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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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In this paper the reflection coefficient of electromagnetic wave incidence on the walls of the buildings and obstacles that occurs in mobile communication path was modified by solving the Riccati nonlinear equations. For this purpose, the building walls are assumed inhomogeneous layers where their permittivity changes as function of the wall thickness. Using this reflection coefficient, a new propagation model based on urn and GID (uniform geometrical theory of diffraction and geometrical theory of diffraction) for multiple diffraction paths is proposed. Using this model, the diffraction loss as well as the path loss for a row of buildings with two in homogeneous faces is calculated and compared with measured data. Comparison of theoretical and measured results reveals that the modified reflection coefficient can adequately predict the reflective properties of the building walls. Moreover, results obtained with the proposed UID model are in good agreement with the measurement data. Therefore, the modified reflection coefficients well as the new UID model can be used for estimation of multipath signals strength, diffraction loss and also path delay in ray tracing algorithms used in mobile communication, radar and radio links.

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Abstract There are numerous applications for gas turbine discs in the aerospace industries such as in turbojet engines. These discs normally work under high temperature while subjected to high angular velocities. Minimizing the weight of such items in aerospace applications results in benefits such as low dead weights and lower costs. Optimization of rotating discs is historically, an area of research due to their vast utilization in industry. The gas turbine disc is one of examples to name. Gas turbine discs work mostly at high temperature gradients and are subjected to high angular velocities. High speed results in large centrifugal forces in disc and simultaneous high temperatures reduces the disc material strength, thus the later increases stress in disc automatically. In order to obtain a reliable disc analysis and arrive at the corresponding correct stress distribution, solution should consider changes in material properties due to the temperature field throughout the disc. To this end, an inhomogeneous disc model with variable thickness is considered. Using the variable material properties method, stresses are obtained for the disc under rotation and a steady temperature field. In this paper this is done by modeling the rotating disc as a series of different rings with constant properties. The optimum disc profile is arrived at by sequentially proportioning the thickness of each ring to satisfy stress requirements. In this paper these are done using the simplex method. Simplex algorithm is applied in Ansys software and the results are presented.

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In this paper, an analytical solution for stress and displacement in an inhomogeneous half space under the action of concentrated normal surface loading is investigated. The Young modulus is considered to vary with the spherical radius R in a power law form of order n, while the Poisson’s ratio is taken to be constant. The problem is solved analytically using an elasticity approach and considering a semi-inverse method in which, based on equilibrium equations on the surface of an arbitrary hemisphere in the half-space and centered at the point of application of load, some stress components are assumed to be proportional to 1/R2. It is then shown that this assumption is valid and all stress components in this axisymmetric problem are proportional to 1/R2, while displacements are proportional to 1/R(n+1). and their variation with azimuthal coordinate φ is in the form of a special function called hyper-geometric function. Illustrative examples are presented, which show variations of stresses and displacements both in R and φ directions. It is seen that the inhomogeneity parameter has a significant effect on both of these field variables.

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