Numeric analysis of elastic plane body static problem by the method of matched sections
Authors: Danylenko Kirill , Orynyak Igor 
Published: December 2024

The paper continues the series of authors' works on the elaboration of a principally new variant of the finite element method, FEM, for the treatment of various problems of mathematical physics, namely the method of matched section, MMS. The elastic plane body under static loading is considered here. As in FEM, the whole body is meshed into the small elements of, preferably, rectangular form. The main peculiarity of the method consists in the introduction of a set of main parameters dependent only on one coordinate variable, i.e. either  or . So, any differential equilibrium equation with two partial derivatives concerning  or  is broken out into two relatively simple equations concerning only one independent variable. This leads to the introduction of one additional constant showing the interchange between these two equations. The introduced constants can be derived from the equation of continuity of kinematic parameters in the center of each element. The main, for example, -dependent parameters are:  and  displacements in vertical (-) and horizontal (-) directions, respectively; normal  and tangential (shear)  forces in  direction, and  direction, respectively; and bending moment  and angle of rotation . Similar parameters are established for -direction. Based on the methodology of the transfer matrix method the analytical matrix-form dependence between these parameters in any point  or  and those at the lower and/or left border of the element are established. For the treatment of oblique and curvilinear boundaries, the right triangular element as a special degenerate case of the rectangular element is derived. The resulting system of linear equations is formulated for unknownThe paper continues the series of authors' works on the elaboration of a principally new variant of the finite element method, FEM, for the treatment of various problems of mathematical physics, namely the method of matched section, MMS. The elastic plane body under static loading is considered here. As in FEM, the whole body is meshed into the small elements of, preferably, rectangular form. The main peculiarity of the method consists in the introduction of a set of main parameters dependent only on one coordinate variable, i.e. either  or . So, any differential equilibrium equation with two partial derivatives concerning  or  is broken out into two relatively simple equations concerning only one independent variable. This leads to the introduction of one additional constant showing the interchange between these two equations. The introduced constants can be derived from the equation of continuity of kinematic parameters in the center of each element. The main, for example, -dependent parameters are:  and  displacements in vertical (-) and horizontal (-) directions, respectively; normal  and tangential (shear)  forces in  direction, and  direction, respectively; and bending moment  and angle of rotation . Similar parameters are established for -direction. Based on the methodology of the transfer matrix method the analytical matrix-form dependence between these parameters in any point  or  and those at the lower and/or left border of the element are established. For the treatment of oblique and curvilinear boundaries, the right triangular element as a special degenerate case of the rectangular element is derived. The resulting system of linear equations is formulated for unknown values of all parameters specified at the border of all elements. The efficiency and the superb accuracy of the MMS are demonstrated in the classical examples of bending of a long rectangular body (beam-like geometry) and tension at infinity of a 2D body with a small circular hole values of all parameters specified at the border of all elements. The efficiency and the superb accuracy of the MMS are demonstrated in the classical examples of bending of a long rectangular body (beam-like geometry) and tension at infinity of a 2D body with a small circular hole.