Hi everyone,
As we know, the formulas of the natural frequencies and their corresponding mode shapes for the uniform linear elastic soil (without damping) on the rigid bed are as follows [1]:
f_n = (2n - 1)vs / (4H)
vs = sqrt(G/ρ)
φ_n = sin((2n - 1)πz / (2H))
where f_n is the natural frequency of the corresponding mode in Hertz, φ_n is the mode shape of the corresponding mode, n is the mode number, H is the depth of the soil column, vs is shear wave speed, and z is the height from the rigid bed.
I create the finite element model (FEM) of soil using the commercial finite element (FE) program Abaqus/CAE 6.14-2 x64. The soil was modeled according to the geometric properties using the C3D8. It has a depth of over 50 meters and a horizontal distance of over 100 meters. The soil element size is 2 meters. The total model consisted of 62500 soil elements. Fig. 1 shows the 3D of the soil model. The unit weight of soil is taken 15 kN/m^3, Young’s modulus is 1000 kN/m^2, Poisson’s ratio is 0.35, Cohesion is 5 kN/m^2.
In order to closely match the theoretical solution, some assumptions have been made for the modeled soil. The soil rests on the rigid bed, modeled as boundary conditions restraining the bottom against translations and rotations at all directions. Further, the outer nodes of the model at the same level (Y equal) are tied using the MPC-Tie constraint to move together.
According to the theoretical formulas mentioned above, the first and second natural frequencies of soil column are obtained 0.0786 Hz and 0.2357 Hz, respectively. Nevertheless, the FE results do not conform to this order as the mode number. The first twenty-two natural frequencies of the soil FEM are shown in Fig. 2. The selected 3D mode shapes of the soil FEM are illustrated in Fig. 3. As can be seen in Fig. 1 and Fig. 2, the 1st frequency of the model occurs at mode numbers 1 and 2 and in two horizontal directions perpendicular to each other. This result is consistent with the theoretical results. However, the 2nd mode shape as the model occurs in modes 11 and 12, which is contrary to the theoretical results. Interestingly, the behavior of other mode shapes of the soil model is also strange.
Why do the order of the fundamental mode shapes and their frequencies of the soil FEM differ with theoretical values? Should further assumptions be made to fully simulate the FEM with the theoretical soil model?
The figures and the Abaqus/CAE FEM file are attached.
Thanks a lot in advance.
References
[1] Steven L. Kramer. Geotechnical Earthquake Engineering. Pearson Prentice Hall, 1 edition, 1996.
I uploaded the input file (*.inp). It is compressed.
It is noted that the soil consists of two layers. At this stage, both specifications were considered the same. A Mohr-Coulomb soil model was used in the FE analysis.
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