## Four-point bending of thin or thick beams

Mathematical Modeling, Vol. 6 (2022), Issue 2, pg(s) 43-46

A paper recently published explains the differential equations for deflection of beams under bending, including the deflection due to transversal force [1]. The present article contains derivation of the main equations, according to the mentioned approach, for deformations of a simply supported beam that is symmetrically loaded with two forces, also known as four-point bending. These deformations are rotation and deflection of the neutral line due to bending moment and transversal force. For thick beams, deflection due to the transverse force is more than 1% from deflection caused by the bending moment. Special attention was paid to the third-point loading test. The presented model is applicable for calculating deflection due to bending moment and transverse force for both thin and thick beams.

## Study of the microstructure evolution during forging in a new design strikers implementing alternating deformation by finite element method

The study of the microstructure evolution during forging in a new design strikers implementing alternating deformation based on computer modeling by the finite element method was carried out. The influence of the flat face inclination angle was studied, for which models with an inclination angle of 0, 15, 30 and 45 degrees were studied. It was established that the most optimal option is the use of strikers with an angle of 30°. On this strikers configuration the influence of the workpiece heating temperatures and the punch movement speeds was studied. The analysis of these technological parameters on the microstructure evolution showed that both parameters affect the structure grinding intensity, while the influence of the heating temperature is more significant.

## Consistent Presentation of the Beam Deflection Theory Including Shear Correction

Mathematical Modeling, Vol. 5 (2021), Issue 4, pg(s) 120-123

This article explains a mathematically consistent approach for solving the equations of Timoshenko’s beam theory for statically loaded beams. Theoretic sections 3.4 – 3.5 give a good description of the shear deformation and the primary approach for calculating deflections of beams under bending, taking into account both causes for deflection: bending moment and shear force. Values for the shear correction factor are discussed in section 4. This work was started to check the validity of an equation for deflection of a symmetrically loaded short rectangular beam with span/height ratio = 3 under four-point bending with upper-span/span ratio = 1/3. The exact solution is not presented here, but we can confirm that the presented theory, when applied for the mentioned loading scheme, leads to thi s equation using a shear correction factor k = 5/6.

## Development and computer modeling of a new forging technology in step-wedge strikers

Machines. Technologies. Materials., Vol. 15 (2021), Issue 6, pg(s) 226-229

A computer simulation of the broaching of the workpiece in step-wedge-shaped strikers of the first and second configurations was carried out by the finite element method. The comparison was made according to the following parameters of the stress-strain state: equivalent strain, average hydrostatic pressure, damage criterion, as well as the deformation force. A comparative analysis of the stress – strain state showed that the second configuration of step-wedge-shaped strikers is the most optimal option for implementing the process of broaching rectangular or square blanks. When using them, a more uniform distribution of the stress state is realized, the deformation processing is sufficiently realized in comparison with the broaching in step-wedge-shaped strikers of the first configuration

## STUDY OF STRESS-STRAIN STATE AND TEMPERATURE FIELD DURING ROLLING UNDER THE NEW SCHEME WITH ALTERNATING AND SHEAR DEFORMATIONS

Mathematical Modeling, Vol. 3 (2019), Issue 2, pg(s) 55-58

In this paper, a computer simulation of a new technology of thick-sheet rolling, including rolling in rolls with a relief surface followed by rolling on rolls with a smooth barrel to the desired size. The analysis of effective plastic deformation, hydrostatic pressure and temperature field was carried out according to the results of modeling. According to the results of the analysis of effective strain, maximum of processing in the first pass receives the ridge area, but after the second pass observed alignment distribution of this parameter over the cross section. The study of the temperature field showed that the greatest temperature difference in the cross section occurs when rolling in relief rolls, in the future when rolling in smooth rolls due to the increase in the contact surface area, this difference decreases. Analysis of hydrostatic pressure showed the presence of both compressive and tensile stresses in the deformation zone. Such distribution is caused by the presence of a relief surface after 1 pass in the further alignment of the strip profile, which occurs both in the longitudinal and transverse directions.