Derive the Impedances Numerically

[Zouatine]

Dear STKO team,

I hope you are doing well.

I would like to kindly request your assistance regarding the following numerical example.

I am trying to learn how to derive impedance functions through a numerical model, so I started with a very simple case: a stiff, massless foundation bonded to a homogeneous half-space soil, where the solution is already available in the literature Gazetas.

I started with the vertical case. I modeled the soil with E = 2e+8, v = 0.25, and rho = 1800 kg/m³. I placed absorbing boundaries at the corners with G = 8e+8, v = 0.25, and rho = 1800 kg/m³. The foundation was modeled with E = 2e+19 and v = 0.4.

To excite a wide range of frequencies, I applied a Ricker wavelet. The procedure I followed to derive the impedance functions is as follows:

1. I extracted the displacement at the foundation node (basically the input) and performed a Fourier transform.
2. I summed all the reactions at the contact nodes between soil and foundation and then performed a Fourier transform.
3. I divided the FFT of the reaction forces by the FFT of the displacement.
4. I normalized the result with respect to (rho * vs**2 / b).

the results of the first and second points are like follow:

fft.png (97.6 KiB) Viewed 246 times

then deriving the two parts of the impedance function real and imaginary :

imaginary.png (60.26 KiB) Viewed 246 times

and compare it Gazetas formula to the proposed solution , the results are like follow:

gesatzt.png (57.9 KiB) Viewed 200 times

I am not sure why, in my simulation, the static stiffness appears to be low ( Re(K) = 1.857862e+05 N/m) compared to the analytical results ( Re(K) = 1.4e+09 N/m).

Here is my model:

Example_For_Impedances_v2_Forum.zip

(1.09 MiB) Downloaded 17 times

Looking forward to your answers.

[STKO Team]

Dear User,

Thank you for your message. Please note that this forum is primarily dedicated to the more technical aspects of the software itself, rather than to general engineering topics.

If possible, we kindly invite you to share the relevant excerpts from the reference you are mentioning and provide a more detailed explanation of how you implemented the same procedure in STKO. This additional information will help us — and other users — better understand your question and provide a more accurate and helpful response.