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Nonlinear Response of Soil
Cyclic Nonlinear Model of Soil
The nonlinear stress-strain behavior of soils can be represented more accurately by cyclic non-linear models that follow the actual stress-strain path during cyclic loading. Shear modulus of soil is considered to be maximum at the shear strain level below 10^{-5}; whereas damping ratio of soil is minimum within this strain range. As shearing strain increases, shear modulus of soil decreases but damping constant increases. Thus non-linear behavior of soil becomes dominant as the strain rate increases.
Dynamic shear modulus corresponds to the shear modulus at the strain amplitude developed under the dynamic loading such as earthquake loading. Strain amplitude within the soil mass under the cyclic loading showing hysteretic behavior changes continuously; thereby the shear modulus, which is defined as the slope of the secant line joining the origin and the point on the stress – strain curve corresponding to the strain amplitude at which dynamic shear modulus (G) is desired as shown in (Fig-4).
At low strain amplitudes, the secant shear modulus is high but it decreases as the strain amplitude increases with numbers of cycles. The locus of points corresponding to the tips of the hysteresis loops of various cyclic strain amplitudes is called “back bone curve” as depicted in (Fig-5). Its slope at the origin corresponding to the cyclic strain amplitude r is termed as the maximum shear modulus, G_{max}or “small strain shear modulus”. Thus,
At greater strain amplitudes, shear modulus value G decreases thus the modulus ratio G/G_{max} drops to values of less than 1. This nonlinear behavior of soil can be accurately represented by cyclic non linear model that follow actual stress – strain path during cyclic loading.
Various non-linear models have been proposed among which “Hardin – Drnevich Model” (Fig-4/5) represents the back bone curve by the hyperbolic function as:
The above relation represents the modulus reduction curve corresponding to the various strain amplitudes.
Fig-4 (Cyclic Non linear model of soil as proposed by Hardin & Drnevich, showing maximum Shear modulus G_{max} and Shear modulus G at various Strain amplitudes)
Fig-5 (Backbone curve showing reduction in Shear modulus G, with increase in Shear strain)
The value of G_{max} corresponds to the shear modulus at strain amplitude less than 10^{-5}, which is generally below the amplitude appropriate for the use in analysis of foundations during seismic excitations. During seismic shaking the soils behave dynamically and undergo relatively large shear strain, more than 10^{-3} in general, when effective stress will decrease with the pore water pressures buildup due to plastic deformation in soil skeleton. In order to obtain the value of Shear Modulus corresponding to the desired strain magnitude, following equation as proposed by Hardin and Drnevich model could be employed;
Where is the desired shear strain magnitude and r is a reference strain defined by,
Fig-6 (Variation of Shear modulus and Damping ratio with Shear strain for Sand (Seed and Idriss 1970, Courtesy of Earthquake Engineering Research Center, University of California at Berkeley))