# Analytical and numerical analysis of shafts’ stress and strain states of the hydro-power unit

## Abstract

The paper presents both the analytical and numerical procedure for the analysis of the stress and strain states of the hydropower unit of the hydropower plant. Numerical analysis allows a comprehensive view of the states of stress and strain of the observed shaft. The analytical procedure is very difficult to apply to structural elements with complex geometry such as the shaft of a hydropower unit. Analytical analysis of stress and strain states serves to test only some of the results obtained on the basis of numerical analysis. A comparative analysis of stress and deflection values was performed on some characteristic shaft cross sections. Analytical determination of deflection, due to the complexity of the shaft construction, was performed on the basis of Mohr’s graph – analytical procedure. The numerical procedure is based on the application of the finite element method. The obtained results confirm the extremely good match of the results obtained by analytical and numerical procedures. Within the numerical analysis based on the given load and boundary conditions, ie. on the basis of the formed computational model of the shaft, the analysis of stress and strain, and displacement, was performed. The analysis of both normal and tangential stresses, as well as equivalent stresses according to the Von-Misses criterion, was performed. The places of maximum stresses are located. Therefore, it is concluded that the numerical analysis of the stress and strain state will give quality results if the computational model is well formed. This means that it is necessary to take into account all the discontinuities in the geometry of the shaft, which is characterized by a variable cross section. Certainly, there is no need to apply a very, very demanding analytical procedure for integrating the differential equation of an elastic line or a procedure based on Mohr’s graph – analytical procedure when calculating the shaft.

## References

1. Katedra za otpornost konstrukcija: "Tablice iz Otpornosti materijala", Mašinski fakultet, Univerzitet u Beogradu, 2015.,
2. Pejović R., Otpornost materijala, Građevinski fakultet, Univerzitet Crne gore, 2015.,
3. Bachschmid N, Cracked Rotors - A survey on static and dynamic behaviour including modeling and diagnosis, Springer, pp 1-13, 2010.,
4. Vullo, V., Vivio, F., Rotors Stress Analysis and Design, Mechanical Engineering Series, Springer, pp. 205, 2013.,
5. Ishida, Y., Cracked Rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcot rotor, Mechanical systems and signal processing 22(4), pp. 805-817, 2008.,
6. Ćulafić, S., Numeričko-eksperimentalna analiza čvrstoće strukturnih elemenata hidroelektrane, doktorska disertacija, Mašinski fakultet, Univerzitet u Beogradu, 2021