ПРО ТОЧНІ РОЗВ’ЯЗКИ РІВНЯНЬ ОСІ Й КУТА ПОВОРОТУ ПЕРЕРІЗУ ПРУЖНОЇ БАЛКИ

Abstract

Calculation of elastic structures for strength and stiffness is important for their safe operation. The need for calculations is caused by structural deformations under the influence of external forces or temperature.The objects of our research are the equation of the axis of an elastic beam during its flat bending, as well as the equation of the angle of rotation of the beam section. These equations are differential. A certain complexity of their solution led to the simplification of equations in classical sources of information. Approximate solutions of these equations are considered there.But when using approximate calculation methods, you need to be able to assess their accuracy, that is, the degree of approximation to the exact result.The goal of our research was to obtain exact solutions. The exact solution of the beam equations is important to avoid its critical deformations.The article presents the exact analytical solutions we obtained for the exact equations of the bent axis of the beam and the angle of rotation of the beam section. The advantage of the exact solution was revealed, in particular, in the fact that the largest value of the deflection and angle of rotation of the beam section can be obtained directly from the properties of thefunctions that describe the solution.Another advantage of the exact solution was the possibility of obtaining an approximate solution with a predetermined accuracy. In the mentioned classical sources of information, the assessment of accuracy was derived from the range of the maximum deflection of the axis during practical calculations of structures.We have obtained a method of analytical assessment of the accuracy of the obtained solutions. The ability to assess the accuracy of calculation results is an important aspect of their practical application. This is important, in particular, for check-ing and clarifying the safe range of movements of beam points during its operation.An example of calculating the deflection and angle of rotation of the beam cross-section with a given load on the axis of the beam is shown