Estimation of geotechnical properties using the Taguchi's design of ...

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Estimation of geotechnical properties using the Taguchi’s design of experiment (DOE) and back analysis method based on the field measurement data Case study: Tehran metro line 7

Abstract Accurate estimation of the geo-mechanical parameter is a challenging task in geotechnical engineering. This paper aims to estimate the geotechnical parameters of the soil layers in the Tehran metro line 7 based on the field measurement data. The Design of experiment (DOE) methodology is utilized for estimation of the geotechnical parameters, which is considered less in the geotechnical engineering. In this paper, the geotechnical parameters such as the modulus of elasticity (E), cohesion (c), and the internal friction angle (φ) are estimated using Taguchi’s DOE and displacement-based back analysis. The results of the DOE experiments are examined by the analysis of range (ANORA) to improve the accuracy of the optimization. Based on the analysis results, the Taguchi’s DOE technique offers more effective and precise estimation of the geotechnical parameters than displacement-based back analysis approach. Finally, the optimal combination of geotechnical parameters is proposed based on analysis results of the both Taguchi’s DOE and back analysis method. It is proved that the combination of the proposed geotechnical parameters by Taguchi’s DOE is more similar to the field measurements data. Keywords: Geotechnical parameters, back analysis technique, design of experiment, 3D FEM modelling, orthogonal array (OA) 1. Introduction Accurate estimation of the geo-mechanical parameter is a challenging task in geotechnical engineering. Accuracy of the obtained parameters from the geotechnical investigations has a vital role for suitable understanding of the ground behavior in tunneling projects. According to the literature, many researches have been conducted back analysis to estimate the geotechnical parameters since 1970, when computer aided to engineering. Back analysis is generally defined as a developed procedure for solving system problems, which can provide the controlling parameters of a system by analysis its output behavior. The first application of the finite element (FEM) back analysis method is conducted by Kavanagh and Clough (1971). Thereafter, many researchers utilized this method to study the various subjects in the soil and rock engineering. Some of these studies summarized in Table 1. Table 1. Review of the performed researches in back analysis methods Subject year researcher Estimation of the soil and rock properties based on the instrument 1980 Gioda, G., et al., (1980) data 1983 Sakurai, S., et al., (1983) 2007 Cai, M., et al., (2007) 2008 Hisatake, M., et al., (2008) Estimation of the Modulus of elasticity and excavation loading 1985 Hisatake, M., et al., (1985) Assessment of the rock mass hydraulically characteristics and 2001 Tonon, F., et al., (2001) boundary conditions Back analysis of the geomechanical parameters 2011 Miranda, M., et al., (2011) Back analysis of shear strength parameters of landslide slip 2013 Zhang, K., et al., (2013) Determination of the time-dependent properties of the tunnel host 2014 Asadollahpour, E., et al., (2014) rock mass Prediction of the mechanical behavior of the soil 2015 Zhao, C., et al., (2015) 2015 Jia, Y., et al., (2015)

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Application of the probabilistic back analysis techniques for 2016 Li, S., et al., (2016) analyzing the high cut rock slope

The displacement-based back analysis technique is one of the most important approach in the existing studies (Sharifzadeh, Tarifard, & Moridi, 2013)]. On the other hand, Taguchi’s DOE method is a statistical method developed by Taguchi and Konishi [(G. Taguchi & Konishi, 1987)] to optimize the process parameters and improve the quality of manufactured goods; later, its application was expanded to many other fields in engineering, biotechnology, marketing, advertising, etc. [(Rao, Kumar, Prakasham, & Hobbs, 2008)]. This method principle was introduced by R. A. Fisher in the 1920s, and the concept was improved in the 1940s by Taguchi(1987). According to the literature, the DOE method has been applied in different industries successfully, includes manufacturing, military, automobile, aerospace and other industries. In the case when one observes many inputs and the simultaneous impact of these inputs on the response is important, the Taguchi’s approach is used for design of experiment [(Roy, 2001); (Casalino, Curcio, & Minutolo, 2005); (Ozcelik & Erzurumlu, 2006)]. Taguchi approach has been used in many fields, because of its ability to optimize the process parameters. Review of the performed studies about of the DOE methodology in various fields summarized in Table 2. Table 2. Review of the done studies in DOE methodology Fields year researcher environmental 2007 Daneshvar, N., et al., (2007) sciences 2009 Delgado-Moreno, L.., et al., (2009) 2016 Sutcu, M., et al., (2016) agricultural sciences 2007 Tasirin, S. M., et al., (2007) 2015 Rasoulifard, M.H., et al., 2(2015) medical sciences 2006 Ng, E.Y. et al., (2006) 2013 Taguchi, K., et al., (2013) 2014 Kurmi, M., et al., (2014) 2016 Sánchez-López, E., et al., (2016) chemistry 2007 Houng, J.Y., et al., (2007) 2012 Rüfer, A., et al., (2012) 2014 Dawud, E.R., et al., (2014) 2016 Siyal, A.A., et al., (2016) Physics 2006 Wu, C.H., et al., (2006) 2014 Tučková, M., et al., (2014) 2016 Tchognia, J.H.N., et al., (2016) statistical technics 2007 Romero-Villafranca, R., et al., (2007) 2010 Kitchen, R.R., et al., (2010) 2015 Li, M., et al., (2015) 2016 Mamourian, M., et al. (2016) management and 2004 Elshennawy, A.K. (2004) business 2011 Azadeh, A., et al., (2011) 2015 Chompu-inwai, R., et al., (2015) optimization of the 2016 Balki, M.K., et al., (2016) operating parameters 2016 Lai, F.M., et al., (2016)

In spite of the current application of DOE methodology in several fields, its use in geotechnical engineering is not popular. This paper aims to estimate the soil mechanical properties based on the field measurement data in Tehran metro line 7 project. In this research, Taguchi’s DOE methodology and displacement-based back analysis method applied for this purpose. The results of the DOE experiments are examined by the analysis of range (ANORA) to improve the accuracy of the optimization. This study also aims to enhance the rules of Taguchi method to estimate the geotechnical parameters. Therefore, the results of Taguchi’s DOE methodology are compared to displacement-based back analysis approach to demonstrate the efficiency of the DOE method. The considered geotechnical parameters are the Modulus of elasticity (E), cohesion (c), and the internal friction angle (φ).

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First, the outputs of back analysis calculations and Taguchi’s experiments are used to simulate the shield mechanized tunneling using the 3D finite element modelling. Field measurement data used to validate the numerical modeling results. The deviation of the numerical analysis of settlement field form real one using the Taguchi’s DOE and back analysis methods are calculated and compared together. In back analysis approach, the five levels are selected for nine geotechnical parameters (E1, E2, E3, c1, c2, c3, φ1, φ2 and φ3) concerning three soil layers surrounding the tunnel. The univariate optimization algorithm optimizes the geotechnical parameters in three steps using the 15 numerical models. The Taguchi’s DOE methodology uses special orthogonal arrays (L27) for the experiments. In this method, three levels are considered for geotechnical parameters. The experiment results are examined by the analysis of range (ANORA). Finally, the optimal combination of geotechnical parameters is proposed based on analysis results of the both Taguchi’s DOE and back analysis method. 2. Geological and geotechnical investigation 2.1. Location The East–West section of Tehran Metro line 7 is designed to be 12 km long, which is under construction in urban areas. The tunnel diameter is about 9 m that excavated with EPB Tunnel Bore Machine (TBM) manufactured by SELI Tecnologie Corporation. The tunnel cover depth to a diameter ratio in section under study is equal to 1.8, representing a shallow tunnel in an urban area. A plan of the proposed tunnel route is illustrated in Figure 1.

Figure 1. Proposed tunnel route line 7, Tehran Metro, Iran

2.2. Geology properties Tehran plain classified into four formations based on geological characteristics that identified as: A (Hezardarreh formation), B (Kahrizak formation), C (Tehran Alluvial formation) and D (Recent Alluvium), as shown in Figure 2(a). The proposed tunnel route passes through the formation D, which consists of non-cemented soil with variable grain sizes (from clay to boulder) (Fakher, Cheshomi, & Khamechiyan, 2007). Geological and geometrical properties of soil layers presented in Figure 2(b). The detailed geological/geotechnical investigations to identify the subsurface stratification, groundwater conditions and geological characteristics of the ground is performed by excavation of 61 boreholes (2487.7m total length) and 16 test pits (296.95m total length). Geotechnical investigations mainly included some field tests and surveying such as Plate loading test (PLT), in situ shear test, Pressuremeter test, standard penetration test (SPT),

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lufran permeability test and In-situ density test; laboratory tests such as direct shear test, Triaxial test, particle size analysis, Atterberg limits test, consolidation, Permeability, Los Angeles Abrasion test and X-Ray Fluorescence (XRF); desk studies including collection of the existing data such as previous reports and drawings, analysis of the laboratory and in situ test results and data processing and analyzing (SAHEL Consulting Engineers (2009)).

a)

b) Figure 2. a) Stratigraphy of Tehran Alluviums formations (Fakher et al., 2007); b) Geological and geometrical properties of soil layers

2.3. In situ measurement data The monitoring activities and field measurements are aimed to a) checking and refining the design choices, b) optimizing the operation of TBM and the excavation modes, and c) minimizing the ground settlements due to the deformations induced by the tunneling, particularly to verify the simulation of the tunneling procedure. The monitoring process is a direct and indirect observation and surveillance of soil movements using the instrument tools. These supervisory data can be applied for controlling the constructed structure behavior in soil as quantitative and qualitative. The monitoring activities in this project is performed by installing the network of ground settlement points 1.2m length arranged on the tunnel axis with and transversally to the route plan (1 to 3 benchmarks per cross section). It done to minimizing the subsidence of existing buildings and facilities due to the deformations induced by the excavation in the top soils. A part of installed instrumentation layout in this project is illustrated in Fig. 3(a). In this project, the settlement trough is monitored through extensive field instrumentation using the settlement point instruments, as shown in Figure 3(b). Figure 3(c) shows the schematic monitoring plan of the section under study.

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a)

b)

c) Figure 3. a) A part of installed instrumentation layout b) Settlement point instruments; c) schematic monitoring plan of the section under study

Figure 4 shows the measured settlement values for five Settlement points. Table 3 summarizes the settlement field measurements from a section of tunnel route that include settlements in seven Settlement points (C1, C2, C2L C2R C3, C4 and C6). 5 0 -5

Settlement (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

C1

-10 -15

C2L

-20

C2

-25

C2R

-30

C3

-35 -40 -45 Figure 4. Reading of settlement in the C1, C2, C2L, C2R and C3 Table 3. The measured settlement values in Settlement points Settlement point C1 C2 C2R C2L C3

Settlement (mm) 39.4 46.7 28.9 29 42.5

C4 C6

60.5 51.2

Distance to centerline (mm) 0 0 7.5 7.5 0 0 0

2.4. Finite element mesh and boundary conditions In this study, the finite element analysis package Plaxis 3D Tunnel is applied for the numerical analyses to simulate the shield mechanized tunneling, including the application of face pressure, shield advancement, grouting and installation of lining.

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Simulation of the tunnel construction process is started with the selection of the model geometry. Considering the symmetry of the tunnel geometry, one-half of the entire domain is created in the analyses shown in Figure 5. Outer boundaries are placed far from the tunnel to eliminate their effects on the analysis results. The numerical model geometry has a 45 m (five Tunnel Radius (TR)) width, 42 m (9.3 TR) height and 105 m (23.3 TR) length generating by 3D wedge iso-parametric elements with 15 nodes. Figure 5 shows the 3D finite element model that is applied in the numerical analyses. The soil layers are modelled as hardening soil material. The Hardening-Soil model is a model for simulating the behavior of different types of soil, both soft soils and stiff soils. When exposed to primary deviatoric loading, soil shows a decreasing stiffness and simultaneously irreversible plastic strains develop. In the special case, the observed relationship between the axial strain and the deviatoric stress can be approximated by a hyperbola. Such a relationship was first formulated by Kondner (1963) and later used in the well-known hyperbolic model (Duncan & Chang, 1970). The Hardening-Soil model, however, supersedes the hyperbolic model by far; firstly, using the theory of plasticity rather than the theory of elasticity. Secondly, by including soil dilatancy and thirdly by introducing a yield cap. The structural characteristics such as lining and shield machine are modelled as linear elastic material. The shield is modelled in 10.5 m length using the solid elements with Bulk weight of 38.15 kN/m/m, Flexural stiffness (EI) of 8.38e4 kNm2/m and Normal stiffness (EA) of 8.2e6kN/m. Excavation round length is 1.5 m equal to segment width. The segments are modelled with Young’s modulus of 30GPa, density of 24 kN/m3 and poison’s ratio of 0.2. In the current analysis, interface elements (Rinter) are applied to permit relative displacement at the soil–lining interface. An interface element is defined by the shear stiffness (Ks) and normal stiffness (Kn) that exist at a point in space that is in contact with a finite plane. 3. Research Methodology This research is conducted to estimate the geotechnical parameters of the soil layers in the Tehran metro line 7 based on the field measurement data by Taguchi’s DOE and back analysis method. Also, the Taguchi’s DOE optimization methodology Compared with back analysis approach to estimate the modulus of elasticity (E), cohesion (c), and the internal friction angle (φ). For this purpose, outputs of back analysis calculations and outputs of Taguchi’s experiments are applied to simulate the shield mechanized tunneling using the 3D finite element modelling. Settlement field measurements are used to validate the numerical models. The numerical results compare with the corresponding field measurement. Considering the results, the most influence parameter on the surface settlement amongst the E, c and φ is determined. Then, optimal combination of the proposed geotechnical parameters by the both methods is suggested. In this research, there are nine geotechnical parameters (E1, E2, E3, C1, C2, C3, φ1, φ2, φ3). Coincident influence of these parameters should be considered. Taguchi’s DOE method is applied, because it provides the possibility of concurrent changes in all input factors (geotechnical parameters) and considers the response changes (surface settlements).

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Figure 5. Finite element mesh and boundary conditions

3.1. The analysis process in the back analysis technique Back analysis is generally defined as a procedure developed for solving system identification problems. It can provide the controlling parameters of a system by analysis its output behavior. The two approaches for the back analysis of geotechnical problems, referred to as “inverse” and “direct” approaches. In the inverse approach, the problem governing equations is rewritten in such a way that geomechanical parameters appear as unknowns and field measured values as input data. In the direct approach, the trial values of the unknown parameters employ as input data, until the discrepancy between measurements and calculated numerical results is minimized. The main components necessary to perform back analysis through the direct approach are the following (Oreste, 2005): 1. A calculation model that can determine the field displacement of the formation; in this study, the 3D numerical modelling is used as a calculation model; 2. An error function; Direct formulation is very flexible and implementing such an approach for complex constitutive models is easier. In addition to, development of the back analysis code is less difficult than development of the code based on an inverse algorithm. The only work is appending an existing program with a module. The applied method is based on optimization of geomechanical properties of soil by trial and error. For this reason, an error function is selected, which minimizes the discrepancy between the quantities measured in the field and computed values through numerical modelling. This discrepancy is expressed as the error function: 𝑛 1 𝑢𝑚 (𝑝) − 𝑢𝑖 𝐸(𝑝) = √ ∑ ( 𝑖 ) 𝑛 𝑢𝑖 𝑖=1

2

Eq. 1

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Where ui and uim(P), i = 1,2,...,n are the measured and corresponding numerical results, respectively and, n is a number of measurement points. Based on the mentioned error function, back analysis aims at the determination of the absolute minimum of E(p). Here, it used a normalized error function to decrease the effect of measurements error. This function provides the best agreement between measured and computed values. 3. The optimization algorithm to reduce the difference between the calculated results and the measured values. The displacement-based back analysis of field measurement data is one of the appropriate approaches for estimating the geotechnical properties correctly. This method used the univariate optimization algorithm, alternative univariate method and pattern search method. The Studies performed by Jeon and Yang (2004) indicated that the univariate optimization algorithm and the alternative univariate method search the design parameters successfully. However, the pattern search method fails in some cases. In univariate optimization algorithm, only one variable is changed at a time and the values of other variables are fixed. After optimizing the one variable, in the next step the value of fixed another variable in previous step is varied and the values of other variables are fixed. This process is continued until the all variables are optimized. The used back analysis algorithm for optimization is shown in Figure 6. Displacement-based back analysis method using the univariate optimization algorithm

Determination of effective factors on the problems; E, C and φ Creation of the five numerical models: E: variable C: constant φ : constant

Decision of levels for factors based on the geotechnical investigations

Creation of the five numerical models: E: optimized C: variable φ : constant

Measuring the field settlement data of levelling points

Creation of the five numerical models: E: optimized C: optimized φ: variable

Calculating the errors of models according to the defined error function No

Whether the error is minimal? For models with E= variable For models with C= variable For models with φ= variable

Yes Optimum E Optimum E, C Optimum E, C, φ

Figure 6. Back analysis algorithm for this problem

3.2. The analysis process in the DOE, Taguchi approach Taguchi is a statistical method developed by Genichi Taguchi to consider the impact of different parameters on variance of performance characteristic that determines the suitable operating conditions of the process [(Kumar, Sureshkumar, & Velraj, 2015)]. It’s making statistical design by using the orthogonal array (OA), also offers an opportunity to reduce the number of experiments (Balki et al., 2016). In this research, Taguchi’s DOE methodology

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is applied to find the optimum geotechnical parameters with a minimum number of the experiments. General steps in Taguchi’s DOE method for such geotechnical subjects are as follows: 1. First, the field measurement data corresponding to the results of numerical modelling are determined. The deviation of the numerical results from the settlement of the Settlement points is defined as Test Index (TI), in order to reach a correct judgment. 2. The influence parameters on the surface settlement are defined and number of possible parameter levels that can vary are determined. 3. An orthogonal array (OA) is arranged, which columns represent parameters and rows represent the number of experiments. 4. The experiments (models) are simulated according to the arrangement of the orthogonal array (OA) and the results are considered for selecting the optimum levels of the factors. 5. Finally, analysis is done from the experimental data to determine the effect of geotechnical parameters on the ground movement and optimal combination of the geotechnical parameters. The results of the DOE experiments are examined by the analysis of range (ANORA) to improve the accuracy of the optimization. The used flowchart in the Taguchi’s DOE for this problem illustrated in Figure 7. Design of Experiment (DOE), Taguchi method

Decision of levels for design factors

Arrangement of the orthogonal array, (OA) matrix

Determination of effective factors on the surface settlement; E, C and φ

Definition of Test Index (TI) factor

Simulation and execution of the experiments (numerical models) according to arraies (OA)

Calculation of TI for experiments (models): Calculation of the error based on the numerical modelling results and the field measurement data comparison Determination of optimal levels for design factors and optimal combination of parameters: 1. Analysis of range (ANORA) for experiment responses 2. Determination of optimal levels for each design factor based on the minimum error in numerical modelling results 3. Combination of the optimal levels of each factor for determining the optimal combination of the parameters Figure 7. Flowchart of the Taguchi’s DOE methodology for this problem

Taguchi’s L27 orthogonal array is applied to determine the optimal geotechnical parameters. Geotechnical parameters, including (E1, E2, E3, c1, c2, c3, φ1, φ2 and φ3) with three levels for all design factors is considered as input parameters. Test Index (TI) factor defined as the deviation of the numerical modelling results from the field settlement of the Settlement points (C1, C2, C2L, C2R and C3) ( Table 3). Input data (e.g. geo-mechanical parameters) for numerical modelling are obtained from soil mechanic tests on the drilled cores in geotechnical investigations. 4. Results of estimating the optimal geotechnical properties

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4.1. Back analysis results Back analysis technique is used to estimate of the Modulus of elasticity, cohesion, and the internal friction angle. Range of geotechnical parameters for soil layers is obtained from the geotechnical investigations, as illustrated in Table 4. Table 4. Geotechnical parameters and their range Layer

Depth (m)

L1

9 - 20.5

L2

20.5-26.5

L3

0–9 > 26.5

parameters E1 (MN/m2) C1 (kN/m2) φ1° E2 (MN/m2) C2 (kN/m2) φ2° E3 (MN/m2) C3 (kN/m2) φ 3°

1 60 10 22 30 22 20 20 22 16

2 70 14 26 40 26 24 30 26 20

Levels 3 80 18 30 50 30 28 40 30 24

4 5 90 100 20 24 34 38 60 70 34 38 32 36 50 60 34 38 28 32

The back analysis is performed in three steps. In first step, five numerical models are created using the E, c and φ parameters (Table 5) to optimize the modulus of elasticity (E), where c and φ are constant and E is variable. Table 5. Used geotechnical parameters for numerical simulation in three steps of back analysis Step 1 Step 2 Step 3 models 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 E1 60 70 80 90 100 70 70 c1 10 10 14 18 20 24 18 φ1 22 28 30 34 36 22 22 E2 30 40 50 60 70 40 40 c2 22 22 26 30 34 38 30 φ2 20 20 20 24 28 32 36 E3 20 30 40 50 60 30 30 c3 22 22 26 30 34 38 30 φ3 16 16 16 20 24 28 32

The computed settlements of the numerical simulations corresponding to the leveling points C1, C2, C2L, C2R and C3 are compared with the field settlement measurements, which are presented in the Table 3. The error values for each model are calculated according to error function using equation, Eq. 1. The numerical results for the first step are presented in Table 6. Table 6. Parameters and numerical results for optimization of E Layer L1 L2 L3

c (kN/m2) 10 22 22 Error (%)

φ° 22 20 16

E1 60 30 20 7.69

E(MN/m2) ranges E2 E3 E4 E5 70 80 90 100 40 50 60 70 30 40 50 60 5.94 11.7 15.49 19.21

The listed error values in Table 6, indicate that the error values are reduced from 19.21% to 5.94%. The model No.2 has a minimum error value of 5.94. Hence, this model is selected as the optimal model and the corresponding E of this model as the optimum value of E parameter (Table 6). The settlement troughs for these numerical models are illustrated in Figure 8(a). Optimization of the cohesion is carried out in the second step. In this step, five models are created, which in these models, E and φ are constant and c is a variable. These parameter values are presented in Table 5. The optimized E value that is obtained in the previous step assigned to the new models, and cohesion parameter is changed.

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Table 7. Parameters and numerical simulating results for optimization of C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Layer L1 L2 L3

E (MN/m2) 70 40 30 Error (%)

φ° 22 20 16

c1 10 22 22 5.94

c(kN/m2) ranges c2 c3 c4 14 18 20 26 30 34 26 30 34 6.91 4.85 13.7

c5 24 38 38 15.11

Similar to the first step, the deviation of the numerical simulating results from the field settlements is calculated. As seen in Table 7, error values are reduced from 15.11% to 4.85%. The model No.8 has a minimum error value, hence, this model is selected as the optimal model, and the corresponding c of this model as the optimum cohesion. The settlement troughs for these numerical models are illustrated in Figure 8(b). Optimization of the internal friction angle (φ) is done similar to the used approach in the previous steps. The values of E, c and φ parameters are presented in Table 5. In step 3, the optimized E and c in previous steps assigned to new models and φ is changed. Similar to the previous steps, error values for models are calculated according to Eq. 1, and shown in Table 8. Table 8. Parameters and numerical simulating results for optimization of φ Layer L1 L2 L3

E c (MN/m2) (kN/m2) φ1 70 18 22 40 30 20 30 30 16 Error (%) 4.85

φ° ranges φ2 φ3 φ4 26 30 34 24 28 32 20 24 28 7.22 4.11 5.41

φ5 38 36 32 8.37

The listed results in Table 8 show that error value is reduced from 8.37% to 4.11%. The model No.13 is selected as the optimal model and its φ as the optimum internal friction angle. The settlement troughs for these numerical models are illustrated in Figure 8(c). The results indicated that minimum deviation of the numerical results from the field measurement data is achieved when that E, c and φ are in accordance to Table 9. Table 9. The optimized geotechnical parameters by back analysis Layer L1 L2 L3

EL1 (MN/m2) EL2 (MN/m2) EL3 (MN/m2)

E (MN/m2) 70 40 30

C (kN/m2) 18 30 30

60 70 80 30 40 50 20 30 40 a) Different Es

11

90 60 50

φ° 30 28 24

100 70 60

cL1 (kN/m2) cL2 (kN/m2) cL3 (kN/m2)

φL1 φL2 φL3

22° 20° 16°

10 22 22

14 18 26 30 26 30 b) Different cs

26° 30° 34° 24° 28° 32° 20° 24° 28° c) Different φs

20 34 34

24 38 38

38° 36° 32°

Figure 8. Settlement troughs for Es, cs and φs

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

220 200 180 160 140 120 100 80 60 40 20 0 ΔE1

Error percentage %

The error values variance versus variation of the geotechnical parameters (E1, E2, E3, c1, c2, c3, φ1, φ2 and φ3) in the numerical models is illustrated in Figure 9.

variation percentage %

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

ΔE2

ΔE3 ΔC1 ΔC2 ΔC3 Δφ1 Δφ2 Δφ3 parameters variations errors variations Figure 9. The error variations percentage versus the variation percentage in each parameter

According to the Figure 9, in spite of the minimum variation of the E1 (67%), the error variation is maximum (13.27%). This means 67% variation from E1 to E5 reduces the error by 13.27% from 19.21 to 5.94. Therefore, the modulus of elasticity has the highest effect on the settlement than cohesion and internal friction angle. In order to have better understanding about the parameters effect on the settlement, Figure 10 shows the deviation of numerical modelling outputs from the corresponding settlement field measurements.

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Level 1 20 15 10 Level 5

Level 2

5 0

Level 4

Level 3

Es

cs

φs

Figure 10. Error values for the different 5 levels

The pentagon larger area means maximum variation of the error versus geotechnical parameters variance, which indicating the higher effect of the related parameter on the settlement. According to Figure 10, the area of the pentagon related to E is larger, which indicates the more effectiveness of the modulus of elasticity on the surface settlement than other parameters. 4.2. Taguchi’s DOE results There are cases where the interaction between experiment factors requires to conduct a multi-factor analysis, in which case DOE based on the Taguchi method can apply because of its ability to change multiple factors at once. 4.2.1. Selection of the Test Index (TI) A suitable TI must be (a) sensitive to the input parameters, and (b) measurable. In this study, test index is defined as the deviation of the settlement of the Settlement points (C1, C2, C2L, C2R, C3) from corresponding to numerical results. In this case, TI is defined as: 2

𝑛 1 𝑢𝑚 (𝑝) − 𝑢𝑖 𝑇𝐼 = 𝐸(𝑝) = √ ∑ ( 𝑖 ) 𝑛 𝑢𝑖 𝑖=1

Eq. 2

where ui and uim(P), i = 1,2,...,n are the measured and corresponding calculated numerical results, respectively, and n is the number of measurement points. Here, we used a normalized error function to decrease the effect of measurements error. 4.2.2. The design factors and orthogonal array (OA) In this section, the Modulus of elasticity (E), cohesion (c), and internal friction angle (φ) are chosen as effective design factors on the settlement. The nine design factors and their levels are summarized in Table 10. Table 10. Design factors with their values at three levels Factor levels

E1

E2

E3

c1

c2

c3

φ1

φ2

φ3

I II III

(MPa) 60 70 80

(MPa) 30 40 50

(MPa) 20 30 40

(kPa) 14 16 18

(kPa) 30 32 34

(kPa) 24 28 32

degree 32 34 36

degree 28 30 32

degree 24 28 32

Design factors

The Taguchi’s DOE methodology uses special orthogonal arrays (OAs) to study all the design factors at the minimum of experiments. In order to study the effects of the design factors, a suitable OA must be chosen. The MINITAB program is used to compose the OA table. In this study, three levels are considered for nine geotechnical parameters. The L27 orthogonal

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array is selected for the experiments. The lower-the-better analysis is applied as the main objective of this methodology is to reduce the deviation of the modeling results from the field measurement data. Taguchi’s L27 orthogonal design is shown in Table 11. Table 11. Experimental layout using L27 orthogonal array Experimental Test E1 c1 φ1 E2 c2 φ2 E3 c3 φ3 Trail Index 1 I I I I I I I I I 5.32 2 I I I I II II II II II 9.72 3 I I I I III III III III III 7.07 4 I II II II I I I II II 5.91 5 I II II II II II II III III 5.34 6 I II II II III III III I I 5.68 7 I III III III I I I III III 6.7 8 I III III III II II II I I 9.89 9 I III III III III III III II II 11.21 10 II I II III I II III I II 9.53 11 II I II III II III I II III 4.78 12 II I II III III I II III I 4.21 13 II II III I I II III II III 5.08 14 II II III I II III I III I 3.82 15 II II III I III I II I II 3.06 16 II III I II I II III III I 6.52 17 II III I II II III I I II 4.71 18 II III I II III I II II III 3.5 19 III I III II I III II I III 4.21 20 III I III II II I III II I 6.76 21 III I III II III II I III II 3.18 22 III II I III I III II II I 3.19 23 III II I III II I III III II 4.97 24 III II I III III II I I III 3.63 25 III III II I I III II III II 3.67 26 III III II I II I III I III 3.43 27 III III II I III II I II I 3.95

The 27 numerical models created and executed according to 27 Taguchi’s experiments. For example, the experiment No.6, shows the geotechnical parameters (E1= 60, E2= 40, E3= 30, c1= 16, c2= 34, c3= 34, φ1= 36, φ2= 28 and φ3= 24). Then, test index is calculated for 27 experiments according to Eq. 2, which is shown shaded in Table 11. Finally, the experiment results are examined by the analysis of range (ANORA). 4.2.3. Analysis of range (ANORA) This section aimed to identify the geotechnical parameters, which minimized the deviation of the field measurement data from the corresponding numerical results. In Taguchi’s DOE, the analysis of range (ANORA) is performed after the collection of experimental results. In ANORA, the relation of each factor and the test index is analyzed statistically. Calculation processes of the mean (Ki) and range (R) of the test indexes, can be found in following. The matrix with calculated Ki (design factors) and R values is shown in Table 11. It’s obtained by computing the average of the test index results. For example, the K11 = 7.42 in the Table 12 for Factor E1 is obtained by adding all TI values for which factor E1 = I, divided by the total number of TI values: TImean,E1,I: K11= (5.32+9.72+7.07+5.91+5.34+5.68+6.7+9.89+11.21)/9= 7.42 (3) Thus: TImean,E1,II: K12= (9.53+4.78+4.21+5.08+3.82+3.06+6.52+4.71+3.5)/9= 5.10 (4) TImean,E1,III:K13=(4.21+6.76+3.18+3.19+4.97+3.63+3.67+3.43+3.95)/9= 3.47 (5) To find the R values, the maximum and minimum K value are found for each of the nine design factors. The R value for each design factor is calculated by finding the difference between maximum and minimum values (Table 12). For example, for Factor E3: R=6.69–

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4.55=2.14. The results of the ANORA calculations are summarized in Table 12 for the selected 27 experiments. Table 12. ANORA calculations for design parameters Level 1 Level 2 Level 3 R E1 K1 7.42 5.10 3.47 3.95 c1 K2 5.98 4.52 5.95 1.46 φ1 K3 5.4 5.16 5.88 0.712 E2 K4 5.01 4.98 6.46 1.48 c2 K5 5.57 5.94 4.94 0.99 φ2 K6 4.87 6.2 5.37 1.33 E3 K7 4.55 5.19 6.69 2.14 c3 K8 5.49 6.01 4.94 1.07 φ3 K9 5.48 6.1 4.86 1.24

The larger the R value for a factor, the stronger is the influence of the design factor on the results. As the R values for E1 are larger than R values for the other factors, it can be concluded that the E1 has a larger influence on the response. In general, modulus of elasticity is the most influencing factor on the surface settlement; because R values for Modulus of elasticity is larger than cohesion and internal friction angle parameters (Figure 11). 4 3.5 3

R values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2.5 2 1.5 1 0.5 0

E1

E2

E3

C1

C2

C3

Phi1

Phi2

Phi3

Design parameters Figure 11. R values for design parameters

c3, 7.45%

φ3, 8.63%

E1, 27.48%

E3, 14.89% c1, 10.16% φ1, c2, E2, 4.95% 6.89% 10.30%

φ2, 9.25%

Figure 12. The percentage of effect of the different parameters on the surface settlement

Considering the test index results, error values is reduced 8.15% from 11.21% to 3.06%. The TI means of the E1 parameter in I, II and III levels is reduced 3.95% from 7.42% to 3.47%. It indicate that percentage of effect (PE) of the E1 parameter on the surface

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settlement is 28.48%. Likewise, the PE of the E2 and E3 parameters are equal 10.3% and 6.9% respectively. For c1, c2 and c3 parameters, the PE values are equal 10.16%, 6.9% and 7.45% respectively. The PE values for φ1, φ2 and φ3 parameters 4.95%, 9.25% and 8.63% respectively. Therefore, it is worth mention that even strength parameters “c” and “Phi” could be have considerable effect on the settlement. The Figure 12 indicates that the order of influence of different parameters on the surface settlement as follows: RE1 > RE3 > RE2 > Rc1 > Rφ2 > Rφ3 > Rc3 > Rc2 > Rφ1 Main Effects Plot for Means Data Means E1

7.5

C1

Phi1

6.0

Mean of Means

4.5 1

2 E2

3

1

2 C2

3

1

2 Phi2

3

1

2 E3

3

1

2 C3

3

1

2 Phi3

3

2

3

1

2

3

1

2

3

7.5 6.0 4.5

7.5 6.0 4.5 1

Figure 13. Mean effect plot for test indexes average

In order to assess the influence of each factor on the R, the means for each design factor is calculated. Figure 13 shows a main effect plot for test indexes average for each design factors. The optimal value for each design factor is obtained set to the minimum value of the test indexes average. For example, c1 parameter is optimum in II level, and φ2 in I level. In order to better finding the optimal level for geotechnical parameters, the TI mean of geotechnical parameters for three levels, I, II and III illustrated in Figure 14. 8 7 6

Test Index Mean

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

5 4 3 2

Level I

1

Level II

0

Level III E1

E2

E3

C1 C2 C3 Phi1 Phi2 Experiment levels Figure 14. The TI Mean for geotechnical parameters

16

Phi3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

According to the Figure 13 and Figure 14, the minimum TI mean for E1 occurs at level III, for E2 at level II, and for E3 at level I. Similarly, the minimum TI mean for parameters c1, c2 and c3 occurs at levels I, II and III respectively, and 𝜑1 , 𝜑2 , and 𝜑3 at levels II, I, and III respectively. Thus, the minimum test index mean is identified, which is selected as optimal level for each geotechnical parameters. Specifically, Table 13 reports values of parameter levels, which minimize R values correspond to the lowest mean. Therefore, the optimal combination of the geotechnical parameters can be proposed as follows: E1=80, E2=40, E3=20, c1=16, c2=34, c3=32, φ1=34, φ2=28 and φ3=32. Table 13. Optimized Levels and values of the geotechnical parameters by Taguchi’s DOE Parameter

E1 (MPa)

c1 (K Pa)

φ1 °

E2 (M Pa)

c2 (K Pa)

φ2 °

E3 (M Pa)

c3 (K Pa)

φ3 °

Level Value

III 80

II 16

II 34

II 40

III 34

I 28

I 20

III 32

III 32

4.3. Comparison of the back analysis method with Taguchi’s DOE In order to verify the results, another section along the tunnel route is simulated using the optimal geotechnical parameters suggested by Taguchi’s DOE and displacement-based back analysis. After executing the models, the deviation of the field data of Settlement points C4 and C6 (Figure 3(c)) from the corresponding numerical results is calculated based on the error function (last column in Table 14). Table 14. Proposed optimal geotechnical parameters by Taguchi’s DOE and Back analysis Parameter

Back analysis DOE

E1 (MPa)

c1 (K Pa)

φ1 °

E2 (M Pa)

c2 (K Pa)

φ2 °

E3 (M Pa)

c3 (K Pa)

φ3 °

error

70 80

18 16

30 34

40 40

30 34

28 28

30 20

30 32

24 32

4.11 2.85

By comparing the presented error values in Table 14 for both approaches, it can be seen that, the error value of the numerical modeling using the suggested Taguchi parameters is lower than those in the back analysis. Thus, the proposed parameters by Taguchi’s DOE are closer to the reality. This can be explained by the fact that: - In displacement-based back analysis using the univariate optimization algorithm, only one variable is changed at a time and the values of other variables are fixed. After optimizing the one variable, in the next step, the value of one variable which was fixed in the previous step is varied while the values of other variables are fixed. This procedure neglects the interaction between factors, which has the adverse effects on the accuracy of the results. However, simultaneous changing of the parameters in DOE methodology leads to the realtime optimizing. - In the back analysis, the response (error value) is considered in limit models. However, in Taguchi approach not only the all input parameters are changed simultaneously, but also the responses of all models (TIs) are considered, and the optimal parameters are suggested. 7. Conclusion In this paper, estimation of the geotechnical parameters of Tehran metro line 7 is conducted using Taguchi’s Design of experiment (DOE) method, which is more effective than some of the existing techniques. Furthermore, the results of Taguchi’s DOE methodology are compared to univariate displacement-based back analysis approach to show the effectiveness of the DOE method. The Taguchi’s experiments and back analysis outputs are used for 3D numerical simulation of the tunnelling procedure by Plaxis 3D tunnel software, and the MINITAB program is used to compose the orthogonal array table. The experiment results are examined by the analysis of range (ANORA). The results of this research are summarized as follows:

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65





The Taguchi’s DOE technique offers more effective and precise estimation of the geotechnical parameters than displacement-based and back analysis approach. The following supporting facts are provided to prove this statement: The displacement-based back analysis uses univariate approach, neglecting interaction effect between parameters, while simultaneous changing of the parameters in Taguchi’s DOE methodology leads to the real-time optimizing. In the back analysis, the response (error value) is considered in just few models. However, the responses of all models (TIs) are considered in Taguchi’s DOE, and the optimal parameters are proposed according to minimum error. The optimal combination of the proposed geotechnical parameters based on the analysis results of the both approaches can be proposed as follows: Back analysis: E1=70, E2=40, E3=30, c1=18, c2=30, c3=30, φ1=30, φ2=28 and φ3=24. DOE method: E1=80, E2=40, E3=20, c1=16, c2=34, c3=32, φ1=34, φ2=28 and φ3=32. It is proved that the combination of the proposed geotechnical parameters by Taguchi’s DOE is more similar to the field measurements data. Considering the results, it is worth mention that geotechnical parameters “E” “c” and “Phi” could be have different weighted effect on the settlement. The percentage of effect (PE) of the E1, E2, E3, c1, c2, c3, φ1, φ2 and φ3 parameters are equal 28.48%, 10.3%, 6.9%, 10.16%, 6.9%, 7.45%, 4.95%, 9.25% and 8.63% respectively. Thus, E1 and φ1 parameters have a most and least percentage of effect on the settlement respectively. The order of the different parameters effect on the settlement as follows: (PE)E1 > (PE)E3 > (PE)E2 > (PE)c1 > (PE)φ2 > (PE)φ3 > (PE)c3 > (PE)c2 > (PE)φ1

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