## 1 Introduction  
The stress-strain relationship of soils plays an important role in performing site-specific ground response analysis [1; 2; 3; 4]. Since the one-dimensional equivalent linear method, which took into account the dynamic property of the soil has been utilized for evaluating the site response under seismic loading, extensive cyclic tests have been carried out in the laboratory to characterize the soil dynamic property for different physical properties and loading conditions. The small-strain shear modulus \(G_{\rm max}\) and the corresponding modulus reduction curve \(G/G_{\rm max}\) are the two basic parameters for ground response analysis [5; 6; 7]. In the laboratory, several cyclic testing apparatuses can be used to measure the soil dynamic property over a wide shear strain (\(\gamma\)) range. For example, [PERSON] and [PERSON] [8] introduced the simple shear device which is capable of testing soils over a strain range of \(10^{-5}\) to \(10^{-2}\). [PERSON] [9] performed a joint use of resonant column with cyclic torsional simple shear to determine the dynamic properties of San Francisco Bay mud. [PERSON] [10] used a modified cyclic triaxial to investigate the influence of the plasticity index on the \(G/G_{\rm max}\) of fine-grained soils. [PERSON] and [PERSON] [11] developed a cyclic double specimen direct simple shear apparatus that allows testing from small shear strains (e.g., \(10^{-5}\)) to shear strains greater than 5%. [PERSON] [12] developed a large-scale, multi-mode, free-free resonant column, to investigate the dynamic properties of gravelly soils. Since the comprehensive reports on the dynamic properties of cohesionless and cohesive soils [2; 6; 7; 13], the number of data on the shear modulus and damping ratio for soils has increased markedly in recent years [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. Besides the dynamic testing apparatus, different determination methods are proposed to calculate the dynamic shear modulus according to the measured hysteresis loop. The conventional method assumes a symmetrical hysteresis loop, thus the secant modulus is computed as the slope of the line through its end points [23; 27; 28]. The form of the modulus reduction curve depends on the amplitude of the strain for which the hysteresis loop is determined. With increasing shear strain, the hysteresis loop progressively becomes asymmetric due to the increased nonlinearity at large strain. However, the conventional method may have no capability of measuring accurately the shear modulus in the large strain range, leading to underestimation or overestimation of the shear modulus. To consider the actual asymmetrical hysteresis loop at large strains, [PERSON] et al. [29] suggested a modified method, which was a variant of the method originally presented by [PERSON] [10]. The average value of compressive and tensile moduli is adopted to represent the asymmetric nature of the hysteresis loop. [PERSON] et al. [30] calculated the dynamic shear modulus based on the theory of correlation function. They stated that the proposed method could effectively eliminate the influence of background noise caused by testing environmental and sensor accuracy on the measured dynamic properties for strain levels less than \(10^{-4}\). [PERSON] and [PERSON] [31] demonstrated that the method proposed by [PERSON] et al. [29] might lead to the tensile modulus being larger than the compressive modulus, due to the accumulated plastic strain. Therefore, the least square method was suggested to calculate the shear modulus to overcome the influence of the irregularity of the hysteresis loop. Successful application of equivalent linear procedures for determining ground response is essentially dependent on the accurate characterization of dynamic soil properties in the analyses, particularly for the forms of the relationships between shear modulus and shear strain [2; 13].