Prediction of uplift capacity using genetic programming

Eleventh International Colloquium on ‫للھندسة‬ ‫عشر‬ ‫الحادى‬ ‫الدولى‬ ‫المؤتمر‬
Structural & Geotechnical Engineering ‫الجيوتقنية‬ ‫و‬ ‫االنشائية‬
17-19 May 2005 17-19‫مايو‬2005
Ain Shams University – Cairo ‫شمس‬ ‫عين‬ ‫جامعة‬-‫القاھرة‬
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 (β). 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

17-19 May 2005 17-19‫مايو‬2005
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-ϕ/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

17-19 May 2005 17-19‫مايو‬2005
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-ϕ) 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,ϕ) 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,ϕ) 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,ϕ) 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,ϕ) 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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‫عشر‬ ‫الحادى‬ ‫ولى‬
‫الجيوتقنية‬ ‫و‬ ‫ئية‬
19‫مايو‬2005
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19‫مايو‬2005
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19‫مايو‬2005
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19‫مايو‬2005
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17-9
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17-19 May 2005 17-19‫مايو‬2005
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(ϕ), γ) 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 n
caln −
×∑
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 / γ.D.tan(ϕ)),
for pure (ϕ-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(ϕ), γ) 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( ))γ ϕ ......(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 (ϕ-soil) (up to 40%) is
higher than in (c-soil) (about 20%).

17-19 May 2005 17-19‫مايو‬2005
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(ϕ), γ) 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 γ ϕ
ϕ
......(4)
=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 (ϕ-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(ϕ), γ) 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 B
B Ln B B C
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B D
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ϕ
γ
.... (5)
=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 (ϕ-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(ϕ), γ,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 B
B B D C
Ln
B D C
B D
= + + + −
− −+( )
.
. . . .tan( ).( )
( .tan( )) .( )
( )
2 2
2
2
2
γ ϕ
ϕ
γ
........ (6)
=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 (ϕ-soil) (up to
25%) is higher than in (c-soil) (about 10%).

Eleventh
Structura
Ain Sh
International
al & Geotechn
17-19 May 20
hams Univers
Colloquium o
nical Engineeri
005
sity – Cairo
on
ing
‫لقاھرة‬
19‫مايو‬2005
17-9
‫جامعة‬

17-19 May 2005 17-19‫مايو‬2005
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(ϕ), γ,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 B
B D
= + + + + + + + +
−
+( . tan( )).( . ) ( )
( )
ϕ
γ γ
2
2 3
2 3
11
− − + + + + −
⎛
⎝
⎜
⎞
⎠
⎟2 2 2
2
γ γ ϕ γ ϕ γ ϕ γ
γ
B B C B
C
.( tan( )) . . . .tan( ). ( ).tan( ) .... (7)
=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 (ϕ-
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 = 1.0045x
R2
= 0.9415
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Experimental capacity (ton)
Ppredictedcapacity(ton)
Generating set
Validation set
Best Fitting
0
20
40
60
80
0 2 4 6 8 10 12 14 16
C / D.γ .tan(φ )
Error%
Generating set
Validation set
0
20
40
60
80
0.00 0.50 1.00 1.50 2.00
B/D
Error%
Generating set
Validation set

17-19 May 2005 17-19‫مايو‬2005
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.

17-19 May 2005 17-19‫مايو‬2005
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 = 0.6377x
R2
= -0.677
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Generating set
Validation set
Best Fitting
y = 0.2812x
R2
= -0.9362
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Generating set
Validation set
Best Fitting

17-19 May 2005 17-19‫مايو‬2005
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

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