تحليل صفحات دايروي گيردار تحت نگره ي تغييرشكل هاي بزرگ با استفاده از سري هاي تواني

Analysis of Clamped Circular Plates with Large Deflections Theory by Exponential Series

تحليل خمشي صفحات نازك در برابر بارهاي جانبي بزرگ، استفاده از نگره ي تغييرشكل هاي بزرگ را مي طلبد؛ زيرا در اين حالت نقاط واقع بر ميان صفحه افزون بر جابجايي عمود بر صفحه، جابجايي درون‌صفحه را نيز تجربه مي كنند. معادلات فون كارمن، پاسخ صفحات خمشي در برابر بارهاي جانبي را از ديدگاه نگره‌ي ناخطي هندسي به‌دست مي دهد. اين مطالعه ، روشي نوين را براي تحليل عددي معادلات فون كارمن ارايه كرده است. بدين منظور با گزينشي منطقي، مقدار تغييرشكل صفحه در حوزه ي ناخطي هندسي به صورت تابعي از خيز صفحه در حالت تغييرشكل هاي كوچك انگاشته شده است. اين تقريب موجب تبديل معادلات ديفرانسيل پاره ‌يي به معادلات جبري ناخطي مي ‌شود، كه به سهولت قابل حل است و سرعت هم‌گرايي فوق العاده‌يي نيز دارد. نتايج به‌‌دست‌آمده با بهره گيري از روش ارايه‌شده در اين نوشتار در مقايسه با ساير روش ها قدرت و توانايي راهكار پيشنهادي را آشكار مي سازد.

The bending analysis of thin plates under large transverse loads requires the use of large deflection theory, because, in this case, points on the middle plane, in addition to perpendicular displacement, experience in-plane displacement. In this paper, the field of the boundary value problem is considered as one circular plate under uniform distributed transverse loads with clamped edges. Von Karmen equations respond to the reaction of bending plates under transverse loads from the viewpoint of geometrical nonlinear theory. These equations for circular plates have been provided as two fourth order partial differential equations in a polar coordinate system. The lack of analytic responses for these equations has compelled researchers to use numerical methods for solving them. In the present paper, Von Karmen equations have been analyzed for circular plates with clamped edges using an exponential series and the expansion of Maclaurin. One of the obvious characteristics of the proposed method is the simplicity of its analytical basis in comparison with existing analytical methods, which proves its superiority. The present study has offered a new method for numerical analysis of Von Karmen equations. Therefore, with a logical selection, the value of plate deflections in a geometric nonlinear district has been viewed as a function of plate deflection in the case of small deflection. This approximation is caused to convert partial differential equations to nonlinear algebraic equations, which are easily solved and have high convergence speed. The introduced method in this research can be generalized for different boundary conditions and loadings in circular plates, which is one of the important characteristics of this method. In the present paper, some parameters have been calculated, such as deflection of various points and radial stresses under uniform distributed load for one circular plate with clamped edges. The obtained results by analysis of Von Karmen nonlinear differential equations, using the provided method in the present paper, reveal the power and capability of the proposed method in comparison with others