In recent years, attention has been given to the construction and development of new educational centers, but their spatial distribution across the cities has received less attention. In this study, the Average Nearest Neighbor (ANN) and the optimized hot spot analysis methods have been used to determine the general spatial distribution of the schools. Also, in order to investigate the spatial distribution of the schools based on the substructure variables, which include the school building area, the results of the general and local Moran and Getis Ord analyses have been investigated. A differential Moran index was also used to study the spatial-temporal variations of the schools’ distribution patterns based on the net per capita variable, which is the amount of school building area per student. The results of the Average Nearest Neighbor (ANN) analysis indicated that the general spatial patterns of the primary schools, the first high schools, and the secondary high schools in the years 2011, 2016, 2018, and 2021 are clustered. Applying the optimized hot spot analysis method also identified the southern areas and the suburbs as cold polygons with less-density. Also, the results of the differential Moran analysis showed the positive trend of the net per capita changes for the primary schools and first high schools. However, the result is different for the secondary high schools.
In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.
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