J. Cent. South Univ. (2015) 22: 1878−1891 DOI: 10.1007/s11771-015-2707-3

Application of artificial neural network for calculating anisotropic friction angle of sands and effect on slope stability Hamed Farshbaf Aghajani, Hossein Salehzadeh, Habib Shahnazari School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O.B.16756-163, Iran © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: The anisotropy effect is one of the most prominent phenomena in soil mechanics. Although many experimental programs have investigated anisotropy in sand, a computational procedure for determining anisotropy is lacking. Thus, this work aims to develop a procedure for connecting the sand friction angle and the loading orientation. All principal stress rotation tests in the literatures were processed via an artificial neural network. Then, with sensitivity analysis, the effect of intrinsic soil properties, consolidation history, and test sample characteristics on enhancing anisotropy was examined. The results imply that decreasing the grain size of the soil increases the effect of anisotropy on soil shear strength. In addition, increasing the angularity of grains increases the anisotropy effect in the sample. The stability of a sandy slope was also examined by considering the anisotropy in shear strength parameters. If the anisotropy effect is neglected, slope safety is overestimated by 5%−25%. This deviation is more apparent in flatter slopes than in steeper ones. However, the critical slip surface in the most slopes is the same in isotropic and anisotropic conditions. Key words: anisotropy; artificial neural network; sand; principal stress rotation; slope stability

1 Introduction In most alluvial deposition in nature, the mechanical properties and strength of soil depend on the direction of the external loads, which is known as the anisotropy effect. When soil particles settle in water, the soil grains lie in the preferred direction, providing that the longitude axis of the grains is almost parallel to the deposition bedding direction. The mechanical response of the soil element thus depends not only on the magnitude of the applied stress but also on the orientation of the applied load relative to the bedding layer [1]. Unfortunately, anisotropy causes the soil mass to be weaker than expected. Thus, if the anisotropy feature of soil is neglected and only the strength characteristics of soil corresponding to the vertical direction are designated for the entire soil mass, the geotechnical structure design may be accompanied by problems. For instance, by supposing that a clay slope is comprised of isotropic cohesive soil and neglecting the anisotropy effect, the safety factor of the slope is overestimated by 5%−25% greater than the actual value [2−6]. Investigation of the anisotropy role in the response of geotechnical structures requires models that consider the effect of anisotropy and the rotation of the stress direction on the mechanical behavior of soil. For the Mohr-Columb failure envelope, CASAGRANDE and

CARILLO [7] presented a closed-form equation for the relation between soil cohesion (c) and the loading orientation (α) based on the exact value of the soil cohesion in horizontal (ch) and vertical (cv) directions as follows. This equation is widely implemented in the stability analysis of anisotropic cohesive soil slopes [3−6]: c ch (cv ch ) cos 2

(1)

Despite the CASAGRANDE and CARILLO equation for anisotropic cohesive soil, there is no mathematical equation for the friction angle parameter concerning the anisotropy effect. In other words, even though many experimental investigations have studied the anisotropic behavior of sands within the past few years, little attention has been paid to processing these test data and developing a computational procedure for the anisotropic friction angle. Therefore, due to this gap in the literature, this work aims to establish a computational procedure for determining the anisotropic friction angle in sand. Numerous experimental records from principal stress rotation tests were processed with an artificial neural network (ANN). After the ANN was trained and the architecture, connected weights, and bias were recognized, sensitivity analyses were performed to determine which parameter influences shear strength anisotropy. Finally, by implementing the ANN, the effect

Received date: 2014−03−17; Accepted date: 2014−12−23 Corresponding author: Hossein Salehzadeh, Associate Professor; Tel: +98−2177240399; E-mail: [email protected]

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of shear strength anisotropy on the stability of a sandy slope is investigated.

2 Anisotropy in sands In the last decade, researchers have utilized the hollow cylinder torsional shear apparatus (HCTSA) to investigate inherent and induced anisotropy in sand and clay soils [8−14]. This apparatus imposes four independent force components in the hollow cylinder sample, including the axial load, torque, and inner and outer pressures. This versatile feature is implemented to control the magnitude and direction of three-dimensional principal stresses within the sample. The stress state can be induced in any desired direction. Thus, the anisotropy behavior of the HCTSA sample is explored by rotating the directions of the principal stress axis in relation to the fixed bedding plane [15]. The stress state components at an arbitrary point within the HCTSA sample are illustrated in Fig. 1. At this point, the maximum (σ1) and minimum (σ3) principal stresses act in the plane of the sample wall, and the maximum principal stress makes an angle against the vertical axis denoted by α. The intermediate principal stress (σ2) is induced in the radial direction and expressed by a non-dimensional parameter called the intermediate principal stress ratio (b) as follows [16]: b

2 3 1 3

(2)

Fig. 1 Stress state in sample of hollow cylinder torsional shear apparatus (HCTSA)

3 Investigation of anisotropy by artificial neural network (ANN) For establishing a mathematical framework for

computing the anisotropic friction angle, all experimental anisotropy test data have been collected from the literature and processed. Because these tests were performed by several researchers under different but comparable experimental conditions, the range of the test data is wide. Therefore, an artificial neural network (ANN) is implemented to statistically process the test data and predict the friction angle of sands based on the soil characteristics and loading orientation. Today, soft computing methods such as the artificial neural network technique are widely used in geotechnical engineering problems as a powerful means for modeling and analyzing events [8, 17−21]. An artificial neural network is a simulation of the biological connection between neurons in the human brain that contributes to the statistical process of the available records of a particular event and find the governing rules that connect the input affecting the variables to the output targets. The multi-layer perceptron (MLP) artificial neural network is comprised of input and output layers and conditionally hidden layers. Each layer contains nodes (or a neuron) that represent a particular variable. Each neuron in the next layer is connected to the neurons of the preceding layer by weighted connections and a threshold bias. To find and adjust the neurons’ connection weights and bias, the network should be trained with a selected set of records until the error between the predicted and measured values of the final target falls within an acceptable range according to the learning rules. Then, by using another set of records called the test set, the network performance is controlled during network learning, and training stops once the error in the testing set increases. After the network training is completed, the performance of the network is assessed by applying the network to the rest of the set of records called the validation set. This procedure for dividing the records into three distinct sets is called the cross-validation technique and is the most effective method for avoiding overfitting of data in ANN [22]. 3.1 Experimental records database The anisotropy experimental data database consists of principal stress rotational test results performed using the HCTSA [12, 14, 23−25]. All tests in the database were conducted in similar conditions, which makes it possible to process them in a unified framework. In all tests, the mean total principal stress (corresponding to the sum of the total principal stresses) is constant during the shearing stage, which is equal to the initial consolidation confining pressure. The direction of the maximum principal stress (denoted by α) and the intermediate principal stress ratio (b) are constant during the test. The fine fraction of all samples (p.p. #200 sieve) is set to be

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zero. In Table 1, information on the database test samples is presented. The total number of records available in the experimental database is 73. 3.2 Input factors affecting anisotropy The effective peak friction angle of the sample (f′) is selected as the target variable in the neural network. This parameter is derived from the minimum and maximum effective principal stresses at the peak of the stress−strain curve with the following equation [14]: sin

1 3 1 3

(3)

The peak friction angle for some of the experimental database records is presented in the original work. However, in others, the friction angle of the samples is recalculated by substituting the peak effective principal stresses in Eq. (3). Based on the findings of experimental investigations, seven input variables are designated as the main factors in the ANN that influence anisotropy in the sands. All input variables are categorized into four groups. Two parameters are related to inherent soil characteristics and include the mean diameter (D50) and the grain shape (angularity). The parameters that describe the sample condition are the void ratio (e) and the sample preparation method. Generally, all samples in the experimental database are prepared with three methods: pluviation in water, pluviation in the air, and tamping. The other input variable is the initial consolidation confining pressure (pc), which is constant during the test. Finally, at the shear loading stage, the angle of the maximum principal stress direction (α) and the intermediate principal stress ratio (b) comprise the other input variables. Because the input variables have different ranges, to improve the efficiency of the network training, all input variables are preprocessed and normalized to fall within a uniform range between 0 and 1. Thus, each input variable (x) is scaled based on the maximum (xmax) and

minimum (xmin) values using the following scale equations. The output target variable is not subjected to normalizing, and a pure value is considered in the neural network: X normalized

x xmin xmax xmin

(4)

In order to adopt the cross-validation technique, all database records are divided into three distinct subsets for training, testing, and validating the neural network, which are 70%, 15% and 15%, respectively, of the records. To divide the data consistently [26], the average and standard deviations of the subsets lie within a range comparable to that of the main set of records. In Table 2, the statistical population of the input variables in three training, validation, and testing subsets together with the database set are presented. 3.3 Development of artificial neural network The artificial neural network implemented here is a feed-forward back propagation type network developed with the neural network toolbox in MATLAB [27]. The architecture of the network, as illustrated in Fig. 2, comprises the input layer with seven nodes, a hidden layer with 14 nodes, and the output layer involving one target neuron. The optimum number of nodes in the hidden layer is attained by several attempts until the neural network output is well fitted to the experimental data with the maximum degree of conformity. The sigmoid logistic function is used as a transfer function to calculate the output of the neuron in the hidden and output layers. In addition, the LevenbergMarquardt back propagation algorithm (the Trainlm function in MATLAB Inc. [27]) is adopted as the governing algorithm for training the network. This algorithm is an approximation of Newton’s method and has been shown to be one of the fastest algorithms for training a moderate-sized multi-layer perceptron type of neural network [27].

Table 1 Information of experimental database of principal stress rotation tests Tests Test Sand D50/ Void ratio, Reference Angularity No. condition name mm e 0.733, [25] 4 Undrained M31 Sand 0.28 Rounded 0.61 Toyoura Sub0.656, [14] 15 Undrained 0.15 Sand angular 0.75, 0.84 Santa Sub[12] 34 Drained Monica 0.25 0.68, 0.84 angular beach sand Ottawa [23] 15 Drained 0.11 Rounded 0.836 sand Toyoura Sub[24] 5 Drained 0.17 0.7 Sand angular

Preparation method Airpluviating Wetpluviating Wetpluviating Tamping Wetpluviating

pc/kPa

α/(°)

b

75, 130, 100, 215

45

0.5

15, 30, 45, 60, 75 0, 14, 13, 196 22, 27, 33, 44, 61 300, 650, 0, 30, 1000, 1350 45, 90 15, 30, 45, 98 60, 75 100

D50−Mean size of grain; pc−Confining pressure (Consolidation stress); α−Angle of maximum principle stress; b−Intermediate principle stress ratio.

0.5 0−1 0, 0.3, 0.5, 1 0.5

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Table 2 Statistics of populations used for training, validation and testing of ANN Data set

Input variable

Average

Mean size of grain, D50/mm

0.20

0.06

Angularity

Total

1

0.11

0.17

22.04

75

25

50

0.71

0.06

0.87

0.61

0.26

Preparation method2

18.47

4.90

30

10

20

Confining pressure, pc/kPa

266.31

278.55

1350.00

75.00

1275.00

Angle of maximum principal stress, α/(°)

37.61

24.61

90

0

90

Intermediate principal stress ratio, b

0.41

0.28

1.00

0.00

1.00

Effective peak friction angle, f′/(°)

39.59

5.25

48.90

27.06

21.84

Mean size of grain, D50/mm

0.20

0.06

0.28

0.11

0.17

61.54

22.18

75

25

50

0.71

0.06

0.87

0.61

0.26

Preparation method

18.85

5.06

30

10

20

1

2

Confining pressure, pc/kPa

267.58

287.39

1350.00

75.00

1275.00

Angle of maximum principal stress, α/(°)

38.23

24.73

90

0

90

Intermediate principal stress ratio, b

0.43

0.27

1.00

0.00

1.00

Effective peak friction angle, f′/(°)

39.44

5.19

48.90

27.06

21.84

Mean size of grain, D50/mm

0.19

0.06

0.25

0.11

0.14

60.00

22.91

75

25

50

0.72

0.06

0.84

0.67

0.17

Preparation method

17.00

4.58

20

10

10

Confining pressure, pc/kPa

347.80

330.46

1000.00

98.00

902.00

Angle of maximum principal stress, α/(°)

27.06

20.94

69.60

0

69.60

Intermediate principal stress ratio, b

0.35

0.28

0.90

0.00

0.90

Effective peak friction angle, f′/(°)

38.88

6.37

45.40

28.69

16.72

Angularity

1

Void ratio, e 2

Mean size of grain, D50/mm

0.18

0.06

0.25

0.11

0.14

Angularity1

65.00

20.00

75

25

50

0.70

0.05

0.84

0.61

0.18

Preparation method

18.00

4.00

20

10

10

Confining pressure, pc/kPa

178.20

74.64

300.00

98.00

202.00

Angle of maximum principal stress, α/(°)

44.94

23.94

90

15

75

Intermediate principal stress ratio, b

0.46

0.25

1.00

0.10

0.90

Effective peak friction angle, f′/(°)

41.09

3.89

47.99

33.49

14.50

Void ratio, e 2

Testing

Range

61.81

Void ratio, e

Validation

0.28

Minimum

Void ratio, e

Angularity

Training

Standard deviation Maximum

1−Angularity, 100: Angular, 75: Sub-angular, 50: Sub-rounded, 25: Rounded; 2−Preparation method, 30: Air-pluviation, 20: Wet-pluviation, 10: Tamping.

The main criteria for checking the performance of the neural network for predicting the target variables well are the coefficient of correlation (r), the coefficient of determination (r2), the root mean squared error (RMSE), and the sum of squared error (SSE). According to SMITH [22], if the value of the |γ| parameter for a network output becomes greater than 0.8, a strong correlation exists between the two sets of variables. After all the connection weights and bias in the neural network are determined, the contribution and the relative importance of each input variable in the target output are

determined by applying the “weights method” proposed by GOH [28]. 3.4 ANN modeling result The neural network is successfully trained and validated for the anisotropy test record subsets until the friction angle of sand is accurately predicted from the sample characteristics and loading orientation. The performance of the network in the training, validation, and testing stages and in modeling whole records is presented in Table 3. As seen, coefficient of correlation

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Fig. 2 Architecture of neural network

(r) for fitting for all records of the database becomes greater than 0.92, which indicates the existence of a good correlation between the predicted and measured values of the sample friction angle. The network prediction for training data sets and the whole database versus the actual value of the records are shown in Figs. 3(a) and (b), respectively.

In Fig. 4, the contribution fraction of each input variable in the target output is presented. The larger contribution belongs to the loading orientation with an importance factor of 18.4%. Other input variables participate with nearly the same fraction in the effective friction angle of sand, and the importance factor is almost 13%−14% for the input variables.

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Table 3 Analysis result of neural network Data set

r

r2

SSE

RMSE

Total data

0.927

0.859

218.906

1.768

Training

0.931

0.866

144.958

1.703

Validation

0.964

0.928

20.454

1.599

Testing

0.881

0.776

41.287

2.272

Fig. 3 Measured versus predicted value by neural network for all available data: (a) Training set; (b) All records in experimental database

Fig. 4 Relative importance for inputs of neural network

4 Sensitivity analysis of anisotropy effect The dependence of sand strength on the loading direction is investigated in various conditions by

changing the input variable in the neural network. In these sensitivity analyses, the variation in the peak effective friction angle of soil with the angle of the maximum principal stress is determined in the ANN by changing one input variable and holding the other variables constant. The angle of maximum principal stress varying from value of 0°, which is perpendicular to the sample bedding direction to 90°, is in coincidence to the sample bedding. 4.1 Effect of confining pressure For sub-angular wet pluviated sand with a mean diameter of 0.2 mm and a void ratio of 0.74, the variation in the effective peak frictional angle versus the angle of the maximum principal stress direction (α) in different confining pressures is illustrated in Fig. 5(b). At any confining pressure level, by rotating the maximum principal stress direction toward the bedding layer, the shear strength of the sample is decreased, which indicates the impact of anisotropy. However, by increasing the confining pressure, the intensity of anisotropy tends to decrease. For instant, under the 75 kPa confining pressure, the decrease in the friction angle due to the maximum stress rotation reaches 10.6°, and the shear strength loss under the higher confining pressure of 1350 kPa becomes 8.27°. In Fig. 5(c), the change of the friction angle versus the angle of maximum principal stress is presented for the loose sample that has the highest void ratio among the experimental database (i.e., e=0.87). In this sample, the anisotropy effect is intensified with confining pressure, and by increasing the applied confining pressure from 75 to 1000 kPa, the total loss of the friction angle due to stress rotation is amplified from 9.6 to 11.47. In the dense sample (with e=0.61), for which the anisotropy effect is presented in Fig. 5(a), the total loss of friction angle due to anisotropy is about 3° for confining pressure less than 300 kPa. Then, by increasing the confine pressure in the sample, the total loss of the sample friction angle is increased to 5°. The effect of confining pressure on the friction angle in the samples with various void ratios and subjected to the vertical loading direction (i.e.,α=0°) is presented in Fig. 6. This type of loading is analogous to axial compression loading in the triaxial test sample where consequent friction angle from such a test is used as the isotropic shear strength parameter of the soil in the stability analysis. The results indicate that although in the medium dense (e=0.74) and dense (e=0.61) samples, increasing confining pressure at the consolidation stage has little influence on the sample strength, the strength of the looser sample (with e=0.87) is affected more by increasing the confining pressure.

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Fig. 6 Effect of confining pressure on friction angle in samples with various void ratios and subjected to vertical maximum principal stress (α=0°) (All samples have sub angular grains with D50=0.2 mm, b=0.5 with wet pluviated preparation method)

Fig. 5 Effective peak friction angle versus maximum stress orientation at various levels of confining pressures in samples with sub angular grains, D50=0.2 mm and b=0.5, prepared by wet pluviation method: (a) e=0.61; (b) e=0.74; (c) e=0.87

4.2 Effect of sample density For samples with a mean diameter of 0.2 mm and consolidated under a confining pressure of 75 kPa, the variation in the effective peak friction angle versus the angle of the maximum principal stress direction is determined at various void ratios, as presented in Fig. 7(a). The void ratios are selected based on the maximum, minimum, and average of the test records in

Fig. 7 Effective peak friction angle versus maximum stress orientation in samples with various void ratios and having sub angular grains, D50=0.2 mm and b=0.5, prepared by wet pluviation method: (a) pc=75 kPa; (b) pc=500 kPa

the experimental database. According to the analysis results, at any loading direction, increasing the void ratio decreases the friction angle of the sample. In addition, the decrease in the

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sample friction angle with the rotation of the loading direction exists in all samples. However, the decrease in shear strength due to anisotropy is influenced by the sample density. In the dense sample, the total decrease in the soil friction angle due to stress rotation from α=0° to α=90° is only 3.4°. In contrast, by increasing the sample void ratio, the decrease in the sample shear strength becomes more profound, but at lower confining pressure, it does not follow any remarkable trend. Although in the medium dense sample with e=0.61 the total decrease in the friction angle due to stress rotation is about 10.6°, in the sample with e=0.87 the anisotropy effect becomes comparatively less than in medium dense sample, and the friction angle decrease is 9.6°. When this phenomenon is found under lower confining pressure, in higher consolidation pressure, the decrease in the friction angle due to anisotropy is increased by increasing the void ratio. In Fig. 7(b), the change of the friction angle versus the angle of α in the sample with various void ratios and subjected to 500 kPa confining pressure is presented. At this confining pressure level, the total decreases in the soil friction angle in samples with a void ratio of 0.61, 0.74, and 0.87 are 3.7°, 10.08°, and 11.4°, respectively. 4.3 Effect of grain size The other important factor affecting soil anisotropy is the grain size of the uniform sample. The decrease in the friction angle due to principal stress rotation in a sample with various grain sizes is presented in Fig. 8. All samples are sub-angular grains with a void ratio of 0.74 and are subjected to constant 300 kPa confining pressure and prepared with the water pluviation method. The range of the sample’s mean diameter (D50) is selected in the bound between 0.1 and 0.24 mm according to the

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experimental ANN records. The strength decrease in the samples due to stress rotation varies with grain size. While the decrease in the friction angle of the sample with smaller particles (i.e. D50=0.11 mm) is about 10.8°, the decrease in soil shear strength in the coarser sample (with D50=0.24 mm) reaches 6.2°. This result means that the anisotropy of strength becomes more dominant in finer soil, and by decreasing the grain size of the soil, the effect of anisotropy in the shear strength of the soil becomes more important. 4.4 Effect of grain angularity For the sample with a void ratio of 0.74 and consolidated under a confining pressure of 300 kPa, the effect of grain shape and angularity on the strength loss due to anisotropy is evaluated, and the result is shown in Fig. 9. In this analysis, three types of grain angularity, including rounded, sub-rounded, and sub-angular, are considered. According to Fig. 9, by changing the grain angularity from rounded to sub-angular, the effect of anisotropy in the soil becomes more distinguished, and the decrease in the sample friction angle due to stress rotation increases from 5.4° for rounded grains to 10° for sub-angular grains. In other words, increasing the angularity of the grains promotes the anisotropy effect in the sample.

Fig. 9 Effective peak friction angle versus maximum stress orientation in looser samples with various grain shapes having grains of D50=0.2 mm, e=0.74, b=0.5 and subjected to pc= 300 kPa, prepared by wet pluviation method

Fig. 8 Effective peak friction angle versus angle of maximum stress orientation in samples with various grain sizes having sub angular grains, e=0.74, b=0.5 and subjected to pc=300 kPa, prepared by wet pluviation method

The effect of grain angularity on soil strength anisotropy has been previously observed in an experimental study. UTHAYAKUMAR and VAID [13] performed the principal stress rotation test on two sands including Fraser River sand with sub-angular to sub-rounded particle and Syncrude sand with angular to sub-angular grains. According to the results, the principal

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stress rotation has more influence on the friction angle of Syncrude sand with angular and coarse grains. 4.5 Effect of preparation method The change of the anisotropic friction angle for samples prepared with various methods is presented in Fig. 10. Two samples are deposited in water and air, and the third sample is prepared with the tamping method. The other characteristics of all samples are the same, and the void ratio, applied confining pressure, intermediate principal stress ratio, and mean grain size are 0.74, 300 kPa, 0.5, and 0.2 mm, respectively. The anisotropy effect is found in all three samples regardless of the preparation procedure; thus, anisotropy is an intrinsic soil property. However, the decrease in the friction angle varies with the preparation method. In the tamped sample, by changing the maximum principal stress from the vertical to the horizontal direction, the peak effective friction angle of the sample varies from 38° to 29°, and the total decrease becomes 8°. In contrast, the total decrease in the friction angle in the wet-deposited sample is about 10.5°. The friction angle decrease due to anisotropy in air-deposited sand is 9.5° and thus lies in an intermediate level between that of the tamped and wet pluviation preparation methods. As a result, the order of the anisotropy effect regarding the preparation method is wet pluviation, air pluviation, and tamping. The evaluation of the effect of the sample preparation method on sand characteristics conducted by MIURA and TOKI [29] indicated that pluviation methods cause more significant anisotropy in sands, and the tamping method gives an intermediate result.

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prominent role in the response of the geotechnical structure especially when the mode of deformation in the whole structure follows the plan strain condition. However, in most cases, the role of intermediate principal stress is neglected in modeling the geotechnical structure due to the limitation of incorporating intermediate stress in most conventional models such as the Mohr-Columb model [30]. For medium dense samples subjected to vertical loading, the variation in the friction angle of the samples at various intermediate principal stress ratios (b) is presented in Fig. 11 for different levels of confining pressures. In all confining pressures, the shear strength of the soil is increased to b=0.5 and then by further increasing the intermediate stress, the friction angle of the sample decreases. Thus, when the intermediate principal stress lies in the middle of the maximum and minimum principal stresses, the soil gains its maximum shear strength.

Fig. 11 Effect of intermediate principal stress ratio (b) on friction angle in samples subjected to vertical maximum principal stress (α=0°) at various confining pressures (All samples have sub angular grains with D50=0.2 mm and e=0.74, prepared with wet pluviated method)

Fig. 10 Effective peak friction angle versus maximum stress orientation in looser samples prepared with different preparation methods having sub angular grains of D50=0.2 mm, e=0.74, b=0.5 and subjected to pc=300 kPa

4.6 Effect of intermediate stress The intermediate principal stress has a very

This phenomenon has been directly reported by other researchers who investigated the response of soil to intermediate stress in true triaxial and hollow cylinder apparatus. Although in some tests especially when the HCTSA is utilized, the b value corresponding to the maximum friction angle is about 0.3 [11, 13, 30], in other tests executed with a true triaxial apparatus, the maximum friction angle is related in b=0.5 [31−32]. Nevertheless, in all experimental evidences, when the maximum shear strength of the soil occurs in the b value between 0.3 and 0.5, the lateral strain of the soil goes to zero. In other words, the intermediate stress from which the soil gains its maximum shear strength is attributed to the plane strain condition.

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5

Application of neural network anisotropic slope stability analysis

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for

By implementing the proposed computational procedure for calculating the soil friction angle via the ANN, the effect of the anisotropy of the shear strength on the stability of the slopes is investigated. For this end, slopes with different geometries are considered, and the stability of the slopes is evaluated with isotropy and anisotropy conditions for the shear strength parameters, and the results of two types of stability analysis are compared to recognize the effect of anisotropy. 5.1 Slope geometry and material The geometry of the slope subjected to stability analysis is presented in Fig. 12 and is composed of an inclined slope face limited between two horizontal surfaces. Various geometric configurations are established for this slope by combining several magnitudes for the slope height (H) and the slope face angle (denoted by β in Fig. 12). Four measurements including H=10, 15, 20, and 25 m are assigned to the slope height. In addition, the face of the slope has various inclination angles: 14°, 21.8°, 26.6°, 35°, 45° and 56.3°.

Fig. 12 Geometrical consideration of slope

In all stability analyses, two conditions are supposed for the shear strength of the soil. In the first analysis, it is assumed that the friction angle of the soil layer is isotropic, and the same frictional angle is assigned to any point on the slope. The second stability analysis is performed by considering the dependence of the soil shear strength on the loading direction and the friction angle for any slice in desired slip is calculated with the ANN. In order to implement an ANN, soil characteristics required as input variables in the ANN are assigned so that they lie within the range of the experimental database. The slope consists of one-layer water-deposited homogenous sand with e=0.74 and has sub-angular grains with a mean diameter of 0.2 mm and specific

gravity of 2.65. The confining stress at the bottom of each slice is assumed to be identical to the normal stress that arises from the projection of the slice weight force acting in a perpendicular direction on the slice bottom and is shown by Ni in Fig. 12. To determine the maximum principal stress direction (αi) in the i-th slice, the following equation proposed by SU and LIAO [4] is used: αi=(90°−ωi)−(45°−

2

)

(5)

where ωi and f′ are the angle of the slice bottom against the vertical axis and the soil friction, respectively. In addition, it is assumed that the pattern of the failure plane is independent of the loading direction, and the isotropic friction angle is substituted in Eq. (5). The friction angle in the isotropic condition is obtained from the ANN by supposing the vertical loading direction (α=0°) and then averaging the outcoming friction angle for a wide range of confining pressures. To avoid slope instability, small cohesion of 5 kPa is assumed for the shear strength of the slope material. 5.2 Analysis method For each slope with specific geometry, the safety factor is calculated using the simplified Bishop method of limit equilibrium [33]. According to WRIGHT et al’s [34] findings, the safety factor calculated with the simplified Bishop procedure has favorable agreement (within about 5%) with the safety factor calculated using stresses computed independently by finite element procedures. In the simplified Bishop method, by satisfying the force equilibrium in all slices included in a circular slip surface, the factor of safety (FS, FOS) for this slip surface can be determined from the following equation: n c li cos i (Wi ui li cos i ) tan i FS cos i (sin i tan i) / FS i 1 n

Wi sin i

(6)

i 1

where Δli, Wi, and ωi are the width of the slice bottom, the weight of a slice, and the inclination angle of slice bottom in the i-th slice, respectively, which are shown in Fig. 12. In addition, f′i, c′i and ui are the effective friction angle, effective cohesion, and pore water pressure of the slice. Each slip surface is divided into n slices. The FOS can be precisely calculated by executing an iterative procedure in such a manner that the relative error between FOS in the left-hand side and FOS in the right-hand side of the equation does not exceed a certain value.

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5.3 Stability analysis result The FOS in the critical slip surface for all analysis cases (that have a minimum FOS) with both shear strength conditions together with the anisotropy ratio of

FOS are presented in Table 4 and Fig. 13. The parameter of the anisotropy ratio of the FOS is defined as the ratio of the FOS in the anisotropic to isotropic condition and used as an indicator for evaluating the anisotropy effect

Table 4 Factor of safety for critical slip surface in each case of analysis

14

Anisotropic FOS 3.773

H=10 m Isotropic FOS 5.014

Anisotropy ratio of FOS 0.752

Anisotropic FOS 3.596

H=15 m Isotropic FOS 4.845

Anisotropy ratio of FOS 0.742

21.8

2.643

3.357

0.787

2.453

3.148

0.779

26.6

2.199

2.729

0.806

2.080

2.586

0.805

35

1.749

2.069

0.845

1.647

1.946

0.846

45

1.368

1.535

0.891

1.285

1.435

0.896

56.3

1.292

1.403

0.921

1.202

1.304

0.922

14

Anisotropic FOS 3.556

H=20 m Isotropic FOS 4.808

Anisotropy ratio of FOS 0.740

Anisotropic FOS 3.561

H=25 m Isotropic FOS 4.813

Anisotropy ratio of FOS 0.740

21.8

2.391

3.081

0.776

2.328

3.018

0.771

26.6

2.000

2.513

0.796

1.959

2.457

0.797

35

1.558

1.868

0.834

1.516

1.811

0.837

Slope face angle, β/(°)

Slope face angle, β/(°)

45

1.247

1.380

0.904

1.224

1.354

0.904

56.3

1.163

1.260

0.923

1.152

1.251

0.921

Fig. 13 Change of FOS with slope face angle for all slopes: (a) FOS with isotropic condition; (b) FOS with anisotropic condition; (c) Ratio of isotropic FOS to anisotropic FOS (anisotropy ratio of FOS)

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on the slope stability. An anisotropy ratio close to one indicates less influence of anisotropy on the slope stability. As expected, by increasing the slope height and the slope face angle, the FOS in both anisotropic and isotropic conditions is decreased. The anisotropy ratio of the FOS varies with the geometric configuration of the slope and depends completely on the face angle. However, in all analysis cases, the anisotropy ratio of the FOS is less than one, which means that the calculated factor of safety in the isotropic condition overestimates the safety of the slope and deviates from the actual state of stability with an error between 5% and 25%. This issue is more critical for steeper and higher slopes where the safety of the slope is more important. For instance, in a slope with H=25 m and β=56.3°, while the calculated value of the FOS in the isotropic condition is about 1.251, the true value of the FOS (when anisotropy is taken into account) becomes 1.152, and thus, the slope is closer to instability. By comparing the FOS value and the anisotropy ratio for various geometric configurations, in slopes with identical height, by increasing the slope face angle, the anisotropy ratio of the FOS is increased and becomes closer to one. Generally, the range of the anisotropy ratio

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of the FOS varies from 0.75 for a flatter slope to about 0.92 for a steeper slope. A survey of the location of the critical slip surfaces in the anisotropic and isotropic conditions implies that the location of critical slip surfaces in most analysis cases is the same in isotropic and anisotropic conditions and not influenced by anisotropy. For instance, the locations of critical slip surfaces in the three analysis cases, 1) H=10 m and β=14°, 2) H=10 m and β=56.3° and 3) H=25 m and β=14°, match in isotropic and anisotropic conditions, as shown in Figs. 14(a)−(c). In the slope with a height of 25 m and a face angle of 56.3° shown in Fig. 14(d), the critical slip surface in the anisotropic condition is located deeper than the isotropic slip surface. This fact that anisotropy does not influence the location of the critical slip surface was previously observed in anisotropic cohesive slopes by ARAI and NAKAGAWA [35] and AL-KARNI and ALSHAMRANI [3]. The other important consequence is that in slopes with the same height, the depth of the critical slip surface (equal to the maximum distance of slope face from slip surface in the perpendicular direction to the slope face denoted by D in Fig. 12) in isotropic and anisotropic conditions is decreased by increasing the slope face

Fig. 14 Location of critical slip in slopes: (a) H=10 m, β=14°; (b) H=10 m, β=56.3°; (c) H=25 m, β=14°; (d) H=25 m, β=56.3°

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inclination angle. In a slope with a height of 25 m, increasing the slope face angle from 14° to 56.4° decreases the critical slip surfaces in the anisotropic condition from 6.62 m to 4.93 m.

[7] [8]

6 Conclusions [9]

1) Seven factors are introduced as input variables to determine the effective peak friction angle in an artificial neural network. The loading orientation contributes more in the sample friction angle. 2) The anisotropy effect is minor in the dense sample. At lower confining pressure, the total loss of the sample friction angle becomes slightly more in the loose sample than in the medium dense sample. However, by increasing the confining pressure, the decrease in the shear strength increases by increasing the sample porosity. 3) Although in the medium dense and dense samples increasing the confining pressure reduces the anisotropy effect, in this loose sample the effect increases as the confining pressure increases. 4) The sensitivity analysis shows that the anisotropy effect is enhanced in sand with fine grains and by decreasing the grain size of the soil, the effect of anisotropy in the shear strength of soil becomes more important. In addition, increasing the angularity of the grains promotes the anisotropy effect in the sample. The order of the anisotropy effect regarding the preparation method is wet pluviation, air pluviation, and tamping. 5) According to the slope stability analysis, the anisotropy effect causes the factor of safety for the critical slip surface to deviate from the isotropic condition and supposing isotropic condition for shear strength parameters of slope material overestimates the factor of safety by 5%−25%. This deviation is found more often in a flatter slope than in a steeper one. However, the critical slip surfaces match in the isotropic and anisotropic conditions.

[10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

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foundations, 2008, 48(6): 805−819. [3]

AL-KARNI A A, AL-SHAMRANI M A. Study of the effect of soil anisotropy on slope stability using method of slices [J]. Computers

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and Geotechnics, 2000, 26(2): 83−103. [4] [5]

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SU S, LIAO H. Effect of strength anisotropy on undrained slope stability in clay [J]. Geotechnique, 1999, 49(2): 215−230. CHEN W F, SNITBHAN N, FANG H Y. Stability of slopes in anisotropic, nonhomogeneous soils [J]. Canadian Geotechnical Journal, 1975, 12(1): 146−152. LO K. Stability of slopes in anistropic soils [J]. Journal of Soil

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Mechanics & Foundations Div, 1965, 91(SM4): 85−106.. CASAGRANDE A, CARILLO N. Shear failure of anisotropic materials [J]. J Boston Soc Civ Eng, 1944, 31(4): 74−87. LADE P, RODRIGUEZ N, VAN DYCK E. Effects of principal stress directions on 3D failure conditions in cross-anisotropic sand [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2014, 140(2): 04013001. CAI Y, YU H S, WANATOWSKI D, LI X. Non-coaxial behavior of sand under various stress paths [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 139(8): 1381−1395. JIANG M, LI L, YANG Q. Experimental investigation on deformation behavior of TJ-1 lunar soil simulant subjected to principal stress rotation [J]. Advances in Space Research, 2013, 52(1): 136−146. KUMRUZZAMAN M, YIN J H. Influences of principal stress direction and intermediate principal stress on the stress-strainstrength behaviour of completely decomposed granite [J]. Canadian Geotechnical Journal, 2010, 47(2): 164−179. LADE P V, NAM J, HONG W P. Shear banding and crossanisotropic behavior observed in laboratory sand tests with stress rotation [J]. Canadian Geotechnical Journal, 2008, 45(1): 74−84. UTHAYAKUMAR M, VAID Y. Static liquefaction of sands under multiaxial loading [J]. Canadian Geotechnical Journal, 1998, 35(2): 273−283. NAKATA Y, HYODO M, MURATA H, YASUFUKU N. Flow deformation of sands subjected to principal stress rotation [J]. Soils and Foundations, 1998, 38(2): 115−128. GENSA H D, SYMES M. The development of a new hollow cylinder apparatus for investigating the effects of principal stress rotation in soils [J]. Geotechnique, 1983, 33(4): 355−383. BISHOP A W. The strength of soils as engineering materials [J]. Rankine Lecture, Geotechnique, 1966, 16(2): 91−130. IKIZLER S B, VEKLI M, DOGAN E, AYTEKIN M, KOCABAS F. Prediction of swelling pressures of expansive soils using soft computing methods [J]. Neural Computing and Applications, 2012, 24(2): 1−13. KANAYAMA M, ROHE A, VAN PAASSEN L. Using and improving neural network models for ground settlement prediction [J]. Geotechnical and Geological Engineering, 2014, 32(3): 687−697. EMAMI M, YASROBI S S. Modeling and interpretation of pressuremeter test results with artificial neural networks [J]. Geotechnical and Geological Engineering, 2014, 32(2): 375−389. SHAHIN M A. Use of evolutionary computing for modelling some complex problems in geotechnical engineering [J]. Geomechanics and Geoengineering, 2014, (ahead-of-print): 1−17. SEZER A. Simple models for the estimation of shearing resistance angle of uniform sands [J]. Neural Computing and Applications, 2013, 22(1): 111−123. SMITH M. Neural networks for statistical modeling [M]. Thomson Learning. New York, NY, USA, John Wiley Sons, Inc, 1993: 59. DAKOULAS P, SUN Y. Fine Ottawa sand: experimental behavior and theoretical predictions [J]. Journal of Geotechnical Engineering, 1992, 118(12): 1906−1923. GUTIERREZ M, ISHIHARA K, TOWHATA I. Flow theory for sand during rotation of principal stress direction [J]. Soils and Foundations, 1991, 31(4): 121−132. TSOMOKOS A, GEORGIANNOU V N. Effect of grain shape and angularity on the undrained response of fine sands [J]. Canadian Geotechnical Journal, 2010, 47(5): 539−551. SHAHIN M A, MAIER H R, JAKSA M B. Data division for developing neural networks applied to geotechnical engineering [J]. Journal of Computing in Civil Engineering, 2004, 18(2): 105−114. DEMUTH H, BEALE M, WORKS M. MATLAB: Neural network toolbox: User’s guide [EB/OL]. MathWorks, Incorporated,

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Application of artificial neural network for calculating anisotropic friction angle of sands and effect on slope stability Hamed Farshbaf Aghajani, Hossein Salehzadeh, Habib Shahnazari School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O.B.16756-163, Iran © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: The anisotropy effect is one of the most prominent phenomena in soil mechanics. Although many experimental programs have investigated anisotropy in sand, a computational procedure for determining anisotropy is lacking. Thus, this work aims to develop a procedure for connecting the sand friction angle and the loading orientation. All principal stress rotation tests in the literatures were processed via an artificial neural network. Then, with sensitivity analysis, the effect of intrinsic soil properties, consolidation history, and test sample characteristics on enhancing anisotropy was examined. The results imply that decreasing the grain size of the soil increases the effect of anisotropy on soil shear strength. In addition, increasing the angularity of grains increases the anisotropy effect in the sample. The stability of a sandy slope was also examined by considering the anisotropy in shear strength parameters. If the anisotropy effect is neglected, slope safety is overestimated by 5%−25%. This deviation is more apparent in flatter slopes than in steeper ones. However, the critical slip surface in the most slopes is the same in isotropic and anisotropic conditions. Key words: anisotropy; artificial neural network; sand; principal stress rotation; slope stability

1 Introduction In most alluvial deposition in nature, the mechanical properties and strength of soil depend on the direction of the external loads, which is known as the anisotropy effect. When soil particles settle in water, the soil grains lie in the preferred direction, providing that the longitude axis of the grains is almost parallel to the deposition bedding direction. The mechanical response of the soil element thus depends not only on the magnitude of the applied stress but also on the orientation of the applied load relative to the bedding layer [1]. Unfortunately, anisotropy causes the soil mass to be weaker than expected. Thus, if the anisotropy feature of soil is neglected and only the strength characteristics of soil corresponding to the vertical direction are designated for the entire soil mass, the geotechnical structure design may be accompanied by problems. For instance, by supposing that a clay slope is comprised of isotropic cohesive soil and neglecting the anisotropy effect, the safety factor of the slope is overestimated by 5%−25% greater than the actual value [2−6]. Investigation of the anisotropy role in the response of geotechnical structures requires models that consider the effect of anisotropy and the rotation of the stress direction on the mechanical behavior of soil. For the Mohr-Columb failure envelope, CASAGRANDE and

CARILLO [7] presented a closed-form equation for the relation between soil cohesion (c) and the loading orientation (α) based on the exact value of the soil cohesion in horizontal (ch) and vertical (cv) directions as follows. This equation is widely implemented in the stability analysis of anisotropic cohesive soil slopes [3−6]: c ch (cv ch ) cos 2

(1)

Despite the CASAGRANDE and CARILLO equation for anisotropic cohesive soil, there is no mathematical equation for the friction angle parameter concerning the anisotropy effect. In other words, even though many experimental investigations have studied the anisotropic behavior of sands within the past few years, little attention has been paid to processing these test data and developing a computational procedure for the anisotropic friction angle. Therefore, due to this gap in the literature, this work aims to establish a computational procedure for determining the anisotropic friction angle in sand. Numerous experimental records from principal stress rotation tests were processed with an artificial neural network (ANN). After the ANN was trained and the architecture, connected weights, and bias were recognized, sensitivity analyses were performed to determine which parameter influences shear strength anisotropy. Finally, by implementing the ANN, the effect

Received date: 2014−03−17; Accepted date: 2014−12−23 Corresponding author: Hossein Salehzadeh, Associate Professor; Tel: +98−2177240399; E-mail: [email protected]

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of shear strength anisotropy on the stability of a sandy slope is investigated.

2 Anisotropy in sands In the last decade, researchers have utilized the hollow cylinder torsional shear apparatus (HCTSA) to investigate inherent and induced anisotropy in sand and clay soils [8−14]. This apparatus imposes four independent force components in the hollow cylinder sample, including the axial load, torque, and inner and outer pressures. This versatile feature is implemented to control the magnitude and direction of three-dimensional principal stresses within the sample. The stress state can be induced in any desired direction. Thus, the anisotropy behavior of the HCTSA sample is explored by rotating the directions of the principal stress axis in relation to the fixed bedding plane [15]. The stress state components at an arbitrary point within the HCTSA sample are illustrated in Fig. 1. At this point, the maximum (σ1) and minimum (σ3) principal stresses act in the plane of the sample wall, and the maximum principal stress makes an angle against the vertical axis denoted by α. The intermediate principal stress (σ2) is induced in the radial direction and expressed by a non-dimensional parameter called the intermediate principal stress ratio (b) as follows [16]: b

2 3 1 3

(2)

Fig. 1 Stress state in sample of hollow cylinder torsional shear apparatus (HCTSA)

3 Investigation of anisotropy by artificial neural network (ANN) For establishing a mathematical framework for

computing the anisotropic friction angle, all experimental anisotropy test data have been collected from the literature and processed. Because these tests were performed by several researchers under different but comparable experimental conditions, the range of the test data is wide. Therefore, an artificial neural network (ANN) is implemented to statistically process the test data and predict the friction angle of sands based on the soil characteristics and loading orientation. Today, soft computing methods such as the artificial neural network technique are widely used in geotechnical engineering problems as a powerful means for modeling and analyzing events [8, 17−21]. An artificial neural network is a simulation of the biological connection between neurons in the human brain that contributes to the statistical process of the available records of a particular event and find the governing rules that connect the input affecting the variables to the output targets. The multi-layer perceptron (MLP) artificial neural network is comprised of input and output layers and conditionally hidden layers. Each layer contains nodes (or a neuron) that represent a particular variable. Each neuron in the next layer is connected to the neurons of the preceding layer by weighted connections and a threshold bias. To find and adjust the neurons’ connection weights and bias, the network should be trained with a selected set of records until the error between the predicted and measured values of the final target falls within an acceptable range according to the learning rules. Then, by using another set of records called the test set, the network performance is controlled during network learning, and training stops once the error in the testing set increases. After the network training is completed, the performance of the network is assessed by applying the network to the rest of the set of records called the validation set. This procedure for dividing the records into three distinct sets is called the cross-validation technique and is the most effective method for avoiding overfitting of data in ANN [22]. 3.1 Experimental records database The anisotropy experimental data database consists of principal stress rotational test results performed using the HCTSA [12, 14, 23−25]. All tests in the database were conducted in similar conditions, which makes it possible to process them in a unified framework. In all tests, the mean total principal stress (corresponding to the sum of the total principal stresses) is constant during the shearing stage, which is equal to the initial consolidation confining pressure. The direction of the maximum principal stress (denoted by α) and the intermediate principal stress ratio (b) are constant during the test. The fine fraction of all samples (p.p. #200 sieve) is set to be

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zero. In Table 1, information on the database test samples is presented. The total number of records available in the experimental database is 73. 3.2 Input factors affecting anisotropy The effective peak friction angle of the sample (f′) is selected as the target variable in the neural network. This parameter is derived from the minimum and maximum effective principal stresses at the peak of the stress−strain curve with the following equation [14]: sin

1 3 1 3

(3)

The peak friction angle for some of the experimental database records is presented in the original work. However, in others, the friction angle of the samples is recalculated by substituting the peak effective principal stresses in Eq. (3). Based on the findings of experimental investigations, seven input variables are designated as the main factors in the ANN that influence anisotropy in the sands. All input variables are categorized into four groups. Two parameters are related to inherent soil characteristics and include the mean diameter (D50) and the grain shape (angularity). The parameters that describe the sample condition are the void ratio (e) and the sample preparation method. Generally, all samples in the experimental database are prepared with three methods: pluviation in water, pluviation in the air, and tamping. The other input variable is the initial consolidation confining pressure (pc), which is constant during the test. Finally, at the shear loading stage, the angle of the maximum principal stress direction (α) and the intermediate principal stress ratio (b) comprise the other input variables. Because the input variables have different ranges, to improve the efficiency of the network training, all input variables are preprocessed and normalized to fall within a uniform range between 0 and 1. Thus, each input variable (x) is scaled based on the maximum (xmax) and

minimum (xmin) values using the following scale equations. The output target variable is not subjected to normalizing, and a pure value is considered in the neural network: X normalized

x xmin xmax xmin

(4)

In order to adopt the cross-validation technique, all database records are divided into three distinct subsets for training, testing, and validating the neural network, which are 70%, 15% and 15%, respectively, of the records. To divide the data consistently [26], the average and standard deviations of the subsets lie within a range comparable to that of the main set of records. In Table 2, the statistical population of the input variables in three training, validation, and testing subsets together with the database set are presented. 3.3 Development of artificial neural network The artificial neural network implemented here is a feed-forward back propagation type network developed with the neural network toolbox in MATLAB [27]. The architecture of the network, as illustrated in Fig. 2, comprises the input layer with seven nodes, a hidden layer with 14 nodes, and the output layer involving one target neuron. The optimum number of nodes in the hidden layer is attained by several attempts until the neural network output is well fitted to the experimental data with the maximum degree of conformity. The sigmoid logistic function is used as a transfer function to calculate the output of the neuron in the hidden and output layers. In addition, the LevenbergMarquardt back propagation algorithm (the Trainlm function in MATLAB Inc. [27]) is adopted as the governing algorithm for training the network. This algorithm is an approximation of Newton’s method and has been shown to be one of the fastest algorithms for training a moderate-sized multi-layer perceptron type of neural network [27].

Table 1 Information of experimental database of principal stress rotation tests Tests Test Sand D50/ Void ratio, Reference Angularity No. condition name mm e 0.733, [25] 4 Undrained M31 Sand 0.28 Rounded 0.61 Toyoura Sub0.656, [14] 15 Undrained 0.15 Sand angular 0.75, 0.84 Santa Sub[12] 34 Drained Monica 0.25 0.68, 0.84 angular beach sand Ottawa [23] 15 Drained 0.11 Rounded 0.836 sand Toyoura Sub[24] 5 Drained 0.17 0.7 Sand angular

Preparation method Airpluviating Wetpluviating Wetpluviating Tamping Wetpluviating

pc/kPa

α/(°)

b

75, 130, 100, 215

45

0.5

15, 30, 45, 60, 75 0, 14, 13, 196 22, 27, 33, 44, 61 300, 650, 0, 30, 1000, 1350 45, 90 15, 30, 45, 98 60, 75 100

D50−Mean size of grain; pc−Confining pressure (Consolidation stress); α−Angle of maximum principle stress; b−Intermediate principle stress ratio.

0.5 0−1 0, 0.3, 0.5, 1 0.5

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Table 2 Statistics of populations used for training, validation and testing of ANN Data set

Input variable

Average

Mean size of grain, D50/mm

0.20

0.06

Angularity

Total

1

0.11

0.17

22.04

75

25

50

0.71

0.06

0.87

0.61

0.26

Preparation method2

18.47

4.90

30

10

20

Confining pressure, pc/kPa

266.31

278.55

1350.00

75.00

1275.00

Angle of maximum principal stress, α/(°)

37.61

24.61

90

0

90

Intermediate principal stress ratio, b

0.41

0.28

1.00

0.00

1.00

Effective peak friction angle, f′/(°)

39.59

5.25

48.90

27.06

21.84

Mean size of grain, D50/mm

0.20

0.06

0.28

0.11

0.17

61.54

22.18

75

25

50

0.71

0.06

0.87

0.61

0.26

Preparation method

18.85

5.06

30

10

20

1

2

Confining pressure, pc/kPa

267.58

287.39

1350.00

75.00

1275.00

Angle of maximum principal stress, α/(°)

38.23

24.73

90

0

90

Intermediate principal stress ratio, b

0.43

0.27

1.00

0.00

1.00

Effective peak friction angle, f′/(°)

39.44

5.19

48.90

27.06

21.84

Mean size of grain, D50/mm

0.19

0.06

0.25

0.11

0.14

60.00

22.91

75

25

50

0.72

0.06

0.84

0.67

0.17

Preparation method

17.00

4.58

20

10

10

Confining pressure, pc/kPa

347.80

330.46

1000.00

98.00

902.00

Angle of maximum principal stress, α/(°)

27.06

20.94

69.60

0

69.60

Intermediate principal stress ratio, b

0.35

0.28

0.90

0.00

0.90

Effective peak friction angle, f′/(°)

38.88

6.37

45.40

28.69

16.72

Angularity

1

Void ratio, e 2

Mean size of grain, D50/mm

0.18

0.06

0.25

0.11

0.14

Angularity1

65.00

20.00

75

25

50

0.70

0.05

0.84

0.61

0.18

Preparation method

18.00

4.00

20

10

10

Confining pressure, pc/kPa

178.20

74.64

300.00

98.00

202.00

Angle of maximum principal stress, α/(°)

44.94

23.94

90

15

75

Intermediate principal stress ratio, b

0.46

0.25

1.00

0.10

0.90

Effective peak friction angle, f′/(°)

41.09

3.89

47.99

33.49

14.50

Void ratio, e 2

Testing

Range

61.81

Void ratio, e

Validation

0.28

Minimum

Void ratio, e

Angularity

Training

Standard deviation Maximum

1−Angularity, 100: Angular, 75: Sub-angular, 50: Sub-rounded, 25: Rounded; 2−Preparation method, 30: Air-pluviation, 20: Wet-pluviation, 10: Tamping.

The main criteria for checking the performance of the neural network for predicting the target variables well are the coefficient of correlation (r), the coefficient of determination (r2), the root mean squared error (RMSE), and the sum of squared error (SSE). According to SMITH [22], if the value of the |γ| parameter for a network output becomes greater than 0.8, a strong correlation exists between the two sets of variables. After all the connection weights and bias in the neural network are determined, the contribution and the relative importance of each input variable in the target output are

determined by applying the “weights method” proposed by GOH [28]. 3.4 ANN modeling result The neural network is successfully trained and validated for the anisotropy test record subsets until the friction angle of sand is accurately predicted from the sample characteristics and loading orientation. The performance of the network in the training, validation, and testing stages and in modeling whole records is presented in Table 3. As seen, coefficient of correlation

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Fig. 2 Architecture of neural network

(r) for fitting for all records of the database becomes greater than 0.92, which indicates the existence of a good correlation between the predicted and measured values of the sample friction angle. The network prediction for training data sets and the whole database versus the actual value of the records are shown in Figs. 3(a) and (b), respectively.

In Fig. 4, the contribution fraction of each input variable in the target output is presented. The larger contribution belongs to the loading orientation with an importance factor of 18.4%. Other input variables participate with nearly the same fraction in the effective friction angle of sand, and the importance factor is almost 13%−14% for the input variables.

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Table 3 Analysis result of neural network Data set

r

r2

SSE

RMSE

Total data

0.927

0.859

218.906

1.768

Training

0.931

0.866

144.958

1.703

Validation

0.964

0.928

20.454

1.599

Testing

0.881

0.776

41.287

2.272

Fig. 3 Measured versus predicted value by neural network for all available data: (a) Training set; (b) All records in experimental database

Fig. 4 Relative importance for inputs of neural network

4 Sensitivity analysis of anisotropy effect The dependence of sand strength on the loading direction is investigated in various conditions by

changing the input variable in the neural network. In these sensitivity analyses, the variation in the peak effective friction angle of soil with the angle of the maximum principal stress is determined in the ANN by changing one input variable and holding the other variables constant. The angle of maximum principal stress varying from value of 0°, which is perpendicular to the sample bedding direction to 90°, is in coincidence to the sample bedding. 4.1 Effect of confining pressure For sub-angular wet pluviated sand with a mean diameter of 0.2 mm and a void ratio of 0.74, the variation in the effective peak frictional angle versus the angle of the maximum principal stress direction (α) in different confining pressures is illustrated in Fig. 5(b). At any confining pressure level, by rotating the maximum principal stress direction toward the bedding layer, the shear strength of the sample is decreased, which indicates the impact of anisotropy. However, by increasing the confining pressure, the intensity of anisotropy tends to decrease. For instant, under the 75 kPa confining pressure, the decrease in the friction angle due to the maximum stress rotation reaches 10.6°, and the shear strength loss under the higher confining pressure of 1350 kPa becomes 8.27°. In Fig. 5(c), the change of the friction angle versus the angle of maximum principal stress is presented for the loose sample that has the highest void ratio among the experimental database (i.e., e=0.87). In this sample, the anisotropy effect is intensified with confining pressure, and by increasing the applied confining pressure from 75 to 1000 kPa, the total loss of the friction angle due to stress rotation is amplified from 9.6 to 11.47. In the dense sample (with e=0.61), for which the anisotropy effect is presented in Fig. 5(a), the total loss of friction angle due to anisotropy is about 3° for confining pressure less than 300 kPa. Then, by increasing the confine pressure in the sample, the total loss of the sample friction angle is increased to 5°. The effect of confining pressure on the friction angle in the samples with various void ratios and subjected to the vertical loading direction (i.e.,α=0°) is presented in Fig. 6. This type of loading is analogous to axial compression loading in the triaxial test sample where consequent friction angle from such a test is used as the isotropic shear strength parameter of the soil in the stability analysis. The results indicate that although in the medium dense (e=0.74) and dense (e=0.61) samples, increasing confining pressure at the consolidation stage has little influence on the sample strength, the strength of the looser sample (with e=0.87) is affected more by increasing the confining pressure.

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Fig. 6 Effect of confining pressure on friction angle in samples with various void ratios and subjected to vertical maximum principal stress (α=0°) (All samples have sub angular grains with D50=0.2 mm, b=0.5 with wet pluviated preparation method)

Fig. 5 Effective peak friction angle versus maximum stress orientation at various levels of confining pressures in samples with sub angular grains, D50=0.2 mm and b=0.5, prepared by wet pluviation method: (a) e=0.61; (b) e=0.74; (c) e=0.87

4.2 Effect of sample density For samples with a mean diameter of 0.2 mm and consolidated under a confining pressure of 75 kPa, the variation in the effective peak friction angle versus the angle of the maximum principal stress direction is determined at various void ratios, as presented in Fig. 7(a). The void ratios are selected based on the maximum, minimum, and average of the test records in

Fig. 7 Effective peak friction angle versus maximum stress orientation in samples with various void ratios and having sub angular grains, D50=0.2 mm and b=0.5, prepared by wet pluviation method: (a) pc=75 kPa; (b) pc=500 kPa

the experimental database. According to the analysis results, at any loading direction, increasing the void ratio decreases the friction angle of the sample. In addition, the decrease in the

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sample friction angle with the rotation of the loading direction exists in all samples. However, the decrease in shear strength due to anisotropy is influenced by the sample density. In the dense sample, the total decrease in the soil friction angle due to stress rotation from α=0° to α=90° is only 3.4°. In contrast, by increasing the sample void ratio, the decrease in the sample shear strength becomes more profound, but at lower confining pressure, it does not follow any remarkable trend. Although in the medium dense sample with e=0.61 the total decrease in the friction angle due to stress rotation is about 10.6°, in the sample with e=0.87 the anisotropy effect becomes comparatively less than in medium dense sample, and the friction angle decrease is 9.6°. When this phenomenon is found under lower confining pressure, in higher consolidation pressure, the decrease in the friction angle due to anisotropy is increased by increasing the void ratio. In Fig. 7(b), the change of the friction angle versus the angle of α in the sample with various void ratios and subjected to 500 kPa confining pressure is presented. At this confining pressure level, the total decreases in the soil friction angle in samples with a void ratio of 0.61, 0.74, and 0.87 are 3.7°, 10.08°, and 11.4°, respectively. 4.3 Effect of grain size The other important factor affecting soil anisotropy is the grain size of the uniform sample. The decrease in the friction angle due to principal stress rotation in a sample with various grain sizes is presented in Fig. 8. All samples are sub-angular grains with a void ratio of 0.74 and are subjected to constant 300 kPa confining pressure and prepared with the water pluviation method. The range of the sample’s mean diameter (D50) is selected in the bound between 0.1 and 0.24 mm according to the

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experimental ANN records. The strength decrease in the samples due to stress rotation varies with grain size. While the decrease in the friction angle of the sample with smaller particles (i.e. D50=0.11 mm) is about 10.8°, the decrease in soil shear strength in the coarser sample (with D50=0.24 mm) reaches 6.2°. This result means that the anisotropy of strength becomes more dominant in finer soil, and by decreasing the grain size of the soil, the effect of anisotropy in the shear strength of the soil becomes more important. 4.4 Effect of grain angularity For the sample with a void ratio of 0.74 and consolidated under a confining pressure of 300 kPa, the effect of grain shape and angularity on the strength loss due to anisotropy is evaluated, and the result is shown in Fig. 9. In this analysis, three types of grain angularity, including rounded, sub-rounded, and sub-angular, are considered. According to Fig. 9, by changing the grain angularity from rounded to sub-angular, the effect of anisotropy in the soil becomes more distinguished, and the decrease in the sample friction angle due to stress rotation increases from 5.4° for rounded grains to 10° for sub-angular grains. In other words, increasing the angularity of the grains promotes the anisotropy effect in the sample.

Fig. 9 Effective peak friction angle versus maximum stress orientation in looser samples with various grain shapes having grains of D50=0.2 mm, e=0.74, b=0.5 and subjected to pc= 300 kPa, prepared by wet pluviation method

Fig. 8 Effective peak friction angle versus angle of maximum stress orientation in samples with various grain sizes having sub angular grains, e=0.74, b=0.5 and subjected to pc=300 kPa, prepared by wet pluviation method

The effect of grain angularity on soil strength anisotropy has been previously observed in an experimental study. UTHAYAKUMAR and VAID [13] performed the principal stress rotation test on two sands including Fraser River sand with sub-angular to sub-rounded particle and Syncrude sand with angular to sub-angular grains. According to the results, the principal

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stress rotation has more influence on the friction angle of Syncrude sand with angular and coarse grains. 4.5 Effect of preparation method The change of the anisotropic friction angle for samples prepared with various methods is presented in Fig. 10. Two samples are deposited in water and air, and the third sample is prepared with the tamping method. The other characteristics of all samples are the same, and the void ratio, applied confining pressure, intermediate principal stress ratio, and mean grain size are 0.74, 300 kPa, 0.5, and 0.2 mm, respectively. The anisotropy effect is found in all three samples regardless of the preparation procedure; thus, anisotropy is an intrinsic soil property. However, the decrease in the friction angle varies with the preparation method. In the tamped sample, by changing the maximum principal stress from the vertical to the horizontal direction, the peak effective friction angle of the sample varies from 38° to 29°, and the total decrease becomes 8°. In contrast, the total decrease in the friction angle in the wet-deposited sample is about 10.5°. The friction angle decrease due to anisotropy in air-deposited sand is 9.5° and thus lies in an intermediate level between that of the tamped and wet pluviation preparation methods. As a result, the order of the anisotropy effect regarding the preparation method is wet pluviation, air pluviation, and tamping. The evaluation of the effect of the sample preparation method on sand characteristics conducted by MIURA and TOKI [29] indicated that pluviation methods cause more significant anisotropy in sands, and the tamping method gives an intermediate result.

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prominent role in the response of the geotechnical structure especially when the mode of deformation in the whole structure follows the plan strain condition. However, in most cases, the role of intermediate principal stress is neglected in modeling the geotechnical structure due to the limitation of incorporating intermediate stress in most conventional models such as the Mohr-Columb model [30]. For medium dense samples subjected to vertical loading, the variation in the friction angle of the samples at various intermediate principal stress ratios (b) is presented in Fig. 11 for different levels of confining pressures. In all confining pressures, the shear strength of the soil is increased to b=0.5 and then by further increasing the intermediate stress, the friction angle of the sample decreases. Thus, when the intermediate principal stress lies in the middle of the maximum and minimum principal stresses, the soil gains its maximum shear strength.

Fig. 11 Effect of intermediate principal stress ratio (b) on friction angle in samples subjected to vertical maximum principal stress (α=0°) at various confining pressures (All samples have sub angular grains with D50=0.2 mm and e=0.74, prepared with wet pluviated method)

Fig. 10 Effective peak friction angle versus maximum stress orientation in looser samples prepared with different preparation methods having sub angular grains of D50=0.2 mm, e=0.74, b=0.5 and subjected to pc=300 kPa

4.6 Effect of intermediate stress The intermediate principal stress has a very

This phenomenon has been directly reported by other researchers who investigated the response of soil to intermediate stress in true triaxial and hollow cylinder apparatus. Although in some tests especially when the HCTSA is utilized, the b value corresponding to the maximum friction angle is about 0.3 [11, 13, 30], in other tests executed with a true triaxial apparatus, the maximum friction angle is related in b=0.5 [31−32]. Nevertheless, in all experimental evidences, when the maximum shear strength of the soil occurs in the b value between 0.3 and 0.5, the lateral strain of the soil goes to zero. In other words, the intermediate stress from which the soil gains its maximum shear strength is attributed to the plane strain condition.

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5

Application of neural network anisotropic slope stability analysis

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for

By implementing the proposed computational procedure for calculating the soil friction angle via the ANN, the effect of the anisotropy of the shear strength on the stability of the slopes is investigated. For this end, slopes with different geometries are considered, and the stability of the slopes is evaluated with isotropy and anisotropy conditions for the shear strength parameters, and the results of two types of stability analysis are compared to recognize the effect of anisotropy. 5.1 Slope geometry and material The geometry of the slope subjected to stability analysis is presented in Fig. 12 and is composed of an inclined slope face limited between two horizontal surfaces. Various geometric configurations are established for this slope by combining several magnitudes for the slope height (H) and the slope face angle (denoted by β in Fig. 12). Four measurements including H=10, 15, 20, and 25 m are assigned to the slope height. In addition, the face of the slope has various inclination angles: 14°, 21.8°, 26.6°, 35°, 45° and 56.3°.

Fig. 12 Geometrical consideration of slope

In all stability analyses, two conditions are supposed for the shear strength of the soil. In the first analysis, it is assumed that the friction angle of the soil layer is isotropic, and the same frictional angle is assigned to any point on the slope. The second stability analysis is performed by considering the dependence of the soil shear strength on the loading direction and the friction angle for any slice in desired slip is calculated with the ANN. In order to implement an ANN, soil characteristics required as input variables in the ANN are assigned so that they lie within the range of the experimental database. The slope consists of one-layer water-deposited homogenous sand with e=0.74 and has sub-angular grains with a mean diameter of 0.2 mm and specific

gravity of 2.65. The confining stress at the bottom of each slice is assumed to be identical to the normal stress that arises from the projection of the slice weight force acting in a perpendicular direction on the slice bottom and is shown by Ni in Fig. 12. To determine the maximum principal stress direction (αi) in the i-th slice, the following equation proposed by SU and LIAO [4] is used: αi=(90°−ωi)−(45°−

2

)

(5)

where ωi and f′ are the angle of the slice bottom against the vertical axis and the soil friction, respectively. In addition, it is assumed that the pattern of the failure plane is independent of the loading direction, and the isotropic friction angle is substituted in Eq. (5). The friction angle in the isotropic condition is obtained from the ANN by supposing the vertical loading direction (α=0°) and then averaging the outcoming friction angle for a wide range of confining pressures. To avoid slope instability, small cohesion of 5 kPa is assumed for the shear strength of the slope material. 5.2 Analysis method For each slope with specific geometry, the safety factor is calculated using the simplified Bishop method of limit equilibrium [33]. According to WRIGHT et al’s [34] findings, the safety factor calculated with the simplified Bishop procedure has favorable agreement (within about 5%) with the safety factor calculated using stresses computed independently by finite element procedures. In the simplified Bishop method, by satisfying the force equilibrium in all slices included in a circular slip surface, the factor of safety (FS, FOS) for this slip surface can be determined from the following equation: n c li cos i (Wi ui li cos i ) tan i FS cos i (sin i tan i) / FS i 1 n

Wi sin i

(6)

i 1

where Δli, Wi, and ωi are the width of the slice bottom, the weight of a slice, and the inclination angle of slice bottom in the i-th slice, respectively, which are shown in Fig. 12. In addition, f′i, c′i and ui are the effective friction angle, effective cohesion, and pore water pressure of the slice. Each slip surface is divided into n slices. The FOS can be precisely calculated by executing an iterative procedure in such a manner that the relative error between FOS in the left-hand side and FOS in the right-hand side of the equation does not exceed a certain value.

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5.3 Stability analysis result The FOS in the critical slip surface for all analysis cases (that have a minimum FOS) with both shear strength conditions together with the anisotropy ratio of

FOS are presented in Table 4 and Fig. 13. The parameter of the anisotropy ratio of the FOS is defined as the ratio of the FOS in the anisotropic to isotropic condition and used as an indicator for evaluating the anisotropy effect

Table 4 Factor of safety for critical slip surface in each case of analysis

14

Anisotropic FOS 3.773

H=10 m Isotropic FOS 5.014

Anisotropy ratio of FOS 0.752

Anisotropic FOS 3.596

H=15 m Isotropic FOS 4.845

Anisotropy ratio of FOS 0.742

21.8

2.643

3.357

0.787

2.453

3.148

0.779

26.6

2.199

2.729

0.806

2.080

2.586

0.805

35

1.749

2.069

0.845

1.647

1.946

0.846

45

1.368

1.535

0.891

1.285

1.435

0.896

56.3

1.292

1.403

0.921

1.202

1.304

0.922

14

Anisotropic FOS 3.556

H=20 m Isotropic FOS 4.808

Anisotropy ratio of FOS 0.740

Anisotropic FOS 3.561

H=25 m Isotropic FOS 4.813

Anisotropy ratio of FOS 0.740

21.8

2.391

3.081

0.776

2.328

3.018

0.771

26.6

2.000

2.513

0.796

1.959

2.457

0.797

35

1.558

1.868

0.834

1.516

1.811

0.837

Slope face angle, β/(°)

Slope face angle, β/(°)

45

1.247

1.380

0.904

1.224

1.354

0.904

56.3

1.163

1.260

0.923

1.152

1.251

0.921

Fig. 13 Change of FOS with slope face angle for all slopes: (a) FOS with isotropic condition; (b) FOS with anisotropic condition; (c) Ratio of isotropic FOS to anisotropic FOS (anisotropy ratio of FOS)

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on the slope stability. An anisotropy ratio close to one indicates less influence of anisotropy on the slope stability. As expected, by increasing the slope height and the slope face angle, the FOS in both anisotropic and isotropic conditions is decreased. The anisotropy ratio of the FOS varies with the geometric configuration of the slope and depends completely on the face angle. However, in all analysis cases, the anisotropy ratio of the FOS is less than one, which means that the calculated factor of safety in the isotropic condition overestimates the safety of the slope and deviates from the actual state of stability with an error between 5% and 25%. This issue is more critical for steeper and higher slopes where the safety of the slope is more important. For instance, in a slope with H=25 m and β=56.3°, while the calculated value of the FOS in the isotropic condition is about 1.251, the true value of the FOS (when anisotropy is taken into account) becomes 1.152, and thus, the slope is closer to instability. By comparing the FOS value and the anisotropy ratio for various geometric configurations, in slopes with identical height, by increasing the slope face angle, the anisotropy ratio of the FOS is increased and becomes closer to one. Generally, the range of the anisotropy ratio

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of the FOS varies from 0.75 for a flatter slope to about 0.92 for a steeper slope. A survey of the location of the critical slip surfaces in the anisotropic and isotropic conditions implies that the location of critical slip surfaces in most analysis cases is the same in isotropic and anisotropic conditions and not influenced by anisotropy. For instance, the locations of critical slip surfaces in the three analysis cases, 1) H=10 m and β=14°, 2) H=10 m and β=56.3° and 3) H=25 m and β=14°, match in isotropic and anisotropic conditions, as shown in Figs. 14(a)−(c). In the slope with a height of 25 m and a face angle of 56.3° shown in Fig. 14(d), the critical slip surface in the anisotropic condition is located deeper than the isotropic slip surface. This fact that anisotropy does not influence the location of the critical slip surface was previously observed in anisotropic cohesive slopes by ARAI and NAKAGAWA [35] and AL-KARNI and ALSHAMRANI [3]. The other important consequence is that in slopes with the same height, the depth of the critical slip surface (equal to the maximum distance of slope face from slip surface in the perpendicular direction to the slope face denoted by D in Fig. 12) in isotropic and anisotropic conditions is decreased by increasing the slope face

Fig. 14 Location of critical slip in slopes: (a) H=10 m, β=14°; (b) H=10 m, β=56.3°; (c) H=25 m, β=14°; (d) H=25 m, β=56.3°

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inclination angle. In a slope with a height of 25 m, increasing the slope face angle from 14° to 56.4° decreases the critical slip surfaces in the anisotropic condition from 6.62 m to 4.93 m.

[7] [8]

6 Conclusions [9]

1) Seven factors are introduced as input variables to determine the effective peak friction angle in an artificial neural network. The loading orientation contributes more in the sample friction angle. 2) The anisotropy effect is minor in the dense sample. At lower confining pressure, the total loss of the sample friction angle becomes slightly more in the loose sample than in the medium dense sample. However, by increasing the confining pressure, the decrease in the shear strength increases by increasing the sample porosity. 3) Although in the medium dense and dense samples increasing the confining pressure reduces the anisotropy effect, in this loose sample the effect increases as the confining pressure increases. 4) The sensitivity analysis shows that the anisotropy effect is enhanced in sand with fine grains and by decreasing the grain size of the soil, the effect of anisotropy in the shear strength of soil becomes more important. In addition, increasing the angularity of the grains promotes the anisotropy effect in the sample. The order of the anisotropy effect regarding the preparation method is wet pluviation, air pluviation, and tamping. 5) According to the slope stability analysis, the anisotropy effect causes the factor of safety for the critical slip surface to deviate from the isotropic condition and supposing isotropic condition for shear strength parameters of slope material overestimates the factor of safety by 5%−25%. This deviation is found more often in a flatter slope than in a steeper one. However, the critical slip surfaces match in the isotropic and anisotropic conditions.

[10]

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