Proposing a new model to approximate the elasticity

Bulletin of Engineering Geology and the Environment https://doi.org/10.1007/s10064-017-1210-5

ORIGINAL PAPER

Proposing a new model to approximate the elasticity modulus of granite rock samples based on laboratory tests results Katayoun Behzadafshar 1 & Mehdi Esfandi Sarafraz 2 & Mahdi Hasanipanah 3 & S. Farid F. Mojtahedi 4 & M. M. Tahir 5 Received: 30 August 2017 / Accepted: 20 November 2017 # Springer-Verlag GmbH Germany, part of Springer Nature 2017

Abstract An accurate examination of deformability of rock samples in response to any change in stresses is deeply dependent on the reliable determination of properties of the rock as analysis inputs. Although Young’s modulus (E) can provide valuable characteristics of the rock material deformation, the direct determination of E is considered a time-consuming and complicated analysis. The present study is aimed to introduce a new hybrid intelligent model to predict the E of granitic rock samples. Hence, a series of granitic block samples were collected from the face of a water transfer tunnel excavated in Malaysia and transferred to laboratory to conduct rock index tests for E prediction. Rock index tests including point load, p-wave velocity and Schmidt hammer together with uniaxial compressive strength (UCS) tests were carried out to prepare a database comprised of 62 datasets for the analysis. Results of simple regression analysis showed that there is a need to develop models with multiple inputs. Then, a hybrid genetic algorithm (GA)-artificial neural network (ANN) model was developed considering parameters with the most impact on the GA. In order to have a fair evaluation, a predeveloped ANN model was also performed to predict E of the rock. As a result, a GA-ANN model with a coefficient of determination (R2) of 0.959 and root mean square error (RMSE) of 0.078 for testing datasets was selected and introduced as a new model for engineering practice; the results obtained were 0.766 and 0.098, respectively, for the developed ANN model. Furthermore, based on sensitivity analysis results, p-wave velocity has the most effect on E of the rock samples. Keywords Rock deformation . Young’s modulus . Genetic algorithm . Artificial neural network

Introduction One of the most important issues in the fields of rock mechanics, geotechnical engineering and geological engineering is appropriate determination of rock material properties, such as uniaxial compressive strength (UCS), Young’s modulus (E) and Poisson's ratio (υ). As suggested in the literature

* Katayoun Behzadafshar [email protected] * Mahdi Hasanipanah [email protected] Mehdi Esfandi Sarafraz [email protected] S. Farid F. Mojtahedi [email protected] M. M. Tahir [email protected]

(e.g. Baykasoğlu et al. 2008; Sarkar et al. 2010; Yagiz et al. 2012; Jahed Armaghani et al. 2016), the unconfined compression test (UCT), which is standardized by the International Society for Rock Mechanics (ISRM), can be directly utilized to determine the elasticity and strength of the rock. However, as highlighted in many studies (Gokceoglu and Zorlu 2004; Zorlu et al. 2008; Sarkar et al. 2010; Beiki et al. 2013),

1

Department of Physics, College of Basic Sciences, Yadegar-e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, Tehran, Iran

2

Department of Civil Engineering, West Tehran Branch, Islamic Azad University, Tehran, Iran

3

Young Researchers and Elite Club, Qom Branch, Islamic Azad University, Qom, Iran

4

Civil Engineering Department, Sharif University of Technology, Tehran, Iran

5

UTM Construction Research Centre, Institute for Smart Infrastructure and Innovative Construction (ISIIC), Faculty of Civil Engineering, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia

K. Behzadafshar et al.

conducting an UCT is expensive, time-consuming and is sometimes complicated. On the other hand, performing some basic rock index tests, such as p-wave velocity (Vp), Schmidt hammer and point load index (Is50) analyses are economical and uncomplicated (Singh et al. 2012). Therefore, the use of these rock index tests is considered an indirect and well-known technique to determine the E and UCS. By reviewing the previous studies (e.g. Dinçer et al. 2004; Moradian and Behnia 2009; Yilmaz and Yuksek 2009; Liu et al. 2015), it was found that several traditional correlations have been proposed to predict the E. Besides, a multiple regression (MR) model is employed as an easy and common method in predicting the E (e.g. Zorlu et al. 2008; Singh et al. 2012; Mohamad et al. 2014; Momeni et al. 2015; Armaghani et al. 2016c; Bejarbaneh et al. 2016; Liang et al. 2016). Nevertheless, the prediction results of MR models is not good enough in some cases and there is a need to develop excellent models in terms of accuracy. In the recent years, the application of soft computing methods in the field of rock engineering and geotechnical engineering have been highlighted by some researchers (Grima et al. 2000; Goh 2000; Verma and Singh 2011, 2013; Tonnizam Mohamad et al. 2012; Khandelwal and Singh 2013; Saadat et al. 2014, 2015; Jahed Armaghani et al. 2015b; Armaghani et al. 2016a; Tonnizam Mohamad et al. 2016; Singh et al. 2016, 2017; Hasanipanah et al. 2016d, e; Sharma et al. 2017a, b, c; Ahmad et al. 2017). A fuzzy inference system (FIS) was proposed to predict E and UCS in the study conducted by Gokceoglu and Zorlu (2004). The values predicted by the FIS were then compared to multiple regression analysis results. Based on the obtained results, the performance capacity of the FIS was better than MR analysis in predicting the E and UCS. In another study by Kahraman et al. (2009), the values of UCS and E for Misis Fault Breccia were determined by using an artificial neural network (ANN) model. Their results show that the ANN model has superior fitting specification in predicting the UCS and E compared to the regression models. ANN and multivariate regression models were employed to examine the effect of slake durability cycles on UCS by Yagiz et al. (2012). They showed that the ANN prediction model was significantly superior to the multivariate regression-based prediction models. In another study, Yilmaz and Yuksek (2008) investigated the ANN model results in predicting the E of gypsum. Their results revealed that the ANN model was an acceptable and reliable model to predict the E of gypsum and had the capacity to generalize. Two soft computing-based models, i.e. the ANN and hybrid neuro-fuzzy models, were offered for estimating the UCS and E in gypsum rock samples in the study carried out by Yilmaz and Yuksek (2009). In their study, several parameters such as Schmidt hammer value, sonic velocity, porosity and water content were assigned as the model's inputs. Their results indicated both methods were capable of predicting the UCS and E with reasonable accuracy. In other studies of hybrid models, Momeni et al. (2015) and Mohamad

et al. (2014) established a hybrid of an ANN and particle swarm optimization (PSO) in estimating the USC. In fact, PSO was used to optimize the ANN model. Their obtained results demonstrated that the hybrid model was a powerful computational tool that can be used to estimate the UCS. ANN networks are good tools for forecasting issues; however, they have several limitations, such as low learning speed and falling into local minima (Lee et al. 1991; Hajihassani et al. 2014; Jahed Armaghani et al. 2015a). As mentioned in the literature (Saemi et al. 2007; Monjezi et al. 2012a; Marto et al. 2014; Hasanipanah et al. 2016a; Jahed Armaghani et al. 2016b, 2017; Mohamad et al. 2016), using efficient optimization algorithms (OAs), these limitations can be overcome. The genetic algorithm (GA) as one of the OAs can be applied to solve ANN problems. Therefore, the present investigation is carried out to assess the potential application of a hybrid ANN and GA model to approximate the E of granite samples collected from a tunnel face excavated in Malaysia. To show the ability of the hybrid model, the proposed GA-ANN model performance is compared with ANN model results.

Intelligent techniques An ANN is one of the most common soft computing models that can solve the real-world problems in different fields. The ANN, which is inspired by the human brain, has ability to establish a non-linear and complex relationship between input (independent) and output (dependent) variables. As stated by some researchers (Khandelwal and Singh 2009; YesilogluGultekin et al. 2013; Hasanipanah et al. 2016c; Jahed Armaghani et al. 2016a, 2016b), back-propagation (BP) is one of the most well-know and common algorithms to train the ANN (Dreyfus 2005). In the other words, BP has a high ability to minimize the error between target and output (obtained by the ANN model). In implementing the ANN, three layers, i.e. input, hidden and output layers, should be considered. The independent and dependent variables are located in input and output layers, respectively. For better understanding of the ANN background and its details, other suggested studies (Ripley 1993; Swingler 1996; Dreyfus 2005; Jahed Armaghani et al. 2014; Zeinali and Story 2016, 2017; Sitton et al. 2017) can be reviewed. The GA, a modern approach to numerical optimization, is considered a part of artificial intelligence and evolutionary algorithms. Although the GA is originated from Charles Darwin’s theory of Bsurvival of the fittest^ and Bnatural selection^, it was first developed by Holland (1992). The GA, as an advantageous method compared to conventional optimization methods, has been applied and modified in many optimization problems. GAs are capable of addressing different optimization problems whether the objective (fitness) function is static or dynamic, linear or non-linear, continuous or discontinuous

Proposing a new model to approximate the elasticity modulus of granite rock samples based on laboratory...

or contains random noise. Although the GA is an advantageous method, as mentioned before, the proper mathematical expression of fitness function and assigning an appropriate selection method could be considered as the GA's disadvantages. Furthermore, it is important to mention that the improper choice for population size and genetic operator rates lead to obtaining inaccurate results or problems in convergence of the algorithm. Considering these benefits and limitations of GAs, it is stated that the GA is still one the best algorithms that has been utilized for non-linear optimization (GOH 2000; Saemi et al. 2007). A random generation of the chromosomes is considered the first step of the GA process. In order to obtain optimum solutions, the selection operator should be applied to the individuals, similar to Darwin’s natural selection. These steps are followed by applying genetic operators (mutation and crossover) on the remaining chromosomes in order to create the next generation of chromosomes. Crossover as the main operator, selects two parent chromosomes randomly and swaps segments of them with each other. The newly created chromosomes are named as children. Moreover, the chromosomes can be selected randomly in a defined range using another genetic operator, referred to as mutation. Once the maximum number of generations or the desired value for the best solution is achieved, these processes can be stopped (Majdi and Beiki 2010; Tonnizam Mohamad et al. 2016). Nowadays, evolutionary algorithms such as the GA are widely employed for optimization aims. Both constrained and unconstrained problems can be optimized by the GA. As mentioned before, the ANN approach has several constraints, including a low learning rate and getting trapped in local minima (Eberhart et al. 1996; Eberhart and Shi 1998; Clerc and Kennedy 2002). In order to get rid of these obstacles, a GA optimization technique can be used. The GA is utilized to adjust the weight and bias of the ANN networks. A combination algorithm of GA with ANN can be seen in Fig. 1. The combination of the GAwith an ANN is valuable to improve the ANN performance, according to several researchers (Majdi and Beiki 2010; Momeni et al. 2014).

Collected datasets and laboratory tests This study focused on investigating properties of rock samples obtained from a transfer tunnel (Pahang Selangor Row Water Transfer Tunnel, PSRWT Tunnel) excavated in Malaysia. The PSRWT Tunnel is aimed to transfer water from Pahang state to Selangor state in Malaysia. A target of transferring 1890 million liters/day of raw water conveyed from the Semantan River is planned for this project. This water is used for both industrial and domestic consumption to reduce future shortage of water. From the connecting basin, the raw water is transferred to an outlet connecting the basin through the tunnel

with gravity flow. In this project, various construction sections including three tunnel boring machine (TBM) sections and four conventional drilling and blasting sections were planned to be excavated. The TBMs with a 5.23-m (17.2 ft) diameter were used to excavate various ground conditions, including mixed ground (11,761 m), very hard ground (11,761 m) and blocky ground (11,218 m), which were excavated by TBM 1, TBM 2 and TBM 3, respectively. The PSRWT Tunnel location in Malaysia is displayed in Fig. 2. In this study, more than 100 block samples were obtained from face of the PSRWT Tunnel in different tunnel distances and then transferred to the laboratory to conduct relevant required tests. Various rock index tests comprising of Schmidt hammer test (Rn), p-wave velocity test (Vp) and point load test (Is50) together with UCS test were carried out according to ISRM (Ulusay and Hudson 2007). In order to obtain E as a system output, a rock stress–strain curve was drawn for each UCS test, and according to the ISRM method, E values were achieved. It is worth mentioning that a linear variable differential transformer (LVDT) was utilized in order to measure axial strain. Sixty-two datasets (including 62 tests results of Vp, Is50 and Rn as inputs and 62 values of E as outputs) were prepared to conduct modeling and analyses of this study. A summary of these inputs and output datasets is shown in Table 1. In the following, simple regression was first analyzed to investigate the possibility for developing intelligent systems and then the two intelligent systems were applied to get a higher performance capacity.

Simple regression analysis In order to investigate a relationship between model inputs (Vp, Is50 and Rn) and system output (E), simple regression models were applied. Different types of equations such as linear, power and exponential equations were tried to evaluate and select the best type in estimating the E of the rock. Evaluation of these equations was performed based on some performance indices (PIs), i.e., coefficient of determination (R2), root mean square error (RMSE) and variance account for (VAF), which were suggested in many studies (e.g. Hasanipanah et al. 2016b; Armaghani et al. 2016b; Mahdiyar et al. 2017). Their formulas can be also found in other studies (e.g., Tonnizam Mohamad et al. 2016). It is important to note that a perfect fit would have an R2 of 1, a RMSE of 0 and a VAF of 100%. Table 2 presents the developed equations for E prediction together with their PIs. These equations were selected based on their PI results compared to other equation types. As it can be seen, values for R2 are obtained as 0.609, 0.582 and 0.625 for Rn, Is50 and Vp, respectively. Graphs of the developed equations to predict E are shown in Figs. 3, 4 and 5. It was found that the obtained results

K. Behzadafshar et al. Fig. 1 Combination algorithm of GA with an ANN (Saemi et al. 2007)

are statistically meaningful, but in order to get higher performance models in practice, developing some new models is needed. In this regard, intelligent models namely the GAANN model and a pre-developed ANN are also built to predict E of the granite rock samples.

Intelligent model development As stated earlier, the GA has an impact on ANN performance (e.g. Lee et al. 1991; Monjezi et al. 2012b; Khandelwal and Armaghani 2016). To obtain the best GA-ANN model, its important factors/parameters should be investigated. Prior to investigation of GA parameters, the ANN architecture should be determined. This has been accomplished considering a trial-and-error process and it was found that an architecture of 3 × 5× 1 (or a model with five hidden neurons) receives better results. Therefore, the mentioned architecture was used for hybrid intelligent systems in this study. The most effective GA parameters that were used to construct hybrid GA-ANN Fig. 2 Location of the PSRWT Tunnel project

models should be selected/determined. Mutation probability values and percentage of recombination were set as 25 and 9%, respectively. As a cross-over operation, a single point with 70% possibility was used. For the next step, in order to determine the maximum number of generation (Gmax) and the optimum population size, a parametric study was conducted as shown in Fig. 6. A value of 500 generations was assigned as the stopping criteria, and the obtained RMSE values were considered for the analyses. In addition, population sizes of 25, 50, 75, 100, 150, 200, 250, 300, 350, 400, 450 and 500 were set in these analyses. As a result, after the number of generations = 250, the network performance was unchanged. Additionally, RMSE results of the population size = 200, showing the minimum system error. Therefore, a Gmax = 250 and a population size = 200 were selected for the best GAANN model and the results will be discussed in detail later. It should be noted that all models were built using 80% of the whole data as training and 20% of the data as testing. Moreover, MatLab software (Demuth and Beale 2000) was utilized to construct the models.

Proposing a new model to approximate the elasticity modulus of granite rock samples based on laboratory... Table 1 Basic statistical description and the range of the dataset used in this study

Table 2 The developed equations for estimating E together with their PIs

Data

Abbreviation

Unit

Type

Min.

Point load index Schmidt hammer rebound number P-wave velocity

Is50 Rn Vp

MPa

Input

– m/s

Input Input

Young’s modulus

E

GPa

Output

Max.

0.89 37 3065

7.1 61 7943

22

156.7

Mean 3/4 49.8 5582.9 84.2

Predictor

Unit

Equation type

Developed equation

R2

RMSE

VAF

Rn Is50 Vp

– MPa

Power Power

E = 0.0005 Rn 3.084 E = 37.01 Is50 0.666

m/s

Power

E = 0.00001 Vp 1.814

0.609 0.582 0.625

21.22 21.04 20.51

60.856 57.685 62.321

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 1  ∑ni¼1 ðM i −P i Þ2 RMSE ¼ n   varðM i −P i Þ  100 VAF ¼ 1− varðM i Þ

Results and discussion In the present study, a hybrid model of an ANN and a GA has been proposed to forecast the E of rock samples. Aside from that, an ANN has been also employed for evaluating the E values predicted by the GA-ANN model. In implementing the ANN model, three neurons were used as input layer, i.e. Rn, Vp and Is50. Furthermore, values of E were used in the output layer. By conducting trial-and-error method, five neurons in the hidden layer received more performance than the other hidden neurons. In order to select the best GA-ANN model, many models were built based on conducting several parametric studies and, finally, a GA-ANN model with a maximum generation of 250 and a population size of 200 outperformed the other implemented models. For checking the performance of the GA-ANN and ANN models, three statistical criteria, i.e. RMSE, VAF and R2 have been utilized. 
2  h i n − ∑ni¼1 ðM i −P i Þ2 ∑i¼1 M i −M  R2 ¼ ð1Þ
2  n ∑i¼1 M i −M

MABE ¼ NS ¼ 1−

ð2Þ ð3Þ

1  ∑ni¼1 jM i −P i j n

ð4Þ

∑ni¼1 ðM i −P i Þ2
2 ∑ni¼1 M i −M

ð5Þ

where Mi is the measured E values, Pi is the predicted E values obtained from the predictors, M is the average of the Mi and var is the variance sign. The values of mentioned statistical criteria for the GA-ANN and ANN models are shown in Table 3. In addition, Figs. 7 and 8 illustrate a comparison between the measured and predicted E values by the predictive models, ANN and GA-ANN, respectively. Observing Table 3 and Figs. 7 and 8, it can be seen that the proposed GA-ANN model outperforms the ANN model in terms of the prediction accuracy level and the generalization capability. In other words, the GA-ANN model with the R2 of 0.959 can estimate E better than the ANN model with the R2 of 0.766 for

180

Fig. 3 Prediction of E using Rn

R² = 0.609

160

Measured E (GPa)

140 120 100 80 60 40 20 0

0

10

20

30

40

50

Schmidt hammer rebound number

60

70

K. Behzadafshar et al. 180

Fig. 4 Prediction of E using Vp

R² = 0.625

160

Measured E (GPa)

140 120 100

80 60 40 20 0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

P-wave velocity (m/s)

180

Fig. 5 Prediction of E using Is50

R² = 0.582

160

Measured E (GPa)

140 120 100 80 60 40 20 0

0

1

2

3

4

5

6

7

point load index (MPa)

0.085

Population Size = 25

Population Size = 50

Population Size = 75

Population Size = 100

Population Size = 150

Population Size = 200

Population Size = 250

Population Size = 300

Population Size = 350

Population Size = 400

Population Size = 450

Population Size = 500

RMSE

0.075

0.065

0.055

0

50

100

150

200

250

Generation

Fig. 6 GA-ANN models with various population sizes

300

350

400

450

500

8

Proposing a new model to approximate the elasticity modulus of granite rock samples based on laboratory... Table 3 The values of statistical criteria obtained from the predictive models

Model

Statistical criteria R2

RMSE Train

ANN

0.767

0.766

0.13

0.098

73.55

70.49

0.101

0.080

0.735

0.697

GA-ANN

0.942

0.959

0.057

0.078

94.18

93.89

0.044

0.070

0.940

0.908

ð4Þ

k¼1

Train

Test

Test

1

R² = 0.768

R² = 0.766

0.8

Predicted E

Predicted E

Test

In the present study, two soft computing-based models, i.e. the ANN and GA-ANN models, were developed to estimate the Young’s modulus of rock. In the first step, a series of rock samples were collected from a tunnel face in Malaysia. Then, a database (62 datasets) of several laboratory tests, Rn, Is50, Vp and E was prepared while Rn, Is50 and Vp were set as inputs to predict the E of the granitic samples. A meaningful

0.6 0.4

0.6 0.4 0.2

0.2 0

Train

Conclusions

Train

0.8

Test

In the above equation, rij is the intensity of input parameters on the output. Based on the obtained results, the values of rij were 0.95, 0.96 and 0.95 for the Rn, Vp and Is50, respectively. Therefore, p-wave velocity is considered as the most effective parameter on the E in the present study.

n

1

Train

NS

Test

∑ ðyik  yok Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rij ¼ rk¼1 n n ∑ yik 2 ∑ yok 2

Test

MABE

Train

testing datasets. Moreover, based on other performance indices, RMSE and VAF, the GA-ANN model is able to provide higher prediction capacity in predicting E. By reviewing previous works in the field of E prediction (Gokceoglu and Zorlu 2004; Yilmaz and Yuksek 2008; Yilmaz and Yuksek 2009; Singh et al. 2012; Beiki et al. 2013; Bejarbaneh et al. 2016), it was found that our presented model (GA-ANN) is considered as the best model in terms of model accuracy level, and it can be introduced as a new model in this field. In this research, a method utilized by Yang and Zang (1997) and Hajihassani et al. (2015) was also performed for the sensitivity analysis, based on the following equation:

k¼1

VAF

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

Actual E

0.6

0.8

1

Actual E

Fig. 7 The performance of the ANN to estimate E of the rock

1

R² = 0.942

R² = 0.960

0.8

Predicted E

0.8 Predicted E

Test

Train

1

0.6 0.4

0.6 0.4 0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

Actual E

Fig. 8 The performance of the GA-ANN model to estimate E of the rock

0

0.2

0.4

Actual E

0.6

0.8

1

K. Behzadafshar et al.

relationship was found between inputs and output by conducting a simple regression analysis. Results of simple regression analysis showed that there is a need to propose models with multiple inputs, i.e., ANN and GA-ANN. In ANN and GA-ANN modeling, the used datasets were divided into training and testing categories. In other words, 50 data-sets were used for training the ANN and GA-ANN models, and then the other 12 data sets were used for verifying the developed models. After modeling, the performance capacity of the constructed models was assessed based on statistical criteria, i.e., R2, RMSE and VAF. The results showed that the GA-ANN model (with R2 of 0.959, RMSE of 0.078 and VAF of 93.89) was significantly superior to the ANN model (with R2 of 0.766, RMSE of 0.098 and VAF of 70.49) for testing datasets. These values proved that GA-ANN model is a powerful computational tool and can be applied with high degree of confidence in this field. In another part of the present study, sensitivity analysis was performed and, according to its results, Vp with an rij of 0.96 was selected as the most influential parameter on the E.

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