Examination of soil-foundation-structure system response and determination of the optimal target frequency in a substructure method considering soil inhomogeneity with depth

Articles in Press

Document Type : Original Research

Author
M. Shamsi
Assistant Professor, Department of Civil Engineering, University of Hormozgan
Abstract
Despite the particular importance of the subject of soil-structure interaction, unfortunately, this issue has received little attention from engineers, and seismic codes have not given much recommendation to consider its effects. Seismic wave frequencies vary continuously, and the stiffness of springs and damping of dampers connected to structural supports also vary with the loading frequency. To simplify time-domain numerical analysis, a constant target frequency can be used to keep stiffness and damping values constant. In the substructure method proposed in this study, the optimal target frequency is the one that yields results that most closely match those of a more accurate nonlinear 3D model analyzed using a direct method. A common simplification is to ignore the foundation’s non-linear response, justified by design requirements to prevent permanent deformation and the complexity of frequency-dependent soil behavior. Though not fully precise, this approach (considering soil heterogeneity and optimal target frequency) offers a forward-looking analysis and a basis for future nonlinear studies. This study presents a three-dimensional (3D) numerical model for analyzing the seismic response of soil-foundation-structure systems embedded in granular soil (with different relative densities) considering the effects of soil heterogeneity (With varying shear modulus with depth and compatible with the practical HSsmall model). The model is capable of accounting for the effects of loading frequency along with the radiation damping of the soil system and can integrate with the widely-used substructuring method considering an optimal target frequency. After verifying the proposed model, the dynamic equilibrium equations of the substructuring system were solved in the time domain using Matlab software. The target frequency was determined using i) Case 1: the fundamental frequency of the soil (or the dominant frequency of the excitations), ii) Case 2: the fundamental frequency of the structural system, iii) Case 3: the fundamental frequency of the soil-foundation-structure system; iv) Case 4: the fundamental frequency of structure with static stiffness and damping support (Case 4); and v) the fundamental frequency of fixed base structure and modified stiffness, and the results were compared together. A comparison of the impedance (dynamic stiffness and damping) of foundations situated on homogeneous and heterogeneous soil, as well as an investigation of the structural response in both cases, is another objective of this research. The analysis results demonstrated the accuracy of the proposed model and the acceptable calculation speed for estimating the dynamic response of structures located on heterogeneous soils under frequent operational earthquakes. The results also showed that with an increase in soil relative density, the seismic behavior of structures on homogeneous and heterogeneous granular soils converges. For instance, the response of the foundation on homogeneous soil bed with relative densities of 55%, 75%, and 95% is on average 23%, 19%, and 15% lower than that of heterogeneous soil, respectively. Additionally, for determining the target frequency, the use of frequency‐independent Kelvin–Voigt models (i.e., Cases 1-5) provides acceptable responses. According to the data presented in Table 4 and Figs. 9 and 10, the following conclusions can be drawn: 1) The soil's fundamental frequency (Case 1) yielded the least precise results. 2) While Case 3 offered the most favorable response, closely matching the direct method, determining the soil-structure system's fundamental frequency through complex integration in numerical software is often impractical. 3) Employing the target frequency in Case 2 produced more satisfactory results than Case 1. 4) Cases 4 and 5 generated nearly identical frequencies. Compared to Case 2, these cases enhanced response accuracy, bringing them closer to the best response (i.e., Case 3). Therefore, for practical applications, it is recommended to utilize the fundamental frequency from either Case 4 or Case 5 instead of the soil-structure system's fundamental frequency (Case 3) to establish the optimal target frequency.

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Available Online from 13 July 2025