As I see from the picture you are maybe familiar with Dr Darendeli's dissertation on equivalent shear modulus and damping of soils. The equilavent elastic method basically does as follows (let say for each layer, or finite element, it's the same):

- on zeroth iteration you have the Gmax and initial Damping, you run a linear elastic time-history, you calculate the peak shear stress; you get 65% of the peak stress as equivalent uniform stress amplitude.

- from the equilavent uniform stress and Gmax you get the equivalent peak strain, from the G-reduction and Damping Curves you calculate the Geq and Damping for 1st iteration

- you rerun the analysis untill some tolerance in strains is achieved 

You may find useful to read the Manual of SHAKE90 or ProSHAKE codes.

Dear Stoyan Andreev,

Thank so much for your help. You mean I will still linear elastic analysis but I using strain to interpolate shear modulus and damping ratio based on Darendeli's curve in iteration?

Best regard,

Quang

Hi,

The secant shear modulus of an element of soil varies with cyclic shear strain amplitude. At low strain amplitudes, the secant shear modulus is high, but it decreases as the strain amplitude increases (what you are seeing in your lower graph). What we obtain from shear wave velocity (Vs) is Gmax, since most seismic geophysical tests measuring Vs induced shear strain lower than about 3x10-4%. According to what you show in your attachment “Gm” is the secant shear modulus and the calculation procedure is also represented. As I understand from your other questions, you are interested in FLAC. Have a look at the “Hysteretic Damping” from “Dynamic Analysis” in FLAC manual.

How do you obtain the upper graph? Strain values are in percent? What is the soil constitutive model in this analysis?

Hi Hossein Khaleghi,

You are right. I am working on FLAC. I used Hysteretic Damping ans strain values are in percent. Soil constitutive model in this analysis is sig3.

Regards,

Quang

Sig3 is a built-in function in FLAC which represent the variation of the shear modulus reduction factor, G/Gmax, with cyclic strain (given in percent). My question is about your constitutive model in the upper graph? elastic, mohr-coulomb, Drucker-Prager or ... ?

Dear Nguyen Van Quang, you are correct. You perform essentially a linear dynamic analysis with interpolated stiffness and damping. This approach is correct for small to medium shear strains in the soil: up to about 0.1%. 

Thank so much Stoyan Andreev.

Dear Hossein Khaleghi,

My constitutive model in the upper graph is Darendeli model.

Regard,

Quang

Dear Stoyan Andreev ,

Thank you for your great sharing! I am still confused about your words and hopefully to hear your explanation!

When we have the equivalent uniform stress at the interface between 2 layers, if we divide this stress by complex shear modulus G* we will have 2 distinct strains, one is above the interface and the other is bellow the interface. What strain is needed to interpolate in the next iteration?

Hieu Nguyen Van, each layer will have its distinctive shear strain time history. You may have different cases for the strain distributions, for example:

- gradually changing strains - within layers with equal or very similar soil properties

- rapidly decreasing strains - when a very soft layer is on top of a stiff rock

- strain peak in an intermediate weak/liquefiable layer surrounded by stiffer/non-liquefiable soil

In order to get credible results, it is of paramount importance to have fine enough discretization in the layers where significant strain gradients are expected.

Hi,

Please see the Ishira book and SHAKE a Computer Program for Earthquake Response Analysis of Horizontally Layered Sites edveloped by Schnabel