A special inverse analysis method is established in order to calibrate soil constitutive models. Taguchi method as a systematic sensitivity analysis is conducted to determine the real values of mechanical parameters. This technique was applied on the hardening soil (as an elastoplastic constitutive model) which is calibrated using the results from pressuremeter test performed on “Le Rheu” clayey sand. Meanwhile, a genetic algorithm (GA) as a well-known optimization technique is used to fit the computed numerical results and observed data of the soil model. This study indicates that the Taguchi method can reasonably calibrate the soil parameters with minimum number of numerical analyses in comparison with GA which needs plenty of analyses. In addition, the contribution of each parameter on mechanical behavior of soil during the test can be determined through the Taguchi method.
One of the most important aspects of geotechnical problems is to adopt a suitable constitutive model for each material. Then, one or more appropriate experimental and/or field tests should be conducted to find the mechanical parameters of each constitutive model. When a set of parameters used in a model is selected so that it creates the most precise coincidence with the soil behavior, then the constitutive model is said to be calibrated. Generally, there are different methods ranging from simple to advanced for calibration of soil constitutive models. Simple conventional calibration techniques typically use stress and strain levels at certain states in which a material undergoes during specific types of laboratory tests. Sometimes, this method of calibration fails to capture the overall behavior of a material, that is, behavior at every point in stress-strain path [
Many researchers adopted inverse analysis method with different modifications. Cekerevac et al. [
In this research, a new systematic search technique is proposed on the basis of genetic algorithm (GA) [
Genichi Taguchi, who first introduced this method during the late 1940s, utilized the conventional statistical tools in a simplified form by identifying a set of stringent guidelines for experiment layout and the analysis of results [
In this paper, the results of pressuremeter tests [
The site is located in the west part of France, in a region called “Le Rheu.” The soil of this site contains reddish sand for tens of meters. Several in situ and laboratory tests have been performed on this soil to identify its mechanical and engineering characteristics. The main reason for selection of this site in current research is the uniformity of the soil type in different depths and the existence of water table at very low levels. These conditions reduce the complexity of modeling process and let all efforts be concentrated on the mathematical solution for inverse calibration.
The results of pressuremeter tests are available at three points of B4, P1, and P2 (Figure
Pressuremeter curves at a depth of 2 m after being modified by lift-off method.
The hardening soil model is an advanced model for simulating the behavior of both soft and stiff soils. When subjected to primary deviatoric loading, the soil shows a decreasing stiffness and simultaneously irreversible plastic strains developing. In the special case of a drained triaxial test, the observed relationship between the axial strain and the deviatoric stress can be well approximated by a hyperbola function, as (Figure
Hyperbolic stress-strain relation in primary loading for a standard drained triaxial test.
In contrast to an elastic-perfectly plastic model, the yield surface of a hardening plasticity model is not fixed in principal stress space, but it can expand due to plastic straining. Distinction can be made between two main types of hardening, namely, shear hardening and compression hardening. Shear hardening is used to model irreversible strains due to primary deviatoric loading. Compression hardening is used to model irreversible plastic strains due to primary compression in oedometer loading and isotropic loading. Both types of hardening are contained in the present model. Table
HS input parameters [
Basic parameters | Explanation | Initial estimates |
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Cohesion |
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Friction angle | Slope of failure line in |
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Dilatancy angle | Function of |
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Secant stiffness in standard drained triaxial test |
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Tangent stiffness for primary oedometer loading |
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Power for stress-level dependency of stiffness | Slope of trend line in |
The first step of numerical modeling is generating the geometry. A mass of soil should be considered in which a borehole is dug to model the pressuremeter test. Then the pressure is induced to the boundary of the soil element adjacent to the middle cell of the probe. Because of the symmetric geometry and loading, only a half of the geometry is modeled (Figure
Geometry of Menard pressuremeter test model.
Region
Region
Region
The boundary conditions and loading position are defined in Figure
Boundary conditions and loading position.
Mesh is generated in the next step as shown in Figure
Mesh generation for pressuremeter model.
For the current pressuremeter test modeling, analysis phases have been defined as follows:
In inverse analysis, a given model is calibrated by iteratively changing input values until the simulated output values match the observed data [
General inverse analysis diagram for calibration of soil constitutive models.
The given constitutive model is calibrated by a repetitive procedure in systematic inverse analysis. In this cycle, input parameters of the constitutive model are changed until the results of numerical simulation match the experimental responses. In this research, the results of Menard pressuremeter tests have been considered as the soil response used for calibration of HS model. A set of input parameters for soil constitutive model which leads to the coincidence of in situ pressuremeter curve and model pressuremeter simulation curve is desired. There is an extreme need for a quantity, which shows the degree of coincidence between the two mentioned curves in order to solve the problem. This quantity which is error function is generally defined as “area between the two curves,” as
Concept of error function.
In this paper, an error function with the following form is used, as
Therefore, the calibration is changed into a familiar optimization problem in which finding a feasible set of soil’s model parameters leads to the least value for error function. Soil constitutive model parameters are those 6 parameters previously introduced in Table
Now, this idealized problem is ready to be solved. The optimization tool used in this research is GA. There are many computer programs written for GA, but none is able to communicate with PLAXIS. To solve this problem, instead of using available programs for GA, a code is written for GA by Visual Basic (VB), which has the ability to interface with the PLAXIS, a useful finite element program which can perform the analysis according to predefined stages. Therefore, this code can change the value of each parameter in that optimization process and obtain the objective function. Figure
The algorithm of the written code for systematic inverse analysis.
The best set of parameters obtained by this method is gained after 496 cycles as shown in Table
The best set of parameters obtained by systematic inverse analysis via GA optimization tool.
Err. fun (kPa) |
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|
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21.8 | 0.76 | 0.23 | 35.11 | 20.14 | 72913 |
In situ pressuremeter curve in comparison with the best simulation curve, pressuremeter simulation curve obtained by inverse analysis (performed after sensitivity analysis), and pressuremeter simulation curve obtained by substituting field attained-parameters (direct method).
Taguchi method is conventionally an approach for sensitivity analysis method, by changing a selected factor in different levels, while the other factors are kept constant. Then, the same process repeats exactly for each of the remaining factors. In Taguchi method, all factors are changed simultaneously according to predefined tables called “orthogonal arrays.” Choosing the appropriate orthogonal array for a given problem is called “experiment design.” The first step to perform a systematic sensitivity analysis is to define experiment design. In order to generate design experiments (i.e., finding the suitable orthogonal array), “degrees of freedom” is needed, which is obtained as follows:
The smallest orthogonal array with the degree of freedom greater than (or equal to) the experiment degree of freedom should be found in this step. Degree of freedom for L16 array is 15:
L16 orthogonal array.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
5 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 |
6 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 |
7 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 |
8 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 |
9 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
10 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |
11 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 |
12 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 |
13 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 |
14 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 |
15 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 2 |
16 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 |
Modified L16 orthogonal array (M16).
1 | 2 | 3 | 4 | 5 | |
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1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 2 | 2 | 2 |
3 | 1 | 3 | 3 | 3 | 3 |
4 | 1 | 4 | 4 | 4 | 4 |
5 | 2 | 1 | 2 | 3 | 4 |
6 | 2 | 2 | 1 | 4 | 3 |
7 | 2 | 3 | 4 | 1 | 2 |
8 | 2 | 4 | 3 | 2 | 1 |
9 | 3 | 1 | 3 | 4 | 2 |
10 | 3 | 2 | 4 | 3 | 1 |
11 | 3 | 3 | 1 | 2 | 4 |
12 | 3 | 4 | 2 | 1 | 3 |
13 | 4 | 1 | 4 | 2 | 3 |
14 | 4 | 2 | 3 | 1 | 4 |
15 | 4 | 3 | 2 | 4 | 1 |
16 | 4 | 4 | 1 | 3 | 2 |
Considered levels for each factor.
Columns | Factors | Level ( |
Level ( |
Level ( |
Level ( |
---|---|---|---|---|---|
1 |
|
20000 | 40000 | 60000 | 80000 |
2 |
|
0.5 | 0.666 | 0.832 | 1 |
3 |
|
1 | 15.66 | 30.32 | 45 |
4 |
|
30 | 33.33 | 36.66 | 40 |
5 |
|
0 | 3.33 | 6.66 | 10 |
Final plan of experiments is shown in Table
Final plan of experiments for this project.
Test number |
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|
|
|
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Result |
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1 | 20000 | 0.5 | 1 | 30 | 0 | 648 |
2 | 20000 | 0.666 | 15.66 | 33.33 | 3.33 | 410 |
3 | 20000 | 0.832 | 30.32 | 36.66 | 6.66 | 281 |
4 | 20000 | 1 | 45 | 40 | 10 | 191 |
5 | 40000 | 0.5 | 15.66 | 36.66 | 10 | 44 |
6 | 40000 | 0.666 | 1 | 40 | 6.66 | 660 |
7 | 40000 | 0.832 | 45 | 30 | 3.33 | 93 |
8 | 40000 | 1 | 30.32 | 33.33 | 0 | 159 |
9 | 60000 | 0.5 | 30.32 | 40 | 3.33 | 315 |
10 | 60000 | 0.666 | 45 | 36.66 | 0 | 256 |
11 | 60000 | 0.832 | 1 | 33.33 | 10 | 910 |
12 | 60000 | 1 | 15.66 | 30 | 6.66 | 203 |
13 | 80000 | 0.5 | 45 | 33.33 | 6.66 | 549 |
14 | 80000 | 0.666 | 30.33 | 30 | 10 | 272 |
15 | 80000 | 0.832 | 15.66 | 40 | 0 | 550 |
16 | 80000 | 1 | 1 | 36.66 | 3.33 | 750 |
Obtained data of Table
Results of ANOVA table.
Col. number | Factor | DOF ( |
Sum of Sqrs. ( |
Variance ( |
|
Pure sum ( |
Percent contribution |
---|---|---|---|---|---|---|---|
1 |
|
3 | 308775.687 | 102925.229 | — | 308775.687 | 14.418 |
2 |
|
3 | 203893.687 | 67964.562 | — | 203893.687 | 9.52 |
3 |
|
3 | 1138383.187 | 379461.062 | — | 1138383.187 | 53.156 |
4 |
|
3 | 386871.687 | 128957.229 | — | 386871.687 | 18.064 |
5 |
|
3 | 103635.187 | 34545.062 | — | 103635.187 | 4.839 |
Other/error | 0 | ||||||
| |||||||
Total: | 15 | 2141559.437 | 100.00% |
The parameter with the degree of importance less than 10% of the most significant factor will be assigned a constant value and removed from the inverse analysis process. As a result, parameter
The best set of soil constitutive model parameters obtained by systematic inverse analysis after performing sensitivity analysis and removing the unimportant parameter.
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Err. fun. |
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60629 | 28 | 34.8 | 2 | 0.98 | 28 |
The HS constitutive model for “Le Rheu” soil has been calibrated directly [
The set of parameters obtained by direct method (experimental method).
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|
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|
Err. fun. |
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55000 | 35 | 32 | 2 | 0.5 | 75.30 |
The main purpose of the paper is to introduce a systematic approach to derive mechanical parameters of a typical soil constitutive model based on an available test data. For adopted example of “Le Rheu” clayey sand, three different test data related to three points of B4, P1, and P2 were available at depth of 2 m. The proposed approach can be applied to the curve of each point independently, and then the corresponding mechanical parameters may be averaged to represent the mean values of the soil mechanical parameters of “Le Rheu” clayey sand at depth of 2 m. According to Figure
Taguchi method was originally proposed to design experiments. However, in this paper it was adopted to derive the mechanical parameters of a soil through systematic inverse analyses. On the other hand, GA is an optimization technique which was utilized here to obtain the optimum parameters fitting to an available soil test data. Though, the above two methods are different tools in engineering and scientific practice, in this paper they were utilized for a single specific application, that is, the calibration of a soil constitutive model. Accordingly, the comparison achieved in the paper between Taguchi and GA methods is only attributed to the precision of the results and the number of analyses needed in each method. In addition, giving the relative significance of each mechanical parameter in soil constitutive model is another ability of the method based on Taguchi approach.
The results of obtained parameters (Table
Taguchi method is a systematic approach for designing experiments which investigates how different parameters affect the mean and variance of a process performance characteristic. However, it is very important to determine the most important parameters (factors) governing the process since the total number of parameters involving the process might be high. In addition, the variation range of each parameter should be introduced as much as limited in order to define minimum number of levels. These considerations may need some experiences and, without such information, the method may not be effective and useful. Having a large number of parameters (factors) with a wide range of variation for each parameter tends to select the orthogonal arrays with numerous tests. This will be time consuming and expensive from computational costs point of view.
Regarding the ability of the method to be applied on other tests or constitutive behaviors, it can be useful to say that we have already utilized the method in order to extract the Mohr-Coulomb perfect plastic parameters of soil from the results of pile load tests [
In this research, a systematic inverse analysis approach is introduced for calibration of soil constitutive models. The capability of this method has been shown in the case of calibrating HS constitutive model for “Le Rheu” soil in pressuremeter stress path. The benefits of using this method are being able to be used for many laboratory or field tests, and also constitutive models, giving the whole parameters simultaneously, automatic procedure of calibration with least interpretation, and considering overall soil behavior (i.e., behavior at every point in stress-strain path).
The Taguchi method is a useful tool for parametric analysis which can be beneficial in geotechnical engineering due to its relatively high precision and low time consumption. Furthermore, the significance of the parameters can be evaluated quantitatively using the Taguchi method. In the current research, it was exhibited that the parameters of soil cohesion and internal friction angle have the most influence on the hardening soil elastoplastic constitutive model and the dilatancy angle has the least influence. This conclusion is probably valid only for clayey sand located in Le Rheu site. For granular soils with large size grains such as gravels in which the dilatancy angle is large, it is possibly expected to observe more contribution of dilatancy.
As illustrated in Tables
Comparison of used methods.
Direct method | GA | Taguchi | |
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Error function (kPa) | 75 | 21.8 | 28 |
Number of analyses | 1 | 496 | 16 |
Importance of parameters | NA | NA |
|