New Gene Expression Programming models for normalized shear modulus and damping ratio of sands
Introduction
The nature and distribution of earthquake hazards are extremely influenced by soil response under cyclic loading. To a large extent, such a response is controlled by the dynamic characteristics of the soil. Dynamic characteristics of geotechnical materials are often represented by shear modulus (G) and damping ratio (D). In order to examine the effects of the dynamic loads on civil engineering systems, it is essential to properly understand soil behavior, attain precision measurements and quantitative models describing the soil dynamic characteristics (Kramer, 1996). Moreover, the description and identification of soil are necessary prior to carrying out any geotechnical design. On the other hand, understanding the dynamic properties of soil allows a geotechnical engineer to more accurately evaluate and monitor the soil parameters. The dynamic response of soils is employed to solve several problems including slope stability, soil-structure interaction, machinery vibrations and seismic stability of structures under sea wave, wind, traffic and other dynamic loads. The relations between soil dynamic parameters are crucial for solving the abovementioned problems. Since there is a wide range of equations associated with the soil damping ratio and shear modulus, the selection of an equation greatly affects the results of engineering analyses. For that reason, the newest and most accurate methods need to be employed so as to achieve minimum error margin.
The experimental evaluation of the shear modulus and damping ratio of soils have so far been measured by numerous researchers through various devices such as Resonant Column (RC) (Hardin and Drnevich, 1972a, Khan et al., 2008, Khan et al., 2011, Senetakis et al., 2012, Wilson, 1988), Cyclic Triaxial (CT) (Khan et al., 2011, Yasuda and Matsumoto, 1993, Yoshimi et al., 1984), Cyclic Simple Shear (CSS) (Lanzo et al., 1997), cyclic simple Torsional Shear (TS) (Yasuda and Matsumoto, 1993) and combined device Resonant Column Torsional Shear (RCTS) (Darendeli, 2001, Lee, 2000, Menq, 2003). The significance of parameters contributing to the dynamic properties of soils has been reported by Hardin and Drnevich (1972b) and Darendeli (2001). The most important parameters contributing to shear modulus include shear strain (γ), mean effective confining pressure () and soil conditions (i.e., D50, etc.) as well as parameters contributing to damping ratio, not to mention the number of loading cycles (N) (Darendeli, 2001, Hardin and Drnevich, 1972b, Ishibashi and Zhang, 1993, Iwasaki et al., 1978, Kokusho, 1980, Stokoe et al., 1999).
Iwasaki et al. (1978) and Kokusho (1980) studied the effect of void ratio, effective confining pressure and shear strain percentage on the dynamic parameters of sands. Moreover, Seed et al. (1986) assessed through field and laboratory experiments the impact of mean effective confining pressure, relative density and shear strain percentage on granular soils. Ishibashi and Zhang (1993) collected different laboratory data for conducting dynamic evaluation of shear modulus and damping ratio in non-plastic sands and high-plasticity clays. They offered a series of equations for soil shear modulus and damping ratio as functions of mean effective confining pressure, plasticity index and shear strain. Their equations are valid within confining pressures between 0.2 and 10 atms (Darendeli, 2001). Darendeli (2001) examined the effect of soil type conditions, loading frequency, loading cycles (f and N) and plasticity index (PI) for a broad range of soils including gravel, sand and clay at various plasticity rates. Zhang et al. (2005) introduced several equations for estimating normalized shear modulus and damping ratio of Quaternary, Tertiary and older, and residual/saprolite soils. The parameters used in equations proposed for shear modulus include shear strain range, plasticity index and confining pressure, where the proposed equation for damping ratio comprises a term as a minimum damping ratio added to a polynomial function of normalized shear modulus. In all the mentioned studies, two or three of these parameters (e, and γ%) have been introduced as contributing factors to dynamic properties of sands.
Nonlinear optimization methods have been used in several problems. In the remediation of soil contamination (Chen et al., in press), an optimization system for surfactant-enhanced aquifer remediation has been developed by Qin et al. (2007). As a powerful method for nonlinear optimization, Genetic Algorithm has been successfully utilized in civil engineering analyses, particularly geotechnical engineering such as numerical modeling of stress–strain behavior under cyclic loading (Shahnazari et al., 2010), liquefaction-induced displacement (Javadi et al., 2006), dynamic soil properties (Cevik and Cabalar, 2009) and simulation of static soil shear strength (Mousavi et al., 2011). The main objective of this study is to provide new mathematical models based on GEP, which can predict normalized shear modulus (G/Gmax) and damping ratio (D%) of sands more accurately with minimum error margin, as compared to previous models. The parameters e, and γ were considered as inputs of the proposed models. Through mathematical models from several researchers including Darendeli (2001), Rollins et al. (1998) and Ishibashi and Zhang (1993) and published experimental data, the validity and performance of these models were compared and demonstrated to be more accurate than previous models. Using the collected data, the accuracy of Darendeli (2001), Rollins et al. (1998) and Ishibashi and Zhang (1993) models was also assessed in the estimation of the normalized shear modulus and damping ratio of sands.
Section snippets
Datasets
The datasets used in this study entailed the valid experimental findings obtained by various researchers which are an outcome of several devices including Resonant Column (RC) (Moayerian, 2012, Saxena and Reddy, 1989, Senetakis et al., 2013), Cyclic Triaxial (CT) (Kokusho, 1980, Kokusho, 2004, Rollins et al., 1998), Cyclic Simple Shear (CSS) (Anderson, 2003, D’Elia et al., 2003, Lanzo et al., 1997), cyclic simple Torsional Shear (TS) (Iwasaki et al., 1978, Uthayakumar, 1992) and combined device
Gene Expression Programming
As an optimization method inspired by the theory of evolution and natural selection, Genetic Algorithm (GA) was recognized as a stochastic optimization technique (Holand, 1975), which has been further developed by Goldberg (1989). Symbolic strings of fixed length (chromosomes) are the unique solutions in Genetic Algorithm; they are assessed through a fitness function, where the algorithm terminates once the output requirement is met. In fact, fitness function is one of the most important
Proposed models
In GEP, the values of setting parameters have important influence on the fitness of the output model. These include the number of genes, number of chromosomes and gene׳s head size, and the rate of genetic operators. Since there was little information available about the genetic parameters, the software was implemented by various adjustments through trial and error. Moreover, the performance of each model was assessed based on statistical measures such as RMSE, MSE and R2. As a result, the most
Accuracy evaluation of the proposed models
The accuracy of the proposed models for training, testing and total data was compared using statistical measures, the results of which can be seen in Table 3, Table 4. These results shed light on the fact that accuracies of the proposed models are almost identical in terms of statistical measures. Hence, both models are sufficiently accurate for estimation of the normalized shear modulus and damping ratio.
In Fig. 2, Fig. 3, a comparison was drawn between the total data for values predicted by
Conclusion
This paper mainly intends to propose several models based on the Gene Expression Programming (GEP), for predicting the normalized strain modulus (G/Gmax) and damping ratio (D%) of sands as functions of mean effective confining pressure, void ratio and shear strain percentage. These models have been measured through valid published experimental data using a number of devices including Resonant Column (RC), Cyclic Triaxial (CT), Cyclic Simple Shear (CSS), cyclic simple Torsional Shear (TS) and
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