Determination of internal efforts in the base finite elements of SAFEM

Abstract

The principles of calculating the internal efforts of a circular finite element in the semi-analytical finite element method (FEM) based on the obtained components of the stress tensor and the peculiarities of the approach associated with the use of the moment scheme of the finite element (FEM) are considered. Formulas for determining longitudinal, shear forces, bending and torque moments have been obtained.
 A special place, among the variety of objects considered with the help of analytical and numerical methods, is occupied by bodies of revolution of complex shape and cross-sectional structure, formed by the movement of some creative surface along a closed or opened line without breaks. The selected geometric class is used as natural structures of nodes and details in construction of mechanical engineering. The sufficiently wide distribution of the specified forms in the construction and machine-building industries, on the one hand, and the possibility of significantly simplifying the solving relationships by taking into account their geometric features, on the other hand, provide a basis for the development and use of various modifications of the finite element method (FEM). The semi-analytical finite element method (SAFEM) is one such approach that has gained widespread use for solving problems whose objects are prismatic bodies and bodies of revolution of complex shape and cross-sectional structure. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation, the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFE), on the one hand, it is possible to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element (FE), on the other hand - maintain the high efficiency of 3D discretization.
 Despite the large number of publications devoted to the semi-analytical method of finite elements, the question of determining internal forces, which are often component factors of the strength criteria laid down in state building codes, is inappropriately neglected. The use of SAFEM in combination with МSFЕ creates some mathematical features of calculating internal longitudinal, shearing forces and moments.