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

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Abstract

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.

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 calculated 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 advantage 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.

Section snippets

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

Modeling of soil deformation moduli

Precise estimation of the soil modulus is an essential criterion in geotechnical design process. In order to provide accurate assessment 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

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

Sensitivity analysis

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, γd) 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

Conclusion

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 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

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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