Empirical modeling of plate load test moduli of soil via gene expression programming

Computers and Geotechnics 38 (2011) 281–286

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Technical Communication

Empirical modeling of plate load test moduli of soil via gene expression programming Ali Mollahasani a, Amir Hossein Alavi b,⇑, Amir Hossein Gandomi c a

Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran c College of Civil Engineering, Tafresh University, Tafresh, Iran b

a r t i c l e

i n f o

Article history: Received 24 June 2010 Received in revised form 22 October 2010 Accepted 24 November 2010 Available online 6 January 2011 Keywords: Soil deformation moduli Soil physical properties Gene expression programming Nonlinear modeling

a b s t r a c t New empirical models were developed to predict the soil deformation moduli using gene expression programming (GEP). The principal soil deformation parameters formulated were secant (Es) and reloading (Er) moduli. The proposed models relate Es and Er obtained from plate load-settlement curves to the basic soil physical properties. The best GEP models were selected after developing and controlling several models with different combinations of the influencing parameters. The experimental database used for developing the models was established upon a series of plate load tests conducted on different soil types at depths of 1–24 m. To verify the applicability of the derived models, they were employed to estimate the soil moduli of a part of test results that were not included in the analysis. The external validation of the models was further verified using several statistical criteria recommended by researchers. A sensitivity analysis was carried out to determine the contributions of the parameters affecting Es and Er. The proposed models give precise estimates of the soil deformation moduli. The Es prediction model provides considerably better results in comparison with the model developed for Er. The simplified formulation for Es significantly outperforms the empirical equations found in the literature. The derived models can reliably be employed for pre-design purposes. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The modulus of soil deformation is an important parameter for the behavior analysis of substructures. The soil modulus can be obtained from a stress–strain curve. The theory of elasticity states that the strains experienced by the soil have a linear relationship with the stresses applied to the soil. This is not in practice true for soils since both elastic and plastic deformations occur during the loading. Because of the elasto-plastic behavior of soils, different moduli can be derived from the stress–strain (load-settlement) curves of laboratory or field test results [1,2]. A typical stress– strain curve is shown in Fig. 1. Referring to this figure, secant (Es), tangent (Et), unloading (Eu), reloading (Er), or cyclic modulus (Ec) can be defined. Es is calculated from the secant slope (Ss) corresponding to the slope from the origin (O) to K1. Et is derived from the tangent slope (St) which is the tangent to the point considered on the curve. If the slope is drawn from K1 to K2, the unloading slope (Su) is derived and the unloading modulus (Eu) is obtained from it. Er corresponds to the slope from K2 to K4 (Sr). Ec is calcu⇑ Corresponding author. E-mail addresses: [email protected] (A.H. Alavi), [email protected] (A.H. Gandomi). 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.11.008

lated from the cyclic slope (Sc) which is the slope from K2 to K3 in Fig. 1 [1]. The soil deformation moduli are usually evaluated by laboratory or field methods. The field test results have been found to be more reliable than those of the laboratory methods [3]. Among different field tests, plate load tests (PLT) has been a traditional in situ method for estimating the soil moduli. Using the results obtained from this test allows minimization of the effects of the scale factor and soil sample disturbance [4]. Several researches have shown that the plate load test provides reliable predictions of the soil modulus [5]. Despite reliability of this testing method, little attention is devoted to developing empirical solutions relating the deformation moduli obtained from the plate load test results to the physical properties of soils. In this context, Reznik [3] proposed analytical expressions describing dependence of the plate load deformation moduli of collapsible soils on void ratio and moisture content. Nearly all of the developed empirical correlations for the soil moduli prediction have been established based on regression analysis [6]. The significant limitations the traditional statistical techniques strongly affect the prediction capabilities of the derived equations. Genetic programming (GP) [7] is a new approach for behavior modeling of geotechnical engineering problems. The main advan-

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Stress

Su

Ss

Sc Sr

St K1

the creation of genetic diversity. GEP has a multi-genic nature. Thus, more complex programs with several subprograms can be generated during the evolutionary process. A gene in GEP is composed of a list of symbols. The symbols are elements from function or terminal sets. Each of the functions takes any value of data type which can be returned by a function or assumed by a terminal [11,12]. A typical GEP gene is as given below:

K4

þ

pffi

a3b

ð1Þ p

K3

K2

O

Strain

Fig. 1. Various definitions of soil modulus (after [1]).

tage of the GP-based approaches over the regression and other soft computing techniques is their ability to generate prediction equations without assuming prior form of the existing relationship (e.g., [8]). Gene expression programming (GEP) [9] is a new variant of GP. This method has good ability to produce computer programs with different sizes and shapes. Unlike classic GP and other soft computing tools like neural networks, the GEP applications to solve problems in civil engineering are restricted to fewer areas (e.g., [10,11]. The main purpose of this paper is to obtain new empirical relationships for determining Es and Er utilizing the GEP method. Various predictor variables included in the analysis were coarse and fine-grained contents, grains size characteristics, liquid limit, moisture content, and soil density. The proposed models were developed based on several plate load tests performed in this study.

where a, b and 3 are elements of the function set; +,  and are the terminal nodes, and ‘‘’’ is the element separator for easy reading. The above expression is called Karva notation or K-expression [9]. A K-expression can be represented by a diagram as an ET. As an example, Fig. 2 illustrates the expression tree of the above sample gene. The first position in the K-expression denotes the root of the ET. The transformation process starts from the root and reads through the string one by one [11,12]. The size of the corresponding ETs changes during the GEP evolutionary process. The valid length of each expression is equal to or less than the length of the gene. The validity of a randomly selected genome is certified by a head–tail method. Each GEP gene has a head and a tail. The head may be composed of both function and terminal symbols. The tail, on the other hand, may only contain terminal symbols [9,11]. Fig. 3 presents a typical representation of the GEP algorithm [12]. A roulette wheel sampling with elitism strategy is employed by GEP to select and copy the individuals. Single or several genetic operators such as crossover, mutation and rotation are used for introducing variations in the population. Note that the rotation operator rotates two subparts of the genome with respect to a randomly chosen point. Further descriptions of GEP can be found in [9,11,12]. 3. Modeling of soil deformation moduli Precise estimation of the soil modulus is an essential criterion in geotechnical design process. In order to provide accurate assess-

2. Gene expression programming GP is a subset of genetic algorithms (GAs). It is a modern regression technique with a great ability to automatically evolve computer programs. The evolutionary process followed by the GP algorithm is inspired from the principle of Darwinian natural selection. GP was introduced by Koza [7] in the late 1980s after experiments on symbolic regression. This classical GP technique is also called tree-based GP [7]. The main difference between the GA and GP approaches is that the evolving programs in GP are parse trees rather than fixed-length binary strings in GA [11]. GEP is a new subarea of GP which was first invented by Ferreira [9]. Function set, terminal set, fitness function, control parameters, and termination condition are the major elements of GEP. The difference between GP and GEP lies in the representation of the solution. GEP creates a fixed length of character strings to represent the solutions. These solutions are further shown as computer models in tree-like structures. These trees are termed expression trees (ETs) [11,12]. On the other hand, the solutions created by GP are represented as tree structures and expressed in a functional programming language [7,11]. In GEP, the genetic operators act on the chromosome level. This leads to an extreme simplification in

Start

Create an Initial Population at Random

Express Chromosome as ET

Execute ET

Evaluate Fitness

Yes

Stop

Termination Criteria Reached? No

Select Chromosome +

Reproduce √

×

Create New Generation a

3

b

Fig. 2. Example of expression trees (ETs).

Fig. 3. A typical representation of the GEP algorithm.

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ment of the soil modulus, the effects of several influencing factors should be incorporated into the model development. The significant influence of the soil physical properties such as particle size distribution, dry density, moisture content, and plasticity on its mechanical properties is well understood [1,6]. For instance, dry density is an indicator of compressibility of a soil. If the soil particles are closely packed, the modulus tends to be high. The moisture content has a major influence on the soil modulus. At low moisture contents, water binds the particles, particularly for fine-grained soils. This causes an increase in the effective stress between the particles through the suction and tensile skin of water phenomenon. Thus, in this case, low moisture content results in high soil modulus. The lubrication effect of water is not considerable at very low water contents. Therefore, the compaction of coarse-grained soils is more efficient at higher moisture contents compared with that in lower moisture contents. As the moisture content increases, water lubrication increases the effect of compaction and, subsequently, the modulus increases [1]. Note that for the moisture content beyond an optimum value, the water occupies more room and pushes the particles apart. Consequently, the compressibility increases and the modulus decreases. The organization of the particles refers to the structure of the soil. A coarse-grained soil can have a loose or dense structure and a fine-grained soil can have a dispersed structure [1]. The main purpose of this study is to derive new relationships for the soil secant (Es) and reloading (Er) moduli using the GEP approach. The most important factors representing the behavior of the soil deformation moduli were detected based on the literature review [1–4,6] and after a trial study. Consequently, Es (kg/cm2) and Er (kg/cm2) were considered to be functions of several parameters as follows:

ðEs ; Er Þ ¼ f ðFC; CC;D10 ; D30 ; D60 ; C u ; C c ; LL;W; c; cd Þ

ð2Þ

where FC (%) is fine-grained content , CC (%) is coarse-grained content, D10 (mm) is grain size for which 10% of the sample is finer, D30 (mm) is grain size for which 30% of the sample is finer, D60 (mm)is grain size for which 60% of the sample is finer, Cu is coefficient of uniformity (D60/D10), Cc is coefficient of curvature ((D30)2/ (D60D10)), LL (%) is liquid limit, W (%) is moisture content, c (g/ cm3) is soil bulk density, and cd (g/cm3) is soil dry density. FC, CC, D10, D30, D60, Cu, Cc, and LL represent the intrinsic soil properties. W, cd, and c carry information on the state of the soil and its compressibility and previous history. Over-consolidation ratio (OCR) could have been included in the analysis. OCR was not used herein as it should be obtained from time-consuming laboratory tests. On the other hand, c and cd can easily be calculated for a soil.

100

Percentage passing (%)

90 80 70 60 50 40 30 20 10 0 0.001

0.01

0.1

1

10

100

Particle size (mm) Fig. 4. Lower and upper limits of the grain size distribution of the soil samples.

283

3.1. Experimental study The experimental program consisted of laboratory and field tests. The field portion of this study included a test pit exploration and plate load tests (PLT). For laboratory testing purposes, several disturbed and undisturbed soil samples were taken from the sites. Extensive geotechnical laboratory test programs were carried out for the basic characterization of soils. These comprised determining the soil moisture content, unit weight, Atterberg limits, specific gravity, and grain size distribution. Fig. 4 illustrates the lower and upper limits of the grain size distribution of the samples tested. Different soil types tested were gravelly silt with sand (ML), silty clay with sand (CL–ML), gravelly lean clay with sand (CL), well-graded sand with silt (SW–SM), and silty gravel with sand (GM). Within the scope of this study, 43 plate load tests were performed to investigate the load-settlement characteristics of soils at some locations in Khorasan Province, Iran. The procedure considered for conducting the plate load tests was in accordance with the standard ASTM D1194-94 [13]. The tests were carried out on leveled surfaces, in shallow pits, or excavations within a small tunnel or adit using 305 mm diameter, 25.4 mm thick, rigid circular steel plates. In all cases, the tests were performed on soil layers situated above the groundwater table. The load was applied through a system comprising a hydraulic jack, a 30 tone ground investigation truck and measured using a calibrated load cell. After placing a high-capacity jack on the top of the loading plate, the test loads were applied to the bearing plates utilizing this hydraulic jack. In the case of performing the test within a small tunnel, bearing plates loaded against the opposite side of the tunnel. The interpretation of the results (deformation properties) is usually made using isotropic elastic theory because of its convenience. In the present study, the elastic moduli, EPLT, was determined using the following relationship [4]:

EPLT ¼ ð1  l2 Þ

pDq 4D

ð3Þ

where l = Possion’s ratio of the soil; D = diameter of the plate; q = applied pressure on the bearing plate; and D= average settlement of the plate. The descriptive statistics of the test results is given in Table 1. The information cited in this table includes FC, CC, D10, D30, D60, Cu, Cc, LL, W, c, and cd. Es and Er are also the measured soil deformation moduli. A major part of the database comprises the test results for fine-grained soil samples. 3.2. Model development using GEP The available database was used for establishing the GEP prediction models relating Es and Er to FC, CC, D10, D30, D60, Cu, Cc, LL, W, c, and cd. For the analysis, the data sets were randomly divided into training and validation subsets. The training data were used for learning (genetic evolution). The validation data were used to measure the performance of the models evolved by GEP on data that played no role in building them. Out of the 43 data for Es and Er, 32 data vectors were used for the training process. The remaining 11 data sets were employed for verifying the validation of the models. The input and output variables were normalized between 0 and 1 after controlling different normalization methods [14]. The normalization values for the predictor variables are presented in Table 1. Several runs were conducted to come up with a parameterization of GEP that provided enough robustness and generalization to solve the problem. Three levels were set for the population size (number of chromosomes) (100, 250, 500). The chromosome architectures of the models evolved by GEP include head size and number of genes. The success of the GEP algorithm usually increases

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Table 1 Descriptive statistics of the variables used in the model development. Parameter

FC (%)

D10 (mm)

D30 (mm)

D60 (mm)

Cu

Cc

LL (%)

W (%)

c (g/cm3)

cd (g/cm3)

Es (kg/cm2)

Er (kg/cm2)

Mean Standard deviation Sample variance Kurtosis Skewness Range Minimum Maximum Normalization form

80.21 20.93 438.07 3.55 1.93 90.32 7.97 98.29 FC/100

0.0092 0.0350 0.0012 40.4770 6.2833 0.2299 0.0001 0.2300 D10/0.25

0.0347 0.0675 0.0046 21.4654 4.3747 0.3993 0.0007 0.4000 D30/0.5

0.3070 1.1174 1.2486 20.2326 4.5220 5.9941 0.0059 6.0000 D60/10

118.80 158.02 24969.1 1.82 1.61 597.04 2.96 600.00 Cu/600

10.32 21.89 479.00 12.75 3.55 109.88 0.33 110.21 Cc/150

25.67 4.68 21.89 0.95 1.03 22.00 18.00 40.00 LL/50

13.50 5.06 25.62 1.08 0.17 19.00 3.00 22.00 W/25

1.84 0.15 0.02 0.97 0.02 0.61 1.54 2.14 c/2.5

1.62 0.12 0.02 0.32 0.74 0.45 1.45 1.90 cd/2.5

316.7 351.3 123423.5 32.7 5.4 2351.9 76.2 2428.0 Es/2500

1514.8 1375.1 1891025.7 2.0 1.4 5960.8 118.9 6079.7 Er/6500

Fig. 5. Experimental versus predicted soil moduli values using the GEP models: (a) Es and (b) Er.

with increasing the initial and maximum program size parameters. In this case, the complexity of the evolved functions increases. Three optimal levels were considered for the head size (3, 5, 8) and two levels were set for the number of genes (2, 3). For the number of genes greater than one, the addition linking function was used to link the mathematical terms encoded in each gene. The mutation rate was set to 0.044. One-point, two-point and gene recombination rates were also set to 0.3, 0.3 and 0.1, respectively. There are 3  3  2 = 18 different combinations of the parameters. All of these combinations were tested and 10 replications for each combination were carried out. This makes 180 runs for each of Es and Er. Therefore, the overall number of runs was equal to 180  2 (number of the considered outputs) = 360. The period of time acceptable for evolution to occur without improvement in best fitness is set via the generations without change parameter. After 2000 generations considered herein, a mass extinction or a neutral gene was automatically added to the model. In order to obtain optimum GEP models, several arithmetic operators and mathp ematical functions (e.g., +, , , /, , exp, log, ln, power) were used. The mean absolute error function was used to calculate the overall fitness of the evolved programs. The program was run until there was no longer significant improvement in the performance of the models. The GEP algorithm was implemented using GeneXproTools [15]. 3.2.1. GEP-based prediction models for soil deformation moduli Several models with different combinations of the input parameters were developed to obtain the optimal models. The best GEP models for predicting Es and Er were built using FC, D10, D30, D60, LL, W, and cd. The GEP-based formulations of Es and Er are as follows:

 Es

kg cm2



h   ¼ 2500 0:368  D60;n þ cd;n e1þD60;n eD10;n D260;n LLn FC n D30;n ð1 þ FC n þ D10;n þ D60;n ÞÞ þ Wn

#

2

 Er

 kg ¼ 6500½D60;n þ LLn cm2 

cd;n þ

FC n W n c2d;n

ð4Þ

!#

pffiffiffiffiffiffiffiffiffiffi 2 FC n þ ðFC n  W n ÞD210;n ð3 þ D60;n Þ ðLLn þ 3Þ= D30;n

ð5Þ where FCn, D10,n, D30,n, D60,n, LLn, Wn and cd,n denote the predictor variables in normalized forms (see Table 1). Correlation coefficient (R), root mean squared error (RMSE) and mean absolute error (MAE) were used to evaluate the performance of the proposed models. Comparisons of the experimental and predicted Es and Er values are respectively shown in Fig. 5. 4. Performance analysis and model validity According to Smith [16], if the R value provided by a model is higher than 0.8 and the error values (e.g., RMSE and MAE) are low, the predicted and measured values are strongly correlated with each other. It can be observed from Fig. 5 that the GEP models with high R and low RMSE and MAE values are able to predict the target values with an acceptable degree of accuracy. The performance of the models on the training and validation data suggests that they have both good predictive abilities and generalization performance. The Es prediction results are better than those for Er. The performance of the Er prediction model on the validation

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A. Mollahasani et al. / Computers and Geotechnics 38 (2011) 281–286 Table 2 Statistical parameters of the GEP models for the external validation. Item

Formula

Condition

Es

1 2

R

0.8 0.5, the condition is satisfied. Furthermore, the squared correlation coefficient between the predicted and measured values (Ro2), and the correlation coefficient between the measured and predicted values (Ro0 2) should be close to 1 [20]. The considered validation criteria and the relevant results obtained

Es (kg/cm2) 0

500

1000

1500

2000

2500

3000

Frequency

100 Es

80

Er

60 40 20 0 FC

D10

D30

D60

LL

W

γd

Fig. 7. Contributions of the predictor variables in the GEP models.

by the models are presented in Table 2. As it is seen, the derived models satisfy nearly all of the required conditions. The exception is for the Rm criterion which is not satisfied by the Er prediction model. The validation phase ensures the derived GEP models for the Es and Er prediction are strongly valid, have the prediction power and are not chance correlations. No rational prediction model for Es and Er has yet been developed that would encompass the influencing variables considered in this study. The existing empirical correlations can solely be applied to the estimation of Es and are developed based on the N-value of SPT or CPT results. Considering the available data, the following correlations in terms of N and plasticity index (PI) were included in the comparative study [21]:

8 3:5N Sand; gravel and other cohesionless soils > > > > > > < 2:5N Low PI ð< 12%Þ Es ¼ 1:5N Medium PI ð12% < PI 6 22%Þ > > > N High PI ð22% 6 PI < 32%Þ > > > : 0:5N Extremely high PI ðPI > 32%Þ

ð6Þ

Fig. 6 visualizes the Es predictions made by the GEP model and the best results obtained by the empirical correlations. As it is seen, the proposed model (R = 0.963, RMSE = 94.36, MAE = 72.5) significantly outperforms the existing correlations (R = 0.490, RMSE = 411.29, MAE = 242.80). Note that the available empirical correlations need N-value or cone point resistance that should be obtained from at least one set of field test. On the other hand, the proposed models for Es and Er were developed using the basic soil physical properties (FC, D10, D30, D60, LL, W, cd). Therefore, they can easily be used for the prediction purposes via hand calculations.

0

5. Sensitivity analysis 4

Experimental Look [21] This work

Depth (m)

8

12

16

Sensitivity analysis is of utmost concern for selecting the important input variables. The contributions of the final predictor variables (FC, D10, D30, D60, LL, W, cd) in the best GEP models were evaluated through a sensitivity analysis. Note that these variables were identified after developing and controlling several models with different combinations of the soil physical properties. To perform the sensitivity analysis, frequency values of the input parameters were obtained. A frequency value equal to 100% for an input indicates that this variable has been appeared in 100% of the best thirty programs evolved by GEP. This is a common approach in the GP-based analyses (e.g., [11]). The frequency values of the predictor variables are presented in Fig. 7. As it is seen, Es and Er are more sensitive to FC, cd and W than other soil properties.

20

6. Conclusion 24 Fig. 6. Experimental versus predicted Es values using different models.

New design equations were derived for predicting the soil deformation moduli (Es and Er) using the GEP method. The proposed relationships were developed based on several plate load

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tests performed in this research. The following conclusions may be drawn based on the results presented: i. The developed models give reliable estimates of the Es and Er values. The validation phases confirm the efficiency of the models for their general application to the estimation of the soil moduli. ii. The developed models are mostly suitable for the finegrained soils with physical properties similar to the soil samples used in this study. The GEP approach only uses the experimental data for specifying the structure of the model. Thus, the derived models are considered to be mostly valid for practical pre-planning and pre-design purposes. iii. The optimal models take into account the role of several important parameters (FC, D10, D30, D60, LL, W, cd) representing the soil moduli behavior. The results indicate that W and cd efficiently represent the initial state and consolidation history of the soil for determining the soil moduli. iv. The soil moduli can easily be estimated from the soil physical properties using the derived models. OCR was not used as a predictor variable since it should be obtained from time-consuming laboratory tests. v. The Es prediction model produces considerably better outcomes than the empirical equations found in the literature. vi. An observation from the results of the sensitivity analysis is that the soil deformation moduli are more affected by FC, cd and W than other soil properties. vii. The predictive capabilities of the derived models are limited to the range of the data used for their calibration. Despite this limitation, the models can easily be retrained and improved to make more accurate predictions for a wider range by including the data for other soil types and test conditions.

Acknowledgements The authors are thankful to Dr. Jafar Boluori Bazaz (Ferdowsi University of Mashhad, Mashhad, Iran) for his support and stimulating discussions.

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