### Dynamic Buckling of Stiffened Shell Structures with Transverse ‎Shears under Linearly Increasing Load

Document Type : Research Paper

Author

Department of Computer Science, Saint Petersburg State University of Architecture and Civil Engineering, 4, 2nd Krasnoarmeyskaya st., Saint-Petersburg, 190005, Russia

Abstract

Thin-walled shell structures are widely used in various fields of engineering, and the process of their deformation is essentially non-linear, which significantly complicates their analysis. The author presents a new mathematical model of shell structure deformation under dynamic loading, taking into account geometrical nonlinearity, transverse shears, and stiffeners. A distinctive feature of the model is the use of the refined discrete method to account for stiffeners, suggested by the author earlier. The method has previously been used only in static problems. It uses adjusting normalizing factors and makes it possible to obtain the most correct critical load values. A technique for numerically investigation of the process of deformation of such structures under dynamic loading is based on the L.V. Kantorovich method and Rosenbrock method for solution of a stiff ODE system. The proposed approach, in contrast to commercial software based on FEM, makes it possible to study in detail the supercritical deformation of structures, to identify patterns of influence of individual parts of the mathematical model on critical loads. Calculations for shallow doubly curved shells and cylindrical panels are provided. It is shown how the number of stiffeners affects the resulting buckling load values. A comparison with the classic discrete method to account for stiffeners is performed. Phase portraits of the system are given for all problems considered.

Keywords

Main Subjects

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[1] Sowiński, K., The Ritz method application for stress and deformation analyses of standard orthotropic pressure vessels, Thin-Walled Structures, 162, 2021, 107585. DOI: 10.1016/j.tws.2021.107585.
[2] Radchenko, P.A., Batuev, S.P., Radchenko, A.V., Plevkov, V.S., Numerical modeling of the destruction of a shell made of concrete and fiber-reinforced concrete under impulse action, Omsk Scientific Bulletin, 3(143), 2015, 345–348. (in Russian)
[3] Raeesi, A., Ghaednia, H., Zohrehheydariha, J., Das, S., Failure analysis of steel silos subject to wind load, Engineering Failure Analysis, 79, 2017, 749–761. DOI: 10.1016/j.engfailanal.2017.04.031.
[4] Amabili, M., Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells, International Journal of Non-Linear Mechanics, 69, 2015, 109–128. DOI: 10.1016/j.ijnonlinmec.2014.11.026.
[5] Xie, K., Chen, M., Zhang, L., Xie, D., Free and forced vibration analysis of non-uniformly supported cylindrical shells through wave based method, International Journal of Mechanical Sciences, 128–129, 2017, 512–526. DOI: 10.1016/j.ijmecsci.2017.05.014.
[6] Dey, T., Ramachandra, L.S., Static and dynamic instability analysis of composite cylindrical shell panels subjected to partial edge loading, International Journal of Non-Linear Mechanics, 64, 2014, 46–56. DOI: 10.1016/j.ijnonlinmec.2014.03.014.
[7] Hao, Y.X., Li, Z.N., Zhang, W., Li, S.B., Yao, M.H., Vibration of functionally graded sandwich doubly curved shells using improved shear deformation theory, Science China Technological Sciences, 61(6), 2018, 791–808. DOI: 10.1007/s11431-016-9097-7.
[8] Krysko, A.V., Awrejcewicz, J., Saltykova, O.A., Vetsel, S.S., Krysko, V.A., Nonlinear dynamics and contact interactions of the structures composed of beam-beam and beam-closed cylindrical shell members, Chaos, Solitons & Fractals, 91, 2016, 622–638. DOI: 10.1016/j.chaos.2016.09.001.
[9] Gonçalves, P.B., Silva, F.M.A., Del Prado, Z.J.G.N., Reduced Order Models for the Nonlinear Dynamic Analysis of Shells, Procedia IUTAM, 19, 2016, 118–125. DOI: 10.1016/j.piutam.2016.03.016.
[10] Leonenko, D.V., Starovoitov, E.I., Vibrations of Cylindrical Sandwich Shells with Elastic Core under Local Loads, International Applied Mechanics, 52(4), 2016, 359–367. DOI: 10.1007/s10778-016-0760-8.
[11] Lee, Y.-S., Kim, Y.-W., Effect of boundary conditions on natural frequencies for rotating composite cylindrical shells with orthogonal stiffeners, Advances in Engineering Software, 30(9-11), 1999, 649–655. DOI: 10.1016/S0965-9978(98)00115-X.
[12] Mustafa, B.A.J., Ali, R., An energy method for free vibration analysis of stiffened circular cylindrical shells, Computers & Structures, 32(2), 1989, 355–363. DOI: 10.1016/0045-7949(89)90047-3.
[13] Talebitooti, M., Ghayour, M., Ziaei-Rad, S., Talebitooti, R., Free vibrations of rotating composite conical shells with stringer and ring stiffeners, Archive of Applied Mechanics, 80(3), 2010, 201–215. DOI: 10.1007/s00419-009-0311-4.
[14] Wang, C.M., Swaddiwudhipong, S., Tian, J., Ritz Method for Vibration Analysis of Cylindrical Shells with Ring Stiffeners, Journal of Engineering Mechanics, 123(2), 1997, 134–142. DOI: 10.1061/(ASCE)0733-9399(1997)123:2(134).
[15] Qu, Y., Wu, S., Chen, Y., Hua, H., Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach, International Journal of Mechanical Sciences, 69, 2013, 72–84. DOI: 10.1016/j.ijmecsci.2013.01.026.
[16] Zhao, X., Liew, K.M., Ng, T.Y., Vibrations of rotating cross-ply laminated circular cylindrical shells with stringer and ring stiffeners, International Journal of Solids and Structures, 39(2), 2002, 529–545. DOI: 10.1016/S0020-7683(01)00194-9.
[17] Jafari, A.A., Bagheri, M., Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods, Thin-Walled Structures, 44(1), 2006, 82–90. DOI: 10.1016/j.tws.2005.08.008.
[18] Khalmuradov, R.I., Ismoilov, E.A., Nonlinear vibrations of a circular plate reinforced by ribs, IOP Conf. Ser.: Earth Environ. Sci., 614, 2020, 012071. DOI: 10.1088/1755-1315/614/1/012071.
[19] Peng-Cheng, S., Dade, H., Zongmu, W., Static, vibration and stability analysis of stiffened plates using B spline functions, Computers & Structures, 27(1), 1987, 73–78. DOI: 10.1016/0045-7949(87)90182-9.
[20] Dung, D.V., Nam, V.H., An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 2: Numerical results and discussion, Vietnam Journal of Mechanics, 36(4), 2014, 255–265. DOI: 10.15625/0866-7136/36/4/3986.
[21] Azarboni, H.R., Ansari, R., Nazarinezhad, A., Chaotic dynamics and stability of functionally graded material doubly curved shallow shells, Chaos, Solitons & Fractals, 109, 2018, 14–25. DOI: 10.1016/j.chaos.2018.02.011.
[22] Bich, D.H., Long, V.D., Non-linear dynamical analysis of imperfect functionally graded material shallow shells, Vietnam Journal of Mechanics, 32(1), 2010, 1–14. DOI: 10.15625/0866-7136/32/1/312.
[23] del Prado, Z., Gonçalves, P.B., Païdoussis, M.P., Non-linear vibrations and instabilities of orthotropic cylindrical shells with internal flowing fluid, International Journal of Mechanical Sciences, 52(11), 2010, 1437–1457. DOI: 10.1016/j.ijmecsci.2010.03.016.
[24] Jansen, E.L., Dynamic Stability Problems of Anisotropic Cylindrical Shells via a Simplified Analysis, Nonlinear Dynamics, 39(4), 2005, 349–367. DOI: 10.1007/s11071-005-4343-1.
[25] Chamis, C.C., Dynamic Buckling and Postbuckling of a Composite Shell, International Journal of Structural Stability and Dynamics, 10(4), 2010, 791–805. DOI: 10.1142/S0219455410003749.
[26] Kubenko, V.D., Koval’chuk, P.S., Nonlinear problems of the dynamics of elastic shells partially filled with a liquid, International Applied Mechanics, 36(4), 2000, 421–448. DOI: 10.1007/BF02681969.
[27] Xin, J., Wang, J., Yao, J., Han, Q., Vibration, Buckling and Dynamic Stability of a Cracked Cylindrical Shell with Time-Varying Rotating Speed, Mechanics Based Design of Structures and Machines, 39(4), 2011, 461–490. DOI: 10.1080/15397734.2011.569301.
[28] Gao, K., Gao, W., Wu, D., Song, C., Nonlinear dynamic stability of the orthotropic functionally graded cylindrical shell surrounded by Winkler-Pasternak elastic foundation subjected to a linearly increasing load, Journal of Sound and Vibration, 415, 2018, 147–168. DOI: 10.1016/j.jsv.2017.11.038.
[29] Lavrenčič, M., Brank, B., Simulation of shell buckling by implicit dynamics and numerically dissipative schemes, Thin-Walled Structures, 132, 2018, 682–699. DOI: 10.1016/j.tws.2018.08.010.
[30] Li, Z.-M., Liu, T., Qiao, P., Nonlinear vibration and dynamic instability analyses of laminated doubly curved panels in thermal environments, Composite Structures, 267, 2021, 113434. DOI: 10.1016/j.compstruct.2020.113434.
[31] Zhou, Y., Stanciulescu, I., Eason, T., Spottswood, M., Fast approximations of dynamic stability boundaries of slender curved structures, International Journal of Non-Linear Mechanics, 95, 2017, 47–58. DOI: 10.1016/j.ijnonlinmec.2017.06.002.
[32] Ren, S., Song, Y., Zhang, A.-M., Wang, S., Li, P., Experimental study on dynamic buckling of submerged grid-stiffened cylindrical shells under intermediate-velocity impact, Applied Ocean Research, 74, 2018, 237–245. DOI: 10.1016/j.apor.2018.02.018.
[33] Patel, S.N., Datta, P.K., Sheikh, A.H., Buckling and dynamic instability analysis of stiffened shell panels, Thin-Walled Structures, 44(3), 2006, 321–333. DOI: 10.1016/j.tws.2006.03.004.
[34] Amiro, I.Ya., Zarutskii, V.A., Stability of ribbed shells, Soviet Applied Mechanics, 19(11), 1983, 925–940. DOI: 10.1007/BF01362647.
[35] Sadeghifar, M., Bagheri, M., Jafari, A.A., Buckling analysis of stringer-stiffened laminated cylindrical shells with nonuniform eccentricity, Archive of Applied Mechanics, 81(7), 2011, 875–886. DOI: 10.1007/s00419-010-0457-0.
[36] Huang, S., Qiao, P., A new semi-analytical method for nonlinear stability analysis of stiffened laminated composite doubly-curved shallow shells, Composite Structures, 251, 2020, 112526. DOI: 10.1016/j.compstruct.2020.112526.
[37] Ghasemi, A.R., Tabatabaeian, A., Hajmohammad, M.H., Tornabene, F., Multi-step buckling optimization analysis of stiffened and unstiffened polymer matrix composite shells: A new experimentally validated method, Composite Structures, 273, 2021, 114280. DOI: 10.1016/j.compstruct.2021.114280.
[38] Dai, Q., Cao, Q., Parametric instability analysis of truncated conical shells using the Haar wavelet method, Mechanical Systems and Signal Processing, 105, 2018, 200–213. DOI: 10.1016/j.ymssp.2017.12.004.
[39] Polat, C., Calayir, Y., Nonlinear static and dynamic analysis of shells of revolution, Mechanics Research Communications, 37(2), 2010, 205–209. DOI: 10.1016/j.mechrescom.2009.12.009.
[40] Storozhuk, E.A., Yatsura, A.V., Analytical-Numerical Solution of Static Problems for Noncircular Cylindrical Shells of Variable Thickness, International Applied Mechanics, 53(3), 2017, 313–325. DOI: 10.1007/s10778-017-0813-7.
[41] Karpov, V., Variational method for derivation of equations of mixed type for shells of a general type, Architecture and Engineering, 1(2), 2016, 43–48. DOI: 10.23968/2500-0055-2016-1-2-43-48.
[42] Abrosimov, N.A., Novosel’tseva, N.A., Computer Modeling of the Dynamic Strength of Metal-Plastic Cylindrical Shells Under Explosive Loading, Mechanics of Composite Materials, 53(2), 2017, 139–148. DOI: 10.1007/s11029-017-9648-x.
[43] Hao, P., Liu, D., Zhang, K., Yuan, Y., Wang, B., Li, G., Zhang, X., Intelligent layout design of curvilinearly stiffened panels via deep learning-based method, Materials & Design, 197, 2021, 109180. DOI: 10.1016/j.matdes.2020.109180.
[44] Hao, P., Yuan, X., Liu, C., Wang, B., Liu, H., Li, G., Niu, F., An integrated framework of exact modeling, isogeometric analysis and optimization for variable-stiffness composite panels, Computer Methods in Applied Mechanics and Engineering, 339, 2018, 205–238. DOI: 10.1016/j.cma.2018.04.046.
[45] Karpov, V.V., Semenov, A.A., Refined model of stiffened shells, International Journal of Solids and Structures, 199, 2020, 43–56. DOI: 10.1016/j.ijsolstr.2020.03.019.
[46] Semenov, A., Buckling of Shell Panels Made of Fiberglass and Reinforced with an Orthogonal Grid of Stiffeners, Journal of Applied and Computational Mechanics, 7(3), 2021, 1856–1861. DOI: 10.22055/jacm.2021.37768.3078.
[47] Semenov, A., Mathematical model of deformation of orthotropic shell structures under dynamic loading with transverse shears, Computers & Structures, 221, 2019, 65–73. DOI: 10.1016/j.compstruc.2019.05.017.
[48] Semenov, A.A., Models of Deformation of Stiffened Orthotropic Shells under Dynamic Loading, Journal of Siberian Federal University, Mathematics & Physics, 9(4), 2016, 485–497. DOI: 10.17516/1997-1397-2016-9-4-485-497.
[49] Li, P., Wang, H., A novel strategy for the crossarm length optimization of PSSCs based on multi-dimensional global optimization algorithms, Engineering Structures, 238, 2021, 112238. DOI: 10.1016/j.engstruct.2021.112238.