Научная электронная библиотека ULRICH'S Periodicals Directory

Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

Сибирский аэрокосмический журнал
ISSN 2712-8970 (Print) ISSN 2782-5760 (on-line)

Сведения об образовательной организации

UDK 539.3 Doi: 10.31772/2712-8970-2021-22-3-432-451
Generalized equivalent strength conditions in the calculations of composite bodies
Matveev A. D.
Institute of Computational Modeling SB RAS. 50/44, Akademgorodok, Krasnoyarsk, 630036, Russian Federation
Structures with an inhomogeneous regular structure (plates, beams, shells) are widely used in engineer-ing, especially in aviation and rocket and space. It is important to know the solution error in the strength elastic calculations for composite structures using the finite element method (FEM),. To analyze the error of the solution, it is necessary to use a sequence of approximate solutions constructed according to the FEM using the grinding procedure for basic discrete models that take into account the non-homogeneous, micro-homogeneous structure of structures (bodies) within the micro-approach. The implementation of the grinding procedure for basic models requires large computer resources. This paper deals with the method of equivalent strength conditions (MESC) for testing the static strength of elastic bodies with an inhomogeneous regular structure, for which sets of different loads are given. According to the MESC, the calculation of the strength of a composite body for which the loading is set is reduced to the calculation of the strength of an isotropic homogeneous body (having the same loading as a composite body) using equivalent strength conditions. In the numerical implementation of the MESC, adjusted equivalent strength conditions are used, which take into account the error of approximate solu-tions. Here, the MESC is implemented on the basis of the FEM. If a set of different loads is specified for a composite body, then generalized equivalent strength conditions are applied in this case. The procedure for constructing generalized equivalent strength conditions is shown. The calculation of the strength of com-posite bodies according to the MESC using multigrid finite elements requires times less com-puter memory than a similar calculation using crushed basic models of composite bodies. The given exam-ple of calculating the strength of a composite beam, for which a number of loads is set with MESC using generalized equivalent strength conditions shows its high efficiency.
Keywords: elasticity, composites, multigrid finite elements, corrected and generalized equivalent strength conditions.
References

1. Pisarenko G. S., Yakovlev A. P., Matveev V. V. Spravochnik po soprotivleniyu materialov [Handbook of resistance materials']. Kiev, Nauk. Dumka Publ., 1975, 704 p.

2. Birger I. A., Shorr B. F., Iosilevich G. B. Raschet na prochnost' detalej mashin [Calculation of the strength of machine parts]. Moscow, Mashinostroenie Publ., 1993, 640 p.

3. Moskvichev V. V. Osnovy konstrukcionnoj prochnosti tekhnicheskih sistem i inzhenernyh so-oruzhenij [Fundamentals of structural strength of technical systems and engineering structures]. Novosibirsk, Nauka Publ., 2002, 106 p.

4. Matveev A. D. [Calculation of elastic structures using the adjusted terms of strength]. Izvestiya AltGU. 2017, No. 4, P. 116–119. Doi: 10.14258/izvasu(2017)4-21.

5.    Norri D., de Friz Zh. Vvedenie v metod konechnykh elementov [Introduction to the finite element method]. Moscow,Mir Publ., 1981, 304 p.

6.    Zenkevich O. Metod konechnykh elementov v tekhnike [Finite element method in engineering]. Moscow, Mir Publ., 1975, 544 p.

7.    Fudzii T., Dzako M. Mekhanika razrusheniya kompozicionnyh materialov [Fracture mechanics of composite materials].Moscow, Mir Publ., 1982, 232 р.

8. Matveev A. D. [The method of multigrid finite elements in the calculations of three-dimensional homogeneous and composite bodies]. Uchen. zap. Kazan. un-ta. Seriia: Fiz.-matem. Nauki. 2016, Vol. 158, No. 4, P. 530–543 (In Russ.).

9. Matveev A. D. [Multigrid method for finite elements in the analysis of composite plates and beams]. Vestnik KrasGAU. 2016, No. 12, P. 93–100 (In Russ.).

10. Matveev A. D. Multigrid finite element method in stress of three-dimensional elastic bodies of heterogeneous structure. IOP Conf, Ser.: Mater. Sci. Eng. 2016, Vol. 158, No. 1, Art. 012067, P. 1–9.

11. Matveev A.D. Metod mnogosetochnyh konechnyh elementov v raschetah kompozitnyh plastin i balok slozhnoj formy [Multigrid finite element Method in the calculations of composite plates and beams of irregular shape]. // The Bulletin of KrasGAU, 2017, No. 11, P. 131–140.

12. Matveev A. D. [Multigrid finite element Method]. The Bulletin of KrasGAU. 2018, No. 2, P. 90–103 (In Russ.).

13. Matveev A. D. [The method of. multigrid finite elements of the composite rotational and bi-curved shell calculations]. The Bulletin of KrasGAU. 2018, No. 3, P. 126–137 (In Russ.).

14. Matveev A. D.  [Method of. multigrid finite elements to solve physical boundary value problems]. Ministry of information technologies and mathematical modeling. Krasnoyarsk, 2017, P. 27–60.

15. Matveev A. D. [Some approaches of designing elastic multigrid finite elements]. VINITI Proceedings. 2000, No. 2990-B00, P. 30.

16. Matveev A. D. [Multigridmodelingof composites of irregular structure with a small filling ratio]. J. Appl. Mech. Tech. Phys.2004, No. 3, P. 161–171 (In Russ.).

17. Matveev A. D. [The construction of complex multigrid finite element heterogeneous and micro-inhomogeneities in structure]. Izvestiya AltGU. 2014, No. 1/1, P. 80–83 (In Russ.). Doi: 10.14258/izvasu(2014)1.1-18.

18. Matveev A. D. [Method of generating finite elements].The Bulletin of KrasGAU. 2018, No. 6, P. 141–154 (In Russ.). 

19. Matveev A. D. [Construction of multigrid finite elements to calculate shells, plates and beams based on generating finite elements]. PNRPU Mechanics Bulletin. 2019, No. 3, P. 48–57 (In Russ.). Doi: 10/15593/perm.mech/2019.3.05.

20.Golushko S. K., Nemirovskij Y. V. Pryamye i obratnye zadachi mekhaniki uprugih kompozitnyh plastin i obolochek vrashcheniya [Direct and inverse problems of mechanics of elastic composite plates and shells of rotation]. Moscow, Fizmatlit Publ., 2008, 432 p.

21.Nemirovskij Y. V., Reznikov B. S. Prochnost' elementov konstrukcij iz kompozitnyh materiallov [Strength of structural elements made of composite materials]. Novosibirsk, Nauka Publ., Sibirskoe ot-delenie. 1984, 164 p.

22.Kravchuk A. S., Majboroda V. P., Urzhumcev Y. S. Mekhanika polimernyh i kompozicionnyh materialov [Mechanics of polymer and composite materials]. Moscow, Nauka Publ., 1985, 201 p.

23.Alfutov N. A., Zinov'ev A. A., Popov B. G. Raschet mnogoslojnyh plastin i obolochek iz kompozicionnyh materialov [Calculation of multilayer plates and shells made of composite materials]. Moscow, Mashinostroenie Publ., 1984, 264 p.

24.Pobedrya B. E. Mekhanika kompozicionnyh materialov [Mechanics of composite materials]. Moscow, MGU Publ., 1984, 336 p.

25.Andreev A. N., Nemirovskij Y. V. Mnogoslojnye anizotropnye obolochki i plastiny. Izgib, ustojchivost’, kolebaniya [Multilayer anisotropic shells and plates. Bending, stability, vibration]. Novosibirsk : Nauka Publ., 2001, 288 p.

26.Vanin G.A. Mikromekhanika kompozicionnyh materialov [Micromechanics of composite materials]. Kiev, Naukova dumka Publ., 1985, 302 p.

27.Vasil’ev V. V. Mekhanika konstrukcij iz kompozicionnyh materialov [Mechanics of structures made of composite materials]. Moscow, Mashinostroenie Publ., 1988, 269 p.

28.Matveev A. D. [Calculation of the strength of composite structures using equivalent strength conditions]. The Bulletin of KrasGAU. 2014, No. 11, P. 68–79 (In Russ.).

29.Matveev A. D. [The method of equivalent strength conditions in calculating composite structures regular structure using multigrid finite elements]. Siberian Journal of Science and Technology. 2019. Vol. 20, No. 4, P. 423-435. Doi: 10.31772/2587-6066-2019-20-4-423-435.

30.Samul’ V. I.Osnovy teorii uprugosti i plastichnosti [Fundamentals of the theory of elasticity and plasticity]. Moscow,Vysshaia shkola Publ., 1982, 264 p.


Matveev Alexander Danilovich – Cand. Sc., associate Professor, senior researcher; Institute of computational modeling SB RAS. E-mail: mtv241@mail.ru.