Abstract
A mathematical model is presented that allows calculating longitudinal deformations of rods having various local changes in the size of cross sections caused by damage or defects. A distinctive feature of the proposed model is that a section of the rod with damage in the form of a smooth or sharp (crack) change in cross-section dimensions is replaced by a section with a constant cross-sectional area. In this area, the size of the cross-section decreases, since the resulting damage reduces the stiffness of the rod. A method is described for determining the geometric parameters of the part of the rod that has a smaller cross-section and is introduced to simulate the damage or defects that have occurred. For a rod with symmetrically arranged cracks in one of its sections and a variable cross-section width in one of the sections, the main provisions of the method and the calculation algorithm are described in detail. The calculation scheme of the proposed model and the ratios on the basis of which calculations are carried out are presented. The main condition on which the calculation method is based is the equality of the deformations of the rod, determined by the formulas of resistance of materials and by the relations of the theory of elasticity. The peculiarity and main advantage of the relations used is that the equation on the basis of which calculations are carried out does not include the modulus of elasticity of the material and the parameters of the stress state or external forces. Therefore, it can be used in modeling a rod with varying thickness and cracks, regardless of what magnitude of stress occurs in the rod. The described method allows you to calculate the deformation (length change) a rod with a crack and with variable cross-section sizes, and then, using this information, determine the stress state. The advantages of such a model are manifested, for example, when a crack occurs in an element of a complex statically indeterminate system. Based on calculations, it is shown that using the described method, it is possible to determine with high accuracy the deformation of rods with defects and with varying cross-sectional dimensions using the formulas of resistance of materials. According to the theory of elasticity, only a fragment of it needs to be calculated.
Published Version
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