Analytical and numerical method of finite bodies for calculation of cylindrical orthotropic shell with rectangular hole

V. N. Bakulin ; V. P. Revenko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2016), pp. 3-14.

Résumé

To solve the boundary-value problem for cylindrical orthotropic shell with sizeable rectangular hole we suggest analytical and numerical method of finite bodies. For determination of the stress state of orthotropic thin-walled cylinder we use a system of equations that exactly satisfies the equilibrium equations of orthotropic cylindrical shell. Representation of the solutions is divided into basic and self-equilibrium state. For some loads of shell we build the basic stress state. We obtain a countable number of resolving functions that exactly satisfy the equation shell and describe the self-equilibrium stress state. We develop the algorithm of the analytical and numerical solutions of boundary-value problem based on approximation of the stress state of the shell by finite sum of resolving functions and propose universal way of reduction of all conditions of the contact parts of the enclosure and the boundary conditions to minimize the generalized quadratic forms. We establish criteria under which the construction of approximate solutions coincides with the exact one.

Détail

Keywords: cylindrical orthotropic shell, rectangular hole, quadratic form.

Bibliographie

[1] Ambartsumyan S. A., Obschaya teoriya anizotropnykh obolochek, Nauka, M., 1974 | MR

[2] Bakulin V. N., Obraztsov I. F., Potopakhin V. A., Dinamicheskie zadachi nelineinoi teorii mnogosloinykh obolochek: deistvie intensivnykh termosilovykh nagruzok, kontsentrirovannykh potokov energii, Fizmatlit, M., 1998

[3] Grigorenko Ya. M., Vlaikov G. G., Grigorenko A. Ya., Chislenno-analiticheskoe reshenie zadach mekhaniki obolochek na osnove razlichnykh modelei, Izdatelskii dom “Akademperiodika”, Kiev, 2006

[4] Guz A. N., Chernyshenko I. S., Chekhov Val. N., Chekhov Vik. N., Shnerenko K. I., Tsilindricheskie obolochki, oslablennye otverstiyami. Metody rascheta obolochek, v. 1, Nauk. dumka, Kiev, 1980 | MR

[5] Novozhilov V. V., Chernykh K. F., Mikhailovskii E. I., Lineinaya teoriya tonkikh obolochek, “Politekhnika”, L., 1991 | MR

[6] Ciarlet PG., Mathematical elasticity, v. 3, Theory of shells, North Holland, 2000 | MR | Zbl

[7] Dikman M., Theory of thin elastic shells, Pitman advanced publ. progr., Boston, 1982 | MR

[8] Guz A. N., Chernyshenko I. S., Chekhov Val. N. i dr., “Issledovaniya po teorii tonkikh obolochek s otverstiyami. Obzor”, Prikl. mekhan., 15:11 (1979), 3–37 | MR | Zbl

[9] Revenko V. P., “Raschet napryazhenno-deformirovannogo sostoyaniya nepologoi ortotropnoi tsilindricheskoi obolochki s ellipticheskim otverstiem”, Prikl. mekhan., 24:4 (1988), 57–63 | Zbl

[10] Bakulin V. N., Martynovich T. L., Revenko V. P., “Raschet napryazhennogo sostoyaniya tsilindricheskoi kompozitnoi obolochki s pryamougolnym otverstiem”, Nauchno-tekhn. konf. “Primenenie kompozitsionnykh materialov na polimernoi i metallicheskoi matritsakh v mashinostroenii”, Ufa, 1985

[11] Salo V.A., Kraevye zadachi statiki obolochek s otverstiyami, NTU “KhPI”, Kharkov, 2003

[12] Bakulin V. N., Repinskii V. V., “O vliyanii razmerov pryamougolnogo vyreza na napryazhenno-deformirovannoe sostoyanie krugovoi tsilindricheskoi obolochki”, Temat. sb. nauchn. tr. “Chisl. metody issled. prochnosti letatelnykh apparatov”, M., 1988, 5–10

[13] Dlugach M. I., Kovalchuk N. V., “Metod konechnykh elementov v primenenii k raschetu tsilindricheskikh obolochek s pryamougolnymi otverstiyami”, Prikl. mekhan., 9:11 (1973), 35–41

[14] Vekua I. N., Nekotorye obschie metody postroeniya razlichnykh variantov teorii obolochek, Nauka, M., 1982 | MR

[15] Bakulin V. N., Revenko V. P., “Raschet metodom konechnykh tel ortotropnoi tsilindricheskoi obolochki s nemalym pryamougolnym otverstiem”, Mater. Kh mezhdunarodn. konf. po neravnovesnym protsessam v soplakh i struyakh, NPNJ 2014 (Alushta, 25–31 maya 2014), Izd-vo MAI, M., 2014, 324–326

[16] Revenko V. P., “O reshenii trekhmernykh uravnenii lineinoi teorii uprugosti”, Prikl. mekhan., 45:7 (2009), 52–65 | MR | Zbl

[17] Revenko V. P., “O chislenno-analiticheskom metode rascheta napryazhennogo sostoyaniya uprugoi pryamougolnoi plastiny”, Prikl. mekhan., 44:1 (2008), 90–98 | MR | Zbl

[18] Bakulin V. N., Revenko V. P., “Method of homogeneous solutions in three-dimensional problems for multilayer cylindrical body”, Materials phys. and mech., 2016 (to appear)

[19] Korn G., Korn T., Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov. Opredeleniya, teoremy, formuly, Nauka, M., 1974 | MR

[20] Meleshko V. V., “Selected topics in the history of the two-dimensional biharmonic problem”, Appl. Mech. Rev., 56:1 (2003), 33–85 | DOI

[21] Sen-Venan B., Memuar o kruchenii prizm. Memuar ob izgibe prizm, Fizmatgiz, M., 1961