Analysis of parametrically excited laminated composite joined conical±cylindrical shells

Transcription

1 Computers and Structures 79 (2001) 65±76 Analysis of parametrically excited laminated composite joined conical±cylindrical shells S. Kamat, M. Ganapathi *, B.P. Patel Institute of Armament Technology, Girinagar, Pune , India Received 1 July 1999; accepted 25 February 2000 Abstract The dynamic instability analysis of a joined conical and cylindrical shell subjected to periodic in-plane load is investigated using C 0 two-noded shear exible shell element. The formulation is based on rst-order shear deformation theory. The present model accounts for in-plane and rotary inertia e ects. The instability regions are determined based on the principle of BolotinÕs method. The boundaries of the principal instability region obtained here are conveniently represented in the non-dimensional excitation frequency ± non-dimensional load amplitude plane. The in uence of various parameters such as orthotropicity, cone angle, lay-up, combinations of di erent sections, thickness ratio, circumferential wave number, static load, and external pressure on the dynamic stability regions of cross-ply laminates is brought out. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Instability; Conical±cylindrical; Laminates; In-plane load; External pressure; Periodic; Finite element 1. Introduction Components such as cylindrical, conical, and joined conical±cylindrical shells are commonly encountered as principal structural elements in the design of aerospace and nuclear structures. Furthermore, the structural sections, in particular, joined conical±cylindrical shells are extensively used as hydraulic nozzles, di users, horn antennae, etc. Although many investigators have developed general nite-di erence or numerical integration techniques that allow the evaluation of dynamic behaviour of the arbitrary shells of revolution, the available studies in the literature are mostly limited to either conical or cylindrical shells. However, no speci c discussion has been found in the literature concerning dynamic analysis, in particular, parametric instability analysis of a composite shell, in the sense, whose meridian contains a geometric discontinuity (conical± cylindrical shell), subjected to periodic in-plane loads. * Corresponding author. Fax: address: gana@iat.ernet.in (M. Ganapathi). Information regarding the dynamic instability characteristics of such joined conical±cylindrical shells for various system parameters may become advantageous and economical in design. In addition, the increasing application of advanced composite materials in the design of such principal structural components makes the investigation of dynamic instability behaviour of laminated composite shell structures like joined conical± cylindrical shells very important. The study of dynamic stability of isotropic circular cylindrical shells by analytical methods has been carried out by many investigators [1±4] and is reviewed by Hsu [5]. Unlike the isotropic case, the dynamic instability analysis of a single layer orthotropic, and laminated anisotropic circular±cylindrical shells has been treated sparsely in the literature [6±8]. In all these studies, analytical methods using assumed mode shapes are employed. In Ref. [6], the investigation is restricted to thin orthotropic circular cylindrical shells whereas a specially orthotropic laminated case wherein bending±stretching e ects are neglected has been considered in Refs. [7,8]. Dynamic instability due to suddenly developed temperature is studied by Ray and Bert [9]. Ganapathi and /01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 66 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 Balamurugan [10] have recently investigated the laminated circular cylindrical shell by employing the nite element procedure. However, to the authorsõ knowledge, no work has been reported in the literature on parametric/dynamic instability of isotropic/orthotropic/ laminated composite conical shell or joined conical± cylindrical shells. This has prompted the authors to investigate the principal instability region of structures having geometric discontinuities such as joined conical± cylindrical shells and provide a detailed information for the design of light-weight structures under pulsating loads. In this paper, the nite element method is used to study parametric instability of laminated anisotropic composite conical±cylindrical shell structure subjected to periodic axial/radial loading. A simple two-noded shear exible axisymmetric shell element based on eld consistency approach [11] is employed. The in-plane and rotary inertia e ects are included in the model. The formulation developed here is validated with the available analytical results, and results are obtained for the isotropic/orthotropic/laminated cross-ply composite joined conical±cylindrical shell structures. A detailed parametric study to highlight the in uences of orthotropicity, cone angles, circumferential wave numbers, lay-up, thickness ratio, static load and external pressures on the principal instability regions are investigated. 2. Formulation An axisymmetric laminated composite joined conical±cylindrical±conical shell is considered with the coordinates s, h along the meridional and circumferential directions, and z along the radial or thickness direction, as shown in Fig. 1. By using the Mindlin formulation, the displacements at a point (s, h, z) are expressed as functions of the mid-plane displacements u 0, v 0 and w, and independent rotations b s and b h of the meridional and hoop sections, respectively, as u s; h; z; t ˆu 0 s; h; t zb s s; h; t ; v s; h; z; t ˆv 0 s; h; t zb h s; h; t ; w s; h; z; t ˆw s; h; t ; 1 where t is the time. By using the semi-analytical approach, u 0, v 0, w, b s, and b h are represented by a Fourier series in the circumferential angle h. For the nth harmonic, these can be written as u 0 ˆ u 0 s; t cosnh; v 0 ˆ v 0 s; t sinnh; w ˆ w s; t cosnh; b s ˆ b s s; t cosnh; 2 b h ˆ b h s; t sinnh: The strains pertaining to truncated cone are de ned as [12] feg ˆ ep ze b : 3a 0 e s The mid-plane strains e p, bending strains e b and shear strains e s, in Eq. (3a) are written as 8 9 < ou 0 =os = fe p gˆ u 0 sin/ ov 0 =oh wcos/ =r : ; ; 3b ou 0 =oh v 0 sin/ =r ov 0 =os 8 9 < ob s =os = fe b gˆ b s sin/ ob h =oh =r : ; ; cos/ov 0 =os ob s =oh b h sin/ =r ob h =os 3c fe s gˆ ow=os b s b h ow=oh v 0 cos / r ; 3d where r(s) is equal to R 1 ssin/ and the subscript comma denotes the partial derivative with respect to spatial co-ordinate succeeding it. R 1 and / are the small circle radius and semi-cone angle, respectively. If {N} represents the membrane stress resultants (N ss, N hh, N sh ) and fmg, the bending stress resultants (M ss, M hh, M sh ), one can relate these to the membrane strains fe p g and bending strains, fe b g, through the constitutive relation as fng ˆ AŠfe p g BŠfe b g and 4 fmg ˆ BŠfe p g DŠfe b g; Fig. 1. Geometry and the loading of a joined conical±cylindircal±conical shell (three sections).

3 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 67 where A ij Š; B ij Š; D ij Š (i, j ˆ 1, 2, 6) are the extensional, bending±extensional coupling, and bending sti ness coe cients of the composite laminate. Similarly, the transverse shear force [Q] representing the quantities (Q xz ; Q yz ) are related to the transverse shear strains fe s g through the constitutive relation as, fqg ˆ EŠfe s g; 5 where E ij Š (i, j ˆ 4, 5) are the transverse shear sti ness coe cients of the laminates. For a composite laminate of thickness, h, consisting of l layers with stacking angles a i i ˆ 1;...; l and the layer thickness, h i i ˆ 1;...; l, the necessary expression to compute the sti ness coe cients, available in the literature [13] are used here. The strain energy functional U is given as Z h U d ˆ 1=2 fe p g T AŠfe p g fe p g T BŠfe b g A fe b g T BŠfe p g fe b g T DŠfe b g i fe s g T EŠfe s g da; 6 where fdg is the vector of the degrees of freedom. The kinetic energy of the shell is given by Z h T d ˆ 1=2 p _u 2 0 _v 2 0 _w2 I b _ 2 s b _ 2 h ida; 7 A where p ˆ R h=2 qdz; I ˆ R h=2 h=2 h=2 qz2 dz and q is the mass density. A dot over the variable represents the partial derivative with respect to time. The conical shell is subjected to internal pressure P m and axial compression P s. The potential energy due to the applied conservative loads for the nth harmonic can be written as [14], V d ˆ 1 Z 2 where A F s r 2 s F h r 2 h F s F h rn 2 da; 8 r s ˆ ow os ; r h ˆ 1 ow r oh v 0 r ; r n ˆ 1 2 F s ˆ P s = 2pR 1 cos/ P m R 1 = 2cos/ ; F h ˆ P m R 1 = cos/ : ov 0 os 1 r ou 0 ; oh Substituting Eqs. (6)±(8) in LagrangeÕs equation of motion, one obtains the governing equation for the shell as MŠf dg KŠfdg K G Šfdg ˆf0g; 9 where [M], [K], and [K G ] are the mass matrix, linear sti ness and geometric sti ness matrices, respectively. 3. Parametric instability analysis The state of the periodic loads, F s and F h is the uniform pulsating one either due to axial force, P s, or external pressure load P m, which may be de ned as for axial force case: F s ˆ P 0 s P 1 s cosxt = 2pR 1 cos/ ˆaN bn cosxt; F h ˆ 0; 10a for external pressure case: F s ˆ R 1 P 0 m P 1 m cosxt = 2cos/ ˆ a=2 N b=2 N cosxt; F h ˆ R 1 P 0 m P 1 m cosxt = cos/ 10b ˆ an bn cosxt; where a ˆ P 0 s = 2pR 1 cos/n or R 1 P 0 m = cos/n ; b ˆ P 1 s = 2pR 1 cos/n or R 1 P 1 m = cos/n ; N ˆ N s cr R 1E L h 3 = 1 V LT V TL =L 2 or N m cr E Lh=R 1 ; and x is the frequency of dynamic in-plane load. The superscripts 0 and l here refer static and dynamic part of the load. The subscripts and superscripts s and m denote the axial and external pressure loads, and the subscript cr refers the critical stress resultant. a and b are the static and dynamic load factors. From Eqs. (9)± (10b), we have the governing equation of the form, MŠfdg KŠ an bn cosxt K G Š fdg ˆ0; 11 where [K G ] is the geometric matrix, due to the axial force or external pressure load. Eq. (11) represents the dynamic stability problem of a system subjected to a periodic in-plane axial force. The dynamic instability boundary is determined using the method suggested in the literature [1]. To obtain points on the boundaries of the instability region, the components fdg are written in the Fourier series as [1,10], fdg ˆ1 2 fbg 0 X fag i sin ixt 2 iˆ2;4;6;... ixt fbg i cos 12 2 with period T, where T ˆ 2p=x, or fdg ˆ X fag i sin ixt ixt fbg 2 i cos 13 2 iˆ1;3;5;... with period 2T. These expressions are substituted in Eq. (11), and the coe cients of each sine and cosine terms are set equal to zero, as well as the sum of the constant terms. For nontrivial solutions, the determinants of the coe cients of these groups of linear homogeneous equations are equal to zero. Solving them for a given value of a, the variation of x with respect to b can be found out. Such a plot shows the instability regions associated with the given shell subjected to harmonically excited in-plane load.

4 68 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 Table 1 Comparison of natural frequencies of a cantilever cylindrical±conical isotropic shell a m b Circumferential wave number (n); f (rad/s) Ref. [16] Present ± ± ± a L c /R 1 ˆ 2, L o /R 1 ˆ 1, h/r 1 ˆ 0.01, m ˆ 0.3, / ˆ 30. b Axial half wave number. 4. Element description The laminated axisymmetric shell element employed here is two-noded C 0 continuous shear exible element with ve nodal degrees of freedom per node (u 0, v 0, w, b s, b h ), as shown in Fig. 1. If the interpolation functions for a two-noded axisymmetric element is used directly to interpolate the ve eld variables, u 0 ;...; b h in deriving the shear strains, the element will lock and show oscillation in the shear stresses. Field consistency requires that the transverse shear strains must be interpolated in a consistent manner. Thus, the b s term in the expression for fe s g given in Eq. (3d) has to be consistent with the eld function w ;s as shown in Ref. [11]. This is achieved by using a eld-redistributed substitute shape function to interpolate those speci c terms which must be consistent. 5. Results and discussion In this section, we use the above formulation to investigate the e ects of parameters like orthotropicity, number of cross-ply layers, ply angle, thickness ratio, static and dynamic load parameters, and circumferential wave number on the dynamic instability of laminated joined conical±cylindrical or conical±cylindrical±conical shells subjected to periodic loads. As the nite element is based on consistency approach, exact integration is used to evaluate all the energy terms. The shear correction factors are evaluated as outlined in Ref. [15], which takes into account of layer properties and stacking sequence of the given laminates. Based on progressive mesh re nement, 40 elements idealisation for two section shell geometry, and 60 element idealisation for three section case are found to be adequate to model the full shell for the present analysis. Firstly, the formulation developed herein is validated considering free vibration analysis of isotropic joined conical±cylindrical shell [16] and laminated circular sandwich conical [17,18], and the results are compared in Tables 1 and 2 along with the available solutions. It is observed from these tables that Table 2 Natural frequencies f (Hz) for the clamped±clamped concial sandwich shell m n Present Ref. [17] Ref. [18] ± ± Top/bottom layer: E f ˆ GPa, m f ˆ 0.20, q f ˆ 2800 kg/m 3, h f ˆ mm; core layer: G 12 ˆ GPa, G 13 ˆ GPa, q c ˆ 36.8 kg/m 3, h c ˆ 7.62 mm; L c ˆ m, small radius ˆ m, / ˆ the present results are fairly in good agreement with those of the existing results. The small discrepancies in the results may be attributed to the di erent theories and approaches used in the literature [16±18]. Next, in view of the computational time involved in the parametric study, one term solution of Eqs. (12) and (13) is employed, which furnishes accurate results for the reasonably low values of load amplitude parameters. Furthermore, the analysis is focused mainly on the determination of boundaries of the primary instability region that occur in the vicinity of 2x i (x i is the exural natural frequency where i ˆ 1; 2; 3...) which is by far the largest one compared to the neighbourhood of combination resonance of rst order, x i x j. This is the most dangerous or critical and has the greatest practical importance [1,7,10]. The e cacy of the present model is also tested for the dynamic instability of laminated cross-ply circular±cylindrical shell, for which solutions are available in the literature [7], and the results are presented in Table 3. It is inferred from Table 3 that

5 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 69 Table 3 Comparison of principal instability boundries of simply supported laminated cross-ply circular cylindrical shell with Ref. [7] a Wave no. Ref. [7] Present m n k 2 Ps 1 ˆ 0 k 2 1 P s 1 ˆ 1 k 2 2 P s 1 ˆ 1 k 2 Ps 1 ˆ 0 k 2 1 P s 1 ˆ 1 k 2 2 P s 1 ˆ 1 R 1 =h ˆ L=R 1 ˆ R 1 =h ˆ L=R 1 ˆ b a k 2 ˆ x 2 qr 4 1 = E Th 2 ; E L =E T ˆ 40; G LT =E T ˆ G LZ =E T ˆ 0:6; G TZ =E T ˆ 0:5; m LT ˆ 0:25. b Multiplying factor for 12 when R 1 /h ˆ 15 case. the performance of the present method is in good agreement with the analytical approach. The material properties of CFRP, unless speci ed, used in the present analysis are, E L =E T ˆ 39:8113; G LT =E T ˆ 0:4906; G TT =E T ˆ 0:2453; c LT ˆ 0:25; q ˆ 1524 Kg=m 3 ; E T ˆ 0: N=m 2 ; where E, G and c are YoungÕs modulus, shear modulus and PoissonÕs ratio. L and T are the longitudinal and transverse directions, respectively, with respect to bres. All the layers are of equal thickness and the ply-angle is measured with respect to the s-axis (longitudinal axis). The simply supported boundary conditions considered in the present analysis are u ˆ w ˆ b h ˆ 0 at s ˆ 0; L: Fig. 2. Dynamic instability regions of isotropic conical±cylindrical shells (two sections); (n ˆ n cr ; m ˆ 1; R 1 =h ˆ 300; a ˆ 0); (a) converging conical ends and (b) diverging conical ends.

6 70 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 Here, a dynamic instability characteristics of isotropic/orthotropic/laminated conical±cylindrical shells L c = R 1 ˆ cosec/ sec / =2; L o =R 1 ˆ 1; R 1 =h ˆ 300 where L c and L o are the meridional length of conical and cylindrical section) are studied for the lowest natural frequency, i.e. n ˆ n cr and m ˆ 1. Here, m and n denote the axial half wave and circumferential full wave numbers, respectively. The plot of primary instability region in the neighbourhood of 2x i, in terms p of non-dimensional excitation frequencies, X ˆ x q 0 hr 4 1 =D ; D ˆ E L h 3 = 12 1 m LT m TL Š versus the dynamic in-plane load parameter b for di erent cone angles / ˆ 0 ; 15 ; 30 and 45 are depicted. The width of primary instability, DX, as shown in Fig. 2, is the separation of the bound- Fig. 3. E ect of semi-cone angle (/) of laminated composite convergent/divergent conical±cylindrical shells (two sections) on dynamic instability: (a) critical circumferential wave no. n cr, (b) origin of instability region, X pertaining to n cr ; (c) dynamic instability width (b ˆ 0:2) and (d) dynamic instability width (b ˆ 0:7). (±h±) 0 layer; (ááá) (0 /90 ); () )))(0 /90 /0 ) and ()áá)áá)áá) (0 =90 =0 =90 s.

7 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 71 aries of the primary instability region for the given shell geometry. This can be used as an instability measure to study the in uence of various structural parameters. The variations of dynamic instability region with respect to amplitude of the load b are described in Fig. 2 for isotropically joined conical±cylindrical shells (two sections) with di erent cone angles. The conical section considered at the end of the cylindrical shell is either convergent or divergent type. It is seen from Fig. 2 that, for the cylindrical shell with convergent conical section, critical wave number n cr corresponding to the lowest fundamental frequency is less in comparison with those of the cylindrical ones with divergent conical end case. Also, it is brought out from this gure that the increase in the cone angle, irrespective of the type of cone, postpones the occurrence of instability to a higher value of frequency. It is mainly due to the increase in the severity of the geometric discontinuity, which, in turn, produces a sti ening e ect around the joint (cone± cylinder) like a sti ened ring. Furthermore, it is noticed from Fig. 2 that the origin of primary instability region shifts to a much higher excitation frequency for the convergent conical±cylindrical shell compared to those of divergent conical±cylindrical case. It is further viewed from this gure that the occurrence of the dynamic instability region, in general, shifts to a higher forcing frequency, and instability width DX also increases with the increase in conical angle, irrespective of the type of the conical end. It is also evident from this gure that, for divergent conical case, the origin for the occurrence of the instability region and the resulting frequency bandwidths (instability width) are less, compared to those of convergent conical case. However, for the cylindrical shell with divergent conical end case, the differences in the frequency for the occurrence of instability and width of the instability zone are insigni cant for higher cone angles considered here. Also, one can notice from this gure that the shell is unstable in wide intervals of frequency with the increase in the amplitude of the dynamic load parameter b, as expected. Lastly, one can conclude from this study that, for the given dynamic load, the shell with only cylindrical section has minimum instability width but the frequency, around which the occurrence of the instability takes place, is much lower than those of conical±cylindrical shell combinations. The in uence of the number of layers in the cross-ply laminated conical±cylindrical shells on the value of n cr, the origin of instability region, and the unstable operating frequency width (instability width) for di erent dynamic load parameters are highlighted in Fig. 3. It is revealed from Fig. 3(a) that, in general, the single-layered orthotropic shell section has the highest critical value for circumferential wave number, n, and then followed by three-layered, two-layered cases, and lastly, by eightlayered symmetric laminate, except at higher cone angle for the convergent conical±cylindrical case. Furthermore, Fig. 4. Dynamic instability regions of isotropic conical±cylindrical±conical shells (three sections) (n ˆ n cr ; m ˆ 1; R 1 =h ˆ 300; a ˆ 0): (a) converging conical ends and (b) diverging conical ends.

8 72 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 it may be observed from Fig. 3(a) that the value n cr is high for the divergent conical±cylindrical shell combination, and then it increases with the increase in the cone angle, whereas the change in the value of n is less for the convergent conical±cylindrical case. However, for the higher cone angle, in the case of convergent conical± cylindrical shell, the critical wave number increases for two-layered and eight-layered shells. For the chosen cone section at the end of the cylindrical shell, the variation in the value of n cr and its associated forcing frequency, for instance, see Fig. 3(b), dealing with the unstable situation highly depend on the directional sti ness provided by the anisotropic properties in the laminates. It is further seen from Fig. 3(b) that the forcing frequency corresponding to the occurrence of instability region is very low for the single-layered orthotropic shell, and next lower cases are those corresponding to three-layered, two-layered, and nally eight-layered symmetric laminates. The behaviours such as variations of n cr and, in turn, change in the frequencies highly depend on the contribution of mem- Fig. 5. E ect of semi-cone angle (f) of laminated composite convergent/divergent conical±cylindrical±conical shells (three sections) on dynamic instability: (a) critical circumferential wave no. n cr, (b) origin of instability region, X pertaining to n cr, (c) dynamic instability width (b ˆ 0:2) and (d) dynamic instability width (b ˆ 0:7). (±h±) 0 layer; (ááá) (0 /90 ); () ) )) (0 /90 /0 ) and ()áá)áá)áá) 0 =90 =0 =90 s.

9 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 73 brane bending and to some extent shear strain energies to the total internal energy of the shell structure. For instance, a higher n cr value for orthotropic case, in general, may lead to a very less contribution of membrane energy to the total strain energy and thus, the system has a very low value of the exural frequency. Also, it is brought out from Fig. 3(c) and (d) that, for the cylindrical section with convergent conical end, the instability width and the origin of the occurrence of instability zone are marginally higher than those of the cylindrical section with divergent conical end. It is evident from these gures that the increase in the dynamic load value enhances the instability width and is similar to that of the isotropic case. Finally, it may be opined that the single-layered orthotropic shell, in general, has the lowest instability width, and the primary instability region occurs early compared to all other cross-ply laminations considered here. Further, the e ect of coupling present in the two-layered laminate is to shift the instability region to lower values of forcing frequency compared to the eight-layered one. Parametric instability study is also made for a joined conical±cylindrical±conical shell (three sections). Keeping the same cylindrical shell geometry as that of two sections and attaching conical section at both ends of the cylindrical shell, three section shell geometry is considered. The length of each conical section is assumed as half of the conical portion pertaining to the two section cases. The dynamic characteristics of both isotropic and laminated shells are predicted and described in Figs. 4 and 5, respectively. It is understood from Fig. 4 that, for the isotropic shell, the critical circumferential wave number for the convergent conical shell combinations, irrespective of the cone angles, is considerably high compared to those of two section case whereas it is slightly more for the divergent conical shell combinations. However, for the divergent conical shell case having higher cone angle, there appears to be no change in the values of n cr while comparing with the two sections case (Fig. 2). Furthermore, it is observed from Fig. 4 that, for the cylindrical shell with divergent conical ends, the origin of instability region is shifted to very high values of forcing frequency compared to the values of two section shells. It is further noticed that, for the case of cylindrical shell with convergent conical ends compared to the corresponding two sections case, the change in the origin of instability zone is less for the higher cone angle, and more so for the deep conical sections. The widths of the instability region for the three section cases, in general, are higher compared to the corresponding two section cases. The instability regions for the higher cone angle considered here are almost coincident. The e ect of the number of layers in the cross-ply conical±cylindrical±conical shell on the value of n cr, the origin of instability region, and the unstable operating frequency width for di erent dynamic load parameters are shown in Fig. 5. It is clear from Fig. 5(a) and (b) that the in uence of the number of layers on the n cr is qualitatively similar to those of two-section shells. However, one can notice here that the value of n cr increases with the increase in the divergent cone angle, up to a certain value. But, unlike in the case of two section shells, here, n cr increases with the cylindrical shell having convergent conical end sections. It can also be opined from Fig. 5(b) that the occurrence of instability region is shifted to slightly higher values for the forcing Fig. 6. Dynamic instability regions of laminated composite conical±cylindrical shells (two sections) against circumferential wave numbers, n (m ˆ 1; R 1 =h ˆ 300, a ˆ 0, (0 /90 /0 /90 ) s ): (a) / ˆ 15 and (b) / ˆ 45.

10 74 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 Fig. 7. Dynamic instability regions of laminated composite conical±cylindrical±conical shells (three sections) against circumferential wave numbers n (m ˆ 1; R 1 =h ˆ 300, a ˆ 0, (0 /90 /0 /90 ) s ): (a) / ˆ 15 and (b) / ˆ 45. frequencies and it shows almost symmetrical behaviour with respect to cone angle. From Fig. 5(c) and (d), one can conclude that the stability strength is low around 30 for the cylindrical shell with divergent conical sections whereas instability width is high around 15 for the cylindrical shell with convergent conical sections. In general, for the shell with three sections considered here, the occurrence of instability region is shifted to higher frequency except for the convergent section cases with a high cone angle, and the instability width is high in comparison with those of two section cases. The e ect of circumferential wave number, n, for eight-layered cross-ply laminates, on the dynamic instability region is presented in Figs. 6 and 7 for both two and three sections having convergent conical end / ˆ 15 ; 30, respectively. It is evident from these gures that, for the given shell, the critical/dangerous instability region always associates with n cr, i.e. a low value of forcing frequency. Also, it occurs at higher values of frequencies for the three section shells compared to two section cases. Furthermore, when the value of n is very less, the value of frequency is mostly dominated by the membrane energy of the structure. The bending energy signi cantly in uences the value of the frequency for higher n. However, the signi cance of all these parameters depends on the shell geometry, material properties and lay-up. The variations of dynamic instability zone under external periodic pressure is analysed considering eightlayered cross-ply joined conical±cylindrical±conical shells and the results are given in Fig. 8. The dynamic instability regions predicted considering di erent cone angles are qualitatively similar to those of in-plane axial load cases. Fig. 9 deals with the dynamic instability characteristics of the shell with two sections having di erent Fig. 8. In uence of periodic external pressure load on dynamic instability of laminated composite convergent conical±cylindrical±conical shells (three sections) with various values of / (semi-cone angle).

11 S. Kamat et al. / Computers and Structures 79 (2001) 65±76 75 Fig. 9. Dynamic instability regions of eight-layered symmetric cross ply conical±cylindrical shells (two sections) with di erent thickness ratios n ˆ n cr ; m ˆ 1 : (a) / ˆ 15 and (b) / ˆ 30. thickness ratio, and static load parameter, a. One can draw the conclusion from Fig. 9 that the increase in the thickness of the shell decreases the instability width and lowers the value of non-dimensional forcing frequency at which instability occurs. One can also see from Fig. 9 that the critical circumferential wave decreases with the increase in the thickness of the shell. The magnitude of the frequency and width of the instability depend on the shell geometry. It can be further noted that the e ect of static load parameter on the parametric excitation is to reduce the frequency at which the structure becomes unstable and also keeps the system unstable over a wide range of operating frequencies for a given amplitude of the dynamic load. 6. Conclusions Parametric instability study of conical±cylindrical shell subjected to periodic in-plane load is examined by considering two-noded axisymmetric shell element based on shear exible theory. Numerical results have been obtained for isotropic/orthotropic/laminated shells with di erent conical sections joined with circular cylindrical shells. From the detailed study, the following observations can be made: 1. The origin of dynamic instability region shifts to higher excitation frequencies with the increase in cone angle of conical±cylindrical shells, irrespective of orthotropicity, and the type and number of conical section, i.e. the stronger the geometric discontinuity in conical±cylindrical shell it produces large sti ening effect around the joints and, in turn, enhances the value of frequency. 2. The occurrence of the instability region and the unstable width depends on the type and the number of conical sections, as they change the frequencies and the associated mode shapes, signi cantly. 3. The dynamic instability characteristics change considerably depending on the directional sti ness provided by the anisotropic properties in the multi-layered laminate. 4. In general, for the shell with three sections considered here, the instability width is high compared to those of two section cases, and it is almost symmetric with respect to cone angle.

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