression tests; CIUE - - isotropically-consolidated undrained extension tests; CIDE - - isotropically-consolidated drained extension tests. ~blim, as determined from the results of isotropically consolidated, monotonic triaxial tests performed for the VELACS project, 11 is presented in Table 2. ~blim is evaluated as a function of the stress ratio r/ at failure, as: q~lim =
sin-1 3lnl
(15)
6+n with the stress ratio for triaxial conditions: r / = (o"1 -- o'3)/p , and p = ½(al + 2o'3) is the effective confining stress. The reliability analysis is performed for each relative density of the sand and, to minimize the type 2 errors discussed in Section 2.1, for each type of experiment separately. The computed values of the friction angle at failure resulting from each of the tests are plotted vs relative density in Fig. 5. Excepting for one test series ( C I D E at D r : 6 0 % ) all the tests are found to provide relatively accurate estimates of the friction angle at failure and half of them can be characterized as offering a relatively high degree of confidence.t It is worthwhile to compare the scatter of the test results presented in Figs 4 and 5, on the one hand, and the results of the reliability analysis performed for the two types of experiments (Tables 1 and 2), on the other hand: although the estimated friction angles are more scattered than the shear moduli values (larger coefficient of variation), the friction angles obtained from the triaxial test results have more reliability and less expected error magnitude than the shear moduli tHere, each predictor should also consider the results of a sensitivity analysis involving the respective soil parameters. For example, an error of 5% in estimating the friction angle can have much larger consequences on the numerical predictions than the same error in estimating the low strain moduli.
obtained from the resonant column tests. This is due to the larger number of triaxial tests performed, a s w a s already illustrated in Fig. 2. 3.2 Centrifuge
experiments
The object of this section is to assess the reliability of the outcome of the centrifuge experiments which were duplicated, in order to point out to those models which offer meaningful results for comparisons with 'class A' predictions. The centrifuge models (Fig. 3) which are discussed are listed in Table 3, along with the laboratories performing the experiments and the number of tests performed for each model available for this study. The assumptions stated in Section 2 are now considered. With reference to the type of errors listed in Section 2.1, it is noticeable that, in the case of centrifuge experiments, type 1 errors do not occur. Moreover, for the particular situation of the VELACS
÷
o floo
I"1
da
t~ta
o
o
1:3 o 0 +
CIUC CIDC CIUE CIDE
(undrained - compression test) (drained - compression test) (undrained - extension test) (drained - extension test)
3o
;o
;o
Relative density D (%)
Fig. 5. Friction angles at failure for Nevada sand, evaluated from the results of the isotropically-consolidated triaxial tests.~I
Reliability assessment of centrifuge soil Table 3. Duplicated centrifuge
Centr. model
4a
4b
12
99
experiments 1°
Test type
Experimenter~
primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating primary duplicating duplicating
RPI UCD CU Boulder RPI UCD Caltech Caltech RPI UCD b UCD RPI Caltech UCD RPI CU Boulder CU Boulder RPI UCD PU (4 tests) RPI UCD (2 tests)
51~
Avail. tests
,n,..~
:
3 b. ~=
i
0 all vslues Included • final results I "
I
3 --
•
•1 •
2
3
;
i
PPT
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i
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i
SSTSiSSTaI134sTn9101ABelABeI12341123 4
I
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o
le I oOO I
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i e I IO I
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acceptsblllty limit: "~ = 45%
I
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7
20%
n
40%
60%
-
project, with each centrifuge experiment duplicated by different soil laboratories, the bias induced by type 3 errors is minimized. However, there are some uncertainties related to the validity of the proposed assumptions, a. Model
a'ansdueerPPT-C
4a -
,:'
L'
[
,,
~
UC Davis
:.--:_.
; ~ ~
xoo,, i I0
0
i 20
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40
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!
~"l'-,~",'tt" ~,'~; ~"~"~1" ''"'~""'"
[
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Mode I 1 I 2 3 I 4a I 4 b l 7 , 12 (3) I [ (3)(3)(3)'(4)(4)(5X4) 92 332132231222222221333133~1~2 2213 34 3 0
aRPI Rensselaer Polytechnic Institute; UCD - - University of California, Davis; CU Boulder - - Colorado University at Boulder; Caltech - - California Institute of Technology; PU - Princeton University. bResults not made available. -
0.3
I ee e I le Oi • 0.2 I I I I I acceptability limit: c ~ m d = 0 , 1 4 • I 03 ! N=3; c=0,7 (upper bound) O o, ,
,
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:
~
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. . . .
~?,~ ',\:.~,~..,~,~#~'.',,':,';'¢" ,'~ ......................... , jlr
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,
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~ne (sec) • Fig. 6. Excess pore pressures recorded during repeated centrifuge experiments: (a) reliable experimental results; (b) scattered excess pore pressure measurements.
80%
100% N - number of experimental results available for each transducer location
Fig. 7. Reliability analysis for the VELACS project centrifuge experiments: (a) quantification of experimental measurements; (b) reliability analysis results evaluated for the centrifuge experiments which were made available: centrifuge models 2, 4a, 4b and 12 are found to provide results which are most appropriate for comparisons with 'class A' predictions. due to the low number of test results available for each model (typically 2 or 3) and to the possibility of a scaling generated bias in the experimental results. As discussed previously, the bias or systematic errors in centrifuge experiments have not been studied and therefore must be deemed to be negligible and are neglected in this study. The study is restricted to recorded excess pore pressures, since they best represent the liquefaction phenomenon. The dynamically induced excess pore pressures, recorded during the experiments, are provided as time histories, as illustrated in Fig. 6. F r o m Fig. 6, it is clear that the mean of the experimental results obtained at the same location by different laboratories is relevant if the records are relatively close to each other (e.g. Fig. 6(a)), but leads to questionable conclusions if the records are scattered (e.g. Fig. 6(b)). The reliability analysis is performed in terms of the integral of the recorded excess pore pressure time histories over a significant time interval (Fig. 7(a)); this is made possible by the fact that all the recorded excess pore pressures are positive. The results are presented in Fig. 7(b), in terms of expected error magnitude - - ~, in
100
R. Popescu, J. H. Prevost
eqn (6) - - and likelihood that the mean of measurements lies within a + 10% interval of the true result - - 7 in eqn (13), with a = 0 " l . The various transducer locations are identified in Fig. 3 for each model. Remarks --The numbers N in Fig. 7(b) represent the actual number of test results used to assess the reliability at every transducer location. This number can be less than the total number of tests reported (Table 3), due to missing or malfunctioning transducers. - - A t locations where one of the records is clearly different from the others, the reliability can be improved by eliminating that record. Results are shown in Fig. 7(b) for such cases: with open dots - using all the records; and with black dots - - for improved reliability. - - F o r reference, a comparison with the expected reliability of conventional laboratory soil tests is provided, using the average values resulting from the previous study (Section 3.1). - - Acceptability limits - - upper bound for the expected error magnitude in and lower bound for the confidence level 7 - - are proposed; they are evaluated for experiments with N = 3 tests. Accounting for the complexity of the centrifuge experiments and comparing the results (in, 7) with those obtained from conventional laboratory soil tests, a relatively high value: c = 0.7 is set in the acceptability relationship (8).
The reliability results shown in Fig. 7(b) lead to the following conclusions: • From the viewpoint of result reliability, there are three groups of models: (1) Models 4a (sand layer only) and 12, with overall highly reliable excess pore pressure results, having a degree of confidence of the same order as some conventional laboratory soil tests. (2) Models 4b (sand layer only) and 2, with acceptably reliable results. (3) Models 1, 3 and 7, deemed as providing results with low reliability, in the sense that the comparisons between numerical predictions and the average values obtained from the recordings should be carefully evaluated and not used for arriving at definite conclusions. • All the pore pressure transducers placed in the silt material provided unreliable results. For models 4a and 4b, one of the causes was found to be the low confining stress environment at the transducer locations. 12 • The importance of repeating tests is again emphasized: - - t h e recordings at PPT7 (model 2) and PPT3
(model 12) have the same expected error magnitude 4, ~ 0.05; however, larger confidence in the mean value is offered by model 12 (7 = 87%, N = 4) than by model 2 (7 = 68%, N = 2); - - t h e average expected error magnitude is almost similar for both triaxial and resonant column tests. However, the reliability of the results obtained from the triaxial tests (which have a larger number of tests performed) is clearly superior to those provided by the resonant column tests. Remarks (1) Two out of the three centrifuge experiments labeled as providing low reliable results (models 3 and 7) have special problems related to the complexity of the soil model involved, e.g. vertical (model 3) or steep sloped (model 7) interfaces between different soil materials, which required special methods for test preparation and resulted in likely deviations from the original specifications. 1° (2) The results of the present study only refer to those centrifuge experiment recordings which have been made available to the authors. Since both reliability indices used in the study (in index and 7 index) are dependent on the number of experiments, a higher reliability of the experimental results and, hence, a more sound basis for verification and validation of analytical procedures (see Ref. 13) might have been possible from a larger number of reported centrifuge experiment results. As shown in Table 3, for most of the centrifuge models, three series of experimental results were made available for this study. However, for model #3 only two experimental results were available, while for model # 12 a total of seven experiments were available. This fact must have influenced the reliability study results shown in Fig. 7(b).
CONCLUSIONS A methodology to assess the reliability of a series of soil test results is presented. Under the random sampling assumption, expected error magnitudes with respect to the 'true value' are derived. Within the context of normally distributed experimental results, confidence intervals for the mean are evaluated, and resulting acceptability limits are proposed. The importance of repeating tests is emphasized through both theoretical considerations and numerical examples. The method is employed to assess the comparative reliability of some results of the centrifuge experiments performed for the VELACS project. The analysis results quite clearly point out to those centrifuge models which offer a meaningful data base for comparisons with 'class A' predictions.
Reliability assessment of centrifuge soil 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 Professor Erik H. Vanmarcke for reviewing the paper and offering valuable suggestions. The help of the VELACS Project Steering Committee in providing the experimental results is al,;o acknowledged.
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