Abstract
This paper investigates the dynamic behavior of laminated composite shells (LCS) under the effect of dynamic loads. The governing equations for composite shells have been derived by using Hamilton’s principle. Then, these governing equations have been obtained by using Navier solution method. Many researchers do not use the term of (\(1+z/R\)) in the stress equations. However, these equations with the term of (\(1+z/R\)) have been obtained to get a more realistic model in the proposed study. It is aimed to emphasize the importance of the term (\(1+z/R\)) to take into account the effect of the radius of curvature in the calculation of the equations of the shell elements. In addition, the truncated series are used for a better numerical stability in the solution of the motion equations. The differential equations governing the system are derived by using the dynamic virtual displacements. Time-dependent ordinary differential equations are transformed into the Laplace space. Equations dependent on the parameters are then solved in this space. Calculations are transformed from the Laplace space into the time space by the help of modified Durbin’s algorithm. In order to verify the methodology, the results obtained in this study are compared with those obtained by Newmark and ANSYS finite element methods. The results of numerical studies for the dynamic response of laminated composite shells (LCS) are demonstrated and compared with previous studies in the literature.
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References
ANSYS: Release 15. Swanson Analysis System Inc. Houston (2014)
Brigham, E.O.: The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs (1974)
Carrera, E., Fazzolari, A.F., Cinefra, M.: Chap. 4—Fundamental of mechanics of beams, plates and shells. In: Thermal Stress Analysis of Beams, Plates and Shells, pp. 91–116 (2017). https://doi.org/10.1016/B978-0-12-420066-1.00006-9
Dogan, A., Arslan, H.M.: Investigation of the effect of shell plan-form dimensions on mode-shapes of the laminated composite cylindrical shallow shells using SDSST and FEM. Steel Compos. Struct. 12(4), 303–324 (2012)
Dogan, A., Arslan, H.M., Yerli Huseyin, R.: Effect of anisotropy and curvature on free vibration characteristics of laminated composite cylindrical shallow shells. Struct. Eng. Mech. 35(4), 493–510 (2010)
Donnel: Stability of thin walled tubes under torsion, I-II, Naca, Report No: 479 (1933)
Durbin, F.: Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. Comput. J. 17(4), 371–376 (1974). https://doi.org/10.1093/comjnl/17.4.371
Ganapathi, M., Patel, B.P., Pawargi, D.S.: Dynamic analysis of laminated cross-ply composite non-circular thick cylindrical shells using higher-order theory. Int. J. Solids Struct. 39(24), 5945–5962 (2002). https://doi.org/10.1016/S0020-7683(02)00495-X
Gong, S.W., Shim, V.P.W., Toh, S.L.: Impact response of laminated shells with orthogonal curvatures. Compos. Eng. 5(3), 257–275 (1995). https://doi.org/10.1016/0961-9526(94)00096-R
Kiani, Y., Sadighi, M., Eslami, M.R.: Dynamic analysis and active control of smart doubly curved FGM panels. Compos. Struct. 102, 205–216 (2013). https://doi.org/10.1016/j.compstruct.2013.02.031
Krishnamurthy, K.S., Mahajan, P., Mittal, R.K.: Impact response and damage in laminated composite cylindrical shells. Compos. Struct. 59(1), 15–36 (2003). https://doi.org/10.1016/S0263-8223(02)00238-6
Leissa, A.W., Chang, J.D.: Elastic deformation of thick, laminated composite shells. Compos. Struct. 35(2), 153–170 (1996). https://doi.org/10.1016/0263-8223(96)00028-1
Leissa, A.W., Qatu, M.S.: Equations of elastic deformation of laminated composite shallow shells. J. Appl. Mech. 58(1), 181–188 (1991a). https://doi.org/10.1115/1.2897146
Leissa, A.W., Qatu, M.S.: Stress and deflection analysis of composite cantilevered doubly-curved shallow shells. J. Eng. Mech. 117(4), 893–906 (1991b). https://doi.org/10.1061/(ASCE)0733-9399(1991)117:4(893)
Leissa, A.W., Qatu, M.S.: Natural frequencies for cantilevered doubly-curved laminated composite shallow shells. Compos. Struct. 17(3), 227–256 (1991c). https://doi.org/10.1016/0263-8223(91)90053-2
Li, J., Hua, H.: Transient vibrations of laminated composite cylindrical shells exposed to underwater shock waves. Eng. Struct. 31(3), 738–748 (2009). https://doi.org/10.1016/j.engstruct.2008.11.018
Marguerre, K.: Zur Theorie der gekrümmten Platte grosser Formanderung. In: Proc. Fifth Int. Congress Applied Mechanics, Cambridge, MA, vol. 93, pp. 93–101 (1938)
MATHEMATICA: Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL (2017)
Mushtari, Kh.M.: On the stability of thin cylindrical shells subjected to torsion. Sb. Nauchn. Trud. Kazan. Aviat. Inst. 2, 3–17 (1934)
Mushtari, Kh.M.: Some generalization of the theory of thin shells with application to the stability of elastic equilibrium (in Russian). Izv. Fiz.-Mat. Obshch. Kazan. Univ. 2, 71–150 (1938)
Narayanan, G.V.: Numerical operational methods in structural dynamics. PhD Thesis, University of Minnesota, Minneapolis, MN, USA (1979)
Nelson, R.B., Lorch, D.R.: A refined theory of laminated orthotropic plates. J. Appl. Mech. 41(1), 177–183 (1974). https://doi.org/10.1115/1.3423219
Park, T., Kim, K., Han, S.: Linear static and dynamic analysis of laminated composite plates and shells using a 4-node quasi-conforming shell element. Composites, Part B, Eng. 37(2–3), 237–248 (2005). https://doi.org/10.1016/j.compositesb.2005.05.007
Prusty, B.G., Satsangi, S.K.: Finite element transient dynamic analysis of laminated stiffened shells. J. Sound Vib. 248(2), 215–233 (2001). https://doi.org/10.1006/jsvi.2001.3678
Qatu, M.S.: Vibration of Laminated Shells and Plates. Elsevier, Amsterdam (2004)
Qatu, M.S., Leissa, A.W.: Free vibrations of completely free doubly-curved laminated composite shallow shells. J. Sound Vib. 151(1), 9–29 (1991). https://doi.org/10.1016/0022-460X(91)90649-5
Reddy, J.N.: Exact solutions of moderately thick laminated shells. J. Eng. Mech. 110(5), 794–809 (1984). https://doi.org/10.1061/(ASCE)0733-9399(1984)110:5(794)
Reddy, J.N.: Mechanics of Laminated Composite Plate and Shells: Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004)
Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69–77 (1945)
Reissner, E.: Stress and displacements of shallow spherical shells, I. J. Math. Phys. 25, 80–85 (1946)
Reissner, E.: Stress and displacements of shallow spherical shells, II. J. Math. Phys. 26, 279–300 (1947)
Sahan, M.F.: Transient analysis of cross-ply laminated shells using FSDT: alternative formulation. Steel Compos. Struct. 18(4), 889–907 (2015). https://doi.org/10.12989/scs.2015.18.4.889
Sahan, M.F.: Dynamic analysis of linear viscoelastic cross-ply laminated shallow spherical shells. Compos. Struct. 149(1), 261–270 (2016). https://doi.org/10.1016/j.compstruct.2016.04.045
Sahan, M.F.: Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads. Mech. Time-Depend. Mater. 21(4), 499–518 (2017). https://doi.org/10.1007/s11043-017-9339-y
Saviz, M.R., Mohammadpourfard, M.: Dynamic analysis of a laminated cylindrical shell with piezoelectric layers under dynamic loads. Finite Elem. Anal. Des. 46(9), 770–781 (2010). https://doi.org/10.1016/j.finel.2010.04.007
Shim, V.P.W., Toh, S.L., Gong, S.W.: The elastic impact response of glass/epoxy laminated ogival shells. Int. J. Impact Eng. 18(6), 633–655 (1996). https://doi.org/10.1016/0734-743X(95)00061-E
Swaddiwudhipong, S., Liu, Z.S.: Response of laminated composite plates and shells. Compos. Struct. 37(1), 21–32 (1997). https://doi.org/10.1016/S0263-8223(97)00051-2
Temel, B., Şahan, M.F.: An alternative solution method for the damped response of laminated Mindlin plates. Composites, Part B, Eng. 47, 107–117 (2013). https://doi.org/10.1016/j.compositesb.2012.10.039
Timoshenko, S.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos. Mag. Ser. 6 41, 742 (1921)
Toh, S.L., Gong, S.W., Shim, V.P.W.: Transient stresses generated by low velocity impact on orthotropic laminated cylindrical shells. Compos. Struct. 31(3), 213–228 (1995). https://doi.org/10.1016/0263-8223(95)00104-2
Vaziri, R., Quan, X., Olson, M.D.: Impact analysis of laminated composite plates and shells by super finite elements. Int. J. Impact Eng. 18(7–8), 765–782 (1996). https://doi.org/10.1016/S0734-743X(96)00030-9
Whitney, J.M.: Shear correction factors for orthotropic laminated under static loading. J. Appl. Mech. 40, 302–304 (1973)
Wu, C.-P., Tarn, J.-Q., Chi, S.-M.: An asymptotic theory for dynamic response of doubly curved laminated shells. Int. J. Solids Struct. 33(26), 3813–3841 (1996). https://doi.org/10.1016/0020-7683(95)00213-8
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Dogan, A. Dynamic response of laminated composite shells under various impact loads. Mech Time-Depend Mater 25, 175–193 (2021). https://doi.org/10.1007/s11043-019-09434-z
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DOI: https://doi.org/10.1007/s11043-019-09434-z