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Estimation of radial deformations around circular tunnels in weak rock masses through numerical modelling

T

A. Sakcalia,∗, H. Yavuzb a b

Ataturk University, Oltu Earth Sciences Faculty, Mining Engineering Department, Erzurum, Turkey Suleyman Demirel University, Engineering Faculty, Mining Engineering Department, Isparta, Turkey

ARTICLE INFO

ABSTRACT

Keywords: Finite difference Geological strength index (GSI) Longitudinal deformation profile (LDP) Numerical modelling Weak rock mass

Tunnelling applications are increasing day by day due to challenging topographic conditions. During tunnel excavation, the primary stresses caused mainly by the overlying weight of rock mass together with the geological factors are replaced by secondary stresses. Thus, tunnel deformations occur around the face but also at and beyond the face in the longitudinal direction as this is truly 3D phenomenon. Determination of deformations behind the advancing face is very important for the design of support systems in tunnel engineering. The researchers emphasized that deformations around the circular tunnels depend on the distance to the face and recommended different equations to construct longitudinal deformation profiles (LDP). In this paper, circular tunnels excavated in weak rock masses with a total of 162 different conditions comprising material properties, tunnel dimensions and depth of cover were modelled by using finite difference method. The data obtained from the models were normalised with different parameters and statistical tests were performed. The equations and response surfaces of maximum and tunnel face deformation for different depths were suggested for weak rock masses. Therewithal, a prediction equation of longitudinal deformation profile (LDP) for weak rock masses was proposed. Assessment of radial deformations around a circular tunnel is possible by using these equations. The suggested equations were verified by means of analysis of variance and previous equations recommended by different researchers.

1. Introduction The need for underground structures for various purposes led to the opening of tunnels using different methods. Modelling studies used as a decision-making tool in solving problems have also increased with the increase of tunnels. There are several studies related to physical and numerical modelling of tunnels by using different methods in the literature. In a few of these studies, the deformations occur around an underground opening were examined.1–4 It is important to determine the radial deformations around the tunnel face for the design of suitable supports to ensure the stability of the tunnels. The analyses of deformations around the underground openings and interaction with applied support systems were carried out by many researchers with analytical methods5–9 or with numerical modelling10–15 by considering two dimensional plain-strain conditions. While 2D plane strain conditions are considered as a suitable tool to predict the Ground Reaction Curve (GRC) and Support Characteristic Curve (SCC) in convergence-confinement method of support design, the effect of face distance on the deformation around the tunnel (Longitudinal

∗

Deformation Profile, LDP) are required to determine the distance behind the face where the load is started to transmit on the support. 3D numerical modelling studies were carried out by some researchers to determine the radial deformations ahead and behind the face of underground opening.16–18 Some researchers recommended LDP prediction equations based on elastic analyses.19–21 Carranza-Torres and Fairhurst22 used a closed form solution based on convergenceconfinement method for determining deformations around an opening and compared the results with 3D numerical modelling. In CarranzaTorres and Fairhurst's paper,22 empirical LDP equation recommended by Hoek23 based on a tunnel project data obtained from Chern et al.‘s24 study is used. Vlachopoulos and Diederichs25 developed equations to predict LDP profiles by considering the normalised plastic zone radius with the excavation radius in addition to distance effect to the face and validated their equations with finite element and finite difference methods. In addition to these proposed analytical and empirical equations, LDP equations based on numerical modelling were also recommended by different researchers in the literature. Unlu and Gercek26 considered the rock mass as an elastic material and examined

Corresponding author. E-mail addresses: [email protected] (A. Sakcali), [email protected] (H. Yavuz).

https://doi.org/10.1016/j.ijrmms.2019.104092 Received 6 December 2018; Received in revised form 22 August 2019; Accepted 1 September 2019 1365-1609/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 1. Pattern of deformation in the rock mass for circular openings (modified from Hoek28).

the deformation changes depending on the distance to the tunnel face for different Poisson's ratios of rock mass. Basarir et al.27 recommended LDP equations for different depths depending on a rock mass classification system (RMR) by using multiple regression analyses data obtained from 3 dimensional numerical modelling. Fig. 1 shows that the deformations of a circular opening start in front of the face and increase toward to the behind of the face. Change in deformations with distance to the face defined as the longitudinal deformation profile (LDP) depends generally on maximum deformation behind the face and the tunnel radius.19–22,25–28 19–22,The equations used or recommended by the researchers for the determination of LDP are given in Table 1. Maximum deformation around the tunnel face in these equations19–22,25,26 is unknown. It should be either measured in the tunnel or estimated from 2-D plain-strain analysis. Maximum deformation changes depending on the quality of the rock mass, in-situ

stress and tunnel radius. Rock mass quality is not directly incorporated into LDP equations. To overcome this problem, a rock mass classification system (RMR value) was incorporated into LDP equations developed after multiple regression analysis of data obtained from arranged numerical models by Basarir et al.27 and given in Table 2 for different depths. Although Basarir et al.27 emphasized that the maximum deformation is reached at 5 times the tunnel radius, there is no recommendation to determine the maximum value of deformation behind the face. In a specific design case, we need to know maximum deformation behind the face of a tunnel to apply current LDP equations. To determine maximum deformation, numerical modelling for each case takes time. Basarir et al.27 incorporated just RMR into their prediction for rock mass property. But, in addition to RMR, uniaxial compressive strength ( ci ) of rocks and rock material constant (mi ) should be taken

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A. Sakcali and H. Yavuz

Table 1 The recommended equations for longitudinal deformation profile (LDP). References

Equations 19

Corbette et al. Panet

21

Carranza-Torres and Fairhurst Unlu and Gercek

22

26

Vlachopoulos and Diederichs25

ur urM

= 0.29 + 0.71 1

e(

ur urM

= 0.25 + 0.75 1

(

ur urM

= 1+e

ur urM ur urM

= ur 0 + Aa [1

ur 0 urM ur urM

= e

(

L /R 1.1

)

2 0.75 0.75 + L / R

, L/ R

0 (1)

, L/ R

0 (2)

1.7

)

= ur 0 + Ab {1

1.5(L / R)0.7)

(3)

e(

Ba ( L / R)) ],

[Bb/(Bb + ( L

L/ R

0 (4)

/R))]2 } ,

L/ R

A a = 0.22 0.19 , ur0 = 0.22 + 0.19 , Ab = 0.22 + 0.81, Bb = 0.39 + 0.65 ,

ur urM

1 3

=

0.15R

ur 0 L / R e urM

=1

1

0 (5)

Ba = 0.73 + 0.81 ,

L/ R = 0 (6) L/ R < 0 (7) ur 0 urM

e

((3L / R)/2R )

L/ R > 0 (8)

R = RP / R , RP : Radius of plastic zone

2. Material and method

Table 2 LDP prediction equations considering RMR of rock mass for different depths (recommended by Basarir et al.27). Depths

100 m 200 m 300 m 400 m

Equations for in front of the face

Equations for behind the face

Coefficients

Coefficients

ur R

= aRMRbc (L / D ) , (L /D < 0) (9)

ur R

2.1. Weak rock mass properties When determining the strength of the rock masses, the sample size representing the rock mass should be very large. It is difficult task to arrange test cells to be used to determine the strength as well as the sampling in these dimensions.29 Rock mass classification systems were developed to overcome these problems. Rock mass classification systems are empirical design approaches and are widely used for characterization of rock masses. Some of these systems are Rock Mass Rating (RMR) proposed by Bieniawski30 and Q proposed by Barton et al.31 The RQD (Rock Quality Designation) developed by Deere32 is needed for determining the rock mass class according to RMR or Q classification system. For weak rock masses, difficulties were encountered in determining the value of RMR and Q because the RQD value is zero or close to zero. Various investigations have been carried out to overcome these problems by different researchers. Geological Strength Index (GSI) is one of these systems developed depending on the characteristics of rock masses such as the surface condition of discontinuities and structure of rock mass.33 GSI classification system can be used even in very weak rock masses.34 Therefore, GSI system was used in this study to determine elastic and mechanical properties of rock masses. The lowest, medium and highest GSI values were selected as 15, 25 and 35, respectively, to represent a range of weak rock mass conditions. The compressive strength of the rock mass is strongly dependent on the uniaxial compressive strength of rock material as well as the discontinuities. Hoek and Brown35 stated that uniaxial compressive strength (UCS) of rock material varies between 5 and 25 MPa for weak and 25–50 MPa for average quality rocks. While the uniaxial compressive strength of intact rocks is determined, no sharp limitation is made with upper and lower values offered by Hoek and Brown.35 In determining the properties of weak rock masses, uniaxial compressive strength of the rock material ( ci ) was taken as 15, 30 and 45 MPa, respectively. The mi values used in previous studies were examined for different weak rock masses. The values of the different rock origins proposed by Hoek and Brown35 were taken into account to represent the weak rock mass. The low, medium and high mi values were preferred as 7, 15, and

= aRMRb (L /D) c , (L /D > 0) (10)

a

b

c

a

b

c

174.587 2018.381 19381.874 120847.542

−2.559 −3.016 −3.514 −3.922

8.624 11.755 15.215 18.197

1271.065 17316.930 103903.104 560042.059

−2.755 −3.269 −3.635 −4.007

0.114 0.127 0.145 0.164

into consideration for correct description of a rock mass. Therefore, there is a need for estimating the maximum deformation and development of LDP equations incorporating the rock mass properties. In this paper, circular tunnels excavated in weak rock masses with a total of 162 different conditions comprising 9 different weak rock mass properties, 3 different tunnel radius and 6 different depth of cover were modelled by using finite difference method. Deformations were recorded at the roof of modelled circular openings at 1 m intervals. Multiple regression analysis was performed to predict the radial deformations around the tunnel. As a result of modelling and multiple regression analyses, models were developed predicting the maximum and advancing face deformation for different depths. In addition, the equation predicting the deformation at any point behind the tunnel face has been suggested. The effect of different parameters on radial deformations was investigated with analyses of variance and can be examined by generated response surfaces and LDP curves. With this study, the radial deformations according to face at any point behind the tunnel can be easily predicted by using the rock mass properties (GSI), rock material properties ( ci , mi ) in-situ stress and tunnel radius without the need for any other tools.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 2. The maximum-minimum principal stress graphs of different weak rock masses.

Fig. 3. The maximum-minimum principal stress graphs of different weak rock masses used in models.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

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Table 3 The material properties for weak rock masses in the models. Models

GSI

Model Model Model Model Model Model Model Model Model

15 25 35 15 25 35 15 25 35

A B C D E F G H I

ci (MPa)

15 15 15 30 30 30 45 45 45

mi

a

mb

s

G (GPa)

K (GPa)

7 7 7 15 15 15 23 23 23

0.561 0.531 0.516 0.561 0.531 0.516 0.561 0.531 0.516

0.336 0.481 0.687 0.721 1.030 1.472 1.105 1.579 2.257

0.000079 0.000240 0.000730 0.000079 0.000240 0.000730 0.000079 0.000240 0.000730

0.21 0.37 0.65 0.29 0.52 0.92 0.36 0.64 1.13

0.34 0.61 1.09 0.49 0.87 1.54 0.60 1.06 1.89

23, respectively. In determining the weak rock mass properties, minimum, average and maximum value of each parameter were preferred. 3 different values for GSI (15, 25, 35), 3 different values for uniaxial compressive strength ( ci ) (15 MPa, 30 MPa, 45 MPa) and 3 different mi values (7, 15, 23) were selected. So, 27 different rock masses were taken into consideration to represent weak rock masses. Maximum-minimum principal stress graphs for these rock masses obtained from Hoek-Brown failure criterion are given in Fig. 2. When maximum-minimum principal stress graphs are examined, it is seen that some of them are overlapping. The modelling of tunnels opened in rock masses showing similar failure characteristics leads to the same results. The overlapping curves were eliminated and it was decided that 9 different rock masses would be suitable representing the properties of the weak rock masses for this parametric modelling study. Maximum-minimum principal stress graphs for these weak rock masses used in the models are given in Fig. 3. The generalized Hoek-Brown failure criterion is given in Eq. (11) which is developed based on the nonlinear Griffith failure criterion36 and empirical equation.37,38 This criterion, introduced by Hoek33 and Hoek et al.,39 can be used for estimating of the rock mass strength properties. According to Hoek-Brown failure criterion, Eqs. (12)–(14) are used to define the strength characteristics of rock mass with mb , s and a which are material constants.40 In this study, a is selected as 0.5 for all cases. 1

=

3

+

ci

mb = mi exp s = exp

a=

mb

3

GSI 100 28 14D

10)/40)

(15)

G=

Em 2(1 + )

(16)

K=

(1 + ) G (3/2)(1 2 )

(17)

Tunnels are opened at different diameters according to their purpose of use. In this study, 3 different excavation diameters, which are widely used in tunnel excavation with TBM, were modelled. Inner diameter of segments in the metro tunnels excavated with TBM in Istanbul is generally 5.7 m.42,43 Inner diameter of segments in waste water tunnels is generally 2.6 m.44 The other preferred tunnel is large diameter Turin-Lyon tunnel where Zhao et al.45 and Hasanpour46 carried out their modelling study. The inner diameters of the segments of selected tunnels opened with TBM are 2.6, 5.7 and 8.1 m, respectively. The excavation diameters of these modelled tunnels are 3.20, 6.63, and 9.44 m, respectively. The tunnels are opened at different depths depending on the intended use and topographic and geological conditions. The metro tunnels for transportation in the cities are opened in shallow depths as possible. But the depths of the tunnels outside the cities have increased even more due to topographic conditions. In order to consider the effect of different depths on the deformation around the face of tunnels, model tunnels were arranged at 50, 100, 150, 200, 250 and 300 m depths.

(12)

20/3)

10 ((GSI

2.2. Conditions of modelled tunnels

(13)

e

ci

100

The average Poisson's ratio ( ) was taken as 0.25 for the weak rock masses and calculations were made accordingly. The Hoek-Brown material parameters for weak rock masses are given in Table 3.

(11)

ci

GSI /15

Em =

a

+s

GSI 100 9 3D

1 1 + (e 2 6

of rock mass (Em ) (for ci < 100 MPa). Eqs. (16) and (17) were used to determine the shear (G) and bulk (K) modulus, respectively.

(14)

where 1 is maximum and 3 is minimum effective principal stresses at failure, mb is the value of the Hoek-Brown constant for the rock mass, s and a are constants which depend upon the rock mass characteristics, ci is the uniaxial compressive strength of the intact rock material, mi is the value of Hoek-Brown constant for intact rock material and GSI is the geological strength index value. D is a factor which depends upon the degree of disturbance due to blast damage and stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses. When excellent quality controlled blasting and mechanical or hand excavation in poor quality rock masses, disturbance factor can be taken 0.28 So, the disturbance factor (D) was used as 0 in the calculations. Eq. (15) proposed by Hoek and Brown35 based on Serafim and Pereira41 equation was used to determine the deformation modulus

2.3. Numerical modelling Besides the rock mass properties and tunnel dimensions, modelling conditions such as model zone numbers, dimensions and boundary conditions should also be determined well. For this purpose, parametric studies were carried out by arranging different model zones. One of these studies is to determine the boundary conditions where edge and corner effects are not observed. It was determined that no edge and corner effects are observed when the boundaries of models set at 7–8 times the tunnel radius. Therefore, at least 10 times the excavation radius by considering the previous works27,37,45,46 is defined as model boundaries in the x and z directions (Fig. 4). But, tunnel excavation

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 4. The model meshes, geometry and boundary conditions.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Table 4 Boundary dimensions of the models. Opening Diameter (D) (m)

x distance (m)

y distance (m)

z distance (m)

Excavation Distance (m)

3.20 6.63 9.44

20 40 55

32 68 94

20 40 55

16 34 47

Fig. 5. The deformations of different depth tunnels excavated 6.63 diameter in a weak rock mass.

radii for modelling conditions are not integer. In order to keep the model boundary conditions about 10 times the radius in x and z directions, cube model meshes were added. The boundaries were placed at 20 times the tunnel radius in the y direction (Fig. 4). Excavated length is chosen as 10 times the tunnel radius to allow the steady state condition to be reached. The length of the tunnel which is not excavated in the y direction is equal to 10 times the tunnel radius. The model boundary dimensions for 3 different excavation radius are given in Table 4. A finer mesh around the periphery of the tunnel opening was formed. Mesh size was increased from the opening towards the model boundaries. The number of zones for tunnel models opened at 3.20,

6.63 and 9.44 m excavation diameters is 91648, 145792 and 201536, respectively. Model boundary conditions are illustrated in Fig. 4. Movement of outer boundaries of models were restricted in the perpendicular direction to boundaries. In situ stresses were applied to the outer boundaries in vertical and horizontal directions (Fig. 4). Vertical stress of model boundaries was assumed to be equal to overburden stress. Brown and Hoek47 emphasized that when determining vertical stress ( v ) at depths 0–3000 m, stress gradient is taken as 0.027 MPa/m and in situ stress depending on the depth can be determined by Eq. (18). v

7

= 0.027. H

(18)

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 6. The LDP curves of different depth tunnels excavated 6.63 diameter in a weak rock mass.

Table 5 The equations of maximum and advancing face deformation at different depths, their coefficients and regression coefficients. Suggested Model

Equation

Maximum deformation (urM / R )

urM / R = a. x1b . x 2c

Advancing face deformation (ur0 /R )

ur 0 /R = a. x1b . x 2c

Depth (m)

Coefficients

50 100 150 200 250 300 50 100 150 200 250 300

where H is excavation depth (m). It is assumed that in situ stresses are hydrostatic. Under hydrostatic conditions, the horizontal stress ( h ) is equal to the vertical stress calculated according to the depth change. The vertical and horizontal stresses for 50, 100, 150, 200, 250 and 300 m depths were calculated as 1.35, 2.7, 4.05, 5.4, 6.05 and 8.1 MPa, respectively.48 The model meshes, geometry and boundary conditions are shown in Fig. 4. Pi in Fig. 4 is the internal support pressure, Pface is the pressure applied to excavation face and these values were taken as 0 in models. In this study, a finite difference method based software FLAC3D was used for modelling. The openings were excavated instantaneously and solving was performed until steady state condition is reached. The model Hoek-Brown was used for determining failure state of material.37 The constitutive behaviour of model material was assumed elasto-

Regression Coefficient (R2 )

a

b

c

113.123 11430.753 100734.554 299603.961 855596.676 1606141.303 0.489 80.183 6664.339 83590.806 702079.067 12521622.06

−2.181 −2.624 −2.771 −2.798 −2.872 −2.905 −1.590 −2.210 −2.746 −2.986 −3.206 −3.650

0.549 0.231 −0.060 −0.321 −0.530 −0.679 0.750 0.623 0.285 −0.118 −0.507 −0.670

0.992 0.996 0.995 0.992 0.996 0.995 0.979 0.867 0.837 0.836 0.806 0.851

plastic since weak rock masses show this behaviour. Shear and Bulk modulus were used for calculating strain increments in model HoekBrown.49 The input data for models are given in Table 3. 3. Analysis of model results Unsupported circular tunnels excavated in weak rock masses with a total of 162 different conditions comprising 9 different weak rock mass properties, 3 different tunnel radius and 6 different depth of cover were modelled. In order to investigate the effect of depth change on deformation around the tunnel, deformation contours of models were plotted. As an example, the deformation contours of different depth tunnels excavated 6.63 m in diameter in a weak rock mass (GSI = 25, ci = 30 MPa, mi = 15) are shown in Fig. 5.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Table 6 The ANOVA tables for equations of maximum and advancing face deformation at different depths. Model

Depth (m)

Source

Degree of Freedom

Sum of Squares

Mean Squares

F Ratio

Prob (F)

Maximum deformation (urM / R )

50

Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total Regression Error Total

2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26 2 24 26

6.075E-4 4.726E-6 6.123E-4 1.705E-2 7.187E-5 1.713E-2 1.330E-1 6.262E-4 1.336E-1 5.633E-1 4.261E-3 5.676E-1 1.929E0 8.687E-3 1.938E0 5.103E0 2.350E-2 5.126E0 6.192E-6 1.345E-7 6.326E-6 1.364E-4 2.095E-5 1.574E-4 1.756E-3 3.418E-4 2.098E-3 1.147E-2 2.245E-3 1.372E-2 6.055E-2 1.455E-2 7.517E-2 3.087E-1 5.393E-2 3.627E-1

3.037E-4 1.969E-7

1542.666

0

8.527E-3 2.995E-6

2847.233

0

6.648E-2 2.609E-5

2548.341

0

2.817E-1 1.775E-4

1586.614

0

9.644E-1 3.620E-4

2664.468

0

2.551E0 9.792E-4

2606.299

0

3.096E-6 5.602E-9

552.604

0

6.822E-5 8.728E-7

78.161

0

8.780E-4 1.424E-5

61.645

0

5.737E-3 9.354E-5

61.333

0

3.027E-2 6.063E-4

49.935

0

1.544E-1 2.246E-3

68.705

0

100 150 200 250 300 Advancing face deformation (ur0 /R )

50 100 150 200 250 300

Radial deformations around the tunnel are used to establish the longitudinal deformation profiles (LDP). LDP is important design component in convergence-confinement method, provides insight into how quickly the support begins to interact with the rock mass behind the face of tunnel.22 The radial deformation data were obtained from the roof of the model tunnel and recorded. The recording points were determined as 1 m intervals according to the tunnel face. The radial deformations (ur ) were calculated using the u x , u y and uz deformation readings at the registration points. LDP curves for different cover depths of the tunnels with an excavation diameter of 6.63 m excavated in weak rock mass (GSI = 25, ci = 30 MPa, mi = 15) are shown in Fig. 6. Deformation of the rock mass around the tunnel starts at 4 times the tunnel radius ahead of the face and continues up to 8 times the tunnel radius behind the face. It was observed that there was a significant increase in deformation around the tunnel with the increase in depth. The radial deformations increase from the face to a point behind the face where maximum deformations occur. The multiple regression analysis was performed to estimate radial deformations depending on the distance to the face, rock mass properties, tunnel radius and depth of cover. The radial deformations obtained from the models were normalised with the excavation radius and this dependent parameter was used in the analysis for prediction. The first independent parameter is taken as GSI. mi which defines the rock mass quality. The second independent parameter is the ratio of uniaxial compressive strength of intact rock to in situ stress ( ci / 0 ). After the

application of multiple regression analysis using various functions for best prediction of the normalised radial deformation at the face and normalised maximum radial deformation behind the face of the opening depending on independent parameters, a three-parameter power function was found to be the best and simple to predict the radial deformations and given in Eqs. (19) and (20).

urM = f GSI . mi , R ur 0 = f GSI . mi , R

ci 0 ci 0

(19) (20

where is the maximum radial deformation behind the tunnel, ur0 is the radial deformation at the tunnel face and R is excavation radius. Deformation changes at different depths were separately evaluated in regression analyses. As a result of multiple regression analyses, equations to predict the maximum radial deformation value behind the tunnel face and the radial deformation at the face of the tunnel were suggested for depths, separately, ranging from 50 m to 300 m at 50 m intervals. These suggested equations with their coefficients and correlation coefficients for each depth are given in Table 5. In these equations, x1 is GSI. mi and x2 is ci / 0 . The high correlation coefficients in the equations show significant relationships. The effect of independent variables on dependent variable can be tested by analysis of variance (ANOVA). The ANOVA tables for assessing maximum radial

urM

9

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 7. The response surfaces of normalised maximum deformation depending on rock mass characteristics and opening conditions for different depths.

deformation behind the face and radial deformation at the face of the tunnel was presented in Table 6. Using suggested models, the maximum radial deformation behind the face and radial deformation at the advancing face of an underground circular opening can be estimated depending on the Geological Strength Index (GSI), mi value and the uniaxial compressive strength of intact rock material ( ci ) and in situ stress ( 0 ). Variation of the normalised maximum radial deformation behind the face and the normalised radial deformation at the advancing face of a circular opening depending on GSI. mi and ci / 0 values are illustrated in Figs. 7 and 8, respectively.

The radial deformations towards to behind of the tunnel face increase in a circular tunnel up to the distance where maximum radial deformation occurs. Multiple regression analysis was also performed to estimate the distance-dependent change of these radial deformations. Radial deformation behind the tunnel face is subtracted from the radial deformation at the tunnel face and this value is normalised with maximum deformation value which is subtracted from the radial deformation at the tunnel face. As seen, both equations proposed in the literature and Eq. (21), the deformation occurs behind the opening face is a function of distance to tunnel face (L) and excavation radius (R).

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 8. Changes of normalised radial deformation at the advancing face depending on the rock mass characteristics and tunnel conditions for different depths.

ur urM

ur 0 L =f R ur 0

equation, x is the ratio of the distance to tunnel face to tunnel radius. The high correlation coefficient shows that the equation suggested for estimation of deformation behind tunnel face can be used. Using the suggested equation given in Table 7, normalised radial deformations behind the tunnel face with maximum radial deformation were calculated depending on the normalised distance behind the face with tunnel radius. Longitudinal deformation profile (LDP) was plotted in Fig. 10. The LDP profile obtained from this equation is compared with the LDP profiles generated by previous equations (1)–(3) and (5) and (8) given in Table 1. As seen in Fig. 10, LDP equation developed in

(21)

As a result of the analyses of the data obtained from the models that radial deformations behind the face increase with increase in distance from the face and relationship is significant. The change of normalised deformations depending on the distance from face normalised with excavation radius is shown in Fig. 9. The suggested longitudinal deformation profile (LDP) equation for weak rock masses, regression coefficient and the results of ANOVA are given in Table 7. In this

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 9. The change of normalised deformations according to tunnel face.

range of rock engineering applications. When estimating deformations towards to behind of a circular opening face for different rock masses, maximum deformation values are needed and these values determined by using plain-strain analysis or other methods. So, it is a time-consuming design tool. In this study, the circular tunnel with 162 different conditions excavated in weak rock masses were modelled by using FLAC3D based on finite difference method. A model condition for a weak rock mass was given as an example. The deformation contours and Longitudinal Deformation Profiles (LDP) using the data obtained from the models for different depths were plotted. Regression analyzes were performed by using the deformation data recorded around the tunnel. The effects of independent variables on prediction equations were checked by analysis of variance. The high correlation coefficients show that the equations give meaningful results. As a result of this study, the equations of maximum and advancing face deformation for circular unsupported tunnels at different depths are suggested. These suggested equations are based on rock mass and intact rock material properties and tunnel conditions. An equation is also developed for determining the radial deformations around the circular tunnel in weak rock masses. By using the equations suggested in this study, radial deformations towards to behind of a circular tunnel can be determined without the need for other tools. Longitudinal Deformation Profile (LDP) for a circular opening condition was plotted using suggested equations in this study and other equations suggested by researchers. Comparing the suggested equation with others, it is seen that the equation gives approximate results with equations obtained from elasto-plastic analyses and different results with equations obtained from elastic analysis. Isotropic rock masses are modelled with this study. When rock mass includes joints such as fault and fracture, these equations should not be used. Non-isotropic metarials can be modelled with discontinuum models.

Table 7 The suggested LDP equation, it's regression coefficient and ANOVA table. Equation Coefficients Regression Coefficient (R2) ANOVA Table

(ur a b

Degree of Freedom Sum of Squares Mean Squares F ratio Prob (F)

ur 0)/(urM

ur 0) = a . x /(b + x )

1.0628 0.571 0.941 Regresion Error Total Regresion Error Total Regresion Error

1 2224 2225 147.349 9.295 156.644 147.349 0.004 35255.260 0

this study is close to recommended equation by Vlachopoulos and Diederichs25 approximately up to 1.5 times the tunnel radius. After 1.5 times the tunnel radius, deformation values coincide with values of Carranza-Torres and Fairhurst's22 equation. Deformations around the tunnel can be determined by recommended LDP equations. With this study, maximum and advancing face deformations could be estimated depending on GSI of rock mass, the uniaxial compressive strength ( ci ) and mi values of intact rock material, tunnel radius (R) and in-situ stress conditions ( 0 ). Using the determined maximum and advancing face deformation values, deformations around the tunnel can be easily estimated with the suggested LDP equation. 4. Conclusions Different rock mass classification systems are used for determining the characteristics of the rock mass on tunnel route. GSI based on basic geological properties is one of these classification systems and has wide 12

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104092

A. Sakcali and H. Yavuz

Fig. 10. The comparison of suggested LDP equation with other equations in the literature.

Acknowledgement

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