A hybrid computational approach to formulate soil deformation moduli obtained from PLT

Abstract

In this study, new empirical equations were developed to predict the soil deformation moduli utilizing a hybrid method coupling genetic programming and simulated annealing, called GP/SA. The proposed models relate secant (E_s), unloading (E_u) and reloading (E_r) moduli obtained from plate load–settlement curves to the basic soil physical properties. Several models with different combinations of the influencing parameters were developed and checked to select the best GP/SA models. The database used for developing the models was established upon a series of plate load tests (PLT) conducted on different soil types at various depths. The validity of the models was tested using parts of the test results that were not included in the analysis. The validation of the models was further verified using several statistical criteria. A traditional GP analysis was performed to benchmark the GP/SA models. The contributions of the parameters affecting E_s, E_u and E_r were analyzed through a sensitivity analysis. The proposed models are able to estimate the soil deformation moduli with an acceptable degree of accuracy. The E_s prediction model has a remarkably better performance than the models developed for predicting E_u and E_r. The simplified formulations for E_s, E_u and E_r provide significantly better results than the GP-based models and empirical models found in the literature.

Introduction

The soil deformation modulus is an essential parameter for the analysis of behavior of structures. The soil modulus can be extracted from a stress–strain curve. According to the theory of elasticity, the strains experienced by the soil are linearly related to the stresses applied. 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 (Briaud, 2001, Briaud et al., 2006). A typical stress–strain curve is shown in Fig. 1. Referring to this figure, different moduli can be defined such as secant (E_s), tangent (E_t), unloading (E_u), reloading (E_r), or cyclic modulus (E_c). For instance, E_s is calculated from the secant slope (S_s) corresponding to the slope from the origin (O) to L₁. If the slope is drawn from L₁ to L₂, the unloading slope (S_u) is derived and the unloading modulus (E_u) is obtained from it. E_r corresponds to the slope from L₂ to L₃ (S_r) in Fig. 1 (Briaud, 2001).

The soil deformation moduli are commonly obtained from laboratory or field methods. The field test results are more reliable than those of the laboratory methods (Reznik, 1995). Among different field tests, plate load tests (PLT) have been a traditional in-situ method for the assessment of the soil moduli. The effects of the scale factor and soil sample disturbance can notably be minimized using the results obtained from PLT (Reznik, 1993). Several researches have proved the reliability of the PLT predictions of the soil modulus (Canadian Geotechnical Society, 1985). However, little attention is devoted to deriving empirical correlations relating the deformation moduli obtained from the PLT results to the physical properties of soils. Reznik (1995) developed analytical expressions describing dependence of the plate load deformation moduli of collapsible soils on void ratio and moisture content. Most of the empirical models developed for the estimation of the soil moduli are established using regression analysis (Reznik, 2007). The significant limitations of the traditional statistical techniques strongly affect the prediction capabilities of the regression-based equations. Most commonly used regression analyses can have large uncertainties. It has major drawbacks pertaining to the idealization of complex processes, approximation, and averaging widely varying prototype conditions. An important limitation of the regression analysis is that it assumes a pre-defined linear or nonlinear equation to model the nature of the corresponding problem. Another major constraint in the application of the regression analysis is the assumption of normality of residuals.

Genetic programming (GP) (Koza, 1992) is a new approach for the modeling of the behavior of geotechnical engineering tasks. GP may generally be defined as a specialization of genetic algorithm (GA). GP is used for function approximation, whereas GA is used for parameter optimization. The main advantage of the GP-based approaches over the regression and other soft computing techniques is their ability to generate greatly simplified equations without assuming prior form of the existing relationship (e.g., Yang et al., 2004, Gandomi et al., 2010, Gandomi and Alavi, 2011.

Simulated annealing (SA) is a general stochastic search algorithm. SA has been successfully applied to many branches in civil engineering optimization (Hasançebi et al., 2009, Vasan and Raju, 2009, Hasançebi et al., 2010, Golafshani et al., 2011, Lou et al., 2011). The Metropolis algorithm, the foundation of SA, was proposed by Metropolis et al. (1953) to simulate the annealing process. Folino et al. (2000) combined GP and SA to make a hybrid algorithm with better efficiency. They showed that introducing this strategy into the GP process improved the profitably of traditional GP. Alavi et al., 2010, Alavi et al., 2011 have recently applied the hybrid GP/SA technique to civil engineering problems.

In this study, new empirical relationships were proposed for determining E_s, E_u and E_r utilizing the GP/SA methodology. The 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.