-
Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo -
Prediction of Uplift Capacity For Shallow Foundations
Using Genetic Programming
Ezzat A. Fattah1, Hossam E.A. Ali2, Ahmed M. Ebid3 ABSTRACT In
most geotechnical problems, it is too difficult to predict soil and
structural behavior accurately, because of the large variation in
soil parameters and the assumptions of numerical solutions. But
recently many geotechnical problems are solved using Artificial
Intelligence (AI) techniques, by presenting new solutions or
developing existing ones. Genetic Programming, (GP), is one of the
most recently developed (AI) techniques based on Genetic Algorithm
(GA) technique. In this research, GP technique is utilized to
develop prediction criteria for uplift capacity of shallow
foundations using collected historical records. The uplift capacity
formula is developed using special software written by the authors
in Visual C++ language. The accuracy of the developed formula was
also compared with earlier prediction methods. Keywords: Uplift
Capacity, GA, GP, and AI
INTRODUCTION Shallow footing ( Pad & Chimney ) is the most
common type of uplift foundation. For wide range of soil types, it
is the easiest, preferred and most economic type of uplift
foundation. There are several methods to design the pad &
chimney footing, these methods can be classified into four groups
based on the concept of design, these groups are Soil Load Methods,
Earth Pressure Methods, Shearing Methods and Constitutive Laws
Methods Soil load methods In these methods the soil resistance to
foundation extraction is represented by means of the weight of a
resisting mass for which it is assumed that it moves together with
the foundation due to the action of the pulling force. The shape
and magnitude of the resisting soil mass have been determined for
calculation by the shape of foundation slab and soil
characteristics. The problem of determination of rupture force with
this method is reduced to the selection rupture surface shape. This
shape is usually given as a function of the type and
characteristics of the soil such as shear characteristics, density,
consistency,etc. Earth cone method (1958): This is the most common
method which has been adopted for design. In Japan the standard
specification for design which was published in 1958 by JEC (
Japanese Electrotechnical Committee ) specified the earth cone
method for the calculation of the uplift resistance. In this method
the ultimate uplift resistance is assumed to be equal to the sum of
the dead weight of the footing and the soil mass contained the
truncated pyramid or cone with the bottom of footing slab as base,
the rupture surface is considered as straight line inclines with
the vertical by certain angle (). In this method the knowledge of
soil mechanics is not taken
1 Prof. of soil mechanics, Ain Shams University, Cairo, Egypt 2
Teacher of soil mechanics, Ain Shams University, Cairo, Egypt 3
Graduate student, Ain Shams University, Cairo, Egypt
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Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - into consideration and therefore the actual
important phenomenon of shear failure in earth body is neglected.
Earth pressure methods In earth pressure methods it is assumed that
the rupture surfaces are vertical , i.e. the soil mass which is
pulling together with the foundation has the shape of upright prism
or cylinder whose cross section is the same as the foundation slab.
The pulling force is determined by the weight of foundation and
soil mass and by the friction along its lateral area. Friction
forces depend on lateral pressure, so the determination of the
intensity of lateral pressure ( for which it is assumed that they
vary linearly depending on the depth ) is one of the basic
problems. Mors Method (1959): In 1959, Mors suggested that the
lateral pressure at the anchor slab level has the value of a
passive earth pressure in accordance with the Ranking equilibrium
theory. The value of the earth pressure in the region between the
ground surface and the anchor slab is varying linearly with the
depth. The fundamental defects of the earth pressure methods (
alike the soil load methods ) are that the shear failure in soil
mass is not taken into consideration and the effect of cohesion is
not considered in the design. Shearing strength methods These
Methods were developed on the basis of experimental and theoretical
results. According to the concept of these methods the ultimate
uplift capacity of the foundation is determined by the weight of
foundation and soil mass within the rupture surface and by the
shearing force (including the friction and cohesion) along that
rupture surface. The shape of the rupture surface is varied from
one method to another according to the experimental and theoretical
bases of the method. Some methods simply assumed the rupture
surface as straight line like Shichiri (1943), and Modified Morse
(1959), methods. On the other hand some methods are very
complicated such as Matsuo (1968), Method which assume that the
rupture surface is a combination of logarithmic spiral curve and
straight line. Shichiri Method (1943): In 1943, Shichiri developed
a method to estimate the uplift capacity of foundation based on
experimental and theoretical results. He suggested a shearing force
acting along a vertical rupture surface. This force expressed in
terms of soil cohesion, angle of internal friction and coefficient
of earth pressure at rest. Sarac method (1961): The method of Sarac
is based on a series of pullout tests in different soils and
variable depths using a circular anchor plate. Sarac noted that the
rupture surface had the shape of convex curve whose tangent at the
contact point with the anchor slab was approximately vertical while
it crossed the ground surface with an angle of (45-/2) . He
approximated the rupture surface by means of a logarithmic spiral
in general form. The ultimate uplift resistance is calculated as
the some of the dead weights of the footing and the soil enclosed
with the rupture surface and the vertical component of the shear
resistance along that surface. Matsuo method (1967,1968): In 1967,
Matsuo developed his method assuming that the rupture surface
consists of a logarithmic spiral curve and its tangential straight
line . He based his assumption on a series of experimental tests on
a circular plate anchor model. For practical design, this method is
very complicated to be applied, so based on the request of IEEJ (
Institute of Electrical Engineering of Japan ) Matsuo simplified
his method in 1968. The estimated loss of accuracy due to this
simplification is about 3% of the ultimate uplift capacity. To
apply his method on the square foots Matsuo suggested to use the
equivalent area concept which means to replace the square footing
with circular foot having the same area taking into consideration
that the perimeter of the square foot is about 10% greater than
the
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Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - perimeter of its equivalent circular foot. So he
increases the uplift capacity by 10%. A series of field tests done
by Matsuo during a 66 kV transmission power line using a square
foots proved that the equivalent area concept is valid to be used
with his method. Constitutive laws methods Gopal and Saran method
(1987): In 1987, Gopal and Sararn developed an analytical method to
predict the Uplift-Displacement characteristic of shallow
foundation in (C-) soil using non-linear constitutive laws. The
method based on assumption that the foundation is rigid and having
a negligible weight, and buried at shallow depth in homogeneous
isotropic medium of semi-infinite extent ( plan strain model ). In
this method the Uplift - displacement curve is divided into four
stages: Stage (1): (Applied load less than Critical load ) The
shear parameters (C,) are considered fully mobilized at the footing
base and have zero value at some level below the ground surface in
linear relationship. Stage (2): (Applied load equal to Critical
load ) The shear parameters (C,) are considered fully mobilized at
the footing base and have zero value at the ground surface level in
linear relationship. Stage (3): (Applied load more than Critical
load and less than the pullout load ) The shear parameters (C,) are
considered fully mobilized at the footing base and partially
mobilized at ground surface level in linear relationship. Stage
(4): (Applied load equal to the pullout load ) The shear parameters
(C,) are considered fully mobilized at the footing base and fully
mobilized at ground surface level in linear relationship. The
physical meaning of the developed equation is similar to Shichiri
method but the Ko (Lateral coefficient at rest) factor is replaced
by (1.0). GENETIC ALGORITHM (GA) The Genetic Algorithm (GA) is an
Artificial Intelligence (AI) technique, based on simulating the
natural reproduction process, following the well-known Darwin's
rule "The fittest survive". The natural selection theory for Darwin
assumes that, for a certain population, there is always some
differences between its members. These differences make some
members more suitable for the surrounding conditions than the
others. Accordingly, they have better chances to survive and
reproduce a next generation with enhanced properties. Generation
after generation most of the population will have these suitable
properties, meanwhile the unsuitable members will eventually be
diminished. In other words, during the reproduction process, the
natural selection increases the fitness of the population, which
means that this population is developed to suite the surrounding
conditions. In the natural reproduction process, certain sequence
of (DNA) characters represent properties of members, each character
is called "Gene", and every set of genes is called "Chromosome"
(Michalewicz, 1992). The theory of biological reproduction process
was first simulated mathematically by John Holland, 1975, where
genes and chromosomes are replaced by a parameters and solutions
respectively, and the surrounding conditions are represented by a
fitting function. Hence, according to Darwin's rule, during the
reproduction process the population is developed to suite the
fitting function (Holland 1975). The most important advantage of GA
technique is its generality and its applicability to very wide
range of engineering problems. This is because GA technique is not
depending on type of data. Encoding the problem parameters in
genetic form is the first and the most important step in the GA
solution.
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19 2005 -
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17-9
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Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - In order to compare predicted and experimental
capacities, the concept of equivalent area was used to find the
pullout load of the equivalent square footing with a width equal to
the diameter of the axi-symmetric footing. During adapted research
program to predict the uplift capacity of shallow foundation using
GP technique, the research program had been conducted using the
last version of GP software. The complexity of the generating
formulas increases gradually from three levels in the first trial
and up to six levels in the last trail. Each trail had been
conducted until the solution error settled at it's minimum value
(which corresponding to the maximum accuracy ) or until the
solution exceeded the practical limits (when the solution takes too
much time). The first three trails had been conducted using only
five variables ( B, D, C, tan(), ) which are footing width in
meters, footing depth in meters, soil cohesion in tons per square
meters , tangent of internal friction angle of soil and effective
unit weight of soil in tons per cubic meters respectively. Where
the last two trails had been conducted using additional five
variables with constant values which are (1, 2, 3, 5, 11). A
summary of the research program and its results are shown in table
(1). The generated formula of each trail is represented in two
charts, the first chart represent the relationship between
predicted and experimental capacities for both generating and
evaluation data sets, where the second chart shows the effect of
shallowness ratio (B/D) and type of soil on the accuracy of
prediction. The average relative error could be calculated from the
following formula:
Average Relative Error % = P P
P ncaln exp
exp1
100 ....... (1)
Accuracy % = 100- Average Relative Error % ....... (2)
Where Pcal , Pexp are the predicted and the experimental uplift
capacities respectively. The soil type is represented by the ratio
between cohesion and friction shear strength ( C / .D.tan()), for
pure (-soil) this ratio is equal to zero and for pure (C-soil) the
ratio yields to infinity. Trial No. (1): Starting with a simple
trail which has only three levels using generating data set
consists of five variables ( B, D, C, tan(), ) produced formula (3)
which corresponding to SSE (Summation of Square Error) equals to
(634). Applying this formula on the evaluation data set produced
SSE equals to (14). The corresponding accuracy of the formula in
case of generating, evaluation and total data sets are (82.30%)
(75.78%) and (81.70%) respectively.
P B C D B D= + + +2 .( ).( . tan( )) ......(3)
The graphical representation of predicated capacities of both
generating and evaluation data sets are shown in fig.(6), the graph
shows that the slope of the best fitting line is (0.9858 1.00) and
the coefficient of determination (R2 =0.8198) which indicate the
good correlation between the predicted and experimental capacities.
Where the upper chart in fig. (7) shows that the footing
shallowness (B/D) has no significant effect on the prediction
accuracy, on the other hand, the lower chart indicates that the
percentage of error in the (-soil) (up to 40%) is higher than in
(c-soil) (about 20%).
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Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - Trial No. (2): Continuing the research program
with the second trail which has four levels using generating data
set consists of five variables ( B, D, C, tan(), ) produced formula
(4) which corresponding to SSE (Summation of Square Error) equals
to (501). Applying this formula on the evaluation data set produced
SSE equals to (26). The corresponding accuracy of the formula in
case of generating, evaluation and total data sets are (84.26%)
(67.00%) and (83.50%) respectively.
P e C B C D eB D e= + + ++( ) tan( )( ) .( tan( )).2 ......(4)
The graphical representation of predicated capacities of both
generating and evaluation data sets are shown in fig.(8), the graph
shows that the slope of the best fitting line is (0.9896 1.00) and
the coefficient of determination (R2 =0.8659) which indicate the
good correlation between the predicted and experimental capacities.
Where the upper chart in fig. (9) shows that the footing
shallowness (B/D) has no significant effect on the prediction
accuracy, on the other hand, the lower chart indicates that the
percentage of error in the (-soil) (up to 60%) is higher than in
(c-soil) (about 10%). Trial No. (3): The conducting of the third
trail which has five levels using generating data set consists of
five variables ( B, D, C, tan(), ) generates formula (5) which
corresponding to SSE (Summation of Square Error) equals to (238).
Applying this formula on the evaluation data set produced SSE
equals to (37). The corresponding accuracy of the formula in case
of generating, evaluation and total data sets are (89.16%) (60.57%)
and (88.08%) respectively.
P e D C B C BB Ln B B C
LnB D C
B D= + + + + +( )
.. . . . tan( ).( )
( . tan( )) .( ( ) )( )
2 22
2 .... (5) The graphical representation of predicated capacities
of both generating and evaluation data sets are shown in fig.(10),
the graph shows that the slope of the best fitting line is (1.0045
1.00) and the coefficient of determination (R2 =0.9415) which
indicate the very good correlation between the predicted and
experimental capacities. Where the upper chart in fig.(11) shows
that the prediction accuracy of deep footings is worst than shallow
ones , on the other hand, the lower chart indicates that the
percentage of error in the (-soil) (up to 30%) is higher than in
(c-soil) (about 10%). Trial No. (4): The forth trail five levels
just like the third one but using generating data set consists of
ten variables ( B, D, C, tan(), ,1,2,3,5,11). Conducting of this
trial produced formula (6) which corresponding to SSE (Summation of
Square Error) equals to (226). Applying this formula on the
evaluation data set produced SSE equals to (35). The corresponding
accuracy of the formula in case of generating, evaluation and total
data sets are (89.44%) (61.57%) and (88.38%) respectively.
P e D C B C BB B D C
LnB D C
B D= + + + +( )
.. . . . tan( ).( )
( . tan( )) .( )( )
2 22
22 ........ (6)
The graphical representation of predicated capacities of both
generating and evaluation data sets are shown in fig.(12), the
graph shows that the slope of the best fitting line is (1.0033
1.00) and the coefficient of determination (R2 =0.9445) which
indicate the very good correlation between the predicted and
experimental capacities. Where the upper chart in fig. (13) shows
that the prediction accuracy of deep footings is worst than shallow
ones , on the other hand, the lower chart indicates that the
percentage of error in the (-soil) (up to 25%) is higher than in
(c-soil) (about 10%).
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Eleventh Structura Ain Sh
International al & Geotechn
17-19 May 20hams Univers
Colloquium onical Engineeri005 sity Cairo
on ing
19 2005 -
17-9
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Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - Trial No. (5): The last trial in the research
program has six levels using generating data set consists of ten
variables ( B, D, C, tan(), ,1,2,3,5,11). Conducting of this trial
generated formula (7) which corresponding to SSE (Summation of
Square Error) equals to (184). Applying this formula on the
evaluation data set produced SSE equals to (4). The corresponding
accuracy of the formula in case of generating, evaluation and total
data sets are (90.46%) (87.24%) and (90.15%) respectively.
P B D C B D D C D D BB D= + + + + + + + + +( . tan( )).( . ) ( )
( )
22 32 3
11
+ + + + 2 2 2
2 B B C BC
.( tan( )) . . . . tan( ). ( ). tan( ) .... (7)
The graphical representation of predicated capacities of both
generating and evaluation data sets are shown in fig.(14), the
graph shows that the slope of the best fitting line is (0.997 1.00)
and the coefficient of determination (R2 =0.9502) which indicate an
excellent correlation between the predicted and experimental
capacities. Where the upper chart in fig. (15) shows that the
footing shallowness (B/D) has no significant effect on the
prediction accuracy, on the other hand, the lower chart indicates
that the percentage of error in the (-soil) (up to 30%) is higher
than in (c-soil) (about 5%).
Figure 10: Representation of the generated formula - trial no.
(3)
Figure 11: Effect of B/D and type of soil on the prediction
accuracy For trail no. (3)
y = 1.0045xR2 = 0.9415
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Experimental capacity (ton)
Ppre
dict
ed c
apac
ity (t
on)
Generating setValidation setBest Fitting
0
20
40
60
80
0 2 4 6 8 10 12 14 16C / D. .tan( )
Erro
r %
Generating set
Validation set
0
20
40
60
80
0.00 0.50 1.00 1.50 2.00B/D
Erro
r %
Generating set
Validation set
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Eleventh Structura Ain Sh
International al & Geotechn
17-19 May 20hams Univers
Colloquium onical Engineeri005 sity Cairo
on ing
19 2005 -
17-9
-
Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo - SUMMARY OF RESULTS The results of the research
program are summarized in table (1), which shows each trail with
its the number of levels and input variables, in addition to its
accuracy percentage and (SSE) value in case of generating,
validation and total data sets. From the summary table, it could be
noted that: a ) For the same data set the accuracy of the generated
formula increases with its complexity ( no. of levels ).
b ) Using constants in the data sets saves the extra levels that
will be used to create these constants, hence they make the
conversion faster.
c ) For second, third and forth trials, it is noticed that the
accuracy of the validation data set is significantly lower than
that of the generating data set, that means that these trials
produced good estimations in case of generating data set and poor
estimations in case of the validation data set. In other words,
these three trails generated a "Memorized" formulas not
"Generalized" formulas.
d ) In spite of the simplicity of first trail formula, it
produced a good estimations in both cases, and due to its
simplicity, it could be used in preliminary designs or rough manual
checking.
e ) The formula generated during the last trial is accurate
enough to be applied in design, the results indicates its validity
in both generating and validation data sets, hens, its generality
and ability to be applied in the mentioned ranges of variables.
COMPARISON WITH EARLIER PREDICTION METHODS In order to compare
the generated formulas with earlier prediction methods, the
capacities of both generating and validation data sets arfe
calculated using six well known methods which are (Earth cone
1958), (Morse 1959), (Shichiri 1943), (Gopal 1987), (Sarac 1961)
and (Matsuo 1967). Figures from (3-23) to (3-27) represent the
relationship between predicted and experimental capacities for both
generating and evaluation data sets for each method of these six
methods. For (Matsuo 1967) method, the chart in Fig.(16) shows that
the slope of the best fitting line is (0.8982) and the coefficient
of determination (R2 =0.778) which indicate a good correlation and
also means that the predicted capacities is about 90% the
experimental ones.
Where Fig.(17) which represents (Sarac 1961) method and Fig.(18)
which represents (Shichiri 1943) method, indicate a fair
correlation and also show that the predicted capacities is about
60-66% the experimental ones. For (Sarac 1961) the slope of the
best fitting line is (0.6566) and the coefficient of determination
(R2 =0.7388) and for (Shichiri 1943) the slope of the best fitting
line is (0.6134) and the coefficient of determination (R2 =0.7402).
For (Gopal 1987) method, the chart in Fig.(19) shows that the slope
of the best fitting line is (0.9863) and the coefficient of
determination (R2 =0.3306) which indicate a poor correlation and
also means that the predicted capacities is almost the same as the
experimental ones. Where Fig.(20) which represents (Morse 1959)
method and Fig.(21) which represents (Earth cone 1958) method,
indicate no correlation and also show a poor relationship between
predicted and experimental capacities.
-
Eleventh Structura Ain Sh
International al & Geotechn
17-19 May 20hams Univers
Colloquium onical Engineeri005 sity Cairo
on ing
19 2005 -
17-9
-
Eleventh Structura Ain Sh
International al & Geotechn
17-19 May 20hams Univers
Colloquium onical Engineeri005 sity Cairo
on ing
19 2005 -
17-9
-
Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo -
Figure 20: Representation of Morse formula - 1959
Figure 21: Representation of Earth cone formula - 1958
The results of the comparison are summarized in table (2), which
shows the method
with its input variables in addition to its accuracy percentage
and (SSE) value in case of generating, validation and total data
sets. From the summary table, it could be noted that:
a ) Earth cone and Morse methods have poor accuracy due to the
neglecting the soil cohesion. where the other earlier predicting
methods shows a fair to good accuracy according to their
complexity.
b ) In spite of the simplicity of trail (1) formula, it shows an
accuracy better than the complicated earlier predicting
methods.
c ) The best predicting method is trail (5) formula, which shows
an excellent accuracy ( about 90%).
y = 0.6377xR2 = -0.677
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Experimental capacity (ton)
Ppre
dict
ed c
apac
ity (t
on)
Generating setValidation setBest Fitting
y = 0.2812xR2 = -0.9362
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Experimental capacity (ton)
Ppre
dict
ed c
apac
ity (t
on)
Generating setValidation setBest Fitting
-
Eleventh Structura Ain Sh
International al & Geotechn
17-19 May 20hams Univers
Colloquium onical Engineeri005 sity Cairo
on ing
19 2005 -
17-9
-
Eleventh International Colloquium on Structural &
Geotechnical Engineering 17-19 May 2005 17-19 2005 Ain Shams
University Cairo -
REFERENCES
1 Ayman Lotfy, (1992). Uplift Resistance of Shallow Foundation,
M.S. Ain Shams University.
2 Dzevad Sarac, (1975). Bearing Capacity of Anchor Foundation as
Loaded by Vertical Force, institute of geotechnics and Foundation
engineering, Sarajevo.
3 Egyptian Ministry of Electricity and Energy, (1981). Design
Standard of Transmission Steel towers, Chapters 2, 12.
4 Holland, J. (1975). "Adaptation in Natural and Artificial
Systems," Ann Arbor, MI, University of Michigan Press.
5 Institute of Electrical Engineering of Japan, (1958). Design
Standard of Transmission Steel towers, JEC.127, pp. 35.39
6 Koza, J. R., (1994). "Genetic Programming-2," MIT Press,
Cambridge, MA.
7 Matsuo M., (1967). Study on the Uplift Resistance of Footing
I, Soils and Foundations, Vol. 7, No. 4, pp. 1.37.
8 Matsuo M., (1968). Study on the Uplift Resistance of Footing
II , Soils and Foundations, Vol. 8, No. 1, pp. 18.48.
9 Michalewicz, Z. (1992)."Genetic Algorithms+Data Structure =
Evaluation Programs", Springer-Verlag Berlin Heidelberg, New
York.
10 Riccardo, P. (1996). "Introduction To Evolutionary
Computation," Collection of Lectures, School of Computer Science,
University of Birmingham, UK.
11 Saran S. and Rajan G. (1987). Soil Anchors and Constitutive
Lows , Journal of Geotechnical Engineering Division, ASCE, Vol.
112, GT(12), pp. 1084.1099