Modelling damping ratio and shear modulus of sand–mica mixtures using genetic programming

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Abstract

This study presents two Genetic Programming (GP) models for damping ratio and shear modulus of sand–mica mixtures based on experimental results. The experimental database used for GP modelling is based on a laboratory study of dynamic properties of saturated coarse rotund sand and mica mixtures with various mix ratios under different effective stresses. In the tests, shear modulus, and damping ratio of the geomaterials have been measured for a strain range of 0.001% up to 0.1% using a Stokoe resonant column testing apparatus. The input variables in the developed NN models are the mica content, effective stress and strain, and the outputs are damping ratio and shear modulus. The performance of accuracies of proposed NN models are quite satisfactory (R2 = 0.95 for damping ratio and R2 = 0.98 for shear modulus).

Introduction

The damping ratio (D) and shear modulus (G) of soils are usually determined using dynamic tests. A widely used way to determine dynamic properties of the soils is the resonant column test method, where the soils are loaded harmonically. In this method the wave propagation in cylindrical soil specimens is studied. The testing procedures and the data reduction for the tests are described in many studies (e.g., ASTM D4015-92, 2000, Drnevich et al., 1978, Toki et al., 1995). In the resonant column technique, a column of cylindrical soil sample is contained within a membrane, and an effective stress is applied during the test. After a specimen in the apparatus is prepared, it is isotropically consolidated, and then a cyclic loading is commenced. The loading frequency is first set at low value, and then increased until the response reaches the maximum value. The frequency is a parameter of the certain characteristics of the apparatus (i.e., interaction between the magnets and coil, fixity between the specimen and drive system, resonance of the other part of the apparatus) as well as the small-strain stiffness of the soils (Richart, Hall, & Woods, 1970).

In many cases it is necessary to obtain data for damping ratios and shear modulus over a range of confining pressures. This can be accomplished by (a) a multistage test, where a specimen is tested at a confining pressure and subsequently retested at other confining pressure values or by (b) several single stage tests, where a specimen is subjected to a confining pressure and then tested at a range of shear strain amplitudes. Although the former procedure is more economical, the question arises if microstructural changes caused by repeated strain cycles, particularly at the higher strain amplitudes, can alter the dynamic properties of the specimen tested (Chung, Yo Kel, & Drnevich, 1984).

This paper provides an alternative approach for the modelling of damping ratio and shear modulus of sand–mica mixtures using Genetic Programming (GP). GP based equations are proposed for damping ratio and shear modulus of sand–mica mixtures. The GP models are based on an experimental database, which was conducted to document the behaviour of saturated Leighton buzzard sand and mica mixtures in Stokoe resonant column apparatus. Following the readily available studies on the influence of particle shape on the behaviour of geomaterials (i.e., Dodds, 2003, Theron, 2004, Vermeulen, 2001), a series of stress-controlled multi stage tests were performed in parallel on various mix ratios. The predictions of GP models developed to explore the cyclic behaviour of the mixtures are found to be quite accurate.

Section snippets

Theoretical background

The first use of the resonant column testing technique dates back to the late 1930s (Iida, 1938, Iida, 1940). The technique was considerably refined by a number of researchers in 1960s and 1970s (e.g., Hardin and Richart, 1963, Drnevich and Richart, 1970). The parameters of G and D determined in the laboratory with resonant column equipment are widely discussed in one of the recent studies by Stokoe, Hwang, Lee, and Andrus, (1994). The Stokoe device is provided with instrumentation to excite

Materials

The aim of the experimental study was to evaluate the influence of various proportions of platy fines on the behaviour of rotund particles. Two different geomaterials were used in all the tests, Leighton buzzard sand and mica.

The Leighton buzzard sand used in the experiments was a fraction B supplied by the David Ball Group, Cambridge, UK, confirming to BS 1881-131:1998. Its specific gravity, minimum and maximum dry densities were found to be 2.65, 1.48 g/cm3 and 1.74 g/cm3 respectively. As it

Presentation and analysis of test results

An approach to the influence of fine platy particles on the behaviour of a rotund sand matrice provides a conceptual basis for the analysis of resonant column tests on samples containing Leighton Buzzard Sand and mica. The observations, first made by Gilboy (1928), that any system of analysis or classification of soil which neglects the presence and effect of the flat-grained constituents will be incomplete and erroneous. An investigation into the behaviour of gold mine tailings by Vermeulen

Overview genetic programming

Genetic algorithm (GA) is an optimization and search technique based on the principles of genetics and natural selection. A GA allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the “fitness” (i.e., minimizes the cost function). The method was developed by Holland (1975) and finally popularized by one of his students, Goldberg (1989), solved a difficult problem involving the control of gas-pipeline transmission for his

Numerical application

The main aim of this study is develop empirical damping ratio and shear modulus equations of sand Mica mixtures using Genetic programming. A total of 99 tests have been carried out shown in Table 1 with ranges of test parameters. The testing (19 tests – % 20) and training (80 tests – % 80) sets for GP training procedure are selected randomly from the experimental database to prevent over fitting. The formulations are based on training sets and are further tested by testing set values to measure

Conclusions

The objective of the study is to develop empirical genetic programming (GP) based models for the prediction of damping ratio and shear modulus of sand–mica mixtures as a function of effective stress, mica content and strain. The database for used for GP modelling is obtained as a results of nine undrained consolidated torsional resonant column tests performed on various mica and Leighton Buzzard Sand mixtures under confining pressures ranging from 350 to 450 kPa while the pore pressure was kept

Acknowledgements

The authors would like to thank Prof. C.R.I. Clayton and Dr. J. Priest of the University of Southampton, and Assoc. Prof. J. Kumar of Indian Institute of Technology for their invaluable helps. The second writer held a UK Overseas Research Students Awards Scheme (ORSAS) and a Ph.D. Scholarship from the University of Southampton. This study was also supported by Gaziantep University Scientific Research Projects Unit.

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