IJMGE Int J Min amp Geo-Eng

Vol49 No2 December 2015 pp187-203

187

Application of tests of goodness of fit in determining the probability

density function for spacing of steel sets in tunnel support system

Farnoosh Basaligheh and Ali Keyhani

Received 25 Jul 2014 Received in revised form 23 Dec 2015 Accepted 3 n 2016 Corresponding author Email f_basalighehyahoocom f_basalighehshahroodutacir

Abstract

One of the conventional methods for temporary support of tunnels is to use steel sets with shotcrete

The nature of a temporary support system demands a quick installation of its structures As a result

the spacing between steel sets is not a fixed amount and it can be considered as a random variable

Hence in the reliability analysis of these types of structures the selection of an appropriate probability

distribution function of spacing of steel sets is essential In the present paper the distances between

steel sets are collected from an under-construction tunnel and the collected data is used to suggest a

proper Probability Distribution Function (PDF) for the spacing of steel sets The tunnel has two

different excavation sections In this regard different distribution functions were investigated and

three common tests of goodness of fit were used for evaluation of each function for each excavation

section Results from all three methods indicate that the Wakeby distribution function can be

suggested as the proper PDF for spacing between the steel sets It is also noted that although the

probability distribution function for two different tunnel sections is the same the parameters of PDF

for the individual sections are different from each other

Keywords probability density function random variable reliability steel sets temporary support

tunnel

1 Introduction

In tunnel engineering a temporary support

structure is needed in many cases Several type

of temporary supports have been proposed and

used in tunnels including wood frames rock

bolts cable bolts steel frames shotcrete lining

and segments [1-3] Generally the most widely

used type of temporary support system is

shotcrete Indeed shotcrete is considered as the

standard temporary support during design and

construction stages of tunnels However if the

magnitude of loads transmitted by the ground to

the support is too large to be carried by shotcrete

alone or if squeezing or raveling behavior

requires complete surface coverage steel sets are

commonly used in combination with shotcrete

This combination can be in the form of a

complete composite annulus or may be a semi-

circular or partial arch configuration [4]

Ja

Department of Civil Engineering Shahrood University of Technology Shahrood Iran

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

188

Sometimes lattice girders are used instead

of steel sets [5-8] In designing steel sets as the

primary support system two important

parameters are the spacing between sets and

the thickness of the shotcrete Wong etal have

proposed and compared different design

methods for steel sets [9] Due to the shortage

of time after the excavation the installation of

the steel sets and the casting of the concrete

layer are generally performed very quickly

Therefore it is observed that the actual

spacing of the steel sets and the thickness of

the shotcrete are unexpectedly different from

the designed values This difference may also

occur as a result of special working

conditions

On the other hand when the New Austrian

Tunneling Method (NATM) is used the

spacing and the thickness of the shotcrete are

changed according to the ground condition

[10] As a result both of these parameters are

not fixed and can be considered as random

variables Therefore a probabilistic and

reliability analysis of the system seems to be

necessary for understanding the behavior of

the support structure

One of the significant problems of reliability

analysis is the selection of an appropriate

Probability Density Function (PDF) for random

variables During construction process of a

project there are many uncertainties such as

natural and human-related uncertainties that

affect the spacing between steel sets and

thickness of the shotcrete The recognition of

each one of these uncertainties needs

experience and one cannot properly know all of

them [11] The objective of the present research

is to study the probability distribution function

for spacing of steel sets used for primary

support of the tunnels

In this regard the Goodness of Fit Tests

are utilized to determine the proper density

function The common methods of goodness

of fit are introduced and discussed in section

2 In section 3 some information about the

spacing of steel sets in a tunnel in Iran is

provided The tunnel is excavated in moderate

to highly weathered Andesite rocks The

nature of rock mass is blocky with maximum

joint spacing equal to 60 cm (classified as fair

rock with RMR between 38 and 53) The

tunnel is excavated by drilling and blasting

methods It has two horseshoe excavation

sections a large section with dimension of 8 x

8 m2 and a small section with dimension of 51

x 5 m2 The large section was excavated in

two stages while the small section was

excavated in only one The spacing of steel

sets was collected in 82 different locations for

the larger tunnel section (8 x 8 m2) and in 108

different locations for the smaller section (51

x 5 m2) Based on this information the proper

density function for the spacing of the steel

sets is selected and discussed Section 4

includes conclusions and summery of the

results

2 Goodness of fit tests and estimation of

probability distribution parameters

A Goodness of Fit Test is used to measure

how well a sample of observed data follows a

distinctive distribution function It is

noteworthy that no statistical distribution can

precisely show a perfect fit with observed

data Therefore one distribution is selected as

the best one based on comparison with the

other distributions [12] In general three tests

of Goodness of Fit are used for determining

the best probability distribution function

These three methods are Chi-Squared

Kolmogorov-Smirnov and Anderson-Darling

Each one of these methods has its

advantages and disadvantages and one cannot

simply prefer one of them Therefore in the

present study all three methods are used to

determine the most suitable density function

Then the desired result is obtained through

scoring each one of these methods and

averaging them [13-15] In the following each

method is concisely introduced and discussed

21 Chi-squared test

The Chi-squared test like other tests uses a

test statistics for comparing the observed data

with a distribution function In this method

the observed data is partitioned to several

categories and the test statistic is estimated

from Equation (1)

2

2

1

NC

i i

i i

f f

f

(1)

In the above equation NC represents the

number of categories refers to the frequency

(number) of the observed (measured) samples

in category i and represents the frequency of

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

189

category i as calculated by the assumed

distribution function

Based on this test the numbers of existing

samples in each category is recommended to

be at least 5 x2 is the Chi-squared distribution

with (NC-NP-1) degree of freedom in which

NP is the number of parameters to be

estimated The above statistic is compared

with the value of Chi-squared distribution with

a definite level of acceptance and degree of

freedom If the calculated value of x2

in the

above equation is less than the value of Chi-

squared distribution the hypothesis that the

data follow the assumed distribution is

accepted [16 17]

22 Kolmogorov-Smirnov test

The Kolmogorov-Smirnov test is another test

used to measure how well a sample follows a

distinctive distribution The Chi-squared test

acts properly when the number of samples is

large enough to have five data in each

category Otherwise it is better to use

Kolmogorov-Smirnov test The statistic of this

test is the highest difference between expected

and real measured frequencies (in absolute

value) in different groups This statistic is

determined through the following equation

D max F x F x (2)

In the above equation F is the observed

relative accumulative frequency and F is the

expected relative accumulative frequency The

following steps are taken to do this test

1 The relative accumulative frequency of

a sample is measured for different

ranges (categories)

2 The relative accumulative frequency for

different categories is obtained through

theoretical statistical distribution or

another instance of information

3 The absolute value of difference

between the two frequencies of the

above two steps is calculated

4 The highest value of difference obtained

from step 3 is defined as the value of

statistic for test D

5 A value of α is selected as error and Dα

is read from the associated tables

6 If the value of D is more than Dα the

hypothesis that the sample follows the

assumed distribution is accepted

Otherwise it is rejected [16]

23 Anderson-Darling test

This test is another method for determining the

fitness of an assumed probability density

function to a given set of data This test

assigns higher weight to sequences in

comparison with other tests as a result it has

higher accuracy For the variable x and

presumed distribution F0() the random

variable nFn(x)is a binomial distribution with

probability of F0(x) The expected value of

nFn(x) is nF0(x) and its variance is nF

0(x)[1-

F0(x)] and the weight function Wn

2 is

calculated as

2 0 2 0 0

n n

0 2 0 0

n

W n [F x F x ] Ψ F x ]dF (x

n [F x F x ] Ψ F x ]f (x dx

(3)

1

1u

u u

(4)

For the values of x we have

0

0 01

nF x F xn

F x F x

(5)

The mean 0 and variance 1 are obtained

when the null hypothesis is correct The

Anderson-Darling statistic is defined in the

following

20

2 0

0 01

n

n

F x F xA n dF x

F x F x

(6)

The above equation can also be written as

2

1

1

12 1 [

1 ]

n

n j

j

n j

A n j logun

log u

(7)

The Anderson-Darling test defines the

distribution limit An for the weight function of

Equation (4)

In the above equation uj=F0(x(j)) and x(1)lt

x(2)lt helliplt x(n) is the sequence of ordered

samples [1819]

3 Random variable of applied spacing for

steel sets

As mentioned before the drilling of tunnels

requires a quick covering and this is done by

shotcreting If the intensity of loads transferred

from ground to bearing structure is high then

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

190

the steel sets are used in combination with

shotcrete for temporary support of the tunnel

The spacing between steel sets is a variable

due to the quick installation As a result the

distance between steel sets can be regarded as

a random variable In the present study the

data gathered from one of the tunnels in Iran is

used to determine the distribution function of

this random variable The tunnel has two

drilling horseshoe section as shown in Figures

1 and 2

Fig 1 Large section of tunnel with dimension of 8 x 8 m2

Fig 2 Small section of tunnel with dimension of 51 x 5 m2

For the larger tunnel section (8 x 8 m2) the

spacing of steel sets are collected in 82

different locations as shown in Table 1 The

spacing of the steel sets for the smaller section

(51 x 5 m2) are collected for 108 different

locations as shown in Table 2

Table 1 Spacing of the steel sets in the large section

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

1 000 70 42 3636 100

2 070 70 43 3736 104

3 140 82 44 3840 100

4 222 80 45 3940 100

5 302 82 46 4040 100

6 384 83 47 4140 100

7 567 82 48 4240 98

8 549 82 49 4338 103

9 631 82 50 4441 102

10 713 82 51 4542 102

11 795 82 52 4645 95

12 877 83 53 4740 100

13 960 82 54 4840 100

14 1042 83 55 4940 101

15 1125 80 56 5041 102

16 1205 85 57 5143 98

17 1290 81 58 5241 99

18 1371 81 59 5340 103

19 1452 101 60 5443 82

20 1553 62 61 5525 100

21 1615 83 62 5625 98

22 1698 82 63 5723 99

23 1780 80 64 5822 100

24 1860 84 65 5922 102

25 1944 80 66 6025 97

26 2024 108 67 6122 101

27 2132 101 68 6223 102

28 2233 100 69 6325 98

29 2333 102 70 6423 103

30 2435 102 71 6526 101

31 2537 104 72 6627 99

32 2641 101 73 6726 104

33 2742 102 74 6830 100

34 2844 100 75 6830 100

35 2944 104 76 7030 101

36 3048 96 77 7131 72

37 3144 95 78 7203 102

38 3239 99 79 7305 95

39 3338 95 80 7400 100

40 3433 102 81 7500 100

41 3535 101 82 7600 118

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

191

Table 2 Spacing of steel sets in small section

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

1 400 42 37 3380 100 73 7042 103

2 442 45 38 3480 100 74 7145 102

3 487 43 39 3580 103 75 7247 103

4 530 73 40 3683 102 76 7350 102

5 603 72 41 3785 106 77 7452 101

6 675 60 42 3891 99 78 7553 101

7 735 50 43 399 100 79 7654 101

8 785 79 44 409 98 80 7755 83

9 864 80 45 4188 102 81 7838 83

10 944 81 46 4290 101 82 7921 82

11 1025 82 47 4391 99 83 8003 80

12 1107 80 48 4490 102 84 8083 87

13 1187 109 49 4592 102 85 8170 80

14 1296 53 50 4694 106 86 8250 80

15 1349 83 51 4800 100 87 8330 84

16 1432 92 52 4900 97 88 8414 79

17 1524 90 53 4997 103 89 8493 83

18 1614 88 54 5100 103 90 8576 89

19 1702 73 55 5203 102 91 8665 91

20 1775 62 56 5305 101 92 8756 91

21 1837 75 57 5406 99 93 8847 101

22 1912 78 58 5305 101 94 8948 102

23 1990 84 59 5605 100 95 9050 105

24 2074 83 60 5705 106 96 9155 87

25 2157 85 61 5811 100 97 9242 81

26 2242 98 62 5911 109 98 9323 85

27 2340 108 63 6020 110 99 9408 80

28 2448 102 64 6130 98 100 9488 82

29 2550 100 65 6228 100 101 9570 84

30 2650 105 66 6328 105 102 9654 85

31 2755 98 67 6433 97 103 9739 81

32 2853 104 68 6530 106 104 9820 84

33 2957 101 69 6636 100 105 9904 82

34 3058 119 70 6736 101 106 9986 80

35 3177 98 71 6837 105 107 10066 82

36 3275 105 72 6942 100 108 10148 84

The probability distribution function of this

variable was obtained by Easy Fit 55

Software and through the above mentioned

three methods From 65 distribution functions

available in the software 32 functions which

best fit the data are selected and their rank

(based on 3 mentioned tests of Goodness of

Fit) are shown in Tables 3 and 4 The sum of

scores for each function in different methods

is provided and is used to define the best fit

distribution function The results are shown in

Tables 5-7 As these tables indicate the

Gumbel Min Function has the first rank in

Kolmogorov-Smirnov and Anderson-Darling

while the Chi-squared method shows rank of

21 for this function This significant difference

between the evaluations by different tests for

Goodness of Fit indicates an irregular data that

requires more examination and understanding

before analysis This difference can be

attributed to the fact that the number of

samples in some categories is low

The Gumbel Min distribution function for

the large section is shown in Figure 3 As it

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

192

can be seen in this Figure and Table 1 the

number of samples in the interval of 86 to 94

centimeters is significantly less than the other

intervals The Gumbel Min distribution

function for the small section is illustrated in

Figure 4 This Figure together with the data

indicated in Table 2 show that the number of

samples in the interval of 86 to 97 centimeters

is much less than the other intervals From

Figures 3 and 4 it is evident that these empty

intervals are at odds with the fitted functions

Table 3 Statistics and ranks of distribution functions for large section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

23 43841 6 49794 8 021754 Beta 1

27 47975 8 52994 9 022023 Burr 2

26 46794 3 47874 5 020889 Burr (4P) 3

5 31318 32 11036 16 025584 Cauchy 4

2 21377 11 6237 11 024138 Chi-Squared 5

20 39714 27 68867 31 027695 Chi-Squared (2P) 6

13 33815 28 69436 29 026923 Erlang 7

10 33524 17 64184 19 026197 Erlang (3P) 8

16 3418 13 63292 13 025443 Error 9

8 32733 23 67096 22 026456 Fatigue Life 10

18 34333 14 63491 14 025448 Fatigue Life (3P) 11

12 33755 26 68443 23 026478 Gamma 12

4 2685 18 65688 28 026773 Gamma (3P) 13

9 33385 19 65998 18 026138 Gen Gamma 14

15 34105 16 6404 17 025873 Gen Gamma (4P) 15

25 46507 2 46156 1 020251 Gumbel Min 16

21 42932 1 45911 2 020293 Kumaraswamy 17

6 3246 24 67841 25 026576 Log-Gamma 18

19 37081 20 66069 24 026528 Log-Logistic 19

28 48387 10 56965 6 021021 Log-Logistic (3P) 20

32 92302 30 71974 30 027007 Logistic 21

7 32711 22 6701 21 026392 Lognormal 22

14 34028 15 6393 15 025582 Lognormal (3P) 23

30 5064 29 70364 20 026375 Nakagami 24

17 34259 12 62615 12 025275 Normal 25

3 24976 25 68315 27 026697 Pearson 5 26

11 33557 21 66461 26 026678 Pearson 5 (3P) 27

31 55147 9 5338 10 022378 Pert 28

1 3473 31 90577 32 033431 Power Function 29

24 45901 7 49889 3 020564 Triangular 30

29 48479 5 49002 7 021481 Weibull 31

22 43445 4 47932 4 020825 Weibull (3P) 32

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

189

category i as calculated by the assumed

distribution function

Based on this test the numbers of existing

samples in each category is recommended to

be at least 5 x2 is the Chi-squared distribution

with (NC-NP-1) degree of freedom in which

NP is the number of parameters to be

estimated The above statistic is compared

with the value of Chi-squared distribution with

a definite level of acceptance and degree of

freedom If the calculated value of x2

in the

above equation is less than the value of Chi-

squared distribution the hypothesis that the

data follow the assumed distribution is

accepted [16 17]

22 Kolmogorov-Smirnov test

The Kolmogorov-Smirnov test is another test

used to measure how well a sample follows a

distinctive distribution The Chi-squared test

acts properly when the number of samples is

large enough to have five data in each

category Otherwise it is better to use

Kolmogorov-Smirnov test The statistic of this

test is the highest difference between expected

and real measured frequencies (in absolute

value) in different groups This statistic is

determined through the following equation

D max F x F x (2)

In the above equation F is the observed

relative accumulative frequency and F is the

expected relative accumulative frequency The

following steps are taken to do this test

1 The relative accumulative frequency of

a sample is measured for different

ranges (categories)

2 The relative accumulative frequency for

different categories is obtained through

theoretical statistical distribution or

another instance of information

3 The absolute value of difference

between the two frequencies of the

above two steps is calculated

4 The highest value of difference obtained

from step 3 is defined as the value of

statistic for test D

5 A value of α is selected as error and Dα

is read from the associated tables

6 If the value of D is more than Dα the

hypothesis that the sample follows the

assumed distribution is accepted

Otherwise it is rejected [16]

23 Anderson-Darling test

This test is another method for determining the

fitness of an assumed probability density

function to a given set of data This test

assigns higher weight to sequences in

comparison with other tests as a result it has

higher accuracy For the variable x and

presumed distribution F0() the random

variable nFn(x)is a binomial distribution with

probability of F0(x) The expected value of

nFn(x) is nF0(x) and its variance is nF

0(x)[1-

F0(x)] and the weight function Wn

2 is

calculated as

2 0 2 0 0

n n

0 2 0 0

n

W n [F x F x ] Ψ F x ]dF (x

n [F x F x ] Ψ F x ]f (x dx

(3)

1

1u

u u

(4)

For the values of x we have

0

0 01

nF x F xn

F x F x

(5)

The mean 0 and variance 1 are obtained

when the null hypothesis is correct The

Anderson-Darling statistic is defined in the

following

20

2 0

0 01

n

n

F x F xA n dF x

F x F x

(6)

The above equation can also be written as

2

1

1

12 1 [

1 ]

n

n j

j

n j

A n j logun

log u

(7)

The Anderson-Darling test defines the

distribution limit An for the weight function of

Equation (4)

In the above equation uj=F0(x(j)) and x(1)lt

x(2)lt helliplt x(n) is the sequence of ordered

samples [1819]

3 Random variable of applied spacing for

steel sets

As mentioned before the drilling of tunnels

requires a quick covering and this is done by

shotcreting If the intensity of loads transferred

from ground to bearing structure is high then

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

190

the steel sets are used in combination with

shotcrete for temporary support of the tunnel

The spacing between steel sets is a variable

due to the quick installation As a result the

distance between steel sets can be regarded as

a random variable In the present study the

data gathered from one of the tunnels in Iran is

used to determine the distribution function of

this random variable The tunnel has two

drilling horseshoe section as shown in Figures

1 and 2

Fig 1 Large section of tunnel with dimension of 8 x 8 m2

Fig 2 Small section of tunnel with dimension of 51 x 5 m2

For the larger tunnel section (8 x 8 m2) the

spacing of steel sets are collected in 82

different locations as shown in Table 1 The

spacing of the steel sets for the smaller section

(51 x 5 m2) are collected for 108 different

locations as shown in Table 2

Table 1 Spacing of the steel sets in the large section

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

1 000 70 42 3636 100

2 070 70 43 3736 104

3 140 82 44 3840 100

4 222 80 45 3940 100

5 302 82 46 4040 100

6 384 83 47 4140 100

7 567 82 48 4240 98

8 549 82 49 4338 103

9 631 82 50 4441 102

10 713 82 51 4542 102

11 795 82 52 4645 95

12 877 83 53 4740 100

13 960 82 54 4840 100

14 1042 83 55 4940 101

15 1125 80 56 5041 102

16 1205 85 57 5143 98

17 1290 81 58 5241 99

18 1371 81 59 5340 103

19 1452 101 60 5443 82

20 1553 62 61 5525 100

21 1615 83 62 5625 98

22 1698 82 63 5723 99

23 1780 80 64 5822 100

24 1860 84 65 5922 102

25 1944 80 66 6025 97

26 2024 108 67 6122 101

27 2132 101 68 6223 102

28 2233 100 69 6325 98

29 2333 102 70 6423 103

30 2435 102 71 6526 101

31 2537 104 72 6627 99

32 2641 101 73 6726 104

33 2742 102 74 6830 100

34 2844 100 75 6830 100

35 2944 104 76 7030 101

36 3048 96 77 7131 72

37 3144 95 78 7203 102

38 3239 99 79 7305 95

39 3338 95 80 7400 100

40 3433 102 81 7500 100

41 3535 101 82 7600 118

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

191

Table 2 Spacing of steel sets in small section

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

1 400 42 37 3380 100 73 7042 103

2 442 45 38 3480 100 74 7145 102

3 487 43 39 3580 103 75 7247 103

4 530 73 40 3683 102 76 7350 102

5 603 72 41 3785 106 77 7452 101

6 675 60 42 3891 99 78 7553 101

7 735 50 43 399 100 79 7654 101

8 785 79 44 409 98 80 7755 83

9 864 80 45 4188 102 81 7838 83

10 944 81 46 4290 101 82 7921 82

11 1025 82 47 4391 99 83 8003 80

12 1107 80 48 4490 102 84 8083 87

13 1187 109 49 4592 102 85 8170 80

14 1296 53 50 4694 106 86 8250 80

15 1349 83 51 4800 100 87 8330 84

16 1432 92 52 4900 97 88 8414 79

17 1524 90 53 4997 103 89 8493 83

18 1614 88 54 5100 103 90 8576 89

19 1702 73 55 5203 102 91 8665 91

20 1775 62 56 5305 101 92 8756 91

21 1837 75 57 5406 99 93 8847 101

22 1912 78 58 5305 101 94 8948 102

23 1990 84 59 5605 100 95 9050 105

24 2074 83 60 5705 106 96 9155 87

25 2157 85 61 5811 100 97 9242 81

26 2242 98 62 5911 109 98 9323 85

27 2340 108 63 6020 110 99 9408 80

28 2448 102 64 6130 98 100 9488 82

29 2550 100 65 6228 100 101 9570 84

30 2650 105 66 6328 105 102 9654 85

31 2755 98 67 6433 97 103 9739 81

32 2853 104 68 6530 106 104 9820 84

33 2957 101 69 6636 100 105 9904 82

34 3058 119 70 6736 101 106 9986 80

35 3177 98 71 6837 105 107 10066 82

36 3275 105 72 6942 100 108 10148 84

The probability distribution function of this

variable was obtained by Easy Fit 55

Software and through the above mentioned

three methods From 65 distribution functions

available in the software 32 functions which

best fit the data are selected and their rank

(based on 3 mentioned tests of Goodness of

Fit) are shown in Tables 3 and 4 The sum of

scores for each function in different methods

is provided and is used to define the best fit

distribution function The results are shown in

Tables 5-7 As these tables indicate the

Gumbel Min Function has the first rank in

Kolmogorov-Smirnov and Anderson-Darling

while the Chi-squared method shows rank of

21 for this function This significant difference

between the evaluations by different tests for

Goodness of Fit indicates an irregular data that

requires more examination and understanding

before analysis This difference can be

attributed to the fact that the number of

samples in some categories is low

The Gumbel Min distribution function for

the large section is shown in Figure 3 As it

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

192

can be seen in this Figure and Table 1 the

number of samples in the interval of 86 to 94

centimeters is significantly less than the other

intervals The Gumbel Min distribution

function for the small section is illustrated in

Figure 4 This Figure together with the data

indicated in Table 2 show that the number of

samples in the interval of 86 to 97 centimeters

is much less than the other intervals From

Figures 3 and 4 it is evident that these empty

intervals are at odds with the fitted functions

Table 3 Statistics and ranks of distribution functions for large section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

23 43841 6 49794 8 021754 Beta 1

27 47975 8 52994 9 022023 Burr 2

26 46794 3 47874 5 020889 Burr (4P) 3

5 31318 32 11036 16 025584 Cauchy 4

2 21377 11 6237 11 024138 Chi-Squared 5

20 39714 27 68867 31 027695 Chi-Squared (2P) 6

13 33815 28 69436 29 026923 Erlang 7

10 33524 17 64184 19 026197 Erlang (3P) 8

16 3418 13 63292 13 025443 Error 9

8 32733 23 67096 22 026456 Fatigue Life 10

18 34333 14 63491 14 025448 Fatigue Life (3P) 11

12 33755 26 68443 23 026478 Gamma 12

4 2685 18 65688 28 026773 Gamma (3P) 13

9 33385 19 65998 18 026138 Gen Gamma 14

15 34105 16 6404 17 025873 Gen Gamma (4P) 15

25 46507 2 46156 1 020251 Gumbel Min 16

21 42932 1 45911 2 020293 Kumaraswamy 17

6 3246 24 67841 25 026576 Log-Gamma 18

19 37081 20 66069 24 026528 Log-Logistic 19

28 48387 10 56965 6 021021 Log-Logistic (3P) 20

32 92302 30 71974 30 027007 Logistic 21

7 32711 22 6701 21 026392 Lognormal 22

14 34028 15 6393 15 025582 Lognormal (3P) 23

30 5064 29 70364 20 026375 Nakagami 24

17 34259 12 62615 12 025275 Normal 25

3 24976 25 68315 27 026697 Pearson 5 26

11 33557 21 66461 26 026678 Pearson 5 (3P) 27

31 55147 9 5338 10 022378 Pert 28

1 3473 31 90577 32 033431 Power Function 29

24 45901 7 49889 3 020564 Triangular 30

29 48479 5 49002 7 021481 Weibull 31

22 43445 4 47932 4 020825 Weibull (3P) 32

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

190

the steel sets are used in combination with

shotcrete for temporary support of the tunnel

The spacing between steel sets is a variable

due to the quick installation As a result the

distance between steel sets can be regarded as

a random variable In the present study the

data gathered from one of the tunnels in Iran is

used to determine the distribution function of

this random variable The tunnel has two

drilling horseshoe section as shown in Figures

1 and 2

Fig 1 Large section of tunnel with dimension of 8 x 8 m2

Fig 2 Small section of tunnel with dimension of 51 x 5 m2

For the larger tunnel section (8 x 8 m2) the

spacing of steel sets are collected in 82

different locations as shown in Table 1 The

spacing of the steel sets for the smaller section

(51 x 5 m2) are collected for 108 different

locations as shown in Table 2

Table 1 Spacing of the steel sets in the large section

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

Fra

me

nu

mb

er

Loca

tion

(m)

Dis

tan

ce t

o n

ext

fram

e (c

m)

1 000 70 42 3636 100

2 070 70 43 3736 104

3 140 82 44 3840 100

4 222 80 45 3940 100

5 302 82 46 4040 100

6 384 83 47 4140 100

7 567 82 48 4240 98

8 549 82 49 4338 103

9 631 82 50 4441 102

10 713 82 51 4542 102

11 795 82 52 4645 95

12 877 83 53 4740 100

13 960 82 54 4840 100

14 1042 83 55 4940 101

15 1125 80 56 5041 102

16 1205 85 57 5143 98

17 1290 81 58 5241 99

18 1371 81 59 5340 103

19 1452 101 60 5443 82

20 1553 62 61 5525 100

21 1615 83 62 5625 98

22 1698 82 63 5723 99

23 1780 80 64 5822 100

24 1860 84 65 5922 102

25 1944 80 66 6025 97

26 2024 108 67 6122 101

27 2132 101 68 6223 102

28 2233 100 69 6325 98

29 2333 102 70 6423 103

30 2435 102 71 6526 101

31 2537 104 72 6627 99

32 2641 101 73 6726 104

33 2742 102 74 6830 100

34 2844 100 75 6830 100

35 2944 104 76 7030 101

36 3048 96 77 7131 72

37 3144 95 78 7203 102

38 3239 99 79 7305 95

39 3338 95 80 7400 100

40 3433 102 81 7500 100

41 3535 101 82 7600 118

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

191

Table 2 Spacing of steel sets in small section

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

1 400 42 37 3380 100 73 7042 103

2 442 45 38 3480 100 74 7145 102

3 487 43 39 3580 103 75 7247 103

4 530 73 40 3683 102 76 7350 102

5 603 72 41 3785 106 77 7452 101

6 675 60 42 3891 99 78 7553 101

7 735 50 43 399 100 79 7654 101

8 785 79 44 409 98 80 7755 83

9 864 80 45 4188 102 81 7838 83

10 944 81 46 4290 101 82 7921 82

11 1025 82 47 4391 99 83 8003 80

12 1107 80 48 4490 102 84 8083 87

13 1187 109 49 4592 102 85 8170 80

14 1296 53 50 4694 106 86 8250 80

15 1349 83 51 4800 100 87 8330 84

16 1432 92 52 4900 97 88 8414 79

17 1524 90 53 4997 103 89 8493 83

18 1614 88 54 5100 103 90 8576 89

19 1702 73 55 5203 102 91 8665 91

20 1775 62 56 5305 101 92 8756 91

21 1837 75 57 5406 99 93 8847 101

22 1912 78 58 5305 101 94 8948 102

23 1990 84 59 5605 100 95 9050 105

24 2074 83 60 5705 106 96 9155 87

25 2157 85 61 5811 100 97 9242 81

26 2242 98 62 5911 109 98 9323 85

27 2340 108 63 6020 110 99 9408 80

28 2448 102 64 6130 98 100 9488 82

29 2550 100 65 6228 100 101 9570 84

30 2650 105 66 6328 105 102 9654 85

31 2755 98 67 6433 97 103 9739 81

32 2853 104 68 6530 106 104 9820 84

33 2957 101 69 6636 100 105 9904 82

34 3058 119 70 6736 101 106 9986 80

35 3177 98 71 6837 105 107 10066 82

36 3275 105 72 6942 100 108 10148 84

The probability distribution function of this

variable was obtained by Easy Fit 55

Software and through the above mentioned

three methods From 65 distribution functions

available in the software 32 functions which

best fit the data are selected and their rank

(based on 3 mentioned tests of Goodness of

Fit) are shown in Tables 3 and 4 The sum of

scores for each function in different methods

is provided and is used to define the best fit

distribution function The results are shown in

Tables 5-7 As these tables indicate the

Gumbel Min Function has the first rank in

Kolmogorov-Smirnov and Anderson-Darling

while the Chi-squared method shows rank of

21 for this function This significant difference

between the evaluations by different tests for

Goodness of Fit indicates an irregular data that

requires more examination and understanding

before analysis This difference can be

attributed to the fact that the number of

samples in some categories is low

The Gumbel Min distribution function for

the large section is shown in Figure 3 As it

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

192

can be seen in this Figure and Table 1 the

number of samples in the interval of 86 to 94

centimeters is significantly less than the other

intervals The Gumbel Min distribution

function for the small section is illustrated in

Figure 4 This Figure together with the data

indicated in Table 2 show that the number of

samples in the interval of 86 to 97 centimeters

is much less than the other intervals From

Figures 3 and 4 it is evident that these empty

intervals are at odds with the fitted functions

Table 3 Statistics and ranks of distribution functions for large section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

23 43841 6 49794 8 021754 Beta 1

27 47975 8 52994 9 022023 Burr 2

26 46794 3 47874 5 020889 Burr (4P) 3

5 31318 32 11036 16 025584 Cauchy 4

2 21377 11 6237 11 024138 Chi-Squared 5

20 39714 27 68867 31 027695 Chi-Squared (2P) 6

13 33815 28 69436 29 026923 Erlang 7

10 33524 17 64184 19 026197 Erlang (3P) 8

16 3418 13 63292 13 025443 Error 9

8 32733 23 67096 22 026456 Fatigue Life 10

18 34333 14 63491 14 025448 Fatigue Life (3P) 11

12 33755 26 68443 23 026478 Gamma 12

4 2685 18 65688 28 026773 Gamma (3P) 13

9 33385 19 65998 18 026138 Gen Gamma 14

15 34105 16 6404 17 025873 Gen Gamma (4P) 15

25 46507 2 46156 1 020251 Gumbel Min 16

21 42932 1 45911 2 020293 Kumaraswamy 17

6 3246 24 67841 25 026576 Log-Gamma 18

19 37081 20 66069 24 026528 Log-Logistic 19

28 48387 10 56965 6 021021 Log-Logistic (3P) 20

32 92302 30 71974 30 027007 Logistic 21

7 32711 22 6701 21 026392 Lognormal 22

14 34028 15 6393 15 025582 Lognormal (3P) 23

30 5064 29 70364 20 026375 Nakagami 24

17 34259 12 62615 12 025275 Normal 25

3 24976 25 68315 27 026697 Pearson 5 26

11 33557 21 66461 26 026678 Pearson 5 (3P) 27

31 55147 9 5338 10 022378 Pert 28

1 3473 31 90577 32 033431 Power Function 29

24 45901 7 49889 3 020564 Triangular 30

29 48479 5 49002 7 021481 Weibull 31

22 43445 4 47932 4 020825 Weibull (3P) 32

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

191

Table 2 Spacing of steel sets in small section

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

Frame

number

Location

(m)

Distance

to next

frame

(cm)

1 400 42 37 3380 100 73 7042 103

2 442 45 38 3480 100 74 7145 102

3 487 43 39 3580 103 75 7247 103

4 530 73 40 3683 102 76 7350 102

5 603 72 41 3785 106 77 7452 101

6 675 60 42 3891 99 78 7553 101

7 735 50 43 399 100 79 7654 101

8 785 79 44 409 98 80 7755 83

9 864 80 45 4188 102 81 7838 83

10 944 81 46 4290 101 82 7921 82

11 1025 82 47 4391 99 83 8003 80

12 1107 80 48 4490 102 84 8083 87

13 1187 109 49 4592 102 85 8170 80

14 1296 53 50 4694 106 86 8250 80

15 1349 83 51 4800 100 87 8330 84

16 1432 92 52 4900 97 88 8414 79

17 1524 90 53 4997 103 89 8493 83

18 1614 88 54 5100 103 90 8576 89

19 1702 73 55 5203 102 91 8665 91

20 1775 62 56 5305 101 92 8756 91

21 1837 75 57 5406 99 93 8847 101

22 1912 78 58 5305 101 94 8948 102

23 1990 84 59 5605 100 95 9050 105

24 2074 83 60 5705 106 96 9155 87

25 2157 85 61 5811 100 97 9242 81

26 2242 98 62 5911 109 98 9323 85

27 2340 108 63 6020 110 99 9408 80

28 2448 102 64 6130 98 100 9488 82

29 2550 100 65 6228 100 101 9570 84

30 2650 105 66 6328 105 102 9654 85

31 2755 98 67 6433 97 103 9739 81

32 2853 104 68 6530 106 104 9820 84

33 2957 101 69 6636 100 105 9904 82

34 3058 119 70 6736 101 106 9986 80

35 3177 98 71 6837 105 107 10066 82

36 3275 105 72 6942 100 108 10148 84

The probability distribution function of this

variable was obtained by Easy Fit 55

Software and through the above mentioned

three methods From 65 distribution functions

available in the software 32 functions which

best fit the data are selected and their rank

(based on 3 mentioned tests of Goodness of

Fit) are shown in Tables 3 and 4 The sum of

scores for each function in different methods

is provided and is used to define the best fit

distribution function The results are shown in

Tables 5-7 As these tables indicate the

Gumbel Min Function has the first rank in

Kolmogorov-Smirnov and Anderson-Darling

while the Chi-squared method shows rank of

21 for this function This significant difference

between the evaluations by different tests for

Goodness of Fit indicates an irregular data that

requires more examination and understanding

before analysis This difference can be

attributed to the fact that the number of

samples in some categories is low

The Gumbel Min distribution function for

the large section is shown in Figure 3 As it

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

192

can be seen in this Figure and Table 1 the

number of samples in the interval of 86 to 94

centimeters is significantly less than the other

intervals The Gumbel Min distribution

function for the small section is illustrated in

Figure 4 This Figure together with the data

indicated in Table 2 show that the number of

samples in the interval of 86 to 97 centimeters

is much less than the other intervals From

Figures 3 and 4 it is evident that these empty

intervals are at odds with the fitted functions

Table 3 Statistics and ranks of distribution functions for large section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

23 43841 6 49794 8 021754 Beta 1

27 47975 8 52994 9 022023 Burr 2

26 46794 3 47874 5 020889 Burr (4P) 3

5 31318 32 11036 16 025584 Cauchy 4

2 21377 11 6237 11 024138 Chi-Squared 5

20 39714 27 68867 31 027695 Chi-Squared (2P) 6

13 33815 28 69436 29 026923 Erlang 7

10 33524 17 64184 19 026197 Erlang (3P) 8

16 3418 13 63292 13 025443 Error 9

8 32733 23 67096 22 026456 Fatigue Life 10

18 34333 14 63491 14 025448 Fatigue Life (3P) 11

12 33755 26 68443 23 026478 Gamma 12

4 2685 18 65688 28 026773 Gamma (3P) 13

9 33385 19 65998 18 026138 Gen Gamma 14

15 34105 16 6404 17 025873 Gen Gamma (4P) 15

25 46507 2 46156 1 020251 Gumbel Min 16

21 42932 1 45911 2 020293 Kumaraswamy 17

6 3246 24 67841 25 026576 Log-Gamma 18

19 37081 20 66069 24 026528 Log-Logistic 19

28 48387 10 56965 6 021021 Log-Logistic (3P) 20

32 92302 30 71974 30 027007 Logistic 21

7 32711 22 6701 21 026392 Lognormal 22

14 34028 15 6393 15 025582 Lognormal (3P) 23

30 5064 29 70364 20 026375 Nakagami 24

17 34259 12 62615 12 025275 Normal 25

3 24976 25 68315 27 026697 Pearson 5 26

11 33557 21 66461 26 026678 Pearson 5 (3P) 27

31 55147 9 5338 10 022378 Pert 28

1 3473 31 90577 32 033431 Power Function 29

24 45901 7 49889 3 020564 Triangular 30

29 48479 5 49002 7 021481 Weibull 31

22 43445 4 47932 4 020825 Weibull (3P) 32

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

192

can be seen in this Figure and Table 1 the

number of samples in the interval of 86 to 94

centimeters is significantly less than the other

intervals The Gumbel Min distribution

function for the small section is illustrated in

Figure 4 This Figure together with the data

indicated in Table 2 show that the number of

samples in the interval of 86 to 97 centimeters

is much less than the other intervals From

Figures 3 and 4 it is evident that these empty

intervals are at odds with the fitted functions

Table 3 Statistics and ranks of distribution functions for large section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

23 43841 6 49794 8 021754 Beta 1

27 47975 8 52994 9 022023 Burr 2

26 46794 3 47874 5 020889 Burr (4P) 3

5 31318 32 11036 16 025584 Cauchy 4

2 21377 11 6237 11 024138 Chi-Squared 5

20 39714 27 68867 31 027695 Chi-Squared (2P) 6

13 33815 28 69436 29 026923 Erlang 7

10 33524 17 64184 19 026197 Erlang (3P) 8

16 3418 13 63292 13 025443 Error 9

8 32733 23 67096 22 026456 Fatigue Life 10

18 34333 14 63491 14 025448 Fatigue Life (3P) 11

12 33755 26 68443 23 026478 Gamma 12

4 2685 18 65688 28 026773 Gamma (3P) 13

9 33385 19 65998 18 026138 Gen Gamma 14

15 34105 16 6404 17 025873 Gen Gamma (4P) 15

25 46507 2 46156 1 020251 Gumbel Min 16

21 42932 1 45911 2 020293 Kumaraswamy 17

6 3246 24 67841 25 026576 Log-Gamma 18

19 37081 20 66069 24 026528 Log-Logistic 19

28 48387 10 56965 6 021021 Log-Logistic (3P) 20

32 92302 30 71974 30 027007 Logistic 21

7 32711 22 6701 21 026392 Lognormal 22

14 34028 15 6393 15 025582 Lognormal (3P) 23

30 5064 29 70364 20 026375 Nakagami 24

17 34259 12 62615 12 025275 Normal 25

3 24976 25 68315 27 026697 Pearson 5 26

11 33557 21 66461 26 026678 Pearson 5 (3P) 27

31 55147 9 5338 10 022378 Pert 28

1 3473 31 90577 32 033431 Power Function 29

24 45901 7 49889 3 020564 Triangular 30

29 48479 5 49002 7 021481 Weibull 31

22 43445 4 47932 4 020825 Weibull (3P) 32

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

193

Table 4 Statistics and ranks of distribution functions for small section

Chi-Squared Anderson-Darling Kolmogorov-Smirnov Distribution No

Rank Statistic Rank Statistic Rank Statistic

9 29952 2 27185 6 016695 Beta 1

19 48294 5 34526 8 018907 Burr 2

21 61609 29 6951 31 024652 Burr (4P) 3

30 90588 32 99515 28 022896 Cauchy 4

2 14388 23 59372 29 022916 Chi-Squared 5

17 45063 21 55468 21 020316 Chi-Squared (2P) 6

3 15592 22 57262 27 022068 Erlang 7

22 64749 15 49525 22 020386 Erlang (3P) 8

11 31334 17 53593 30 02417 Error 9

31 10624 26 65355 17 020113 Fatigue Life 10

25 66017 9 42818 10 019633 Fatigue Life (3P) 11

1 13718 18 54334 23 021317 Gamma 12

27 67938 14 4827 18 020174 Gamma (3P) 13

10 3018 20 55434 19 02018 Gen Gamma 14

23 65302 8 4191 15 020093 Gen Gamma (4P) 15

14 38057 1 2585 2 01582 Gumbel Min 16

8 28513 4 27641 5 016621 Kumaraswamy 17

12 33235 28 66463 20 020189 Log-Gamma 18

15 41547 27 65374 9 019429 Log-Logistic 19

28 69704 6 38033 7 017756 Log-Logistic (3P) 20

6 20992 12 44279 26 021811 Logistic 21

32 11475 25 63035 13 019865 Lognormal 22

26 66377 13 4535 16 020098 Lognormal (3P) 23

5 20133 16 51817 25 021773 Nakagami 24

24 65998 10 43282 11 019794 Normal 25

16 42835 30 73541 12 019835 Pearson 5 26

4 18496 19 54897 24 021628 Pearson 5 (3P) 27

13 3788 7 40332 3 016006 Pert 28

29 8349 31 81503 32 025049 Power Function 29

20 49583 24 62101 14 01998 Triangular 30

18 4757 11 43586 1 015709 Weibull 31

7 2851 3 27592 4 016521 Weibull (3P) 32

Table 5 Rank of distribution functions in both sections by Kolmogorov-Smirnov test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 14 7

Burr 17 8

Burr (4P) 36 17

Cauchy 44 23

Chi-Squared 30 12

Chi-Squared (2P) 52 29

Erlang 56 30

Erlang (3P) 41 21

Error 43 22

Fatigue Life 39 19

Fatigue Life (3P) 24 11

Gamma 46 26

Gamma (3P) 46 26

Gen Gamma 37 18

Gen Gamma (4P) 32 14

Gumbel Min 3 1

Distribution

Function

Sum of Scores

for Rank Rank

Kumaraswamy 7 2

Log-Gamma 45 24

Log-Logistic 33 15

Log-Logistic (3P) 13 5

Logistic 56 29

Lognormal 34 16

Lognormal (3P) 31 13

Nakagami 45 24

Normal 23 10

Pearson 5 39 19

Pearson 5 (3P) 50 28

Pert 13 5

Power Function 64 32

Triangular 17 8

Weibull 8 3

Weibull (3P) 8 3

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

194

Table 6 Rank of distribution functions in both

sections by Anderson-Darling test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 8 4

Burr 13 5

Burr (4P) 32 15

Cauchy 64 32

Chi-Squared 34 18

Chi-Squared (2P) 48 26

Erlang 55 29

Erlang (3P) 32 15

Error 30 13

Fatigue Life 49 27

Fatigue Life (3P) 23 10

Gamma 44 22

Gamma (3P) 32 15

Gen Gamma 39 19

Gen Gamma (4P) 24 11

Gumbel Min 3 1

Kumaraswamy 5 2

Log-Gamma 52 28

Log-Logistic 47 24

Log-Logistic (3P) 16 6

Logistic 42 21

Lognormal 47 24

Lognormal (3P) 28 12

Nakagami 45 23

Normal 22 9

Pearson 5 55 29

Pearson 5 (3P) 40 20

Pert 16 6

Power Function 62 31

Triangular 31 14

Weibull 16 6

Weibull (3P) 7 3

Table 7 Rank of distribution functions in both

sections by Chi-squared test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 32 13

Burr 46 29

Burr (4P) 47 30

Cauchy 35 16

Chi-Squared 4 1

Chi-Squared (2P) 37 18

Erlang 16 4

Erlang (3P) 32 13

Error 27 8

Fatigue Life 39 21

Fatigue Life (3P) 43 26

Gamma 13 2

Gamma (3P) 31 12

Gen Gamma 19 6

Gen Gamma (4P) 38 19

Gumbel Min 39 21

Kumaraswamy 29 9

Log-Gamma 18 5

Log-Logistic 34 15

Log-Logistic (3P) 56 32

Logistic 38 19

Lognormal 39 21

Lognormal (3P) 40 24

Nakagami 35 16

Normal 41 25

Pearson 5 19 6

Pearson 5 (3P) 15 3

Pert 44 27

Power Function 30 11

Triangular 44 27

Weibull 47 30

Weibull (3P) 29 9

Fig 3 Fitted diagram of gumbel min for large section

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

195

Fig 4 Fitted diagram of gumbel min for small section

To examine whether these almost empty

intervals are just created randomly or if there

are any reasons for this phenomena the

observed data was investigated to find whether

any order can be discovered is the data

In Table 1 for the large section it was

observed that in the first part of the tunnel

(from Frame No 1 to Frame No 25) the

distances of steel sets are generally under 85

cm and in the second part of the tunnel (from

Frame No 26 to Frame No 82) the distance of

the frames are generally more than 95 cm

It can also be observed from Table 2 that in

the first part of the tunnel in the small section

(from Frame No 1 to Frame No 26) the

distances of steel sets are generally under 90

cm in the second part (from Frame No 27 to

Frame No 79) these distances are generally

more than 95 cm and in the third part (from

Frame No 80 to Frame No 108) the distances

are generally under 90 cm

Therefore it can be concluded that the

empty intervals in the data are not random and

there should be a reason behind the lack of

data in the empty intervals The reason may be

a change in the working condition including

1 The condition of the ground has

changed along the tunnel and the

spacing of steel sets has altered

accordingly

2 The different construction teams have

done their job with different levels of

risk and decisions

3 Different judgment of supervisors from

the safety point of view may lead to

variations in distance between steel sets

Regardless of which reason is correct one

can divide the observed (measured) data into

two categories of A (low spacing of steel sets)

and B (high spacing of steel sets) With such

analyses the difference between Chi-squared

method and other methods can be justified

Based on the above analysis the data for both

large and small sections of the tunnel (Tables

1 and 2) was divided into two types of A and

B and the distribution function for each type

of data was obtained separately by Easy Fit

55 Software From 65 distribution functions

available in the software 32 best fitted

functions were selected The list of these

distribution functions the statistic values and

the rank of each function by three methods of

Goodness of Fit are shown in Tables 8-11

Then the summation of scores of each method

was obtained and finally the total score was

used to determine the best function

A summary of these values is shown in

Tables 12-14 As these tables show the

Wakeby Distribution Function has rank (1) for

all three methods The fitted Wakeby

Distribution Function for all cases are shown in

Figures 5-8 From these Figures it can be seen

that Wakeby function which is an advanced

distribution function with 5 parameters is the

appropriate distribution function for the random

variable of spacing of the steel sets

From the above discussions it can also be

concluded that the significant difference

between the evaluation results performed by

different tests for Goodness of Fit indicates an

irregularity in the data In order to find the

source of this irregularity and possible

correction of data more examinations and

understanding is needed

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

196

Table 8 Statistics and rank of distribution function for large section (type A)

No Distribution Kolmogorov-Smirnov Anderson-Darling Chi-Squared

Statistic Rank Statistic Rank Statistic Rank

1 Beta 059703 32 12444 32 1528 16

2 Cauchy 022061 2 10596 2 17305 24

3 Chi-Squared 040103 30 46 29 16645 22

4 Chi-Squared (2P) 038767 28 38098 26 56276 27

5 Erlang 036026 25 34525 24 1515 12

6 Erlang (3P) 033775 19 32912 15 15122 11

7 Error 028749 5 2446 4 5963 30

8 Fatigue Life 033902 21 33452 20 154 17

9 Fatigue Life (3P) 032065 10 32015 8 14999 5

10 Frechet 043752 31 46834 30 85242 31

11 Frechet (3P) 036591 26 38488 27 46539 25

12 Gamma 033308 17 32892 13 15239 14

13 Gamma (3P) 033593 18 32899 14 15167 13

14 Gumbel Max 038869 29 47556 31 56253 26

15 Gumbel Min 031483 8 38652 28 17102 23

16 Hypersecant 030828 6 27547 6 1511 9

17 Inv Gaussian 032215 11 32914 16 1544 19

18 Inv Gaussian (3P) 032035 9 32035 9 15013 6

19 Laplace 028749 4 2446 3 5963 29

20 Log-Gamma 034189 23 33731 21 15551 20

21 Log-Logistic 038319 27 36705 25 56739 28

22 Log-Logistic (3P) 024813 3 25858 5 90273 32

23 Logistic 031478 7 2926 7 15063 7

24 Lognormal 033854 20 33406 19 15401 18

25 Lognormal (3P) 03273 13 32419 11 15111 10

26 Nakagami 032784 14 32465 12 15099 8

27 Normal 032249 12 3213 10 14977 4

28 Pearson 5 034459 24 34031 22 15622 21

29 Pearson 5 (3P) 034036 22 33224 17 15255 15

30 Wakeby 019716 1 074762 1 13338 2

31 Weibull 032828 15 34233 23 13069 1

32 Weibull (3P) 032884 16 33289 18 14717 3

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

196

Table 8 Statistics and rank of distribution function for large section (type A)

No Distribution Kolmogorov-Smirnov Anderson-Darling Chi-Squared

Statistic Rank Statistic Rank Statistic Rank

1 Beta 059703 32 12444 32 1528 16

2 Cauchy 022061 2 10596 2 17305 24

3 Chi-Squared 040103 30 46 29 16645 22

4 Chi-Squared (2P) 038767 28 38098 26 56276 27

5 Erlang 036026 25 34525 24 1515 12

6 Erlang (3P) 033775 19 32912 15 15122 11

7 Error 028749 5 2446 4 5963 30

8 Fatigue Life 033902 21 33452 20 154 17

9 Fatigue Life (3P) 032065 10 32015 8 14999 5

10 Frechet 043752 31 46834 30 85242 31

11 Frechet (3P) 036591 26 38488 27 46539 25

12 Gamma 033308 17 32892 13 15239 14

13 Gamma (3P) 033593 18 32899 14 15167 13

14 Gumbel Max 038869 29 47556 31 56253 26

15 Gumbel Min 031483 8 38652 28 17102 23

16 Hypersecant 030828 6 27547 6 1511 9

17 Inv Gaussian 032215 11 32914 16 1544 19

18 Inv Gaussian (3P) 032035 9 32035 9 15013 6

19 Laplace 028749 4 2446 3 5963 29

20 Log-Gamma 034189 23 33731 21 15551 20

21 Log-Logistic 038319 27 36705 25 56739 28

22 Log-Logistic (3P) 024813 3 25858 5 90273 32

23 Logistic 031478 7 2926 7 15063 7

24 Lognormal 033854 20 33406 19 15401 18

25 Lognormal (3P) 03273 13 32419 11 15111 10

26 Nakagami 032784 14 32465 12 15099 8

27 Normal 032249 12 3213 10 14977 4

28 Pearson 5 034459 24 34031 22 15622 21

29 Pearson 5 (3P) 034036 22 33224 17 15255 15

30 Wakeby 019716 1 074762 1 13338 2

31 Weibull 032828 15 34233 23 13069 1

32 Weibull (3P) 032884 16 33289 18 14717 3

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

197

Table 9 Statistics and rank of distribution function for large section (type B)

No Distribution Kolmogorov-Smirnov Anderson-Darling Chi-Squared

Statistic Rank Statistic Rank Statistic Rank

1 Beta 02074 4 48111 8 34266 8

2 Cauchy 015086 2 081021 1 095324 1

3 Chi-Squared 036983 32 12251 31 13031 32

4 Chi-Squared (2P) 02955 28 88653 28 65408 30

5 Erlang 024217 19 56122 18 59634 17

6 Erlang (3P) 023443 17 58147 19 59735 21

7 Error 023232 14 31364 5 5207 3

8 Fatigue Life 024751 23 60574 23 59805 24

9 Fatigue Life (3P) 022843 10 5301 12 5959 15

10 Frechet 035595 31 11288 30 6276 28

11 Frechet (3P) 03208 30 99406 29 94606 31

12 Gamma 02345 18 54993 17 59641 18

13 Gamma (3P) 024592 21 58719 20 59689 20

14 Gumbel Max 02973 29 13146 32 38731 11

15 Gumbel Min 022871 12 49164 9 20844 6

16 Hypersecant 022862 11 38525 6 22108 7

17 Inv Gaussian 022696 6 54727 16 59686 19

18 Inv Gaussian (3P) 023236 16 53572 14 5959 14

19 Laplace 023232 15 31364 4 5207 4

20 Log-Gamma 024906 24 61295 24 59851 26

21 Log-Logistic 028051 27 73507 27 62917 29

22 Log-Logistic (3P) 017945 3 21616 3 15542 5

23 Logistic 022776 7 43658 7 59839 25

24 Lognormal 024653 22 60286 22 59803 23

25 Lognormal (3P) 021409 5 52976 11 34346 10

26 Nakagami 022961 13 50416 10 59572 13

27 Normal 022818 9 53229 13 59594 16

28 Pearson 5 02536 25 63294 25 60963 27

29 Pearson 5 (3P) 02452 20 59466 21 59747 22

30 Wakeby 014792 1 098525 2 12573 2

31 Weibull 025526 26 64967 26 46281 12

32 Weibull (3P) 022812 8 54545 15 34276 9

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

198

Table 10 Statistics and rank of distribution function for small section (type A)

No Distribution Kolmogorov-Smirnov Anderson-Darling Chi-Squared

Statistic Rank Statistic Rank Statistic Rank

1 Beta 021199 6 28112 8 11887 10

2 Cauchy 007607 2 053788 2 21722 6

3 Chi-Squared 029894 29 51518 26 31632 17

4 Chi-Squared (2P) 028843 25 452 21 48678 21

5 Erlang 028241 22 45923 23 49836 22

6 Erlang (3P) 026405 20 38826 18 51219 28

7 Error 024156 8 27296 7 2441 15

8 Fatigue Life 029247 26 50929 25 03688 2

9 Fatigue Life (3P) 024479 11 34882 11 50842 24

10 Frechet 035623 32 84124 32 86158 32

11 Frechet (3P) 028254 23 56972 30 066478 4

12 Gamma 027184 21 44063 20 49965 23

13 Gamma (3P) 025932 19 38438 16 51505 29

14 Gumbel Max 031812 31 84001 31 12216 13

15 Gumbel Min 018491 4 26527 5 94236 8

16 Hypersecant 0242 10 28445 9 38078 18

17 Inv Gaussian 024902 13 45891 22 52185 31

18 Inv Gaussian (3P) 025228 17 35832 13 50933 26

19 Laplace 02417 9 26242 4 23138 14

20 Log-Gamma 029409 27 52505 28 036155 1

21 Log-Logistic 029863 28 5185 27 10664 9

22 Log-Logistic (3P) 017615 3 24925 3 12115 12

23 Logistic 02453 12 30722 10 39536 20

24 Lognormal 028812 24 49858 24 29995 7

25 Lognormal (3P) 025558 18 36985 14 50942 27

26 Nakagami 025039 15 39165 19 38496 19

27 Normal 024919 14 35302 12 50863 25

28 Pearson 5 030139 30 56013 29 037492 3

29 Pearson 5 (3P) 025217 16 38459 17 51776 30

30 Wakeby 007214 1 026787 1 20788 5

31 Weibull 02368 7 37601 15 29367 16

32 Weibull (3P) 020699 5 27058 6 11924 11

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

199

Table 11 Statistics and rank of distribution function for small section (type B)

No Distribution Kolmogorov-Smirnov Anderson-Darling Chi-Squared

Statistic Rank Statistic Rank Statistic Rank

1 Beta 013295 10 058419 9 42656 8

2 Cauchy 01413 13 15249 15 53011 14

3 Chi-Squared 037863 32 1179 32 11297 32

4 Chi-Squared (2P) 02387 30 25615 28 62896 15

5 Erlang 016974 18 1677 21 97442 23

6 Erlang (3P) 020935 29 29752 29 76403 16

7 Error 018653 26 17549 24 89962 17

8 Fatigue Life 016979 19 15503 20 94049 21

9 Fatigue Life (3P) 013199 9 05606 8 43182 9

10 Frechet 013163 7 05478 6 29949 6

11 Frechet (3P) 012417 3 05094 3 2893 5

12 Gamma 017154 20 16969 22 1483 29

13 Gamma (3P) 013723 12 060896 10 42479 7

14 Gumbel Max 013352 11 070654 11 4847 13

15 Gumbel Min 024705 31 55136 31 15403 31

16 Hypersecant 017893 25 1536 16 14431 26

17 Inv Gaussian 017646 23 17388 23 13955 24

18 Inv Gaussian (3P) 013187 8 055587 7 43229 10

19 Laplace 018653 27 17549 25 89962 18

20 Log-Gamma 016859 16 15482 19 94534 22

21 Log-Logistic 013 6 097782 13 47364 11

22 Log-Logistic (3P) 011206 2 049141 2 27435 3

23 Logistic 017779 24 15461 17 14555 28

24 Lognormal 016964 17 15475 18 94046 20

25 Lognormal (3P) 012971 5 053476 5 12767 1

26 Nakagami 017295 21 18473 27 14453 27

27 Normal 017646 22 17978 26 14167 25

28 Pearson 5 016734 15 14795 14 93157 19

29 Pearson 5 (3P) 012617 4 051629 4 12812 2

30 Wakeby 010513 1 045257 1 28143 4

31 Weibull 019563 28 47543 30 1508 30

32 Weibull (3P) 015192 14 077512 12 47765 12

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

200

Table 12 Rank of distribution functions in all cases

by Kolmogorov-Smirnov test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 52 9

Cauchy 19 3

Chi-Squared 123 32

Chi-Squared (2P) 111 31

Erlang 84 23

Erlang (3P) 85 24

Error 53 11

Fatigue Life 89 26

Fatigue Life (3P) 40 4

Frechet 101 30

Frechet (3P) 82 21

Gamma 76 19

Gamma (3P) 70 18

Gumbel Max 100 29

Gumbel Min 55 13

Hypersecant 52 9

Inv Gaussian 53 11

Inv Gaussian (3P) 50 7

Laplace 55 13

Log-Gamma 90 27

Log-Logistic 88 25

Log-Logistic (3P) 11 2

Logistic 50 7

Lognormal 83 22

Lognormal (3P) 41 5

Nakagami 63 17

Normal 57 15

Pearson 5 94 28

Pearson 5 (3P) 62 16

Wakeby 4 1

Weibull 76 19

Weibull (3P) 43 6

Table 13 Rank of distribution functions in all cases

by Anderson-Darling test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 57 12

Cauchy 20 3

Chi-Squared 118 32

Chi-Squared (2P) 103 30

Erlang 86 22

Erlang (3P) 81 20

Error 40 7

Fatigue Life 88 23

Fatigue Life (3P) 39 6

Frechet 98 29

Frechet (3P) 89 24

Gamma 72 17

Gamma (3P) 60 14

Gumbel Max 105 31

Gumbel Min 73 18

Hypersecant 37 5

Inv Gaussian 77 19

Inv Gaussian (3P) 43 10

Laplace 36 4

Log-Gamma 92 26

Log-Logistic 92 26

Log-Logistic (3P) 13 2

Logistic 41 8

Lognormal 83 21

Lognormal (3P) 41 8

Nakagami 68 16

Normal 61 15

Pearson 5 90 25

Pearson 5 (3P) 59 13

Wakeby 5 1

Weibull 94 28

Weibull (3P) 51 11

Table 14 Rank of distribution functions in all cases by Chi-squared test

Distribution

Function

Sum of Scores

for Rank Rank

Beta 42 3

Cauchy 45 4

Chi-Squared 103 32

Chi-Squared (2P) 93 29

Erlang 74 24

Erlang (3P) 76 25

Error 65 13

Fatigue Life 64 12

Fatigue Life (3P) 53 7

Frechet 97 31

Frechet (3P) 65 13

Gamma 84 28

Gamma (3P) 69 19

Gumbel Max 63 11

Gumbel Min 68 17

Hypersecant 60 10

Inv Gaussian 93 29

Distribution

Function

Sum of Scores

for Rank Rank

Inv Gaussian 93 29

Inv Gaussian (3P) 56 8

Laplace 65 13

Log-Gamma 69 19

Log-Logistic 77 26

Log-Logistic (3P) 52 6

Logistic 80 27

Lognormal 68 17

Lognormal (3P) 48 5

Nakagami 67 16

Normal 70 22

Pearson 5 70 22

Pearson 5 (3P) 69 19

Wakeby 13 1

Weibull 59 9

Weibull (3P) 35 2

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

201

Fig 5 Fitted diagram of Wakeby for large section (type A)

Fig 6 Fitted diagram of Wakeby for large section (type B)

Fig 7 Fitted diagram of Wakeby for small section (type A)

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

202

Fig 8 Fitted diagram of Wakeby for small section (type B)

4 Conclusion

In the present study the spacing of the steel

sets for temporary support of tunnels was

considered as a random variable To determine

the distribution function of this random

variable the real data gathered from a tunnel

in Iran with two drilling sections was used

The proper distribution function for both

sections was evaluated by three methods of

Goodness of Fit (Kolmogorov-Smirnov

Anderson-Darling and Chi-squared) The

analysis of data was performed by the Easy Fit

Software and the following results were

obtained

Wakeby Distribution function is the

best probability distribution function

that can be fitted to the spacing of the

steel sets

A significant difference between

evaluations by different tests for

Goodness of Fit indicates an irregular

data that requires more examination and

understanding in order to lead us to find

the source of irregularity and possible

corrections

The studies in this research show that a

change in the tunnel section or a change

in working conditions does not affect

the proper distribution function

However it affects the parameters of

the PDF Of course this result needs to

be verified by more case studies

References [1] Madani H (2005) Tunneling (Volume 4)

Design and construction of support systems

ISBN 964-463-135-8

[2] Biron C Ariogulu E amp Lucas J R (1983)

Design of supports in mines Wiley ISBN-10

0471867268 ISBN-13 978-463-135-8

0471867265

[3] Hjaacutelmarsson EH (2011) Tunnel Support Use

of lattice girders in sedimentary rock MSc

Thesis University of Iceland

[4] Carranza-Torres C amp Diederichs M (2009)

Mechanical analysis of circular liners with

particular reference to composite supports For

example liners consisting of shotcrete and steel

sets Tunnelling and Underground Space

Technology 24 506ndash532 DOI 101016jtust

200902001

[5] Basaligheh F (2003) The advantages of using

lattice girder with shotcrete instead of steel sets

for temporary support of tunnels (in Persian)

Proceedings of the Sixth International

Conference on Civil Engineering 5 265-270

[6] Basaligheh F (2006) Evaluation of two

methods for the temporary support of the

tunnels from economic point of view (in

Persian) Journal of Science and technology

12 11-17

[7] Basaligheh F amp Keyhani A (2013) The

advantages of using the method of ldquoequivalent

sectionrdquo in design of shotcrete and steel sets for

temporary support of tunnels (in Persian)

Proceedings of Seventh National Congress on

Civil Engineering Shahid Nikbakht Faculty of

Engineering Zahedan-Iran

[8] Basaligheh F amp Keyhani A (2013) New

approaches for modelling of two common

methods of temporary support for tunnels

Proceedings of Tenth Iranian Tunnelling

Conference

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

203

[9] Wong LNY Fang Q amp Zhang D (2013)

Mechanical analysis of circular tunnels

supported by steel sets embedded in primary

linings Tunnelling and Underground Space

Technology 37 80ndash88 DOI

101016jtust201303011

[10] Schwingenschloegl R amp Lehmann C

(2009) Swelling rock behaviour in a tunnel

NATM-support vs Q-support ndash A comparison

Tunnelling and Underground Space

Technology 24 356ndash362 DOI

101016jtust200808007

[11] Nowak A a (2000) Reliability of structures

McGraw Hill International Edition ISBN 0-

07-116354-9

[12] Rezaee-Pajand H (2002) Application of

statistics and probability for water reservoirs

Sokhan-Gostar Press ISBN 964-6932-35-5

[13] Sanchidrian J O A (2014) Size distribution

functions for rock fragments International

Journal of Rock Mechanics amp Mining Sciences

71 381-394 DOI101016jijrmms

201408007

[14] Milford R (1987) Annual maximum wind

speeds from parent distribution functions

Journal of wind Engineering and Industrial

Aerodynamics 25 pp 163-178

[15] Ghamishon R amp MalekianA (2011)

Determination of the most suitable statistical

distribution functions for regional floods (Case

study of southwest Kerman Province)

Proceedings of Sixth National Watershed

Sciences and Engineering Conference and forth

national congress on soil erosion and sediment

Tarbiat-Modares University

[16] Meftah-Halghi M Zangale M E amp Aghili

R (2012) A Comparison of the most suitable

statistical distribution functions for maximum

daily flow rate and rainfall (Case study

Hydrometry station of Gonbad Kavos) Fifth

National Watershed Sciences and Engineering

Conference Iran University of Agricultural

Sciences of Gorgan COI

WATERSHED05_224httpwwwcivilicacom

Paper-WATERSHED05-

WATERSHED05_224html

[17] Cochran WG (1952) The χ^2 test of

goodness of fit Institute of Mathematical

Statistics Vol 23 No 3 pp 315-345

httpwwwjstororgstable2236678

[18] Anderson T A (1952) Asymptotic theory of

certain goodness of fit criteria based on

stochastic processes Ann Math Statist 23 pp

193-212

[19] Darbandi S Mahmudi S amp Ebrahimi S amp

Shoeybi-Nobarian M R (2012) Introduction

and application of Anderson-Darling in river

engineering of East Azerbaijan Province Fifth

National Conference on Watersheds and

management of soil and water resources

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

202

Fig 8 Fitted diagram of Wakeby for small section (type B)

4 Conclusion

In the present study the spacing of the steel

sets for temporary support of tunnels was

considered as a random variable To determine

the distribution function of this random

variable the real data gathered from a tunnel

in Iran with two drilling sections was used

The proper distribution function for both

sections was evaluated by three methods of

Goodness of Fit (Kolmogorov-Smirnov

Anderson-Darling and Chi-squared) The

analysis of data was performed by the Easy Fit

Software and the following results were

obtained

Wakeby Distribution function is the

best probability distribution function

that can be fitted to the spacing of the

steel sets

A significant difference between

evaluations by different tests for

Goodness of Fit indicates an irregular

data that requires more examination and

understanding in order to lead us to find

the source of irregularity and possible

corrections

The studies in this research show that a

change in the tunnel section or a change

in working conditions does not affect

the proper distribution function

However it affects the parameters of

the PDF Of course this result needs to

be verified by more case studies

References [1] Madani H (2005) Tunneling (Volume 4)

Design and construction of support systems

ISBN 964-463-135-8

[2] Biron C Ariogulu E amp Lucas J R (1983)

Design of supports in mines Wiley ISBN-10

0471867268 ISBN-13 978-463-135-8

0471867265

[3] Hjaacutelmarsson EH (2011) Tunnel Support Use

of lattice girders in sedimentary rock MSc

Thesis University of Iceland

[4] Carranza-Torres C amp Diederichs M (2009)

Mechanical analysis of circular liners with

particular reference to composite supports For

example liners consisting of shotcrete and steel

sets Tunnelling and Underground Space

Technology 24 506ndash532 DOI 101016jtust

200902001

[5] Basaligheh F (2003) The advantages of using

lattice girder with shotcrete instead of steel sets

for temporary support of tunnels (in Persian)

Proceedings of the Sixth International

Conference on Civil Engineering 5 265-270

[6] Basaligheh F (2006) Evaluation of two

methods for the temporary support of the

tunnels from economic point of view (in

Persian) Journal of Science and technology

12 11-17

[7] Basaligheh F amp Keyhani A (2013) The

advantages of using the method of ldquoequivalent

sectionrdquo in design of shotcrete and steel sets for

temporary support of tunnels (in Persian)

Proceedings of Seventh National Congress on

Civil Engineering Shahid Nikbakht Faculty of

Engineering Zahedan-Iran

[8] Basaligheh F amp Keyhani A (2013) New

approaches for modelling of two common

methods of temporary support for tunnels

Proceedings of Tenth Iranian Tunnelling

Conference

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

203

[9] Wong LNY Fang Q amp Zhang D (2013)

Mechanical analysis of circular tunnels

supported by steel sets embedded in primary

linings Tunnelling and Underground Space

Technology 37 80ndash88 DOI

101016jtust201303011

[10] Schwingenschloegl R amp Lehmann C

(2009) Swelling rock behaviour in a tunnel

NATM-support vs Q-support ndash A comparison

Tunnelling and Underground Space

Technology 24 356ndash362 DOI

101016jtust200808007

[11] Nowak A a (2000) Reliability of structures

McGraw Hill International Edition ISBN 0-

07-116354-9

[12] Rezaee-Pajand H (2002) Application of

statistics and probability for water reservoirs

Sokhan-Gostar Press ISBN 964-6932-35-5

[13] Sanchidrian J O A (2014) Size distribution

functions for rock fragments International

Journal of Rock Mechanics amp Mining Sciences

71 381-394 DOI101016jijrmms

201408007

[14] Milford R (1987) Annual maximum wind

speeds from parent distribution functions

Journal of wind Engineering and Industrial

Aerodynamics 25 pp 163-178

[15] Ghamishon R amp MalekianA (2011)

Determination of the most suitable statistical

distribution functions for regional floods (Case

study of southwest Kerman Province)

Proceedings of Sixth National Watershed

Sciences and Engineering Conference and forth

national congress on soil erosion and sediment

Tarbiat-Modares University

[16] Meftah-Halghi M Zangale M E amp Aghili

R (2012) A Comparison of the most suitable

statistical distribution functions for maximum

daily flow rate and rainfall (Case study

Hydrometry station of Gonbad Kavos) Fifth

National Watershed Sciences and Engineering

Conference Iran University of Agricultural

Sciences of Gorgan COI

WATERSHED05_224httpwwwcivilicacom

Paper-WATERSHED05-

WATERSHED05_224html

[17] Cochran WG (1952) The χ^2 test of

goodness of fit Institute of Mathematical

Statistics Vol 23 No 3 pp 315-345

httpwwwjstororgstable2236678

[18] Anderson T A (1952) Asymptotic theory of

certain goodness of fit criteria based on

stochastic processes Ann Math Statist 23 pp

193-212

[19] Darbandi S Mahmudi S amp Ebrahimi S amp

Shoeybi-Nobarian M R (2012) Introduction

and application of Anderson-Darling in river

engineering of East Azerbaijan Province Fifth

National Conference on Watersheds and

management of soil and water resources

Basaligheh and Keyhani Int J Min amp Geo-Eng Vol49 No2 December 2015

203

[9] Wong LNY Fang Q amp Zhang D (2013)

Mechanical analysis of circular tunnels

supported by steel sets embedded in primary

linings Tunnelling and Underground Space

Technology 37 80ndash88 DOI

101016jtust201303011

[10] Schwingenschloegl R amp Lehmann C

(2009) Swelling rock behaviour in a tunnel

NATM-support vs Q-support ndash A comparison

Tunnelling and Underground Space

Technology 24 356ndash362 DOI

101016jtust200808007

[11] Nowak A a (2000) Reliability of structures

McGraw Hill International Edition ISBN 0-

07-116354-9

[12] Rezaee-Pajand H (2002) Application of

statistics and probability for water reservoirs

Sokhan-Gostar Press ISBN 964-6932-35-5

[13] Sanchidrian J O A (2014) Size distribution

functions for rock fragments International

Journal of Rock Mechanics amp Mining Sciences

71 381-394 DOI101016jijrmms

201408007

[14] Milford R (1987) Annual maximum wind

speeds from parent distribution functions

Journal of wind Engineering and Industrial

Aerodynamics 25 pp 163-178

[15] Ghamishon R amp MalekianA (2011)

Determination of the most suitable statistical

distribution functions for regional floods (Case

study of southwest Kerman Province)

Proceedings of Sixth National Watershed

Sciences and Engineering Conference and forth

national congress on soil erosion and sediment

Tarbiat-Modares University

[16] Meftah-Halghi M Zangale M E amp Aghili

R (2012) A Comparison of the most suitable

statistical distribution functions for maximum

daily flow rate and rainfall (Case study

Hydrometry station of Gonbad Kavos) Fifth

National Watershed Sciences and Engineering

Conference Iran University of Agricultural

Sciences of Gorgan COI

WATERSHED05_224httpwwwcivilicacom

Paper-WATERSHED05-

WATERSHED05_224html

[17] Cochran WG (1952) The χ^2 test of

goodness of fit Institute of Mathematical

Statistics Vol 23 No 3 pp 315-345

httpwwwjstororgstable2236678

[18] Anderson T A (1952) Asymptotic theory of

certain goodness of fit criteria based on

stochastic processes Ann Math Statist 23 pp

193-212

[19] Darbandi S Mahmudi S amp Ebrahimi S amp

Shoeybi-Nobarian M R (2012) Introduction

and application of Anderson-Darling in river

engineering of East Azerbaijan Province Fifth

National Conference on Watersheds and

management of soil and water resources