MOHR'S FORMULA WHEN USED SYNTHESIS OF THE THEORY OF RODS AND THE THEORY OF ELASTICITY
Authors:
1.
Kazan National Research Technological University
(Department "Fundamentals of Design and Applied Mechanics", Professor)
employee from 01.01.1923 to 01.01.1927
Kazan National Research Technological University (Department "Fundamentals of Design and Applied Mechanics", Professor)
from 01.01.1923 to 01.01.1927
Kazan, Kazan, Russian Federation
employee from 01.01.1923 to 01.01.1927
Kazan National Research Technological University (Department "Fundamentals of Design and Applied Mechanics", Professor)
from 01.01.1923 to 01.01.1927
Kazan, Kazan, Russian Federation
Type:
Article
eLocation:
Status:
Published
Received:
07.08.2025
Accepted:
07.08.2025
Published:
07.08.2025
Language:
Russian
Keywords:
BEAM, BENDING, DAMAGE, DEFECTS, MATHEMATICAL MODEL, CALCULATION METHOD, THEORY OF RODS, THEORY OF ELASTICITY
Abstract (English):
The paper presents a mathematical model and a method for calculating the stress-strain state (SSS) of beams under bending based on the synthesis of the theory of rods and theory. The method is focused on solving problems of calculating rod elements of structures that have damage - local changes in geometric and mechanical characteristics. Such damage includes cracks, cavities and inhomogeneities in the material of the structure, variables of the dimensions of the cross sections of the rod. When finding a solution, the beam is divided into sections, for each of which the calculation of the SSS and displacements is found according to the theory of rods or the theory of elasticity. The relations of the theory of elasticity are used for those parts of the rod for which it is impossible to obtain reliable calculation data according to the theory of rods. The displacements arising in the beam are calculated by the Mohr method. The derivation of the Mohr formula using the relations of the theory of elasticity is given. It is noted that the choice of this method for determining displacements is associated with the simplicity of calculations and the widespread use in calculation practice of various methods, the relations of which are based on the Mohr formula. When calculating a separate section of a rod according to the theory of elasticity, it is necessary to take into account external forces acting on other sections and the conditions of fixing the beam. The paper proposes a method for taking into account such forces and fixing conditions. For this, the basics of the calculation scheme used in calculating beams under bending were used. The method is relatively simple to implement, since solutions are found separately for each of the selected sections. In this case, various analytical or numerical calculation methods can be used. An example of solving the problem is given, illustrating the features of the implementation of the described method, the reliability and accuracy of the obtained calculation data. Calculations show that, using the described models of beam deformation, it is possible to determine the stress-strain state of rod elements with defects and with changing cross-sectional dimensions quite simply and with high accuracy.
The paper presents a mathematical model and a method for calculating the stress-strain state (SSS) of beams under bending based on the synthesis of the theory of rods and theory. The method is focused on solving problems of calculating rod elements of structures that have damage - local changes in geometric and mechanical characteristics. Such damage includes cracks, cavities and inhomogeneities in the material of the structure, variables of the dimensions of the cross sections of the rod. When finding a solution, the beam is divided into sections, for each of which the calculation of the SSS and displacements is found according to the theory of rods or the theory of elasticity. The relations of the theory of elasticity are used for those parts of the rod for which it is impossible to obtain reliable calculation data according to the theory of rods. The displacements arising in the beam are calculated by the Mohr method. The derivation of the Mohr formula using the relations of the theory of elasticity is given. It is noted that the choice of this method for determining displacements is associated with the simplicity of calculations and the widespread use in calculation practice of various methods, the relations of which are based on the Mohr formula. When calculating a separate section of a rod according to the theory of elasticity, it is necessary to take into account external forces acting on other sections and the conditions of fixing the beam. The paper proposes a method for taking into account such forces and fixing conditions. For this, the basics of the calculation scheme used in calculating beams under bending were used. The method is relatively simple to implement, since solutions are found separately for each of the selected sections. In this case, various analytical or numerical calculation methods can be used. An example of solving the problem is given, illustrating the features of the implementation of the described method, the reliability and accuracy of the obtained calculation data. Calculations show that, using the described models of beam deformation, it is possible to determine the stress-strain state of rod elements with defects and with changing cross-sectional dimensions quite simply and with high accuracy.
Keywords:
BEAM, BENDING, DAMAGE, DEFECTS, MATHEMATICAL MODEL, CALCULATION METHOD, THEORY OF RODS, THEORY OF ELASTICITY
BEAM, BENDING, DAMAGE, DEFECTS, MATHEMATICAL MODEL, CALCULATION METHOD, THEORY OF RODS, THEORY OF ELASTICITY
