On the solution of eigenvalue problems of laminated non-homogeneous orthotropic circular shells with clamped edges subjected to hydrostatic pressure. (English) Zbl 1351.74062
Summary: This article presents a method to study the free vibration and stability of laminated homogeneous and non-homogeneous orthotropic cylindrical, truncated and complete conical shells of general staking with clamped edges under a hydrostatic pressure. Based on the Love first approximation theory, the basic relations, the modified Donnell-type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the material properties of which vary piecewise continuously in the thickness direction. To solve this problem an unknown parameter \(\lambda \) was included in the approximation functions. Applying Galerkin methods, the buckling pressures and fundamental natural frequencies of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The parameter \(\lambda \) which is included in the obtained formulas is obtained from the minimum conditions of critical stresses and frequencies. The different generalized values are obtained for the parameter \(\lambda \) for buckling pressures and frequencies of cylindrical shells, truncated and complete conical shells. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of conical shell characteristics on the critical hydrostatic pressure and natural frequencies are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.
References:
| [1] | Wilkins D.J., Bert J.C.W., Egle D.M.: Free vibrations of orthotropic sandwich conical shells with various boundary conditions. J. Sound Vib. 13, 211–228 (1970) · Zbl 0218.73103 · doi:10.1016/S0022-460X(70)81175-0 |
| [2] | Chandrasekaran K., Ramamurti V.: Asymmetric free vibrations of layered conical shells. J. Mech. Des. 104, 453–462 (1982) · doi:10.1115/1.3256367 |
| [3] | Kayran A., Vinson J.R.: Free vibration analysis of laminated composite truncated circular conical shells. AIAA J. 28, 1259–1269 (1990) · doi:10.2514/3.25203 |
| [4] | Wang H., Wang T.K.: Stability of laminated composite circular conical shells under external pressure. J. Appl. Math. Mech. 12, 1153–1161 (1991) · Zbl 0747.73028 · doi:10.1007/BF02456054 |
| [5] | Tong L.: Free vibration of composite laminated conical shells. Int. J. Mech. Sci. 35, 47–61 (1993) · Zbl 0825.73343 · doi:10.1016/0020-7403(93)90064-2 |
| [6] | Shu C.: Free vibration analysis of composite laminated conical shells by generalized differential quadrature. J. Sound Vib. 194, 587–604 (1996) · Zbl 1232.74034 · doi:10.1006/jsvi.1996.0379 |
| [7] | Korjakin A., Rikards R., Chate A., Altenbach H.: Analysis of free damped vibrations of laminated composite conical shells. Compos. Struct. 41, 39–47 (1998) · doi:10.1016/S0263-8223(98)00024-5 |
| [8] | Correia I.F.P., Soares C.M.M., Soares C.A.J., Herskovits J.: Analysis of laminated conical shell structures using higher order models. Compos. Struct. 62, 383–390 (2003) · doi:10.1016/j.compstruct.2003.09.009 |
| [9] | Ng T.Y., Li H., Lam K.Y.: Generalized differential quadrature for free vibration of rotating composite laminated conical shell with various boundary conditions. Int. J. Mech. Sci. 45, 567–587 (2003) · Zbl 1116.74439 · doi:10.1016/S0020-7403(03)00042-0 |
| [10] | Reddy J.N.: Mechanics of Laminated Composite Plates and Shells, Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004) · Zbl 1075.74001 |
| [11] | Heuer R., Ziegler F.: Thermoelastic stability of layered shallow shells. Int. J. Solid Struct. 41, 2111–2120 (2004) · Zbl 1072.74029 · doi:10.1016/j.ijsolstr.2003.11.032 |
| [12] | Heuer R.: Thermo-piezoelectric flexural vibrations of viscoelastic panel-type laminates with interlayer slip. Acta Mech. 181, 129–138 (2006) · Zbl 1096.74026 · doi:10.1007/s00707-005-0299-y |
| [13] | Liang S., Chen H.L., Chen T., Michael Y., Wang M.Y.: The natural vibration of a symmetric cross-ply laminated composite conical-plate shell. Compos. Struct. 80, 265–278 (2007) · doi:10.1016/j.compstruct.2006.05.014 |
| [14] | Civalek O.: Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: discrete singular convolution (DSC) approach. J. Comput. Appl. Math. 205, 251–271 (2007) · Zbl 1115.74058 · doi:10.1016/j.cam.2006.05.001 |
| [15] | Bargmann H.W.: On the stability of thin-walled, corrugated, circular cylindrical shells under external pressure. Acta Mech. 195, 117–128 (2008) · Zbl 1136.74016 · doi:10.1007/s00707-007-0553-6 |
| [16] | Gu B., Mai Y.-W., Ru C.Q.: Mechanics of microtubules modeled as orthotropic elastic shells with transverse shearing. Acta Mech. 207, 195–209 (2009) · Zbl 1173.74027 · doi:10.1007/s00707-008-0121-8 |
| [17] | Goldfeld Y.: An alternative formulation in linear bifurcation analysis of laminated shells. Thin-Walled Struct. 47, 44–52 (2009) · doi:10.1016/j.tws.2008.05.001 |
| [18] | Volmir, A.S.: Stability of Elastic Systems. Moscow, Nauka (English Translation) Foreign Technology Division, Air Force Systems Command. Wright-Patterson Air Force Base, Ohio, AD628508 (1967) |
| [19] | Leissa A.W.: Vibration of Shells. NASA SP-288, Washington (1973) · Zbl 0268.73033 |
| [20] | Delale F., Erdogan F.: The crack problem for a non-homogeneous plane. ASME J. Appl. Mech. 50, 609–614 (1983) · Zbl 0542.73118 · doi:10.1115/1.3167098 |
| [21] | Khoroshun, L.P., Kozlov, S.Y., Ivanov, Y.A., Koshevoi, I.K.: The Generalized Theory of Plates and Shells Non-Homogeneous in Thickness Direction. Naukova Dumka, Kiev (1988) (in Russian) |
| [22] | Baruch, M., Arbocz, J., Zhang, G.Q.: Laminated conical shells–considerations for the variations of stiffness coefficients. Report LR-671, Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands (1992) |
| [23] | Gutierrez R.H., Laura P.A.A., Bambill D.V., Jederlinic V.A., Hodges D.H.: Axisymmetric vibrations of solid circular and annular membranes with continuously varying density. J. Sound Vib. 212, 611–622 (1998) · doi:10.1006/jsvi.1997.1418 |
| [24] | Elishakoff I.: Inverse buckling problem for inhomogeneous columns. Int. J. Solids Struct. 38, 457–464 (2001) · Zbl 1009.74021 · doi:10.1016/S0020-7683(00)00049-4 |
| [25] | Sofiyev A.H., Schnack E.: The buckling of cross-ply laminated non-homogeneous orthotropic composite conical thin shells under a dynamic external pressure. Acta Mech. 162, 29–40 (2003) · Zbl 1064.74101 · doi:10.1007/s00707-002-1001-2 |
| [26] | Weng G.J.: Effective bulk moduli of two functionally graded composites. Acta Mech. 166, 57–67 (2003) · Zbl 1064.74147 · doi:10.1007/s00707-003-0063-0 |
| [27] | Massalas C., Dalamanagas D., Tzivanidis G.: Dynamic instability of truncated conical shells with variable modulus of elasticity under periodic compressive forces. J. Sound Vib. 79, 519–528 (1981) · Zbl 0483.73038 · doi:10.1016/0022-460X(81)90463-6 |
| [28] | Heyliger P.R., Julani A.: The free vibrations of inhomogeneous elastic cylinders and spheres. Int. J. Solid Struct. 29, 2689–2708 (1992) · Zbl 0775.73140 · doi:10.1016/0020-7683(92)90112-7 |
| [29] | Zhang X., Hasebe N.: Elasticity solution for a radially non-homogeneous hollow circular cylinder. J. Appl. Mech. 66, 598–606 (1999) · doi:10.1115/1.2791477 |
| [30] | Jabareen M., Eisenberge M.: Free vibrations of non-homogeneous circular and annular membranes. J. Sound Vib. 240, 409–429 (2001) · doi:10.1006/jsvi.2000.3249 |
| [31] | Ding H.J., Wang H.M., Chen W.Q.: A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermo-elastic problems. J. Sound Vib. 263, 815–829 (2003) · doi:10.1016/S0022-460X(02)01075-1 |
| [32] | Gupta U.S., Lal R., Sharma S.: Vibration of non-homogeneous circular Mindlin plates with variable thickness. J. Sound Vib. 302, 1–17 (2007) · doi:10.1016/j.jsv.2006.07.005 |
| [33] | Lal R.: Transverse vibrations of non-homogeneous orthotropic rectangular plates of variable thickness, A spline technique. J. Sound Vib. 306, 203–214 (2007) · Zbl 1242.74210 · doi:10.1016/j.jsv.2007.05.014 |
| [34] | Sofiyev A.H., Omurtag M.H., Schnack E.: The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure. J. Sound Vib. 319, 963–983 (2009) · doi:10.1016/j.jsv.2008.06.033 |
| [35] | Chalivendra V.B.: Mixed-mode crack-tip stress fields for orthotropic functionally graded materials. Acta Mech. 204, 51–60 (2009) · Zbl 1162.74015 · doi:10.1007/s00707-008-0047-1 |
| [36] | Mecitoglu Z.: Governing equations of a stiffened laminated inhomogeneous conical shell. AIAA J. 34, 2118–2225 (1996) · Zbl 0910.73044 · doi:10.2514/3.13360 |
| [37] | Zenkour A.M., Fares M.E.: Bending, buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory. Compos. B Eng. 32, 237–247 (2001) · doi:10.1016/S1359-8368(00)00060-3 |
| [38] | Wu C.-P., Lee C.-Y.: Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int. J. Mech. Sci. 43, 1853–1869 (2001) · Zbl 1048.74603 · doi:10.1016/S0020-7403(01)00010-8 |
| [39] | Goldfeld Y., Arbocz J.: Buckling of laminated conical shells given the variations of the stiffness coefficients. AIAA J. 42, 642–649 (2004) · doi:10.2514/1.2765 |
| [40] | Kitipornchai S., Yang J., Liew K.M.: Random vibration of the functionally graded laminates in thermal environments. Comput. Meth. Appl. Mech. Eng. 195, 1075–1095 (2006) · Zbl 1115.74031 · doi:10.1016/j.cma.2005.01.016 |
| [41] | Tripath V., Singh B.N., Shukla K.K.: Free vibration of laminated composite conical shells with random material properties. Compos. Struct. 81, 96–104 (2007) · doi:10.1016/j.compstruct.2006.08.002 |
| [42] | Sofiyev A.H., Karaca Z.: The vibration and buckling of laminated non-homogeneous orthotropic conical shells subjected to external pressure. Eur. J. Mech. Solid A Solids 28, 317–328 (2009) · Zbl 1156.74334 · doi:10.1016/j.euromechsol.2008.06.002 |
| [43] | Singer J.: Buckling of clamped conical shells under external pressure. AIAA J. 4, 328–337 (1966) · doi:10.2514/3.3436 |
| [44] | Singer J., Baruch M., Reichenthal J.: Influence of in plane boundary conditions on the buckling of clamped conical shells. Isr. J. Tech. 9, 127–139 (1971) · Zbl 0214.52006 |
| [45] | Tani, J.: Free transverse vibrations of truncated conical shells. The Report Institute of High Speed Mechanics, Tohoku University, Japan, vol. 24. pp. 87–116 (1971) |
| [46] | Tani, J.: Influence of pre-buckling deformations on the buckling of truncated conical shells under external pressure. The Report Institute of High Speed Mechanics, Tohoku University, Japan, vol. 28. pp. 149–164 (1973) |
| [47] | Irie T., Yamada G., Tanaka K.: Natural frequencies of truncated conical shells. J. Sound Vib. 92, 447–453 (1984) · doi:10.1016/0022-460X(84)90391-2 |
| [48] | Lam K.Y., Li H.: On free vibration of a rotating truncated circular orthotropic conical shell. Compos. Part B Eng. 30, 135–144 (1999) · doi:10.1016/S1359-8368(98)00049-3 |
| [49] | Wang Y., Liu R., Wang X.: Free vibration analysis of truncated conical shells by the differential quadrature method. J. Sound Vib. 224, 387–394 (1999) · doi:10.1006/jsvi.1999.2218 |
| [50] | Liew K.M., Ng T.Y., Zhao X.: Free vibration analysis of conical shells via the element-free kp-Ritz method. J. Sound Vib. 281, 627–645 (2005) · doi:10.1016/j.jsv.2004.01.005 |
| [51] | Weingarten V.I.: Free vibration of thin cylindrical shells. AIAA J. 2, 717–722 (1964) · doi:10.2514/3.2405 |
| [52] | Agenesov, L.G., Sachenkov, A.V.: Free vibration and stability of conical shells with arbitrary transverse section. Research on the Theory of Plates and Shells, Kazan State University, Kazan 4, pp. 342–355 (1966) (in Russian) |
| [53] | Soedel W.: A new frequency formula for closed circular cylindrical shells for a large variety of boundary conditions. J. Sound Vib. 70, 209–217 (1980) · Zbl 0463.73067 · doi:10.1016/0022-460X(80)90325-9 |
| [54] | Koga T.: Effects of boundary conditions on the free vibrations of circular cylindrical shells. AIAA J. 26, 1387–1394 (1988) · Zbl 0676.73035 · doi:10.2514/3.10052 |
| [55] | Lam K.Y., Loy C.T.: Effect of boundary conditions on frequencies of a multi-layered cylindrical shell. J. Sound Vib. 188, 363–384 (1995) · doi:10.1006/jsvi.1995.0599 |
| [56] | Loy C.T., Lam K.Y., Shu C.: Analysis of cylindrical shells using generalized differential quadrature. Shock Vib. 4, 193–198 (1997) |
| [57] | Naem M.N., Sharma B.: Prediction of natural frequencies for thin circular cylindrical shells. Proc. Inst. Mech. Eng. 214, 1313–1328 (2000) |
| [58] | Zhang X.M., Liu G.R., Lam K.Y.: Vibration analysis of thin cylindrical shells using wave propagation approach. J. Sound Vib. 239, 397–403 (2001) · doi:10.1006/jsvi.2000.3139 |
| [59] | El-Mously M.: Fundamental natural frequencies of thin cylindrical shells, a comparative study. J. Sound Vib. 264, 1167–1186 (2003) · doi:10.1016/S0022-460X(02)01391-3 |
| [60] | Han B., Simitses G.J.: Analysis of anisotropic laminated cylindrical shells subjected to destabilizing loads. Part II, Numerical results. Compos. Struct. 19, 183–205 (1991) · doi:10.1016/0263-8223(91)90022-Q |
| [61] | Shen H.S., Li Q.S.: Postbuckling of cross-ply laminated cylindrical shells with piezoelectric actuators under complex loading conditions. Int. J. Mech. Sci. 44, 1731–1754 (2002) · Zbl 1032.74559 · doi:10.1016/S0020-7403(02)00056-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
