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International Journal of Scientific & Engineering Research Volume 10, Issue 3, March-2019 613

ISSN 2229-5518

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Predicting (Nk) factor of (CPT) test using (GP):

Comparative Study of MEPX & GN7

Ahmed H. ELbosraty1, Ahmed M. Ebid2, Ayman L. Fayed3

Abstract— Static cone penetration test (CPT) is a broadly satisfactory and dependable geotechnical in-situ apparatus that gives brisk and

honest substantial measure of data about soil classification, stratification and properties. Un-drained shear strength of clay (cu) is one of

the principle soil parameters that could be sensibly evaluated from the (CPT) results, as it is specifically connected to the tip resistance

through the experimental cone factor (Nk). Earlier researches showed that (Nk) value depends on type of soil, nature and stress history

conditions and many other variables. Construction development in some locations with thick deposits of soft to very soft clays motivates

extensive researches to define the reasonable value of the (Nk) factor for such types of clay. The performed study concentrated on utilizing

the genetic programming technique (GP) to predict (Nk) value of clay using the consistency limits that can be easily determined in the

laboratory. A set of 102 records were gathered from the CPT site investigations and corresponding consistency limits and other physical

properties experiments, were divided into training set of 72 records and validation set of 30 records. Both (GN7) & (MEPX) software were

used to apply (GP) on the available data. Four trials for each software with different chromosome lengths were performed to correlate the

(Nk) factor with the clay consistency limits, water content (wc) and unit weight (γ) using training data set, then, the produced relations were

tested using the validation data set. The four generated formulas using (GN7) showed accuracies ranging between 93% and 97% and

coefficient of determination (R2) ranging between 0.7 and 0.9, while the other four formulas form (MEPX) showed accuracy not exceeding

95% and coefficient of determination (R2) ranging between 0.45 and 0.75.

Index Terms— CPT, Consistency Limits, Genetic Programming (GP), Multi Expression Programming (MEP), Cone Factor (Nk).

—————————— ——————————

1 INTRODUCTION

Lassic static cone penetration test (CPT) is one of the well-

known site tests which carried out to characterize the soil

formations and estimate their mechanical proprieties

based on their penetration resistance. Today, modern (CPT)

equipment is capable to measure many more parameters than

penetration resistance such as pore water pressure, lateral soil

pressure at rest, lateral elastic modulus of soil. Figure (1)

shows (CPT) test overview and sample of its output. [3, 10,

15].

Many theories were introduced to simulate the behavior of the

soil during static penetration process such as the bearing ca-

pacity theory (Meyerhof 1961, Durgunoglu and Mitchell 1975),

cavity expansion theory (Vesic 1972 and Yu and Houlsby

1991), the strain path method proposed by Baligh (1985), cali-

bration chamber testing and the finite element analysis (Walk-

er and Yu 2006). [1, 8, 9, 12, 17,19 and 20].

Although, many previous researches were carried out to corre-

late tip resistance from (CPT) with other soil properties spe-

cially the un-drained shear strength of clay (cu), but none of

them derives a proper correlation due to the sophisticated be-

havior of the clay which depends on many parameters such as

initial stresses, pore-water pressure, penetration rate and over

consolidation ratio. In addition, uncertainties in measured

values make the correlation more difficult. [4, 14].

Previously suggested formulas to correlate (CPT) results with

the un-drained shear strength of clay (cu) are summarized in

many publications [3, 6, 9, 10, 15, and 16]. Many of those re-

searches considered that (cu) proportional linearly with the

corrected tip resistance of the cone as shown in equation (1)

Nk

q

voc

u

σ

−

=c

….... (1)

Where,

cu : Un-drained shear strength of clay.

qc : Tip resistance of the cone.

σvo : Total overburden pressure.

Nk : Empirical cone factor.

Accordingly, most of the previous researches were concerned

in estimating (Nk) value which correlates (CPT) with (cu).

As summarized by Zsolt Rémai (2013) [17], typical values for

(Nk) for different soil types has been suggested by many re-

searchers. Lunne and Kleven (1981) [13] suggested that (Nk)

varies between 11 and 19 for normally consolidated, Scandi-

navian marine clays. Jörss (1998) [7] suggested that (Nk)

equals 20 for marine clays and 15 for boulder clays. Gebre-

selassie (2003) [5] proposed that (NK) value is ranged between

7.6 and 28.4 for different soil types. Finally, Chen (2001) [4]

recommended (Nk) values varying between 5 and 12.

C

————————————————

1 Graduate Student, Department of Structural Engineering, Faculty of Engi-

neering, Ain Shams Universit

y E-Mail: eng_elbosraty@yahoo.com

2 Lecturer,

Department of Structural Engineering, Faculty of Engineering

&

Tech

.., Future University, Egypt. E-Mail: ahmed.abdelkhaleq@fue.edu.eg

3 Associate Professor, Department of Structural Engineering, Faculty of

Engineering, Ain Shams University E

-Mail: ayman_fayed@eng.asu.edu.eg

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2 (GP) & (MEP)

2.1 Genetic programming (GP)

(GP) is a direct application of genetic algorithm (GA) optimi-

zation technique on a population of mathematical formulas to

generate the most fitting formula for certain given points in a

hyper-space. Accordingly, (GP) may be described as Multivar-

iable Regression Procedure. (Koza,1994)

(GP) is big title includes several techniques such as Linear GP,

Cartesian GP, Compacted GP and many others. [2 ,11,18].

Classic (GP) procedure starts with randomly generating a

population of mathematical formulas which are encoded in

genetic form (chromosome form) and testing each formula

using the training data set to calculate its fitness. Only the

most fitting formulas (survivors) will be selected to generate

the next cycle (or generation) using crossover and mutation

operators, then the new population to be tested again to calcu-

late their fitness and so on until accepted accuracy is achieved.

2.2 Multi Expression Programming (MEP)

(MEP) is a technique to automatic generation of computer

programs. Accordingly, it could be used to generate fitting

mathematical formulas for certain data set. MEP differentiates

from classic (GP) techniques by encoding multiple solutions in

the same chromosome. Same as classic (GP), crossover is ap-

plied in (MEP) using one Point Crossover technique, where

one crossover point is randomly chosen and the parent chro-

mosomes exchange the sequences at the right side of the

crossover point. Also, both classic (GP) and (MEP) are sharing

the same mutation technique where randomly selected gens

(or symbols) are changed. Unlike classic (GP), the output of

the (MEP) is a series of programming commands, if all these

commands are mathematical expressions, then the output

could be simplified in one mathematical expression just like

classical (GP).

3 (GN7) & (MEPX)

3.1 (GN7) software

(GN7) is the 7th version of classic (GP) software which was

developed by the author in (2004) in C++[2]. Figure (2) shows

the encoding technique and the principal of tree levels to

measure the complexity of the mathematical formula. It is

clear that complex of the formulas needs more levels to repre-

sent it than simple ones. As shown in Figure (2). The chromo-

some consists of two parts, “operators” and “variables”. The

“operators” part contains the entire tree except the level 0 and

has (2No. of levels - 1) genes. The “variables” part contains only the

level 0 of the tree and has (2No of levels) genes. Therefore, the total

number of genes in the chromosome is (2No. of levels + 1) genes [2].

(GN7) supports eight operators which are (=, +, -, x, /,Xy , e^,

Ln) and support up to 7 levels of complexity. Regarding

crossover procedure, it doesn’t support the classic one-point

crossover technique, instated, it supports random crossover

technique which was proposed by author, 2004 [2] to generate

the new chromosomes by randomly selecting each gene from

similar surviving chromosomes as shown from figure (3). Mu-

tation is applied by replacing some randomly selected genes

with random operator (in the “operators” part) or variable (in

“variables” part). Since most mathematical formula have con-

stant values, hence variables with constant values are used to

present those constants. Usually, the following set of constants

is used (1, 3, 5, 7 and 11). (GN7) uses the sum of squared errors

(SSR) method to measure the fitness.

Figure (2) Mathematical and Genetic Representation of Binary

Tree (after A. Ebid 2004)

Figure (3) Random Crossover Technique (after A. Ebid 2004)

3.2 (MEPX) software

(MEPX) is free and open source software that uses (MEP)

technique. This project started in 2001 and the first end-user

for windows is released in 2015. Unlike (GN7), current version

of (MEPX) has a graphical user interface (GUI). Both source

code and compiled software could be freely downloaded from

http://www.mepx.org. The software is easy to learn and of-

fers many options to control the searching process as shown in

figure (4), these options could be summarized in the following

points:

- Three types of problems (regression, binary classifica-

tion and multi-class classification)

- Two methods to measure error (mean absolute error

and mean squared error)

- 26 different mathematical, logical, statistical and trig-

onometrical operators.

- Two methods of crossover (uniform and one point

crossover)

- Two methods to generate constants (user defined and

automatically generated)

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- Code length, is the number of genes on each chromo-

some, it is a measurement for the complexity of the

solution which is equivalent to number of levels in

(GN7).

Figure (4) screenshot of (GUI) of (MEPX) software

3.3 Comparison Bases

In order to fairly compare the results of the two programs, the

following points were considered:

- Using same set of variables, liquid limit (L.L), plastic

limit (P.L), plasticity index (P.I), water content (wc)

and unit weight of clay (γ).

- Using same constant values (1,3,5,7,11)

- Using the same training and validation data sets

- Using the same population size

- Using same number of generations

- Using same method to measure error (SSR)

- Using same complexity level (code length)

- Since the output should be mathematical formula, on-

ly mathematical operators were used in (MEPX)

- Unaccepted too complicated expressions such as mul-

ti-power (x^(y^z)) and multi logarithms (log(log(x))

were eliminated from both programs.

- For the best fitting formula of each trial, its accuracy

was determine using equation (2) , the predicted val-

ues of (Nk) were plotted against the experimental

ones and the coefficient of determination (R2) was de-

termined.

rec

cal NNk

NkNk 100

100 = (%)Accuracy exp

exp ×

−

−

∑

…(2)

4 PREDICTION OF (NK) USING (GN7)

Four trials were carried out using (GN7) to predict the value

of (Nk) factor using the training data set as follows:

- 1st trial had only two levels of complexity (chromo-

some length is 8 genes), Population size was 5000

chromosome, number of generations was 50 and the

best formula was equation (3)

LP LL

..6.3

7.32 =Nk ×

−

…(3)

- 2nd trial had three levels of complexity (chromosome

length is 16 genes), Population size was 10000 chro-

mosome, number of generations was 50 and the best

formula was equation (4)

56.2

..12

)-(PILn .161 =Nk −

×LL LP

γ

…(4)

- 3rd trial had four levels of complexity (chromosome

length is 32 genes), Population size was 20000 chro-

mosome, number of generations was 50 and the best

formula was equation (5)

79.1

.).(

20

..21

.111 =Nk

2

−

+

+

×LPLnLL LP

…(5)

- 4th trial had five levels of complexity (chromosome

length is 64 genes), Population size was 40000 chro-

mosome, number of generations was 50 and the best

formula was equation (6)

19.5)5).()(2(

)14.04( 3)/5(.

.31 =Nk −

−−−

+

++

×LLLnPILn

PILn

LP

γ

γ

…(6)

Accuracies and coefficient of determination (R2) of training

and validations sets for each one of the four trials are summa-

rized in table (1). Figure (5) represent the correlation between

the predicted (Nk) values using the equations (3),(4),(5),(6)

and the measured ones.

TABLE (1): SUMMARY OF ACCURACIES AND (R2) VALUES FOR

EQUATIONS (3),(4),(5),(6)

Trial No.

No. of Levels

Proposed

Formula

Accuracy % R2

Training

Validation

Total

Training

Validation

Total

1 2 Eq. (3) 93 95 94 0.72 0.71 0.71

2 3 Eq. (4) 96 97 96 0.87 0.89 0.87

3 4 Eq. (5) 96 96 96 0.82 0.88 0.84

4 5 Eq. (6) 96 97 97 0.91 0.88 0.87

The following points could be noted from table (1):

- Accuracies of all proposed formulas are ranged be-

tween 93% to 97%, while (R2) values are ranged be-

tween 0.71 to 0.91 which indicates good fitting

- The enhancement in fitting between equations

(4),(5),(6) is negligible, on other hand, the remarkable

complexity difference between them makes equation

(4) more favorable than the others.

- None of the four proposed formulas contains water con-

tent (wc) which indicates that (Nk) doesn’t depend on

it.

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a) Trial(1) – Eq. 3

b) Trial(2) – Eq. 4

c) Trial(3) – Eq. 5

d) Trial(4) – Eq. 6

Figure (5) Relation between the Predicted and Measured (Nk) values for Developed Correlations using (GN7)

5 PREDICTION OF (NK) USING (MEPX)

Four equivalent trials were carried out using (MEPX) to pre-

dict the value of (Nk) factor using the training data set as fol-

lows:

- 1st trial had chromosome length of 8 genes, Popula-

tion size was 5000 chromosome, number of genera-

tions was 50 and the best formula was equation (7)

IP

LL .55.11

=Nk −

…(7)

- 2nd trial had chromosome length is 16 genes, Popula-

tion size was 10000 chromosome, number of genera-

tions was 50 and the best formula was equation (8)

[ ]

(P.I)Ln P.IP.L11

.1111

=Nk +++

+IP

…(8)

- 3rd trial had chromosome length is 32 genes, Popula-

tion size was 20000 chromosome, number of genera-

tions was 50 and the best formula was equation (9)

( )( )

[ ]

).ln(7).(6.1

.

7

)./7( ./7

=Nk IPIPLn

IPIPLn IP −+++

…(9)

- 4th trial had chromosome length is 64 genes, Popula-

tion size was 40000 chromosome, number of genera-

tions was 50 and the best formula was equation (10)

11. )5(

P.I

)5(11L.L 11

=Nk

2

−

+

+

IP

Ln

γ

γ

…(10)

Accuracies and coefficient of determination (R2) of training

and validations sets for each one of the four trials are summa-

rized in table (2). Figure (6) represent the correlation between

the predicted (Nk) values using the equations (7),(8),(9),(10)

and the measured ones.

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a) Trial(1) – Eq. 7

b) Trial(2) – Eq. 8

c) Trial(3) – Eq. 9

d) Trial(4) – Eq. 10

Figure (6) Relation between the Predicted and Measured (Nk) values for Developed Correlations using (MEPX)

TABLE (2): SUMMARY OF ACCURACIES AND (R2) VALUES FOR

EQUATIONS (7),(8),(9),(10)

Trial No.

Code Length

Proposed

Formula

Accuracy % R2

Training

Validation

Total

Training

Validation

Total

1 8 Eq. (7) 93 93 93 0.44 0.54 0.52

2 16 Eq. (8) 94 94 94 0.53 0.63 0.58

3 32 Eq. (9) 93 94 94 0.53 0.58 0.56

4 64 Eq. (10) 95 95 95 0.63 0.76 0.67

The following points could be noted from table (2):

- Accuracies of all proposed formulas are ranged be-

tween 93% to 95%, while (R2) values are ranged be-

tween 0.44 to 0.76 which indicates fair fitting

- Equation (10) is the most accurate one and the only one

that used unit weight (γ) variable which indicates the

importance and the impact of this variable.

- None of the four proposed formulas contains water con-

tent (wc) which indicates that (Nk) doesn’t depend on

it.

6 CCONCLUSIONS

By comparing the summarized results in tables (1),(2), the fol-

lowing points could be noted:

- Although equation (4) is not the most accurate pro-

posed formula, but considering its simplicity, it is still

the most favorable one.

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- Formulas contains unit weight (γ) variable are more ac-

curate than others regardless the used software, this re-

flects the high correlations between (Nk) and (γ).

- None of the proposed formulas regardless the used

software contains water content (wc) which indicates

that (Nk) doesn’t depend on it.

- Although proposed formulas from (GN7) & (MEPX)

almost have same accuracies for same level of com-

plexity (code length), but coefficients of determination

(R2) of (GN7) formulas are higher than those of

(MEPX) which indicates the random crossover tech-

nique of (GN7) is more efficient than the one point

crossover technique of (MEPX).

- It is also noted that (MEPX) is almost twice faster

than (GN7), this may be because (MEPX) uses multi

threads.

REFERENCES

[1] Aas, Gunnar, Suzanne Lacasse, Tom Lunne, and Kaare Hoeg. "Use of

in situ tests for foundation design on clay." In Use of In Situ Tests in

Geotechnical Engineering, pp. 1-30. ASCE, (1986).

[2] Ahmed M. Ebid, Ezzat A. Fattah, Hossam E.A. Ali. "Applications of

genetic programming in geotechnical engineering. " DOI:

10.13140/RG.2.1.1967.9203 (2004).

[3] Boufrina, T., Bouafia, A., Panfilov, A. V., & Ter-Sarkisova, L. A.

"Numerical modeling of CPT in clay to evaluate bearing capacity for

shallow foundations." Проблемы и перспективы развития

сельского хозяйства и сельских территорий. (2014).

[4] Chen, C. "Evaluating un-drained shear strength of Klang clay from

Cone penetration test." In International Conference on In situ Meas-

urement of Soil Properties and Case Histories, In: Proceedings of the

International Conference on In Situ Measurement of Soil Properties

and Case Histories, Graduate Program, Parahyangan Catholic Uni-

versity, pp. 141-148. (2001).

[5] Gebreselassie, Berhane. "Experimental, Analytical and Numerical

Investigations of Excavations in Normally Consolidated Soft Soils."

(2003).

[6] Hossain, Md Imran. "Evaluation of Un-drained Shear Strength and

Soil Classification from Cone Penetration Test." (2018).

[7] Jörß, O. "Erfahrungen bei der Ermittlung von cu-Werten mit der

Hilfe von Drucksondierungen in bindigen Böden." Geotechnik 21,

no. 1 (1998): 26-27.

[8] Kim, Daehyeon, Younjin Shin, and Nayyar Siddiki. "Geotechnical

design based on CPT and PMT." (2010).

[9] Kim, Hobi, Monica Prezzi, and Rodrigo Salgado. "Use of dynamic

cone penetration and clegg hammer tests for quality control of road-

way compaction and construction." (2010).

[10] Kim, Kwang Kyun, Monica Prezzi, and Rodrigo Salgado. "Interpreta-

tion of cone penetration tests in cohesive soils." (2006).

[11] Koza, John R. Genetic programming II, automatic discovery of reus-

able subprograms. MIT Press. Cambridge, MA, (1992).

[12] Larsson, R., and M. Mulabdic. Piezocone tests in clay. Swedish Ge-

otechnical Institute, Linköping, Sweden. No. 42. Report, (1991).

[13] Lunne, Tom, and Arne Kleven. "Role of CPT in North Sea foundation

engineering". In Cone penetration testing and experience, pp. 76-

107. ASCE, (1981).

[14] Mayne, Paul W. Cone penetration testing. Vol. 368. Transportation

Research Board,(2007).

[15] Muduli, Pradyut Kumar, and Sarat Kumar Das. "CPT-based seismic

liquefaction potential evaluation using multi-gene genetic program-

ming approach". Indian Geotechnical Journal44, no. 1 (2014): 86-93.

[16] Otoko, George R., Isoteim Fubara-Manuel, Mike Igwagu, and Clem-

ent Edoh, "Empirical cone factor for estimation of un-drained shear

strength." Electronic Journal of Geotechnical Engineering 21 (2016):

6069-6076.

[17] Rémai, Zsolt. "Correlation of un-drained shear strength and CPT

resistance", Periodica Polytechnica Civil Engineering 57, no. 1 (2013):

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[18] Rezania, Mohammad, and Akbar A. Javadi, "A new genetic pro-

gramming model for predicting settlement of shallow foundations".

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[19] Robertson, P. K., and K. L. Cabal. "Guide to Cone Penetration Testing

for Geo-Environmental Engineering." (2008).

[20] Robertson, P. K. "Soil classification using the cone penetration test."

Canadian Geotechnical Journal 27, no. 1 (1990): 151-158.

APPENDIX: DATA SETS

Validation data set

L.L.

(%)

P.L.

(%)

P.I.

(%)

wc

(%)

γ

(t/m3) Nk

60

26

34

46

1.7

18.6

134

33

101

69

1.6

14.8

95

30

65

66

1.6

16.4

83

28

54

56

1.7

16.7

109

31

78

64

1.6

15.2

136

33

103

69

1.6

14.5

40

21

19

41

1.8

17.3

82

34

47

60

1.6

19.7

86

34

52

58

1.8

18.4

58

27

31

59

1.8

18.6

84

34

50

59

1.6

19.6

51

26

26

57

1.7

17.9

118

40

78

68

1.5

18.8

53

26

27

49

1.8

18.5

128

32

96

57

1.7

13.8

84

26

58

67

1.6

15.3

146

34

112

57

1.7

13.1

43

24

20

40

1.7

18.9

49

23

26

43

1.8

17.9

54

26

28

62

1.7

18.5

72

32

40

60

1.7

19.5

101

30

71

59

1.6

15.5

128

33

95

66

1.6

14.0

81

33

48

56

1.6

20.1

82

34

49

55

1.6

19.6

102

30

72

74

1.6

15.1

87

34

53

36

1.8

18.2

43

21

22

33

1.9

19.3

38

21

17

36

1.8

16.3

126

33

93

72

1.6

14.2

156

36

120

69

1.6

15.0

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Training data set

L.L.

(%)

P.L.

(%)

P.I.

(%)

wc

(%)

γ

(t/m3) Nk

94

30

64

53

1.7

16.9

132

34

98

57

1.7

13.7

93

29

63

36

1.8

13.4

109

31

79

66

1.6

15.3

136

34

103

83

1.5

14.3

76

28

48

35

1.8

16.6

100

30

70

47

1.7

15.5

90

29

61

63

1.6

16.4

141

34

107

56

1.7

12.7

92

36

56

59

1.9

18.4

41

21

20

41

1.7

17.8

72

27

45

52

1.7

16.6

73

31

43

58

1.8

18.8

93

35

58

57

1.7

18.5

97

38

59

61

1.6

19.8

80

28

52

52

1.7

16.5

140

33

107

59

1.6

13.2

142

34

108

59

1.6

12.8

95

30

66

57

1.7

16.9

120

32

88

59

1.6

14.1

53

26

27

60

1.7

18.4

72

29

43

64

1.7

18.3

112

42

71

68

1.6

20.2

41

21

20

49

1.7

17.5

53

25

28

49

1.7

18.6

72

29

42

58

1.6

18.3

53

25

28

48

1.8

17.3

48

24

24

52

1.7

18.6

68

29

40

51

1.7

18.8

49

22

28

26

1.8

17.0

51

26

26

46

1.7

19.1

65

29

35

47

1.7

20.2

72

31

41

59

1.6

21.3

131

33

98

62

1.6

14.8

41

22

20

44

1.7

17.2

48

25

24

49

1.7

18.8

117

41

77

58

1.6

18.1

77

31

46

45

1.7

18.8

97

35

62

57

1.6

18.9

93

29

64

59

1.6

16.1

111

32

79

72

1.6

15.4

104

30

74

63

1.6

14.9

91

29

62

70

1.6

16.7

92

29

63

70

1.6

16.2

107

31

76

76

1.7

15.4

57

25

32

67

1.6

17.8

78

32

46

37

1.8

18.3

157

39

118

54

1.7

13.4

72

29

43

38

1.8

17.9

75

30

45

37

1.8

18.4

85

34

50

36

1.8

18.2

56

24

31

41

1.7

17.8

70

28

41

66

1.6

18.8

48

20

28

44

1.8

16.1

65

27

39

33

1.9

17.9

87

30

57

37

1.8

18.8

48

22

25

50

1.6

17.5

56

26

31

47

1.7

17.3

73

31

46

38

1.8

20.4

122

32

90

50

1.7

14.3

88

29

59

67

1.5

15.2

104

31

73

63

1.6

15.1

86

29

57

69

1.6

16.1

111

32

80

69

1.6

15.6

122

32

90

71

1.6

14.5

101

31

70

73

1.6

17.1

105

31

74

55

1.7

15.5

91

29

62

69

1.6

15.8

123

32

91

68

1.6

14.0

101

29

72

62

1.6

15.0

117

32

86

70

1.6

13.6

IJSER