Sensitive Analysis of Nodal Force Function in the Elastostatic Problems in Decoupled Equations Method

The elastostatic problems are a significant subject 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 assessment 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 to 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 ξ, 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, however the “redistribution” of the stresses at the LCO and resolving the problem for each DOF, represents the connection between all DOFs of the domain. In the solution procedure, the order of displacement function u(ξ) depends on nodal force function F^b (ξ). To analysis of elastostatic problems in the classical Decoupled Equations Method, F^b (ξ) varies in the undertaken domain like a body force. Therefore, F^b (ξ) 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 to get optimum results, proposing an appropriate nodal force function corresponding to physical concept is necessary. For example in the cantilever beam which is subjected to a shear force at its free end, by considering the linear form for nodal force function results in minimum error. In 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.