Cam Clay Parameters

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Computers and Geotechnics 34 (2007) 524–531 www.elsevier.com/locate/compgeo

Optimisation procedure for choosing Cam clay parameters V. Navarro *, M. Candel, A. Barenca, A. Yustres, B. Garcı´a Geoenvironmental Group, Civil Engineering School, University of Castilla-La Mancha, Avda. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain Received 27 February 2006; received in revised form 24 January 2007; accepted 25 January 2007 Available online 23 March 2007

Abstract This paper analyzes the application of a grid-search approach for the estimation of modified Cam clay parameters from triaxial tests. By means of the systematic sampling of the error, in addition to locating the area presenting the smallest error, its ‘‘roughness’’, is also characterized. This is a valuable information to evaluate the quality of the identification that has been carried out. The methodology proposed here does not aspire to be ‘‘the solution’’ to the problem of parameter identification. The aim is simply to provide a tool which may aid users with criteria.  2007 Elsevier Ltd. All rights reserved. Keywords: Parameter estimation; Modified Cam clay model; Triaxial; Grid-search; Error topology

1. Introduction Since their first introduction (see [1,2]), critical state models have successful in describing many of the most important features of the mechanical behavior of soil [3]. Although several different formulations have been put forth to improve the quality of the predictions of this behavior, the modified Cam clay model (MCC) is still widely referenced and used in solving boundary value problems in geotechnical engineering [4,5]. The overwhelming acceptance of this model has even led to it being applied to simulate compaction, tillage, stresses around growing roots and other deformation events in agricultural engineering [6]. Like other soil models, the MCC model improves its performance if parameter estimation is carried out by choosing experimental data obtained from tests with stress levels, stress states and stress paths close to those for which numerical predictions are subsequently required [7]. Moreover, predictions are also improved if, when estimating model parameters, inverse analysis or identification techniques are applied to the interpretation of the experimental data [8,6]. However, this gives rise to a minimiza*

Corresponding author. Tel.: +34 926 295 453; fax: +34 926 295 391. E-mail address: [email protected] (V. Navarro).

0266-352X/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.01.007

tion problem which is generally not too easy to solve. To collaborate on the resolution of this problem, in this paper we present a tool aimed to support the identification of MCC parameters from triaxial tests. 2. Triaxial tests: error topology for the MCC model The definition of a critical state model requires the determination five constitutive parameters: (i) slope M of the critical state line in the p 0 –q effective stress space (where p 0 = (r1 + r2 + r3)/3 and q = r1  r3, being r1, r2 and r3 the principal stresses); (ii, iii) slopes j and k of virgin compression and unload–reload in the e  ln p 0 space (where e is the void ratio); (iv) the location of the normal consolidation line in the compression plane defined by the void ratio e1 at p 0 equal to 1 kPa; and, (v) some elastic property, such as the Poisson’s ratio m. When the results of the triaxial tests to identify these parameters are analyzed, all the experimental data are usually equally reliable. Therefore, the usual procedure is to select a least-squares fitting criteria, adopting square error SE as the objective or merit function: SE ¼

n X 2 ðeaT;i  eaM;i Þ i¼1

ð1Þ

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531

where n is the number of data points, and ea is the axial strain. Subscripts ‘‘T’’ and ‘‘M’’ define Test and Model values respectively. In certain cases, instead of using the axial strain, the deviatoric strain eS is used. It is defined for triaxial conditions as eS = 2 Æ (e1  e3)/3, where e1 and e3 are the principal strains. The root of the mean square error RMSE = (SE/n)1/2 is also commonly used as the merit function. In any event, experience tells us that these types of objective functions have a very irregular shape for the MCC model, and therefore may be sensitive to the starting point, and converge on a local minimum rather than the global minimum of the function [7]. This can be seen in Fig. 1, which synthesizes a comprehensive simulation exercise. Here, parameters from column 2 in Table 1 were used

1

a

6

11

16

21

26

31

36

41

46

RMSE / RMSE M

1

0.01

0.001

RMSE / RMSE M

1

Search space

GD

Max

Min

0.95 0.093 0.035 1.100 0.3

1.2 0.2 0.04 1.2 0.45

0.8 0.04 0.01 0.8 0.2

10 10 10 10 5

Kaolin M k j e1 (p 0 = 1 kPa) m0

1.02 0.26 0.05 2.913 0.3

1.2 0.3 0.06 2.9 0.4

0.8 0.1 0.03 2.5 0.2

10 10 10 10 5

London Clay M k j e1 (p 0 = 1 kPa) m0

0.888 0.161 0.062 1.828 0.3

1 0.3 0.05 2 0.4

0.6 0.04 0.01 1.6 0.2

10 10 10 10 5

Weald Clay M k j e1 (p 0 = 1 kPa) m0

D-OC D-NC U-OC U-NC

0.01

p0CSD (kPa)

p0O (kPa)

Drained

eO W

K

L

400 400 400 400

100 400 100 400

Yes Yes No No

0.592 0.543 0.592 0.543

1.424 1.355 1.424 1.355

0.949 0.863 0.949 0.863

The identification of the tests is shown in column 1. They were all conventional triaxial tests: after isotropic consolidation, the deviatoric stress was increased with the chamber pressure remaining constant. It was assumed that the effective preconsolidation of the soil, p0CSD , was always equal to 400 kPa. After consolidation, the mean effective stress, p0o was assumed to be equal to 100 kPa (overconsolidated samples), and 400 kPa (normally consolidated samples). Both drained and undrained tests were simulated. Columns 5–7 indicate the void ratio, eO, after consolidation for the three soils under consideration (W, Weald Clay; K, Kaolin; L, London Clay; see Table 1).

0.001

RMSE / RMSE M

Value

Table 2 Definition of the tests simulated to illustrate error topology

0.1

c

Table 1 Description of the examples used to illustrate the ‘‘roughness’’ in the error

Column 2 shows the parameters used for the three soils analyzed. Columns 3 and 4 indicate the space where error topology is described. Column 5 defines the number of grid divisions employed to characterize the above topology.

0.1

b

525

1

0.1

0.01

D-OC

D-NC

U-OC

U-NC

combined

Fig. 1. Curves of minimums: variation of the RMSE in the grids under study (see Table 1). A graph was drawn of the lowest RMSE values which are found as you advance along each one of the grids, starting at the point where the parameters exhibit minimum values, passing through the point at which the RMSE reaches its minimum and ending at the point where all the parameters show their maximum values. (a) Weald Clay, (b) Kaolin and (c) London Clay from Table 1. The 12 curves have been made dimensionless by dividing them by the maximum of the root of the mean square error, RMSEM, recorded in each one. The numerical tests are identified in the figure legend (see Table 2).

to simulate the conventional triaxial tests defined in Table 2. For each numerical test we obtained deformations eaT = [eaT,1, . . . , eaT,n], which were adopted as a reference. We then carried out the simulations associated with each one of the vectors of parameters x = [M, k, j, e1, m] generated after discretizing the search space defined in columns 3 and 4 of Table 1 by means of the grid defined in column 5. So, each numerical test has associated 87,846 (11 · 11 · 11 · 11 · 6) vectors eaM = [eaM,1, . . . , eaM,n], which, when compared with eaT, allowed us to obtain a ‘‘systematic sampling’’ (SyS) of the RMSE. A graph of the variation in the RMSE was made by drawing the lowest RMSE values which are found as one advance along the grid, starting

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531 9000 8196 8000 7000 6000 frecuency

5000 3659

4000

3222

3000

2560

2000

1510

1265

957

191

100

199

1.1E-02

1.5E-02

1000

7.5E-03

2.1E-01

1.4E-01

9.7E-02

6.7E-02

4.6E-02

3.1E-02

0 2.2E-02

at the point where the parameters exhibit minimum values, passing through the point at which the RMSE reaches its minimum, and ending at the point where all the parameters show their maximum values. This resulted in the ‘‘curve of minimums’’ shown in Fig. 1, which illustrates that different local minimum values do exist. The local minimums can also be seen in Fig. 2. In this figure, for the D-NC test (Table 2) on Weald Clay (see Table 1), the variation in the RMSE has been represented with M and k, keeping constants j, e1 and m at the values at which the RMSE is the lowest. Fig. 2 also allows us to corroborate the irregular or ‘‘rough’’ shape of the RMSE. Moreover both Figs. 1 and 2 reveal that parameter identification is often an ‘‘ill-posed’’ problem in the sense of Hadamard [9], as indicated by the ‘‘plateaus’’ that can be seen in the area around the minimum values in both figures. To assess the extent to which these plateaus are important, it is useful to obtain a histogram, like the one portrayed in Fig. 3, for each identification process. Here, the range of the RMSEs recorded in the simulation of test D-NC (Table 2) on Weald Clay (Table 1) has been divided into 10 intervals of exponentially increasing amplitude. As can be seen, a relatively substantial number of points (191) on the grid in which the search space has been discretized, define RMSE values which are similar to the minimum. Consequently, some degree of uncertainty arises when defining the optimum value of x. Kirby et al. [6] made a proposal to reduce uncertainty by conducting a multiobjective identification. This was done as follows. In drained tests, in addition to entering the data relative to the axial or deviatoric deformation, we used data related to the evolution of the void ratio. Merit function FD = RMSEeRMSEe was defined where the root mean square error associated with the axial or deviatoric strain (RMSEe) is equal to the RMSE defined up to now, while the RMSEe was calculated

RMSE

Fig. 3. Histogram of the RMSE (its range has been divided into 10 intervals of exponentially increasing amplitude) related to test D-NC (Table 2) on Weald Clay (Table 1).

1

1

6

11

16

21

26

31

RSME

0.1

FD

RMSE

526

0.01

0.001

grid steps

Fig. 4. A comparison of the evolution of the RMSE and FD associated with test D-NC (Table 2) on Weald Clay (Table 1).

Fig. 2. Variation in the RMSE with M and k, maintaining constants j, e1 and m at the values where the RMSE is the lowest, for test D-NC (Table 2) on Weald Clay (see Table 1).

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531 1

6

11

16

21

26

31

36

41

46

RMSE/ RMSE M

1

0.1

0.01

0.001

grid steps D-OC

D-NC

U-OC

U-NC

combined

Fig. 5. A comparison of the four curves of the minimum values of the RMSE related to Weald Clay (Fig. 1a) with the minimums obtained by jointly analyzing the four tests on the same clay (see Table 2) associated with the four curves.

by replacing e with e. In undrained cases, merit function FU = RMSEeRMSEq was used. Here the RMSEq error associated with the stress path was defined as: 0 !2 11=2 n X qT;i  qM;i 1 A RMSEq ¼ @ ð2Þ n i¼1 qT;AV

where subscripts T and M define, again, Test and Model values, and qT,AV is the average value of q throughout the test. In Fig. 4, we have compared the evolution of the RMSE and FD associated with the numerical test in Fig. 2. Similar to what occurred when performing the same exercise with other tests and soils, the uncertainty has been reduced by very little. This would seem to indicate that we have not succeeding in introducing very much independent information by using the multiobjective analysis. Therefore, an analysis was carried out to determine the effectiveness of combining data from several tests. Fig. 5 presents a comparison of the curves of minimums of Weald Clay (Fig. 1a), with the minimum values obtained after jointly analyzing the four tests on the same clay described in Table 2. As was also demonstrated in the other soils under study, although the ‘‘plateaus’’ seem to be reduced, the roughness remains. While in practice, the joint analysis of several triaxial tests should allow us to clarify the reliability of experimental data [7], in ‘‘ideally reliable’’ numerical tests the results show no significant improvement. The basic question probably lies in analyzing the extent to which a conventional triaxial test is actually able to contribute sufficient information to carry out the identification proposed here. This analysis, however, goes beyond the scope of this paper. Here we will focus on putting forth a method to support the interpretation of the triaxial tests that are usually carried out in Soil Mechanics laboratories.

εa

test SyS

600

600

500

500

400

400

300

300

200

200

100

100

test SyS local

local 400 425 450 475 500 525 550

q (kPa)

q (kPa)

p ' (kPa)

350 375

527

0 575 600

0 0

0.04

0.08 0.12

0.16

0.2

0.24

0.28 0.32 0.36

350 375 400 425 450 475 500 525 550 575 600 0.520 0.500

void ratio

0.480 0.460 0.440 test SyS local

0.420 0.400 0.380

p ' (kPa) Fig. 6. Trajectories followed in spaces p 0 –q, ea–q, and p 0 –e when test D-NC (Table 2) was simulated on Weald Clay (Table 1) by using the parameters identified after performing the SyS and the grid-search (SyS + local identification).

528

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531

3. Proposed identification process We might not have been aware of the uncertainty associated with the determination of the optimal if we had only performed a ‘‘fit-by-eye’’ (expression adapted from Press et al. [10]) by means of curves p 0 –q, q–ea, yea–Du (where Du is the increase in pore water pressure). The good fit seen in Fig. 6 for the simulation of the test associated with Fig. 2 (numerical test D-NC, Table 2, on Weald Clay, Table 1) does not give us any indication of the uncertainty revealed in Figs. 2–4. Moreover, as can be seen in Fig. 6, the fit is even better if we use the optimal value found with the SyS as the initial value for local minimization using gradient-based techniques (see [11,12,8]). Fig. 7 confirms the identification of parameters having errors of generally less than 1% in the cases analyzed with this double global/local strategy. If, however, in the problem associated with Fig. 2, we carry out a simple sensitivity analysis based on the optiM

λ

κ

e1

γ

relative error (%)

100 10 1 0.1 0.01 0.001 0.0001 0.00001

relative error (%)

0.000001 100 10 1 0.1 0.01 0.001 0.0001 0.00001

mal value obtained after local identification, we will once again be able to see that the topology of the error is considerably irregular (see Fig. 8). Therefore, although in the exercises carried out on the soils shown in Table 1 the identification was satisfactory, we cannot be sure that by applying the same methodology, the results will be as successful in other cases. The methodology proposed here, i.e., SyS/global identification + local identification, is commonly referred to as a ‘‘grid-search’’. From a numerical standpoint, it is probably the simplest complete method for bound constrained problems [13]. To identify the minimum, it would be more efficient to use one of the global search algorithms currently available (see, for example Horst and Pardalos [14], or Pinte´r [15]). In the problems analyzed here, however, it took us no more than 4 min on a laptop computer with a 1500 MHz Intel Pentium processor to obtain the 87,846 error values associated with each test (each of them having 121 data points). Therefore, given the fact that the CPU time spent is acceptable, and that with this it is possible to obtain a good characterization of the roughness of the RMSE, we recommend the use of the SyS to analyze triaxial tests. However, grid-search is not appropriate for another constitutive model with more parameters than Cam clay. In this case, from a computational point of view, it may be excessively expensive. Nevertheless, parameter identification should not focus only on determining the optimal value. It should also provide us with information on the existence of other combinations of parameters that are capable of reproducing, with comparable quality, the experimental behaviors observed. It is not a question of mere minimization, but also of offering criteria to be able to evaluate the likelihood of the identification. Therefore, the application of a global identification procedure which can offer us an indication of the error topology is strongly recommended. To assist with this task we have designed a Microsoft Excel spreadsheet which uses a macro to automatically

0.000001 100

4.0E-01 3.5E-01

νΞ

0.1

3.0E-01

e(p'=1) e (p’=1) κ kappa

0.01

2.5E-01

1

RMSE

relative error (%)

10

0.001 0.0001

lambda λ M

2.0E-01 1.5E-01

0.00001

1.0E-01

0.000001 D-OCG U-OCG

D-OCL U-OCL

D-NCG U-NCG

D-NCL U-NCL

Fig. 7. Relative error in each parameter of the model. The error was defined for each parameter and each type of material ((a) Weald Clay; (b) Kaolin; (c) London Clay) using the values defined in column 2 of Table 1 as a reference. The numerical tests are identified in the figure legend (see Table 2). Also indicated is whether the identification was based on the SyS alone (identified by the letter ‘‘G’’), or if a local search was also used (‘‘L’’).

5.0E-02 0.0E+00 -10%

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

10%

variation Fig. 8. Sensitivity analysis of the error around the optimal value (column 2 of Table 4) resulting from the grid-search associated with test D-NC (Table 2) on Weald Clay (see Table 1). The percentage of variation corresponding to each parameter is depicted on the x-axis.

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531

carry out both the calculations associated with the SyS as well as the illustration of the graphs shown in Figs. 1–3 and 6. The spreadsheet includes modules which, assuming that the samples analyzed have a perfectly homogeneous behavior, allow us to characterize their behavior for triaxials tests in which the ratio Dp/Dq is constant (where p is the mean total stress). For undrained tests, since the effective stress path is a monotonic function controlled by the initial void ratio, it was relatively easy to integrate the basic equations of the MCC model (described, for example, in Schofield and Wroth [16], or Wood [17]) to obtain closed-form expressions of q and ea (equal to eS for undrained conditions) as functions of p 0 . For drained tests, the effective stress path may not be a monotonic function, as occurs with the softening of an over-consolidated clay. So, a numerical integration (second order Runge–Kutta) is carried out to compute both the void ratio and the deviatoric strain. Both types of modules, undrained and drained, have been duly verified. Although the systematic sampling needs no user intervention once the analysis has been started, the user must define the search space to be considered, as well as the density of the grid on which it is discretized. In short, the user must ‘‘direct’’ the search. The local identification was carried out with the help of the minimization algorithms included in the ‘‘Solver’’ utility on the spreadsheet. The analysis of the sensitivity of the solution found through local identification (Fig. 8) was also done automatically with a macro. The spreadsheet can be obtained by submitting a request to the first author. 4. Examples of application This section includes two examples of the application of the method. First of all, we analyzed the tests performed by Bishop and Henkel [18] on Weald Clay, using the results reported by Atkinson and Bransby [19] as a reference. As can be seen in Table 3, only tests on normally consolidated samples were analyzed. By simulating the tests on overconsolidated samples (tests with an overconsolidation ratio of 24 were studied) it was found that the MCC model significantly overestimates yield stresses. This is a well known

529

consequence that occurs if yielding takes place to the left of the maximum of the Cam clay ellipse (‘‘dry’’ or supercritical side) [3]. In this article, we will not analyze this question. An excellent study for consultation can be found in Gens and Potts [3]. Here, we have limited ourselves to adopting an MCC model, which, despite its shortcomings on the supercritical side, remains the most widely used critical state model (Borja and Andrade [20]). Both combined and individual analyses of data from tests on normally consolidated samples have allowed us to obtain, by means of the SyS, a good grouping of the parameters identified with regard to the parameters proposed by Schofield and Wroth [16] for Weald Clay (see Tables 1 and 4). However, as can be seen in Fig. 9, error roughness is not negligible. Moreover, the RMSE value obtained after simulating both tests using the parameters of Schofield and Wroth [16] is higher than the value resulting from the parameters identified as optimal (Table 4). What this does is to emphasize the uncertainty associated with the estimation of optimal parameters. To finish the analysis of this example, it is interesting to note that, while joint identification has not eliminated error roughness (local minimums still exist), it has, in fact, reduced the ‘‘plateau’’ around the minimum (see Fig. 9). It has gone from 100 (drained case) and 67 (undrained case) vectors of parameters that define an RMSE similar to that of the minimum, to a mere 20. Secondly, included here is the application of the procedure proposed to obtain the parameters of Spestone Kaolin analysed by Wood et al. [7]. The control maintained in both the making of this soil and the performance of the tests makes the use of these experiments, as an exer-

Table 3 Tests on normally consolidated samples performed by Bishop and Henkel [18] on Weald Clay

D-NC U-NC

p0CSD (kPa)

p0O (kPa)

Drained

eO

207 207

207 207

Yes No

0.632 0.632

As in Table 2, the p0CSD indicates the effective preconsolidation, and p0O the mean effective stress after consolidation.

Table 4 Optimal values found for tests D-NC (column 2) and U-NC (column 3) carried out by Bishop and Henkel [18] on Weald Clay (see Table 3), in addition to the RMSE of both grid-search procedures

M k j e1 (p 0 = 1 kPa) m0 RMSE

O D-NC

O U-NC

O (U&D)-NC

Max

Min

GD

0.88 0.095 0.03 1.139 0.28 2.57 · 103

0.92 0.095 0.035 1.139 0.45 1.39 · 102

0.92 0.106 0.03 1.200 0.4 1.25 · 102

1.4 0.15 0.06 1.2 0.45

0.8 0.04 0.01 0.8 0.2

10 10 10 10 5

D-NC S&W

U-NC S&W 0.95 0.093 0.035 1.100 0.3

3.19 · 102

2.56 · 102

Column 4 shows the parameters and the RMSE associated with the joint identification of the parameters, using the results of the two tests, D-NC and U-NC simultaneously. The search space defined in columns 5 and 6, as well as the number of grid divisions in column 7, were employed in the three identifications. The last two cells indicate the RMSE obtained with each test provided that it is simulated following the parameters proposed by Schofield and Wroth [16], assuming that m = 0.3.

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531 0.18 0.16

combined

0.14 D-NC

RMSE

0.12 0.1

U-NC

0.08 0.06 0.04 0.02 0 1

5

9

13

17

21

25

29

33

grid steps Fig. 9. A comparison of the curves of minimums of the RMSE obtained after performing the grid-search using data from the tests carried out by Schofield and Wroth [16] on normally consolidated samples. A graph is drawn of the minimum values obtained by using data from a drained test (D-NC), an undrained test (U-NC), and by the identification carried out joining together the data from both tests (curve labeled as ‘‘combined’’).

cise in validation, particularly advantageous. Similar to the work carried out by Wood et al. [7] for optimization, among the different tests done, we analyzed the drained cycle of constant p 0 = 150 kPa loading from q = 0 to q = 100 kPa of test L1. Although this soil was tested in a true triaxial apparatus, the whole test was performed with r2 = r3 [7], and can be interpreted as a conventional triaxial test. The parameters were identified by using the search space and grid defined in Table 5. After conducting the SyS, it was found that the minimum was close to the values assigned in column xG of Table 5. The same table shows the parameters identified after the local search, xL. In Fig. 10 it is possible to see that the fit obtained after the local search appears to improve upon the one reported by Wood et al. [7]. Thus, the RMSE values resulting from the parameters proposed by Wood et al. [7] when analyzing both the whole cycle of loading, as well as only the initial part of the shearing, are higher (see Table 5). Since the local search applied in our paper is not of better quality than the one applied by Wood et al. [7], the results would seem to indicate that the

SyS made it possible to start at a more consistent initial value than the initial value used by these authors, thereby improving the end result of the search. It is interesting to take into account that, in our model, we have introduced the effective stress path as known action, whereas Wood et al. [7] have used the observed strain path as input control. Provided that the mean stress paths are very sensitive to the erratic changes experienced by the experimental strain paths, lower variations of the mean stress are obtained as higher values of j and k are considered. In consequence, an extremely high value of j was identified by Wood et al. [7] (see Table 5, and Holtz and Kovacs [21], for instance, to look up typical values typical values of j). Hence, for the shear modulus considered by Wood et al. [7] (4090 kPa and 5000 kPa, for W1 and W2, respectively), negative values of the Poisson’s ratio were obtained. In this sense, the parameters identified by Wood et al. [7] are perhaps less consistent than those that we have obtained.

εs 125

100

75

q (kPa)

530

test

50

grid-search

25

Wood et al. (1992)

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 10. Results in space q–eS of the test carried out by Wood et al. [7] on Spestone Kaolin. The final drained-shearing cycle of constant p 0 = 150 kPa loading from q = 0 to q = 100 kPa was analyzed. The experimental results are compared with those obtained using both parameters resulting from the grid-search (column 6 of Table 5), as well as the optimal parameters identified by Wood et al. [7] after analyzing the complete loading cycle (column ‘‘W1’’ of Table 5).

Table 5 Search space (columns 2 and 3) and grid (column 4) used to identify the parameters of Spestone Kaolin tested by Wood et al. [7] (p0CSD ¼ 150 kPa, p0O ¼ 150 kPa, eO = 1.479) M k j e1 (p 0 = 1 kPa) m0 RMSE

Max

Min

GD

xG

xL

W1

W2

0.9 0.3 0.03 2.9 0.45

0.7 0.1 0.01 2.5 0.20

10 10 10 10 5

0.78 0.22 0.018 2.62 0.35 2.860 · 103

0.73 0.18 0.043 2.404 0.34 2.475 · 103

0.82 0.62 0.350

0.78 0.58 0.430

0.42

0.42

Column xG indicates the parameters identified after carrying out the SyS. The parameters identified after the local search are given in column xL. Columns 6 and 7 contain the parameters proposed by Wood et al. [7], after analyzing both the complete cycle of loading, W1, as well as only the initial part of the shearing (q/p 0 6 0.5), W2.

V. Navarro et al. / Computers and Geotechnics 34 (2007) 524–531

5. Conclusions The application of the grid-search to the estimation of modified Cam clay parameters from triaxial tests has produced good quality results at a lower computational cost. Moreover, the systematic sampling of the error has made it possible to characterize its ‘‘roughness’’, which is a reflection of the uncertainty associated with the location of the optimal value and a way to measure the quality of the identification carried out. To make its application more user-friendly, a free-access spreadsheet was developed. However, the tool is not the one trying to locate the optimal parameters. Although it is of great help, it is the user and his or her criteria, who actually solve the identification problem. Acknowledgements This research was financed in part by a Research Grant awarded to the third and fourth authors by the Education and Research Department of the Castilla-La Mancha Regional Government and the European Social Fund within the framework of the Integrated Operative Programme for Castilla-La Mancha 2000–2006, approved by Commission Decision C(2001) 525/1. References [1] Roscoe KH, Schofield AN. Mechanical behaviour of an idealised ‘‘wet’’ clay. In: Proceedings second European conference on soil mechanics and foundation engineering, Wiesbaden, vol. 1; 1963, p. 47–54. [2] Roscoe KH, Burland JB. On the generalised stress–strain behaviour of ‘‘wet’’ clay. In: Heyman J, Leckie FA, editors. Engineering plasticity. Cambridge University Press; 1968. p. 535–609. [3] Gens A, Potts DM. Critical state models in computational geomechanics. Eng Comput 1988;5:178–97. [4] Potts DM, Zdravkovic L. Finite element analysis in geotechnical engineering: theory. London: Thomas; 1999. [5] Liu MD, Carter JP. A structured Cam clay model. Can Geotech J 2002;39(6):1313–32.

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