Prediction of Uplift Capacity Using Genetic Programming

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

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

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.

Eleventh Structura Ain Sh The stanpopulatifitting faccordindestroyecrossovcycle staccurac GENET GP is onto the GconceptGP tech

First, fisurface minimiz SS

International al & Geotechn

17-19 May 20hams Univers ndard GA pion of solut

function, anng to thereed. Finally,

ver operator tarted againy is accepte

TIC PROG

ne of the moGA, which ct of GP is tohnique. In or

tness evaluis represen

zed for best

SE = [ GP

Colloquium onical Engineeri005 sity Cairo

procedure ctions are ge evaluating e fitness ar producing on the surv

n by evaluated (Michale

GRAMMIN

ost recent dcan be defino find the berder to use G

Fig

uation methonted herein t fitting surf

P prediction

on ing

consists of fenerated anof the fitne

re arrangednew solutio

viving solutting the fitnewicz 1992)

NG (GP)

developed knned as Mulest fitting suGA, the pre

gure 1: Flow

od (functionby the sum

face. The S

n - Target ou

four main snd encoded ess of each d and the uons to keeptions. Mutatness of the n).

nowledge-bltivariable Iurface in hyevious steps

w chart for

n to be optmmation ofSE is calcul

utput ]2

steps as depin genetic solution is unsuitable (p the population operatonew solutio

base techniqInterpolationyper-space fs will be fol

GA procedu

timized) haf squared elated by

picted in Figform. Secoconducted. (least qualiation size coor may be ans and so o

ques and it in Procedurefor a certainlowed, (Ko

ure

s to be deteerrors (SSE

(1)

19 2005 -

g.1. First, aond, using aThen, the s

ified) solutonstant by aapplied and on until the

is next devee (MLP). Tn given poinza, 1994).

ermined. FiE), which ha

17-9

a random a certain solutions ions are applying then the solution

elopment The basic nts using

itness of as to be

Eleventh Structura Ain Sh Then, c(i.e., dethe chrohave to

1. Any surfaces

2. Any such as

3. The ma polyno

4. The fhas one Therefousing thgraphica Using tcomplexrepresen

As showOperatogenes. Tgenes. Tgenes. After cooperatioto repla(in varicompon On thefeatureswere tw(1996), two paroperatorsame op


17-19 May 20hams Univers

conducting eterminationomosome) tbe consider

set of pois with defercomplicated(=, +, -, x, /

most simpleomial equatfive basic oinput and o

ore, to creathe aforemeally represe

the previousxity of the fnting formu

wn in Fig. 3ors part repThe variablTherefore, t

onducting eons (crossovace some raniables part)nents of the e other hans from theirwo ways to

is called twrts of the crs part of operation wil


the most n of numberto representred:

ints in certrent accuracd equation /, sin, cos.e case is to tion. perators hav

one output.

te a formulentioned fivented in Fig

s operators,formula, the

ulas in a tree

Fig

3, each chroresents all

les part reprthe total nu

encoding over and mutndomly sele). Oppositetwo parts o

nd, the newr parents.

o apply the wo-point crochromosomne parent wll be applied

on ing

important r of genes ft a formula

ain domaincy dependincan be con

etc. ). use only th

ve two inpu

la in genetive basic op. 2.

, any polyne more levee form is sh

gure 2: The

omosome cothe tree exresent only umber of ge

f chromosotation) haveected genesely, the croof the chromw chromosoThat meanscrossover,

ossover. In me independwill be swapd on the var

step in Gfor each chin genetic f

n of hyper ng on the costructed fro

he five basic

uts and one

ic form, a perators (i.e

nomial can els of tree aown in Fig.

five basic o

onsists of twxcept the lev

the level 0enes on eve

ome procede to be perfs with randoossover pro

mosome musomes geners that they

the first mthis techniqdently. Thupped with thriables part t

GP which ihromosome,form. By do

space can mplexity of

om certain b

c operators

output exce

binary treee. =, +, -,

be represenare needed t. 3.

operators in

wo parts; opvel 0 and i0 of the treeery chromo

dure, the prformed. Muom operatorocedure is st not be mirated durincannot be

method, whque, crossovus, a certaiheir image too as show

is encoding, and arrangoing so, som

be represef these surfabasic functi

(=, +, -, x,

ept the oper

e structure w x, /). Th

nted in a trto represent

GP

perators partit consists oe and consisome is ( 2

rocedures toutation is ver (in operatonot that si

ixed during ng crossove

generated rhich is suggver procedun number from the ot

wn in Fig. 4.

19 2005 -

g of chromgement of gme importan

ented by maces. ons (operat

, /) to const

rator (=) wh

will be conhis tree stru

ree form. Tt it. An exam

t and variabof (2 No. of lists of (2 No2 No. of levels

o apply thery simple oors part) or imple, becacrossover.

er must havrandomly. gested by Rure is applie

of genes fther parent,

17-9

mosomes genes on nt points

many

tors)

truct

hich

nstructed ucture is

he more mple for

bles part. levels - 1) o. of levels) + 1 - 1)

e genetic operation

variable ause the

ve some So there Riccardo ed on the from the , and the

Eleventh Structura Ain Sh

The secgeneratisurvivinselectedso do th



Figu

cond way toion of chrong chromosd randomly he next gene


ure 3: Math

Fi

o apply croomosomes isomes. In from the fir

es. This proc

on ing

ematical an

gure 4: Two

ossover wass generatedother word

rst genes ofcess is depi

nd genetic re

o-point cros

s proposed d by randomds, the firsf the whole icted in Fig.

epresentatio

ssover meth

by the authmly selectint gene of tsurviving s. 5 for three

on of binary

hod

hors. In thing each genthe new chet of parent parents and

19 2005 -

y tree

is techniquene from thehromosome t chromosomd one child.

17-9

e, a new e similar

will be mes, and .


In randocan be cbe half chosen Practicafrom thminimaconversAfter thgeneratiSSE). Aminimusurface accurac1996). PREDI AI technwith nousing Ggenerateused to The genpower foundatincludesincludes(C), tanThe evadiameteconstanbulk cas



om selectionchosen. On

f of the popin this res

ally, the fashe populatioa. On othersion will be he previouion. GeneraAfter a certum SSE). A

for this nuy of this su

ICTION OF

niques had oisy and inaGP technique the formuevaluate thenerating datransmissio

tion depths s (C-ss footing w

ngent of intealuation dataer of the font and equal se.


Figure

n crossoverthe other h

pulation. Fosearch to cstest converon. For lessr hand, if thvery slow. s three ste

ation after gtain numberAt this stagumber of trurface is no

F UPLIFT

been wildlyaccurate date, the result

ulas, where ae generated

ata set conson line, fou

are ranged oil, -soil, idth (B) and

ernal frictiona set consis

ooting is ranto (1.00m)

on ing

e 5: Random

r technique and, in twoor this reascarry out thrsion occurs number o

he number o

eps, GA cageneration, r of genera

ge, the corrree levels (wt enough, l

CAPACIT

y used in geta. In order ts of (31) panother set

d formulas.sists of (31undation dibetween (C- soil) id depth (D)n angle (tansts of (4) punged betwe). All the se

m selection

number of -point crossson, the ranhe crossovers when theof survivorsof survivors

an be applthe fitness

ations the firesponding which meanarger numb

TY USING

eotechnical to predict

pullout testsof (4) pullo

) pullout fiimensions 1.05 to 1.80in both bulk) beside thren()) and effullout laboraeen (0.40 toets had been

crossover m

survivors frsover technndom selecer operatione number os the solutis is more th

lied on thewill increasitness will chromosomns for this

ber of tree l

(GP)

field recentthe uplift c

s carried outout tests car

ield tests oare ranged0m). The sk and submee soil para

ffective unit atory tests oo 1.00m), wn carried ou

method

from one geique numbe

ction crosson in the d

of survivorson may be

han 50% of

e first and se (which msettle at a

mes represedegree of

levels must

tly because capacity of t by (Matsu

rried out by

n square fod between oil conditio

merged caseameters whi

weight of son axi-symmwhere the fut using the

19 2005 -

neration to er of survivoover techniqdeveloped ss equals to trapped in

f the popula

randomly means a deccertain valu

ent the moscomplexitybe used (R

of its abilitshallow fou

uo, 1967) is(Sarac, 197

ootings of (1.20 to 2

on of these s. Each datich are the csoil (). metric footinfoundation e same (C-

17-9

the next ors must que was software.

30-40% n a local ation, the

created crease in ue (with st fitting y). If the Riccardo,

ty to dial undation s used to 75) were

66 K.V. 2.00m ),

data set ta record cohesion

ngs. The depth is soil) in

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%).

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%).





on ing

19 2005 -

17-9

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





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.





on ing

19 2005 -

17-9


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


Ppre

dict

ed c

apac

ity (t

on)


y = 0.2812xR2 = -0.9362

0

10

20

30

40

50

60

0 10 20 30 40 50 60


Ppre

dict

ed c

apac

ity (t

on)






on ing

19 2005 -

17-9


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

Documents

Prediction of Uplift Capacity Using Genetic Programming