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International Journal of Scientific & Engineering Research Volume 10, Issue 3, March-2019 613
ISSN 2229-5518
IJSER © 2019
http://www.ijser.org
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
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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
