DOI: https://doi.org/10.15587/1729-4061.2019.169631

Calculation of stress concentrations in orthotropic cylindrical shells with holes on the basis of a variational method

Valentin Salo, Valeriia Rakivnenko, Vladimir Nechiporenko, Aleksandr Kirichenko, Serhii Horielyshev, Dmytro Onopreichuk, Volodymyr Stefanov

Abstract


A variational numerical-analytical method (called the RVR method) is suggested for calculating the strength and stiffness of statically loaded non-thin orthotropic shell structures weakened by holes (stress concentrators) of arbitrary shapes and sizes. The theoretically substantiated new method is based on the Reissner variational principle and the method of I. N. Vekua (the method of decomposing the desired functions into the Fourier series of the orthogonal Legendre polynomials with respect to the coordinate along the constant shell thickness). In this case, the use in the proposed RVR method of the general equations of three-dimensional problems of the linear theory of elasticity makes it possible to determine the total stress-strained state of an elastic shell (in particular, a plate) with holes. At the same time, using the R-functions, at the analytical level, the geometric information of boundary-value problems for multiply connected domains is taken into account and solutions structures are constructed that exactly satisfy different variants of boundary conditions. The use of a software-implemented algorithm for the two-sided integral assessment of the accuracy of approximate solutions in the study of mixed variational problems helps automate the search for such a number of approximations in which the process of convergence of solutions becomes stable.

For orthotropic and isotropic materials, the possibilities of the RVR method are shown in numerical examples of solving the corresponding boundary value problems of calculating the stress concentration in a cylindrical shell with an elliptical or rectangular hole under axial load. The results of the performed tests are discussed, and the features characteristic of the new method prove that it can be effectively used in the design of critical lamellar and shell elements of structures in various fields of modern technology.


Keywords


orthotropic shell with holes; stress concentration; Reissner principle; R-functions theory.

References


Rezaeepazhand, J., Jafari, M. (2010). Stress concentration in metallic plates with special shaped cutout. International Journal of Mechanical Sciences, 52 (1), 96–102. doi: https://doi.org/10.1016/j.ijmecsci.2009.10.013

Guz', A. N., Chernyshenko, I. S., Chekhov, Val. N., Chekhov, Vik. N., Shnerenko, K. I. (1980). Teoriya tonkih obolochek, oslablennyh otverstiyami. Vol. 1. Metody rascheta obolochek. Kyiv: Naukova dumka, 636.

Washizu, K. (1982). Variational methods in elasticity and plasticity. New York, 542.

Klochkov, Y. V., Nikolaev, A. P., Sobolevskaya, T. A., Klochkov, M. Y. (2018). Comparative analysis of efficiency of use of finite elements of different dimensionality in the analysis of the stress-strain state of thin shells. Structural Mechanics of Engineering Constructions and Buildings, 14 (6), 459–466. doi: https://doi.org/10.22363/1815-5235-2018-14-6-459-466

Li, J., Shi, Z., Liu, L. (2019). A scaled boundary finite element method for static and dynamic analyses of cylindrical shells. Engineering Analysis with Boundary Elements, 98, 217–231. doi: https://doi.org/10.1016/j.enganabound.2018.10.024

Ádány, S. (2016). Shell element for constrained finite element analysis of thin-walled structural members. Thin-Walled Structures, 105, 135–146. doi: https://doi.org/10.1016/j.tws.2016.04.012

Ádány, S. (2017). Constrained shell finite element method for thin-walled members with holes. Thin-Walled Structures, 121, 41–56. doi: https://doi.org/10.1016/j.tws.2017.09.021

Salo, V. A. (2003). Kraevye zadachi statiki obolochek s otverstiyami. Kharkiv: NTU «KhPI», 216.

Reissner, E. (1950). On a Variational Theorem in Elasticity. Journal of Mathematics and Physics, 29 (1-4), 90–95. doi: https://doi.org/10.1002/sapm195029190

Pramod, A. L. N., Natarajan, S., Ferreira, A. J. M., Carrera, E., Cinefra, M. (2017). Static and free vibration analysis of cross-ply laminated plates using the Reissner-mixed variational theorem and the cell based smoothed finite element method. European Journal of Mechanics - A/Solids, 62, 14–21. doi: https://doi.org/10.1016/j.euromechsol.2016.10.006

Faghidian, S. A. (2018). Reissner stationary variational principle for nonlocal strain gradient theory of elasticity. European Journal of Mechanics - A/Solids, 70, 115–126. doi: https://doi.org/10.1016/j.euromechsol.2018.02.009

Vekua, I. N. (1965). Teoriya tonkih pologih obolochek peremennoy tolschiny. Vol. 30. Tbilisi, 3–103.

Timoshenko, S., Woinowsky-Krieger, S. (1987). Theory of Plates and Shells. New York: McGraw-Hill Book Company, 580.

Salo, V. A., Nechiporenko, V. M. (2017). Research of durability of the elastic cylindrical structure affected by the local loading. Zbirnyk naukovykh prats Natsionalnoi akademiyi Natsionalnoi hvardiyi Ukrainy, 2, 76–82.

Salo, V. A., Litovchenko, P. I., Chizhikov, I. V. (2011). Napryazhenno-deformirovannoe sostoyanie uprugoy tsilindricheskoy paneli s otverstiem. Voprosy proektirovaniya i proizvodstva konstruktsiy letatel'nyh apparatov, 1, 63–70.

Salo, V. A. (2004). O kontsentratsii napryazheniy okolo otverstiya v uprugoy sfericheskoy obolochke. Voprosy proektirovaniya i proizvodstva konstruktsiy letatel'nyh apparatov, 2, 66–72.

Salo, V. A. (2000). Dokazatel'stvo dostatochnogo priznaka skhodimosti metoda Rittsa dlya smeshannogo variatsionnogo printsipa Reyssnera. Vestnik Har'kov. gos. politekh. un-ta, 95, 70–75.

Awrejcewicz, J., Kurpa, L., Shmatko, T. (2015). Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory. Composite Structures, 125, 575–585. doi: https://doi.org/10.1016/j.compstruct.2015.02.054

Nechyporenko, V. M., Salo, V. A., Litovchenko, P. I., Kovbaska, B. V., Verkhorubov, D. O. (2016). Vykorystannia teoriyi R-funktsiy dlia stvorennia ratsionalnykh posadok z natiahom. Zbirnyk naukovykh prats Natsionalnoi akademiyi Natsionalnoi hvardiyi Ukrainy, 2, 72–76.

Rodionova, V. A., Titaev, B. F., Chernyh, K. F. (1996). Prikladnaya teoriya anizotropnyh plastin i obolochek. Sankt-Peterburg: SpbGU, 278.

Rodionova, V. A. (1983). Teoriya tonkih anizotropnyh obolochek s uchetom poperechnyh sdvigov i obzhatiya. Leningrad: LGU, 116.

Salo, V. A. (2003). O dvustoronney otsenke tochnosti priblizhennyh resheniy zadach teorii obolochek, poluchennyh metodom Rittsa dlya neekstremal'nogo funktsionala Reyssnera. Dopovidi NAN Ukrainy, 1, 53–57.

Tennyson, R. S., Roberts, D. K., Zimcik, D. (1968). Analysis of the stress distribution around unreinforced cutouts in circular cylindrical shells under axial compressions. NRC, NASA, Annyal Progress Report, UTIAS.

Zirka, A. I., Chernopiskiy, D. I. (2001). Eksperimental'nye issledovaniya kontsentratsii napryazheniy v tolstyh tsilindricheskih obolochkah s pryamougol'nymi otverstiyami pri osevom szhatii. Prikladnaya mekhanika, 5, 133–135.


GOST Style Citations


Rezaeepazhand J., Jafari M. Stress concentration in metallic plates with special shaped cutout // International Journal of Mechanical Sciences. 2010. Vol. 52, Issue 1. P. 96–102. doi: https://doi.org/10.1016/j.ijmecsci.2009.10.013 

Teoriya tonkih obolochek, oslablennyh otverstiyami. Vol. 1. Metody rascheta obolochek / Guz' A. N., Chernyshenko I. S., Chekhov Val. N., Chekhov Vik. N., Shnerenko K. I. Kyiv: Naukova dumka, 1980. 636 p.

Washizu K. Variational methods in elasticity and plasticity. New York, 1982. 542 p.

Comparative analysis of efficiency of use of finite elements of different dimensionality in the analysis of the stress-strain state of thin shells / Klochkov Y. V., Nikolaev A. P., Sobolevskaya T. A., Klochkov M. Y. // Structural Mechanics of Engineering Constructions and Buildings. 2018. Vol. 14, Issue 6. P. 459–466. doi: https://doi.org/10.22363/1815-5235-2018-14-6-459-466 

Li J., Shi Z., Liu L. A scaled boundary finite element method for static and dynamic analyses of cylindrical shells // Engineering Analysis with Boundary Elements. 2019. Vol. 98. P. 217–231. doi: https://doi.org/10.1016/j.enganabound.2018.10.024 

Ádány S. Shell element for constrained finite element analysis of thin-walled structural members // Thin-Walled Structures. 2016. Vol. 105. P. 135–146. doi: https://doi.org/10.1016/j.tws.2016.04.012 

Ádány S. Constrained shell finite element method for thin-walled members with holes // Thin-Walled Structures. 2017. Vol. 121. P. 41–56. doi: https://doi.org/10.1016/j.tws.2017.09.021 

Salo V. A. Kraevye zadachi statiki obolochek s otverstiyami. Kharkiv: NTU «KhPI», 2003. 216 p.

Reissner E. On a Variational Theorem in Elasticity // Journal of Mathematics and Physics. 1950. Vol. 29, Issue 1-4. P. 90–95. doi: https://doi.org/10.1002/sapm195029190 

Static and free vibration analysis of cross-ply laminated plates using the Reissner-mixed variational theorem and the cell based smoothed finite element method / Pramod A. L. N., Natarajan S., Ferreira A. J. M., Carrera E., Cinefra M. // European Journal of Mechanics - A/Solids. 2017. Vol. 62. P. 14–21. doi: https://doi.org/10.1016/j.euromechsol.2016.10.006 

Faghidian S. A. Reissner stationary variational principle for nonlocal strain gradient theory of elasticity // European Journal of Mechanics - A/Solids. 2018. Vol. 70. P. 115–126. doi: https://doi.org/10.1016/j.euromechsol.2018.02.009 

Vekua I. N. Teoriya tonkih pologih obolochek peremennoy tolschiny. Vol. 30. Tbilisi, 1965. P. 3–103.

Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. New York: McGraw-Hill Book Company, 1987. 580 p.

Salo V. A., Nechiporenko V. M. Research of durability of the elastic cylindrical structure affected by the local loading // Zbirnyk naukovykh prats Natsionalnoi akademiyi Natsionalnoi hvardiyi Ukrainy. 2017. Issue 2. P. 76–82.

Salo V. A., Litovchenko P. I., Chizhikov I. V. Napryazhenno-deformirovannoe sostoyanie uprugoy tsilindricheskoy paneli s otverstiem // Voprosy proektirovaniya i proizvodstva konstruktsiy letatel'nyh apparatov. 2011. Issue 1. P. 63–70.

Salo V. A. O kontsentratsii napryazheniy okolo otverstiya v uprugoy sfericheskoy obolochke // Voprosy proektirovaniya i proizvodstva konstruktsiy letatel'nyh apparatov. 2004. Issue 2. P. 66–72.

Salo V. A. Dokazatel'stvo dostatochnogo priznaka skhodimosti metoda Rittsa dlya smeshannogo variatsionnogo printsipa Reyssnera // Vestnik Har'kov. gos. politekh. un-ta. 2000. Issue 95. P. 70–75.

Awrejcewicz J., Kurpa L., Shmatko T. Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory // Composite Structures. 2015. Vol. 125. P. 575–585. doi: https://doi.org/10.1016/j.compstruct.2015.02.054 

Vykorystannia teoriyi R-funktsiy dlia stvorennia ratsionalnykh posadok z natiahom / Nechyporenko V. M., Salo V. A., Litovchenko P. I., Kovbaska B. V., Verkhorubov D. O. // Zbirnyk naukovykh prats Natsionalnoi akademiyi Natsionalnoi hvardiyi Ukrainy. 2016. Issue 2. P. 72–76.

Rodionova V. A., Titaev B. F., Chernyh K. F. Prikladnaya teoriya anizotropnyh plastin i obolochek. Sankt-Peterburg: SpbGU, 1996. 278 p.

Rodionova V. A. Teoriya tonkih anizotropnyh obolochek s uchetom poperechnyh sdvigov i obzhatiya. Leningrad: LGU, 1983. 116 p.

Salo V. A. O dvustoronney otsenke tochnosti priblizhennyh resheniy zadach teorii obolochek, poluchennyh metodom Rittsa dlya neekstremal'nogo funktsionala Reyssnera // Dopovidi NAN Ukrainy. 2003. Issue 1. P. 53–57.

Tennyson R. S., Roberts D. K., Zimcik D. Analysis of the stress distribution around unreinforced cutouts in circular cylindrical shells under axial compressions // NRC, NASA, Annyal Progress Report, UTIAS. 1968.

Zirka A. I., Chernopiskiy D. I. Eksperimental'nye issledovaniya kontsentratsii napryazheniy v tolstyh tsilindricheskih obolochkah s pryamougol'nymi otverstiyami pri osevom szhatii // Prikladnaya mekhanika. 2001. Issue 5. P. 133–135.







Copyright (c) 2019 Valentin Salo, Valeriia Rakivnenko, Vladimir Nechiporenko, Aleksandr Kirichenko, Serhii Horielyshev, Dmytro Onopreichuk, Volodymyr Stefanov

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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061