كليدواژه :
بسامد طبيعي , ارتعاش آزاد، تير عميق , بستر ارتجاعي , توابع پتانسيل
چكيده فارسي :
در اين نوشتار، حل دقيق ارتعاش آزاد تير مستطيلي همگن و ايزوتروپ بر بستر ارتجاعي دو پارامتري در صفحه تحتاني ارائه شده است. مدل دو پارامتري پاسترناك به منظور مدل سازي اندركنش تير با بستر در سطح تماس انتخاب شده است. معادلات بر اساس تئوري دوبعدي ارتجاعي فرمول بندي شده و با استفاده از تابع پتانسيل تغيير مكان معادلات حاكم به يك معادله مرتبه چهار كاهش و توسط روش جداسازي متغييرها و اعمال دقيق شرايط مرزي حل شده است. روش ارائه شده در اين پژوهش بدون فرضيات ساده كننده و برخلاف تئوري هاي متداول تير، محدوديتي در انتخاب ضخامت نداشته و براي نسبت هاي مختلف ضخامت به طول تير داراي اعتبار است. به منظور اعتبارسنجي، نتايج به دست آمده از اين پژوهش با ساير كارهاي تحليلي مقايسه شده است. نتايج نشان ميدهد كه افزايش ضرايب بستر با افزايش بسامد طبيعي تير همراه است كه شدت آن با افزايش نسبت ضخامت به طول و در مقادير بزرگتر از 0/2 و در مودهاي بالاي ارتعاشي داراي كاهش قابل ملاحظه اي است.
چكيده لاتين :
Beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation is vastly applied in many branches of engineering problems namely geo-technics, road, railroad, marine engineering and bio-mechanics. Foundation is often a rather complex medium, e.g. a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Herein, the response of the foundation at the contact area is of interest, and not the stresses or displacements inside the foundation material. In most cases, the contact is presented by replacing elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs, which produce distributed reactions in the direction of the deflection of the beam. However, since the model does not take either continuity or cohesion of the bed into account, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model. Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering. With increase in thickness, the existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, the application of determination coefficient in first-order shear theory and the expression of one or few unknown functions based on other functions in higher-order shear theories are accompanied by reduction in the accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving partial differential equations of fourth order through the separation of variables and the application of boundary conditions. The major characteristics of present study include lack of limitations for thickness and its validity for beams of low, medium and large thicknesses is quite reliable. To verify, the results of present study were compared with those of other studies. Results show that increases in foundation parameters are associated with increases in natural frequency. The intensity is reduced considerably by increase in the ratio of thickness to length, for the values larger than 0.2 and in the higher modes of vibration .
