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368 R. Rikards et al. / Composite Structures 51 (2001) 361±370 Table 4 Natural frequencies [Hz] of laminated composite plate Mode Ref. [31] ANSYS Present With rib No rib With rib No rib With rib No rib 1 213.8 85.1 213.1 85 215 85.6 2 229.4 134 220.8 133.9 235.5 135.7 3 270.2 207.4 270.3 206.3 274.5 208.1 4 313.8 216.1 308.8 215.5 315.4 219.9 5 354 252.5 353.5 251.4 361.4 256.3 Exploring the mode shapes of the sti€ened plate it should be noted that, for example, for the ®rst frequency there is a coupled skin±rib antisymmetric vibration mode. This is due to low torsional sti€ness of the rib since the shear modulus G 23 of the composite is very low (see Table 3). 5.4. Buckling of sti€ened isotropic plate under axial compression In order to verify the present element SH36 for solution of buckling problems a clamped sti€ened plate under axial compression is considered (see Fig. 9). The reference solution of the present numerical example was obtained in [32]. The critical stress of the plate is given by r cr ˆ k ; D ˆ p 2 D Eh 3 a 2 h 12…1 m 2 † : …24† Here critical parameter k is function of non-dimensional quantities. b ˆ b a ; j ˆ EI aD ; e ˆ A ah ; …25† where A is area, I the second moment of area of the sti€ener and h is a skin-plate thickness. The boundary conditions of the clamped plate are as follows: u; v; w; c x ; c y ; cj xˆ0;a; yˆ0 ˆ 0; u; w; c x ; c y ; cj yˆb ˆ 0: …26† The shell model was used in order to calculate the critical stress by linearized buckling approach. In analysis, a 18 18 mesh was employed assuming the values of nondimensional parameters: b ˆ 1; e ˆ 0:2; j ˆ 20; h s ˆ 10:483h; h s =t s ˆ 2:75; a=h ˆ 200. The present analysis gives the value of critical parameter k ˆ 24:85 in Eq. (24). The reference solution is k ˆ 25:46 [32]. Note that reference solution was obtained employing the beam model for the sti€ener. The result obtained by the ®nite element code ANSYS is k ˆ 23:44. In this case, the mesh 20 20 was employed. The buckling mode is antisymmetric in x-direction, however the sti€ener's aspect ratio h s =t s ˆ 2:75 is rather low and it has enough torsional rigidity. 5.5. Buckling of composite cylindrical panel under axial compression Buckling load of composite cylindrical panel under axial compression (see Fig. 10) is compared with the reference solution [33], where the ®rst-order shear deformation theory is also used for buckling analysis. The laminate of 12 plies is considered. The ply thickness is 0.018 in and the laminate ply stacking sequence is ‰45=0=90= 45Š s . The nominal ply mechanical properties for the composite material used are given by (see [33]) E 1 ˆ 13:75 Msi; E 2 ˆ 1:03 Msi; G 12 ˆ G 13 ˆ G 23 ˆ 0:420 Msi; m 12 ˆ 0:25: Geometric parameters of the shell are as follows: R ˆ 40 in; L ˆ 22 in; h ˆ 0:216 in; a ˆ 180°: Fig. 9. Clamped sti€ened plate under axial compression. Fig. 10. Cylindrical composite panel under axial compression.
R. Rikards et al. / Composite Structures 51 (2001) 361±370 369 Table 5 Buckling loads [lbs/in] of composite cylindrical panel Ref. [33] ANSYS Present STAGS Segment approach 3328 3278 3285 3313 Acknowledgements The investigations concerning the development of ®- nite elements for analysis of laminated shells are sponsored through Contract No. G4RD-CT-1999-00103. The author are pleased to acknowledge the ®nancial support by the Commission of European Union. Thanks are also due to Latvian Council of Science for ®nancial support through Grant No. 6430.2. References Fig. 11. Buckling mode of cylindrical panel under axial compression. The boundary conditions for a simply supported cylindrical panel are given by u ˆ wj aˆ0°;180° ˆ 0; v; wj xˆ0 ˆ 0; wj xˆL ˆ 0: Table 5 shows the results for cylindrical panel. The present analysis (see Fig. 11) is performed by a mesh 14 30 elements in the axial and circumferential direction, respectively, while the STAGS and ANSYS ®nite element model consists of a mesh 20 40 elements. The results obtained by the present element SH36 are found to be in good agreement with the reference solutions. 6. Conclusions The ®nite element for vibration and buckling analysis of laminated composite sti€ened shells has been developed. The element is based on ®rst-order shear deformation theory with six degrees of freedom of normal line. For analysis of sti€ened structures two models are considered: the beam and the shell model. It was shown that for torsionally rigid sti€eners the beam model can be employed. Natural frequencies and buckling loads for sti€ened structures obtained from the present analysis are found to be in good agreement with the reference solutions and experiment. [1] Zienkiewicz OC, Taylor RL. The ®nite element method. Vol. 1. Basic formulations and linear problems. London: McGraw-Hill; 1989. [2] Zienkiewicz OC, Taylor RL. The ®nite element method. Vol. 2. Solid and ¯uid mechanics, dynamics and nonlinearity, 4th ed. London: McGraw-Hill; 1991. [3] Huang H-C. Static and dynamic analyses of plates and shells. London: Springer; 1989. [4] Bernadou M. Finite element methods for thin shell problems. Chichester and Paris: Wiley; 1996. [5] Mackerle J. Finite and boundary element analysis of shells ± A bibliography (1990±1992). Finite Elements Anal Design 1993;14:73±83. [6] Gilewski W, Radwanska M. A survey of ®nite element models for the analysis of moderately thick shells. Finite Elements Anal Design 1991;9:1±21. [7] Reddy JN. A review of re®ned theories of laminated composite plates. Shock Vib Digest 1990;22:3±17. [8] Reddy JN, Robbins Jr DH. Theories and computational models for composite laminates. Appl Mech Rev 1994;47:147±69. [9] Timoshenko S. On the correction for shear of the di€erential equation for transverse vibration of prismatic bars. Philos Mag 1921;41:744±6. [10] Reissner E. The e€ect of transverse shear deformations on the bending of elastic plates. J Appl Mech 1945;12:A69±77. [11] Hencky H. Uber die Berucksichtigung der Schubverzerung in ebenen Platten. Ingenieur Archiv 1947;16:72±6. [12] Mindlin RD. In¯uence of rotatory inertia and shear on ¯exural motions of isotropic elastic plates. J Appl Mech 1951;18:31±8. [13] Naghdi PM. On the theory of thin elastic shells. Quart Appl Math 1957;14:369±80. [14] Naghdi, PM. Foundations of elastic shell theory. In: Sneddon IN, Hill R, editors. Progress in solid mechanics, vol. 1. Amsterdam: North-Holland; 1963. p. 1±90 [Chapter 4]. [15] Galimov KZ. Fundamentals of the non-linear theory of thin shells. Khazan: Khazan University Publishing House; 1975 [in Russian]. [16] Bedair OK. A contribution to the stability of sti€ened plates under uniform compression. Comput Struct 1998;66:535±70. [17] Rikards R, Chate AK. Isoparametric triangular ®nite element of a multilayer shell after Timoshenko's shear model. 1. Sti€ness mass and geometric element sti€ness matrices. Mech Compos Mater [Mekhanika Kompozitnykh Materialov] 1981;17:302±9 [trans.; Russian by Plenum Publishing Corporation, New York]. [18] Rikards R, Chate AK. Vibration and damping analysis of laminated composite and sandwich shells. Mech Compos Mater Struct 1997;4:209±32. [19] Luri'e AI. Theory of elasticity. Moscow: Nauka Publishing House; 1970 [in Russian]. [20] Wempner G. Mechanics of solids with application to thin bodies. Alphen aan den Rijn, The Netherlands: Sijtho€ & Noordho€; 1981.
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