To equistability problem of the reinforced shell structure under combined loading

1Degtyarenko, PG, 2Grishchak, VZ, 3Grishchak, DD, 4Dyachenko, NM
1Yangel Yuzhnoye State Design Office, Dnipro, Ukraine
2Dnipro University of Technology, Dnipro, Ukraine
3Central Scientific Research Institute for Armament and Military Equipment of the Armed Forces of Ukraine, Kyiv, Ukraine
4Zaporizhzhia National University, Zaporizhzhia, Ukraine
Space Sci. & Technol. 2019, 25 ;(6):03-14
https://doi.org/10.15407/knit2019.06.003
Publication Language: Russian
Abstract: 
The stability of a cylinder-cone-type shell structure of the launch vehicle is studied under the combined loading of external pressure, axial compression, and torque. The resolving equations for the problem are solved analytically using the asymptotic approach. In the case of the conical compartment, two analytical methods, such as the WKB and the hybrid WKB-Galerkin methods, are used. To analyze the stability of the reinforced shells, we use the matrix method that allows determining the change in the shell stressstrain state through the ring. The characteristic equation for determining critical loads is obtained based on the matrix method and the conjugation equations. Particular attention is paid to the selection of ring stiffness coefficients for the conic al and cylindrical parts providing equal critical pressures. It is obtained that the values of the critical pressure in the equistable structure are lower than in its parts.
            The stability of the reinforced equistable structure under the combined loading is studied. The results of numerical calculations for different types of reinforced structures are discussed. It is shown that in the extreme cases, for cylinder or conical shells, numerical results are well correlated with data of known publications.
Keywords: combined loading, cylinder-cone-type shell structure, equistable shell, ring stiffness, shell stability
References: 
1. Avramov K. V., Chernobryvko M. V., Batutina T. Ya., Degtyarenko P. G., Tonkonozhenko A. M. (2015). Dynamic instability of rockets fairings. Space Sci. Technol., 21, No. 1, 10—14 [in Russian].
2. Akimov D. V., Gristchak V. Z., Grebeniuk S. M., Gomeniuk S. I. (2016). Comparative analysis of the calculation methods of the stress-strain state of the launch vehicle structural elements. Novi materialy i technologii v metalurgii ta mashynobuduvanni, No. 2, 116—120 [in Russian]. URL: http://nbuv.gov.ua/j-pdf/Nmt_2016_2_22.pdf (Last accessed 01.07.2019).
3. Volmir A. S. (1967). Stability of deformable systems. Moscow: Nauka [in Russian].
4. Gristchak V. Z., Dyachenko N. N. (2017). Stability areas determination of the conical shell at combined loading on a hybrid asymptotic approach basis. Visnyk of Zaporizhzhya National University. Physical and Mathematical Science, No. 2, 33—46.
5. Gristchak V. Z., Manevich A. I. (1972). Influence of a ring stiffness on a bend out of a plane on the stability of a reinforced cylindrical shell. Gidroaeromekhanika i teoriya uprugosti, No. 14, 121—130 [in Russian].
6. Makarenko I. N. (2001). Stability of conjugate shells of rotation. Vestik. SPGU. Ser. 1., No. 3 (17), 61—69 [in Russian].
7. Pechnikov V. P. (1968). Investigation on the basis of a semi-momentless theory of the stability of a conical shell reinforced by elastic rings. Izv. vuzov. Mashinostroenie, No. 10, 37—42 [in Russian].
8. Postnov V. A., Tumashik I. V., Moskvina I. V. (2007). About stability of a reinforced cylindrical shell. Problem of Strenght and Plasticity, No. 69, 18—23 [in Russian].
9. Preobrazhensky I. N., Gristchak V. Z. (1986). Stability and oscillations of conical shells. Moscow: Mashinostroenie [in Russian].
10. Sachenkov A. V. (1964). On the stability of a circular conical shell under the joint action of loads. Studies on the theory of plates and shells. Kazan: Publishing house of Kazan University, No. 2, 57—70 [in Russian].
11. Andres M., Harte R. (2006). Buckling of concret shells: A simplified numerical approach. Journal of the International Association for Shell and Spatial Structures, 47, No. 3, December n. 152, 279—290.
12. Bai X., Xu W., Ren H., Li J. (2017). Analysis of the influence of stiffness reduction on the load carrying capacity of ring-stiffened cylindrical shell. Ocean Engineering, 135, 52—62.
13. Geer J. F., Andersen C. M. (1989). A hybrid perturbation-Galerkin technique with application to slender body theory. SIAM J. Appl. Mech., 49, 344 — 361.
14. Gristchak V. Z., Dimitrijeva E. M. (1998). A Hybrid WKBGalerkin Method and its Using to Applied Mechanics Problems. FACTA UNIVERSITATIS. Ser.: Mechanics, Automatic Control and Robotics, 2, No. 8, 709—713.
15. Gristchak V. Z., Gristchak D. D., Fatieieva Yu. A. (2016). Hybrid asymptotic methods. Theory and applications. Zaporizhzhya: Zaporizhzhya National University.
16. Gristchak V. Z., Pogrebitskaya A. M. (2011). On approximate analytical solution of nonlinear thermal emission problems. Technische Mechanik, 31, No. 2,112—120.
17. Pimenta P. M., Wriggers P. (Eds.). (2010). New Trends in Thin Structures: Formulation, Optimization and Coupled Problems. CISM International Centre for Mechanical Sciences, Springer, Vol. 519.
18. Seide P., Weingarten V. L. (1965). Elastic stability of thinwalled cylindrical and conical shells under combined external pressure and axial compression. AIAA Journal, 3 (5), 913—920.
19. Simo J. C., Hughes T. J. R. (1986). On the Variational Formulation of Assumed Strain Methods. J. Appl. Mech., 53, 51—54.
20. Stein M. (1968). Some recent advances in the investigation of shell buckling. AIAA Journal, 6, 2239—2245.
21. Ramm E. (Ed.). (1982). Buckling of shells. Berlin: Springer-Verlag.
22. Tafreshi A., Bailey C. G. (2007). Instability of imperfect composite cylindrical shells under combined loading.Composite Structures, 80 (1), 49—64.
23. Teng J. G., Barbagallo M. (1997). Shell restraint to ring buckling at c one-cyl inder intersections. Engineering Structure, 19 (6), 425—431.
24. Teng J. G., Rotter J. M. (2004). Buckling of Thin Metal Shells. London and New York: CRC Press.
25. Xue J., Hoo Fatt M. S. (2002). Buckling of non-uniform, long cylindrical shell subjected to external hydrostatic pressure. Engineering structures, 24 (8), 1027—1034.