Forced vibrations of an elastic rectangular plate supported by a unilateral two-parameter foundation via the Chebyshev polynomials expansion
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- Structural Engineering and Mechanics
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- v.90 no.6
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- pp.551-568
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- 2024
The present study deals with static and dynamic behaviors including forced vibrations of an elastic rectangular nano plate on the two-parameter foundation. Firstly, the rectangular plate is assumed to be subjected to uniformly distributed and eccentrically applied concentrated loads. The governing equations of the problem are derived by considering the dynamic response of the plate, employing a series of the Chebyshev polynomials for the displacement function and applying the Galerkin method. Then, effects of the non-essential boundary conditions of the plate, i.e., the boundary conditions related to the shearing forces, the bending moments and the corner forces, are included in the governing equation of motion to compensate for the non-satisfied boundary conditions and increase the accuracy of the Galerkin method. The approximate numerical solution is accomplished using an iterative process due to the non-linearity of the unilateral property of the two-parameter foundation. The plate under static concentrated load is investigated in detail numerically by considering a wide range of parameters of the plate and the foundation stiffnesses. Numerical treatment of the problem in the time domain is carried out by assuming a stepwise variation of the concentrated load and the linear acceleration procedure is employed in the solution of the system of governing differential equations derived from the equation of motion. Time variations of the contact region and those of the displacements of the plate are presented in the figures for various numbers of the two-parameter of the foundation, as well as the classical and nano parameters of the plate particularly focusing on the non-linearity of the problem due to the plate lift-off from the unilateral foundation. The effects of classical and nonlocal parameters and loading are investigated in detail. Definition of the separation between the plate and the two-parameter foundation is presented and applied to the given problem. The effect of the lift-off on the static and dynamic behavior of the rectangular plate is studied in detail by considering various loading conditions. The numerical study shows that the effect of nonlocal parameters on the behavior of the plate becomes significant, when nonlinearity becomes more profound, due to the lift-off of the plate. It is seen that the size effects are significant in static and dynamic analysis of nano-scaled rectangular plates and need to be included in the mechanical analyses. Furthermore, the corner displacement of the plate is affected more significantly from the lift-off, whereas it is less marked in the time variation of the middle displacement of the plate. Several numerical examples are presented to examine the sensibility of various parameters associated with nonlocal parameters of the plate and foundation. Both stiffening and softening nonlocal parameters behavior of the plate are identified in the numerical solutions which show that increasing the foundation stiffness decreases the extent of the contact region, whereas the stiffness of the shear layer increases the contact region and reduces the foundation settlement considerably.
A statistical model to predict soil temperature from the ambient meteorological factors including mean, maximum and minimum air temperatures, precipitation, wind speed and snow depth combined with Fourier time series expansion was developed with the data measured at the Suwon Meteorolical Service from 1979 to 1988. The stepwise elimination technique was used for statistical analysis. For the yearly oscillation model for soil temperature with 8 terms of Fourier expansion, the mean square error was decreased with soil depth showing 2.30 for the surface temperature, and 1.34-0.42 for 5 to 500-cm soil temperatures. The
Franchising is one of the fastest growing types of business. It is already popular and well-known in the U.S., and has been growing in many other countries including Korea. Furthermore, many Korean franchising companies have expanded their business overseas actively. According to the data by the Ministry of Industry and Resource, 82 companies out of a sample of 500 franchising companies are already operating in many foreign countries and 48% of them have started their foreign business since 2006. This clearly indicates the fast growing current trend of foreign operation by Korean franchising companies. In spite of the fast growing trend of foreign expansion in the industry, academic research on internationalization of franchising companies is extremely difficult to find. Accordingly, academic research on the issue is necessary and urgent in Korea. Among the various research questions on internationalization of franchising business, this study intends to investigate the difference in organizational factors between the franchising companies doing foreign operation and those doing business only domestically. More specifically, this research has the following purposes. First, considering the lack of theoretical basis of previous studies, resource-based theory and agency theory are employed as the theoretical bases. Second, this study explains the difference in internationalization based on organizational factors such as company size, history and growth rate. Third, the five hypotheses regarding the difference in organizational factors are presented and tested empirically, which is the first attempt in the area of this topic. Finally, the study attempts to clarify the conflicting implications among theories regarding some organizational factos such as growth rate. As the theoretical background, resource-based theory and agency theory are discussed. According to resource-based theory, a firm can grow continuously when it has competence and resource, and also the ability to develop them. The competence and resource can include capital, human resource, management skill, market information, ability to manage risk, etc. Meanwhile, agency theory views the relationship between franchisor and franchisee as an agency relationship. In agency theory, bonding capability and monitoring capability are the two key factors which promote internationalization of franchising companies. Based on the two theories, a conceptual model is designed. The model consists of two groups of variables. One is organizational factors including size, history, growth rate, price bonding and geographic dispersion. The other is whether a franchising company is operating overseas or not. We developed the following five research hypotheses basically describing the relationship between organizational factors and internationalization of franchising companies. H1: The size of franchising companies operating overseas is larger than that of franchising companies operating domestically. H2: The history of franchising companies operating overseas is longer than that of franchising companies operating domestically. H3: The growth rate of franchising companies operating overseas is higher than that of franchising companies operating domestically. H4: The price bonding of franchising companies operating overseas is higher than that of franchising companies operating domestically. H5: The geographic dispersion of franchising companies operating overseas is wider than that of franchising companies operating domestically. Data for the analyses are obtained from 2005 Korea Franchise Survey data co-generated by Ministry of Industry and Resource, GS1 Korea, and Korea Franchise Association. Out of 2,804 population companies, 2,489 companies are excluded for various reasons and 315 companies are selected as the final sample. Prior to hypotheses tests, validity and reliability of the measures of size, history, growth rate and price bonding are examined for further analyses. Geographic dispersion is not validated since it is measured using nominal data. A series of independent sample T-tests is used to find out whether there exists any significant difference between the companies internationalized and those operating only domestically for each organizational factor. Among the five factors, size and geographic dispersion show significant difference, growth rate and price bonding do not reveal any difference and, finally, history factor shows conflicting results in the difference depending on how to measure it.