РАСЧЕТ НЕСУЩЕЙ СПОСОБНОСТИ ИЗГИБАЕМЫХ СЛОИСТО-ВОЛОКНИСТЫХ КОЛЬЦЕВЫХ ПЛАСТИН, КОНТАКТИРУЮЩИХ С АГРЕССИВНОЙ НЕСЖИМАЕМОЙ ЖИДКОСТЬЮ. 1. ПОСТАНОВКА ЗАДАЧИ И МЕТОД РЕШЕНИЯ
Ключевые слова:
кольцевая слоистая пластина, армирование, составная конструкция, контакт с агрессивной жидкостью, жесткопластические материалы, кусочно-линейные условия текучести, пластическая анизотропия, предельное равновесие, деградация пластических свойств
Аннотация
Сформулирована задача на условный экстремум, позволяющая рассчитывать верхнюю границу предельной нагрузки, выдерживаемой изгибаемой слоистой кольцевой пластиной, контактирующей с агрессивной несжимаемой жидкостью. Внутреннее отверстие в составной конструкции плотно закрыто абсолютно жесткой вставкой. Пластина может быть соединена со вставкой шарнирно или жестко. Внешняя кромка конструкции защемлена или шарнирно оперта; допустимо ее вертикальное смещение: подвижная заделка или шарнир. В предельном состоянии пластины жидкость не может перетекать с одной ее лицевой поверхности на другую. Слои в пластине имеют переменную толщину и по-разному усилены волокнами. В каждом слое структура армирования является радиально- и осесимметричной. Материалы фаз композиции жесткопластические, имеющие разные пределы текучести при растяжении и сжатии; связующее может быть ортотропным. Пластическое течение в компонентах композиции ассоциировано с модифицированными условиями текучести Треска. В предельном состоянии кинематика составной конструкции описывается уравнениями классической теории изгиба пластин. Рассмотрены схемы усиления каждого слоя: 1) два семейства арматуры из одного и того же материала уложены по радиально симметричным спиралям; 2) в радиальном и/или окружном направлениях (материалы волокон при этом могут быть различны). Выполнена дискретизация поставленной задачи, что позволило свести ее к проблеме теории линейного программирования. Показано, что получающееся численное решение является неполным, так как не позволяет определить величину отпора жидкости в предельном состоянии конструкции; несмотря на это, несущую способность пластины все-таки удается рассчитать с точностью порядка 1%.
Поддерживающие организации
Выполнено в рамках государственного задания (№ госрегистрации 124021400036-7).
Литература
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15. Nemirovsky Yu.V., Romanova T.P. Nesushchaya sposobnost' usilennykh ledyanykh kruglykh plastin [Load-bearing capacity of reinforced ice round plates]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2011. Iss. 73. P. 25–34 (In Russian).
16. Jahangirov A.A. Nesushchaya sposobnost' usilennoy trekhsloynoy voloknistoy krugloy plastinki, zashchemlennoy po konturu i nakhodyashcheysya na neszhimayemoy srede [Carrying capacity of reinforced three layers circular composite plate clamped on edge and lying on non-compressible foundation]. Mekhanika mashin, mekhanizmov i materialov [Mechanics of Machines, Mechanisms and Materials]. 2015. No 4 (33). P. 50–54 (In Russian).
17. Cheng Wang, Tonghui Yang, Wan Li, Li Tao, Abuziarov M.Kh., Kochetkov A.V. Modelirovaniye uprugoplasticheskogo deformirovaniya elementov prostranstvennykh konstruktsiy pri impul'snom vzaimodeystvii s zhidkost'yu na osnove metoda Godunova povyshennoy tochnosti [Modeling of elastic-plastic deformation of elements of spatial structures during pulse interaction with fluid based on the Godunov's method of increased accuracy]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2019. Vol. 81. No 4. P. 489–500 (In Russian).
18. Yankovskii A.P. Opredeleniye verkhney granitsy nesushchey sposobnosti izgibayemykh armirovannykh metallokompozitnykh kol'tsevykh plastin, kontaktiruyushchikh s zhidkoy neszhimayemoy sredoy. 1. Postanovka zadachi i metod rascheta [Determination of the upper limit of the load-loading capacity of bending reinforced metal-composite ring plates in contact with a liquid incompressible medium. 1. Problem formulation and calculation method]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2024. Vol. 86. No 3. P. 270–284 (In Russian).
19. Yankovskii A.P. Opredeleniye verkhney granitsy nesushchey sposobnosti izgibayemykh armirovannykh metallokompozitnykh kol'tsevykh plastin, kontaktiruyushchikh s zhidkoy neszhimayemoy sredoy. 2. Analiz rezultatov raschetov [Determination of the upper limit of the load-loading capacity of bending reinforced metal-composite ring plates in contact with a liquid incompressible medium. 2. Analysis calculation results]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2024. Vol. 86. No 4. P. 445–460 (In Russian).
20. Yankovskii A.P. Chislennoye opredeleniye nesushchey sposobnosti armirovannykh kol'tsevykh plastin, pokoyashchikhsya na neszhimayemom zhidkom osnovanii i po-raznomu soprotivlyayushchikhsya rastyazheniyu i szhatiyu [Numerical determination of the bearing capacity of reinforced annular plates resting on an incompressible liquid base and differently resistant to tension and compression]. Vychislitelnaya mekhanika sploshnykh sred [Comput. Contin. Mech]. 2025. Vol. 18. No 2. P. 155–172 (In Russian).
21. Zuhovickii S.I., Avdeeva L.I. Linejnoe i vypukloe programmirovanie [Linear and Convex Programming]. Moscow. Nauka Publ. 1964. 348 p. (In Russian).
22. Grigorenko Ya.M. Izotropnyye i anizotropnyye sloistyye obolochki vrashcheniya peremennoy zhestkosti [Isotropic and Anisotropic Layered Shells of Rotation of Variable Rigidity]. Kiev. Naukova dumka Publ. 1973. 228 p. (In Russian).
23. Lokoshchenko A.M. Polzuchest i dlitelnaya prochnost metallov [Creep and Long-Term Strength of Metals]. Moscow. Fizmatlit Publ. 2016. 504 p. (In Russian).
24. Hu L.W. Modified Tresks's yield condition and associated flow rules for anisotropic materials and applications. J. Franclin Inst. 1958. Vol. 265. No 3. P. 187–204.
25. Ramu S.A., Iyengar K.J. Plastic response of orthotropic spherical shells under blast loading. Nuclear Engineering and Design. 1979. Vol. 55. No 3. P. 363–373.
26. Chakrabarty J. Applied Plasticity. New York. Springer. 2010. 755 p.
27. Ishlinskiy A.Yu., Ivlev D.D. Matematicheskaya teoriya plastichnosti [Mathematical Theory of Plasticity]. Moscow. Fizmatlit Publ. 2001. 707 p. (In Russian).
28. Romanova T.P., Yankovskii A.P. Piecewise-linear yield loci of angleply reinforced medium of different-resisting rigid-plastic materials at 2D stress state. Mechanics of Solids. 2020. Vol. 55. No 8. P. 1235–1252.
2. Kompozitsionnye materialy. Spravochnik [Composite Materials. Reference Book]. Ed. D.M. Karpinos. Kiev. Naukova Dumka Publ. 1985. 592 p. (In Russian).
3. Braestrup M.W. Concrete plasticity – A historical perspective. Struct. Concr. 2021. Vol. 22. No 5. P. 2508–2525. DOI:10.1002/suco.202100444.
4. Khazov P.A., Vedyaykina O.I., Pomazov A.P., Kozhanov D.A. Uprugoplasticheskoe deformirovanie stalebetonnykh balok s lokalnym smyatiem pri trekhtochechnom izgibe [Elastic-plastic deformation of steel-concrete beams with local crumpling during three-point bending]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2024. Vol. 86. No 1. P. 71–82 (In Russian).
5. Mroz Z., Shamiev F.G. Simplified yield conditions for fibre-reinforced plates and shells. Arch. Inz. Lad. 1979. Vol. 25. No 3. P. 463–476.
6. Brassart L., Stainier L., Doghri I., Delannay L. Homogenization of elasto-(visco) plastic composites based on an incremental variational principle. Int. J. Plast. 2012. Vol. 36. P. 86–112. https://doi.org/10.1016/J.IJPLAS.2012.03.010.
7. Erkhov M.I. Teoriya idealno plasticheskikh tel i konstruktsiy [Theory of Ideally Plastic Bodies and Structures]. Moscow. Nauka Publ. 1978. 352 p. (In Russian).
8. Vasiliev V.V., Morozov E. Advanced Mechanics of Composite Materials and Structural Elements. Amsterdam. Elsevier. 2013. 412 p.
9. Romanova T.P. Modelirovanie dinamicheskogo izgiba zhestkoplasticheskikh armirovannykh sloistykh kruglykh plastin s proizvolnym otverstiem na vyazkom osnovanii pri vzryvnykh nagruzkakh [Modeling of rigid-plastic dynamic bending of reinforced layered circular plates with arbitrary hole on viscous foundation under explosive loads]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2017. Vol. 79. No 3. P. 267–284 (In Russian).
10. Ho P.L.H., Le C.V., Tran-Cong T. Limit state analysis of reinforced concrete stabs using an integrated radial basis function based mesh-free method. Appl. Math. Model. 2018. Vol. 53. P. 2168-1–2168-11 DOI: 10.1016/j.apm.2017.08.006.
11. Hosseini H.R., Amoushahi H., Azhari M. Nonlinear effect of low-velocity impact on tapered laminated composite structures using spline finite strip method. Structure. 2022. Vol. 39. P. 739–764. https://doi.org/10.1016/j.istruc.2022.03.051.
12. Kumar S., Rai A., Jain N. Yield line analysis of two way reinforced concrete stab of three-sides supported with various loading conditions. IOP Conference Series: Earth and Environmental Science. 2024. Vol. 1326. No 1. Article No 012034. DOI: 10.1088/1755-1315/1326/1/012034.
13. Romanova T.P. Mathematical modeling the limit and dynamic behavior of rigid plastic reinforced laminated annular plates with free parts of contour under explosive loads. Mechanics of Advanced Materials and Structures. 2026. Vol. 33. Iss. 1. Article No 2588655. DOI: 10.1080/15376494.2025.2588655.
14. Hodge P.G., Chang-Kuei Sun. Yield-point load of a circular plate seating an incompressible fluid. Int. J. Mech. Sci. 1967. Vol. 9. Iss. 7. P. 405–414.
15. Nemirovsky Yu.V., Romanova T.P. Nesushchaya sposobnost' usilennykh ledyanykh kruglykh plastin [Load-bearing capacity of reinforced ice round plates]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2011. Iss. 73. P. 25–34 (In Russian).
16. Jahangirov A.A. Nesushchaya sposobnost' usilennoy trekhsloynoy voloknistoy krugloy plastinki, zashchemlennoy po konturu i nakhodyashcheysya na neszhimayemoy srede [Carrying capacity of reinforced three layers circular composite plate clamped on edge and lying on non-compressible foundation]. Mekhanika mashin, mekhanizmov i materialov [Mechanics of Machines, Mechanisms and Materials]. 2015. No 4 (33). P. 50–54 (In Russian).
17. Cheng Wang, Tonghui Yang, Wan Li, Li Tao, Abuziarov M.Kh., Kochetkov A.V. Modelirovaniye uprugoplasticheskogo deformirovaniya elementov prostranstvennykh konstruktsiy pri impul'snom vzaimodeystvii s zhidkost'yu na osnove metoda Godunova povyshennoy tochnosti [Modeling of elastic-plastic deformation of elements of spatial structures during pulse interaction with fluid based on the Godunov's method of increased accuracy]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2019. Vol. 81. No 4. P. 489–500 (In Russian).
18. Yankovskii A.P. Opredeleniye verkhney granitsy nesushchey sposobnosti izgibayemykh armirovannykh metallokompozitnykh kol'tsevykh plastin, kontaktiruyushchikh s zhidkoy neszhimayemoy sredoy. 1. Postanovka zadachi i metod rascheta [Determination of the upper limit of the load-loading capacity of bending reinforced metal-composite ring plates in contact with a liquid incompressible medium. 1. Problem formulation and calculation method]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2024. Vol. 86. No 3. P. 270–284 (In Russian).
19. Yankovskii A.P. Opredeleniye verkhney granitsy nesushchey sposobnosti izgibayemykh armirovannykh metallokompozitnykh kol'tsevykh plastin, kontaktiruyushchikh s zhidkoy neszhimayemoy sredoy. 2. Analiz rezultatov raschetov [Determination of the upper limit of the load-loading capacity of bending reinforced metal-composite ring plates in contact with a liquid incompressible medium. 2. Analysis calculation results]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2024. Vol. 86. No 4. P. 445–460 (In Russian).
20. Yankovskii A.P. Chislennoye opredeleniye nesushchey sposobnosti armirovannykh kol'tsevykh plastin, pokoyashchikhsya na neszhimayemom zhidkom osnovanii i po-raznomu soprotivlyayushchikhsya rastyazheniyu i szhatiyu [Numerical determination of the bearing capacity of reinforced annular plates resting on an incompressible liquid base and differently resistant to tension and compression]. Vychislitelnaya mekhanika sploshnykh sred [Comput. Contin. Mech]. 2025. Vol. 18. No 2. P. 155–172 (In Russian).
21. Zuhovickii S.I., Avdeeva L.I. Linejnoe i vypukloe programmirovanie [Linear and Convex Programming]. Moscow. Nauka Publ. 1964. 348 p. (In Russian).
22. Grigorenko Ya.M. Izotropnyye i anizotropnyye sloistyye obolochki vrashcheniya peremennoy zhestkosti [Isotropic and Anisotropic Layered Shells of Rotation of Variable Rigidity]. Kiev. Naukova dumka Publ. 1973. 228 p. (In Russian).
23. Lokoshchenko A.M. Polzuchest i dlitelnaya prochnost metallov [Creep and Long-Term Strength of Metals]. Moscow. Fizmatlit Publ. 2016. 504 p. (In Russian).
24. Hu L.W. Modified Tresks's yield condition and associated flow rules for anisotropic materials and applications. J. Franclin Inst. 1958. Vol. 265. No 3. P. 187–204.
25. Ramu S.A., Iyengar K.J. Plastic response of orthotropic spherical shells under blast loading. Nuclear Engineering and Design. 1979. Vol. 55. No 3. P. 363–373.
26. Chakrabarty J. Applied Plasticity. New York. Springer. 2010. 755 p.
27. Ishlinskiy A.Yu., Ivlev D.D. Matematicheskaya teoriya plastichnosti [Mathematical Theory of Plasticity]. Moscow. Fizmatlit Publ. 2001. 707 p. (In Russian).
28. Romanova T.P., Yankovskii A.P. Piecewise-linear yield loci of angleply reinforced medium of different-resisting rigid-plastic materials at 2D stress state. Mechanics of Solids. 2020. Vol. 55. No 8. P. 1235–1252.
Опубликован
2026-10-07
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Статьи
Copyright (c) 2026 А.П. Янковский Yankovskii
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