كليدواژه :
روش معادلات مجزا , مسائل الاستواستاتيك , بردار نيروهاي گره اي , بازتوزيع تنش , مسائل دوبعدي
چكيده فارسي :
يكي از پركاربردترين مسائل مربوط به مهندسي در مكانيك جامدات، مسائل الاستواستاتيك است. بسياري از مسائل الاستواستاتيك به صورت تحليلي قابل حل نيستند؛ از اين رو، حل اين مسائل با روش هاي عددي به يكي از مسائل مهم تبديل گشته است. مقاله حاضر به بررسي يك روش جديد به نام روش معادلات مجزا در مسائل الاستواستاتيك مي پردازد. روش معادلات مجزا يك روش نيمه تحليلي با ماتريس ضرايب قطري است. در اين روش، تنها مرز مسئله با استفاده از توابع شكل مرتبه بالا و توابع نگاشت چبيشفي گسسته سازي مي گردد. در اين روش، با استفاده از روش باقيمانده هاي وزن دار و روش انتگرالي كلينشا-كورتيز، معادلات ديفرانسيل اويلري به صورت مجزا ايجاد مي گردند و در ادامه با تعريف بردار نيروهاي گره اي، فرايند حل با استفاده از مفهوم بازتوزيع تنش صورت مي گيرد. در مقاله حاضر با تعريف فرم هاي جديدي از بردار نيروهاي گره اي، معادله ديفرانسيل حاكم بر مسئله الاستواستاتيك استخراج مي گردد و در نهايت، با حل دو مثال عددي، روش معادلات مجزا در برابر فرم توزيع تنش در فضاي مسئله مورد حساسيت سنجي قرار گرفته است. نتايج نشان مي دهد كه در روش معادلات مجزا پاسخ مسائل الاستواستاتيك كاملاً وابسته به فرم توابع جديد بردار نيروهاي گره اي مي باشد و بسته به نوع مسئله، در هر مسئله يك فرم خاص از بردار نيروهاي گره اي داراي دقت بهينه است.
چكيده لاتين :
The elastostatic problems are significant subjects in the analysis and design of solids and structures. As most of the complicated elastostatic problems do not have closed-form solutions, numerical methods such as finite element method (FEM), boundary element method (BEM), discrete element method (DEM), meshless methods, scaled boundary finite element method (SBFEM), and hybrid methods are the current approaches dealing with these types of engineering problems. This study presents a novel application of the decoupled equations method (DEM) to assess elastostatic issues. In the present method, the so-called local coordinate's origin (LCO) is selected at a point, from which the entire domain boundary may be observed. For the bounded domains, the LCO may be chosen on the boundary or inside the domain. Furthermore, only the boundaries which are visible from the LCO need to be discretized, while other remaining boundaries passing through the LCO are not required to be discretized. In this method, only the boundaries of problems are discretized using specific higher-order sub-parametric elements and higher-order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw-Curtis quadrature result in diagonal Euler’s differential equations. So, the coefficient matrices are diagonal, which provide a system of single Euler’s differential equations for the ith degree of freedom (DOF). If n indicates the number of DOFs of the problem assumed to be analyzed by the proposed method, only n Euler’s differential equations (with only one unknown differential equation for each DOF) should be solved. In the proposed method, the LCO is the same for all nodes, for which the LCO has the same displacement components. Therefore, the physical concept of this fact may be considered as some semi-parallel springs adjoining each other at the LCO. Therefore, the proposed procedure is called “redistribution” of the stresses in the present method. At the final step, using the calculated displacement field along , the displacement at any point of the problem’s domain is interpolated by using the proposed special shape functions. Although the governing equation of each DOF is decoupled from those of other DOFs, the “redistribution” of the stresses at the LCO and resolving the problem for each DOF represent the connection between all DOFs of the domain. In the solution procedure, the order of displacement function depends on nodal force function . To analyze the elastostatic problems in the classical Decoupled Equations Method, varies in an undertaken domain like a body force. Therefore, is defined as a linear function. In this study by proposing new forms of force function, the response of elastostatic problems is assessed. In the following, sensitivity of this method via proposed nodal force functions is fully demonstrated through two benchmark problems. The results show that stress and displacement fields totally depend on the form of force function. Also, the results show that in order to get optimum results, it is necessary to propose an appropriate nodal force function corresponding to physical concept. For example in the cantilever beam - which is subjected to a shear force at its free end - minimum error occurs if the nodal force function is presented by linear form. On the other hands, in the Kirsch’s problem - with a central small circular hole - considering the nonlinear form for nodal force function leads to minimum error.
