VARIATIONS OF THE ENERGY METHOD FOR STUDYING CONSTRUCTION STABILITY

Objectives. The aim of the work is to find the most rational form of expression of the potential energy of a nonlinear system with the subsequent use of algebraic means and geometric images of catastrophe theory for studying the behaviour of a construction under load. Various forms of stability criteria for the equilibrium states of constructions are investigated. Some aspects of the using various forms of expression of the system’s total energy are considered, oriented to the subsequent use of the catastrophe theory methods for solving the nonlinear problems of construction calculation associated with discontinuous phenomena.

Methods. According to the form of the potential energy expression, the mathematical description of the problem being solved is linked to a specific catastrophe of a universal character from the list of catastrophes. After this, the behaviour of the system can be predicted on the basis of the fundamental propositions formulated in catastrophe theory without integrating the corresponding system of nonlinear differential equations of high order in partial derivatives, to which the solution of such problems is reduced.

Results. The result is presented in the form of uniform geometric images containing all the necessary qualitative and quantitative information about the deformation of whole construction classes under load for a wide range of changes in the values of external (control) and internal (behavioural) parameters.

Conclusion. Methods based on catastrophe theory are an effective mathematical tool for solving non-linear boundary-value problems with parameters associated with discontinuous phenomena, which are poorly analysable by conventional methods. However, they have not yet received due attention from researchers, especially in the field of stability calculations, which remains a complex, relevant and attractive problem within structural mechanics. To solve a concrete nonlinear boundary value problem for calculating structures by algebraic means and using geometric images of catastrophe theory, it is necessary to establish the connection between the mathematical description of the problem being solved, characterised by the functional of the variety of the energy method and universal problems solved on the basis of the fundamental provisions of catastrophe theory. Present work is an effort to revive interest in the methods of catastrophe theory and their use for solving various problems. 

1. Poston T., Styuart I. Teoriya katastrof i ee prilozheniya. M.: Mir; 1980. 607 s. [Poston T., Styuart I. Catastrophe theory and its applications. Moscow: Mir; 1980. 607 p. (in Russ.)]

2. Koiter W.T. The non-linear buckling problem of a complete spherical shell under uniform external pressure. Proc. K. ned. Akad. Wet., Ser. B. 1969;72:40.

3. Vol'mir A.S. Ustoychivost' deformiruemykh sistem. M.: Nauka; 1967. 984 s. [Vol'mir A.S. Stability of deforming systems. Moscow: Nauka; 1967. 984 s. (in Russ.)]

4. Thomson J. M. T., Hunt G. W. Elastic Instability Phenomena. London: Wiley; 1984.

5. Alfutov N.A. Osnovy rascheta na ustoychivost' uprugikh sistem. M.: Mashinostroenie; 1991. 336 s. [Alfutov N.A. Calculation fundamentals for stability of elastic systems Osnovy rascheta na ustoychivost' uprugikh sistem. Moscow: Mashinostroenie; 1991. 336 p. (in Russ.)]

6. Murtazaliev G.M. Metody teorii katastrof v zadachakh ustoychivosti obolochek. DGTU. Makhachkala; 2004. 200 s. [Murtazaliev G.M. Methods of catastrophe theory in shell stability problems. DGTU. Makhachkala; 2004. 200 p. (in Russ.)]

7. Mukharlyamov R. G., Amabili M., Garziera R., Riabova K. Ustoychivost' nelineynykh kolebaniy pologikh obolochek dvoynoy krivizny. Vestnik Rossiiskogo universiteta druzhby narodov. Seriya: Matematika. Informatika. Fizika. 2016;2:53-63. [Mukharlyamov R. G., Amabili M., Garziera R., Riabova K. Stability of non-linear vibrations of doubly curved shallow shells. RUDN Journal of Mathematics, Information Sciences and Physics. 2016;2:53-63. (in Russ.)]

8. Bazhenov V. A., Krivenko O. P., Solovey N. A. Nelineynoe deformirovanie i ustoychivost' uprugikh obolochek neodnorodnoy struktury. Modeli, metody, algoritmy, maloizuchennye i novye zadachi. Moskva: Knizhnyy dom "LIBROKOM"; 2013. 329 s. [Bazhenov V. A., Krivenko O. P., Solovey N. A. Non-linear deformation and stability of elastic shells with heterogeneous structure. Models, methods, algorithms, poorly-studied and new problems. Moscow: Knizhnyy dom "LIBROKOM"; 2013. 329 p. (in Russ.)]

9. Bazhenov V. G., Gonik E. G., Kibets A. I., Shoshin D. V. Ustoychivost' i predel'nye sostoyaniya uprugoplasticheskikh sfericheskikh obolochek pri staticheskikh i dinamicheskikh nagruzheniyakh. Prikladnaya mekhanika i tekhnicheskaya fizika. 2014;55(1):13-22. [Bazhenov V. G., Gonik E. G., Kibets A. I., Shoshin D. V. Stability and ultimate states of elastic-plastic spherical shells at static and dynamic loadings. Journal of Applied Mechanics and Technical Physics. 2014;55(1):13-22. (in Russ.)]

10. Ganeeva M. S., Moiseeva V. E. Nelineynyy izgib i ustoychivost' sfericheskikh i ellipsoidal'nykh obolochek pri neosesimmetrichnom nagruzhenii. Probl. prochn. i plastich. 2013;75(2):105-114. [Ganeeva M. S., Moiseeva V. E. Non-linear bend and stability of spherical and ellipsoidal shells at non-axissymmetrical loading. Problems of Strength and Plasticity. 2013;75(2):105-114. (in Russ.)]

11. Malykh K. S., Novichkov A. A., Pridat'ko I. S. Ustoychivost' sfericheskikh obolochek s uchetom nachal'nykh nepravil'nostey formy. Trudy 6 Obshcherossiyskoy molodezhnoy nauchno-tekhnicheskoy konferentsii ―Molodezh'. Tekhnika. Kosmos‖. Sankt-Peterburg. 2014. S. 62-64. [Malykh K. S., Novichkov A. A., Pridat'ko I. S. stability of spherical shells accouting for initial form irregularities. Proccedings of the 6th All-Russian scientific-technical conference ―Youth. Technics. Cosmos‖. SanktPeterburg. 2014. P. 62-64. (in Russ.)]

12. Petrov V. V., Krivoshein I. V. Vliyanie neodnorodnosti materiala na ustoychivost' nelineyno deformiruemykh pologikh obolochek dvoyakoy krivizny. Vestnik SGTU. 2014;4:20-25. [Petrov V. V., Krivoshein I. V. Material heterogeneity influence on stability of non-linear deformed shallow shells of double curvature. Vestnik Saratov State Technical University. 2014;4:20-25. (in Russ.)]

13. Pikul' V. V. Ustoychivost' obolochek. Probl. mashinostr. i avtomatiz. 2012;2:81-87. [Pikul' V. V. Shell stability. Engineering and Automation Problems. 2012;2:81-87. (in Russ.)]

14. Semko V. V., Krivoshein I. V. Modelirovanie vliyaniya vida granichnykh usloviy na ustoychivost' nelineyno deformiruemykh pologikh obolochek. Sbornik trudov 26 Mezhdunarodnoy nauchnoy konferentsii ―Matematicheskie metody v tekhnike i tekhnologiyakh (MMTT-26)‖. Nizhniy Novgorod; 2013. S. 53-55. [Semko V. V., Krivoshein I. V. Modelirovanie vliyaniya vida granichnykh usloviy na ustoychivost' nelineyno deformiruemykh pologikh obolochek. Proceedings of the 26 International scientific conference ―Mathematical modeling in technics and technologies (MMTT-26)‖. Nizhniy Novgorod; 2013. P. 53-55. (in Russ.)]

15. Arnol'd V.I. Teoriya katastrof. Moskva: Lenand; 2016. 134 s. [Arnol'd V.I. Catastrophe theory. Moscow: Lenand; 2016. 134 p. (in Russ.)]

16. Ostreykovskiy V. A. Analiz ustoychivosti i upravlyaemosti dinamicheskikh sistem metodami teorii katastrof: Uchebnoe posobie dlya studentov vuzov. Moskva: Izdatel'stvo "Vysshaya shkola"; 2005. 327 s. [Ostreykovskiy V. A. Analysis of stability and controllability of dynamic systems by catastrophe theory methods: a tutorial for students of higher education institutions. Moscow: Izdatel'stvo "Vysshaya shkola"; 2005. 327 p. (in Russ.)]

17. Postle D. Calastrophe Theory. London: Fontana; 1980.

18. Gilmor R. Prikladnaya teoriya katastrof. M.: Nauka; 1990. 350 s. [Gilmor R. Applied calastrophe theory. Moscow: Nauka; 1990. 350 p. (in Russ.)]

19. Tompson D.M.T. Neustoychivosti i katastrofy v nauke i tekhnike. M.: Mir; 1985. 256 s. [Tompson D.M.T. Non-stabilities and calastrophes in science and technics. Moscow: Mir; 1985. 256 p. (in Russ.)]

20. Keller Dzh. B., Antman S. Teoriya vetvleniya i nelineynye zadachi na sobstvennye znacheniya. M.: Mir; 1974. 254 s. [Keller Dzh. B., Antman S. Branching theory and non-linear eigenproblem. Moscow: Mir; 1974. 254 p. (in Russ.)]

21. Murtazaliev G.M. K raschetu gibkikh obolochek metodami teorii katastrof. Prochnost' i nadezhnost' sooruzheniy: Sb. nauchnykh tr. TsNIISK im. V.A. Kucherenko. M: Stroyizdat; 1989. S. 34-41. [Murtazaliev G.M. On the calculation of flexible shells by calastrophe theory methods. Durability and reliability of constructions: scientific work collection of TSNIISK named after V.A. Koucherenko. Moscow: Stroyizdat; 1989. P. 34-41. (in Russ.)]

22. Murtazaliev G.M. K ispol'zovaniyu metodov teorii katastrof dlya analiza povedeniya tsilindricheskikh paneley peremennoy tolshchiny. Dep. v VINITI 24.09.92, N 2839 – V92. [Murtazaliev G.M. On the application of calastrophe theory methods in behavior analysis of cilindric planes of variable thikness. Dep. in VINITI 24.09.92, N 2839 – V92. (in Russ.)]

23. Murtazaliev G.M., Payzulaev M.M., Guseynova S.V. Geometricheskie obrazy teorii katastrof v nelineynykh zadachakh. Sbornik statey po materialam vserossiyskoy nauchno-prakticheskoy konferentsii ―Teoriya sooruzheniy: dostizheniya i problemy‖. Makhachkala; 2012. 126 s. [Murtazaliev G.M., Payzulaev M.M., Guseynova S.V. Geometric images of calastrophe theory in non-linear problems. Matrials of All-Russian scientific-practical conference ―Theory of construction: acheivments and problems‖. Makhachkala; 2012. 126 p. (in Russ.)]

24. Murtazaliev G.M., Payzulaev M.M. Metody teorii katastrof v mekhanike konstruktsiy. Sbornik statey po materialam II Vserossiyskoy nauchno-prakticheskoy konferentsii ―Teoriya sooruzheniy: dostizheniya i problemy‖. Makhachkala; 2015. 132 s. [Murtazaliev G.M., Payzulaev M.M. Calastrophe theory methods in the mechanics of constructions. Matrials of All-Russian scientific-practical conference ―Theory of construction: acheivments and problems‖. Makhachkala; 2015. 132 p. (in Russ.)]

25. Borodin A.I., Novikova N.N., Shash N.N. Primenenie sinergeticheskikh metodov i teorii katastrof. Effektivnoe antikrizisnoe upravlenie. 2015;2(89):84-90. [Borodin A.I., Novikova N.N., Shash N.N. Application of synergetic methods and calastrophe theory. Effective Crisis Management Journal. 2015;2(89):84-90. (in Russ.)]