prediction of uplift capacity using genetic programming

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

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

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

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

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

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 (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 BB Ln B B C

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+ −+( ).

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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 (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 BB B D C

LnB D C

B D= + + + −

− −+( ).

. . . . tan( ).( )( . tan( )) .( )

( )2 2

22

2γ ϕϕ

γ ........ (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 (ϕ-soil) (up to 25%) is higher than in (c-soil) (about 10%).





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Eleventh International Colloquium on المؤتمر الدولى الحادى عشر للھندسة Structural & Geotechnical Engineering االنشائية و الجيوتقنية 17-19 May 2005 17-19 2005مايو Ain Shams University – Cairo القاھرة -جامعة عين شمس 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 BB D

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11

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2γ γ ϕ γ ϕ γ ϕ γ

γB B C B

C.( 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 (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 = 1.0045xR2 = 0.9415

0

10

20

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





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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

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y = 0.2812xR2 = -0.9362

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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