Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 69- 75 (2008) Dynamic Analysis of Thin Composite Cylindrical and Spherical Shells Qusai H. Jebur Technical College-Baghdad Foundation of Technical Education Email: kusay74hj@yahoo.com (Received 10 Decemper 2006; accepted 2 January 2008) Abstract In this work, an investigation for the dynamic analysis of thin composite cylindrical and spherical shells is presented. The analytical solution is based upon the higher order shear deformation theory of elastic shells from which the developed equations are derived to deal with orthotropic layers. This will cover the determination of the fundamental natural frequencies and mode shapes for simply supported composites cylindrical and spherical shells. The analytical results obtained by using the derived equations were confirmed by the finite element technique using the well known Ansys package. The results have shown a good agreement with a maximum percentage of discrepancy, which gives a confidence of using this solution in prediction the dynamic analysis of cylindrical and spherical shells. Keywords: Dynamic, Composite, Cylindrical, Spherical, Shells. 1. Introduction A shell is a three-dimensional body which is bounded by two closely spaced curved surfaces, the distance between the surfaces, being small in comparison with the other dimensions [1]. A shell is considered to be a thin shell when the shell thickness is less than 1/20 of the wavelength of the deformation mode and/or radius of curvature and at the small time, it is assumed that both shear deformation and rotary inertia are ignored. The dynamic analysis of shells has been expanding rapidly due to the importance of shell structures in civil, mechanical and aerospace engineering. The thin composite cylindrical and spherical shells are found in many aerospace and aircraft industrial applications such as aircraft wings and fuselage radomes, EWACS and fuel tanks. Also, the composite cylindrical shells are found in compressor blades, ships and rocket. Humayun R.H. Kabir [2] investigated analytically the free vibration of composite shallow cylindrical shells with simply supported boundary conditions using Kirchoff-Love theory. H.M. Wang [3] investigated the dynamic solution of a multi layered hollow cylinder in a state of axisymmetric plane strain. The solution is divided into two parts: one is quasi-static and the other is dynamic. The qusai static is solved by the state space method, and the dynamic part is obtained by the separation of variables coupled with the initial parameter method . Rong-Tyai & Zung-Xian Lin [4] presented the formulation of governing equations for a symmetric cross-ply laminated cylindrical shell with a circumferential stiffener. Penzes and Burgin [5] were the first to solve the problem of the free vibrations of thin isotropic spheroid al shells by using Galerkins method using membrane theory and harmonic axisymmetric exitation. Al-Najim F. [6] used the Rayliegh method to obtain the natural frequencies and mode shapes of axisymmetric vibrations of thin elastic oblate spheroidal shells theoretically and experimentally. He showed that the Rayleigh's method is suitable to deal with such types of problems. From the previous literatures, it is found that there is still a chance to work in this field especially for the analysis of composite cylindrical and spherical shells and therefore this work will be devoted in this respect analytically using the higher order shear deformation theory mailto:kusay74hj@yahoo.com Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 69-75 (2008) 70 and numerically using the finite element method. Mathematical Analysis The review of literature reveals that the governing equations for thin spherical shells are not available. The following formulation gives the free formulation for the thin shells of revolution which can be applied for both cylindrical and spherical shells. Based upon the Third-order theory of Reddy using the displacement field Ref. [7]:                   x w H z tyxztyxutzyxu 0 12 3 10 3 4 ),,(),,(),,,(   ),,(),,,( 3 4 ),,(),,(),,,( 0 2 0 22 3 20 tyxwtzyxw x w H z tyxztyxvtzyxv                        The resulting strain-displacement relations are:                          2 1 0 2 1 1 2 3 1 1 1 0 1 3 4 x w xH z x z x u                                     2 2 0 2 2 2 2 3 2 2 2 0 2 0 2 3 4 x w x H z x z R w x v    0 3                                                  21 0 2 2 1 1 2 2 3 1 2 2 1 2 0 1 0 6 2 3 4 xx w xx H z xx z x u x v                               2 0 22 2 2 0 24 4 x w H z x w                      1 0 12 2 1 0 15 4 x w H z x w  ...(1) Appling Hamilton’s principles, the resulting equations contain double and triple integrals and because of the variation principle of displacements, the coefficient of the variations displacements is zero from which the following equations of motion are obtained:   0 1 2  t t tKU  …(2) where: dzdxRd U zA      4455 66332211         t V dtdzdxdwwvvuuRK  ...... Carrying out the above integrations will give the followings: x w R I I H R I I HR I I u R I x N x N                                                       .. 1 5 42 2 .. 1 5 42 1 3 2 .. 1 1 2 6 1 1 3 4 3 4 2  …(3) 2 .. 2 5 42 .. 2 2 1 2 .. 2 5 42 2 3 2 1 6 2 2 3 42 3 4 x w R I I H v R I I R I I HR I I x N x N                                                                        …(4) .. 1 2 .. 2 4 5 2 4 2 5 1 4 2 5 1 4 2 2 2 1 16 2 2 1 1 2 2 2 2 2 2 9 16 3 4 4 2 3 4 wI x v RH I H I q x Q x Q x K x K H R N R N x S x S x S H                                                                                  x u R I I Hx H I H I R x w x w H I H I xH I H I                                                                                                           .. 1 5 42 1 1 .. 2 5 4 7 2 2 .. 2 2 1 .. 2 4 7 2 4 2 2 .. 2 5 4 7 3 4 3 4 9 16 9 16 3 4 3 4 9 16   …(5) Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 69-75 (2008) 71 1 .. 4 7 2 5 .. 14 7 2 5 3 .. 1 5 42 1 3 252 5 2 6 2 2 6 1 1 2 1 1 9 16 3 8 9 16 3 8 3 44 3 4 3 4 x w H I H I H I H I I u R I I HR I IK H Q x S Hx M x S Hx M                                                                 …(6) 2 .. 4 7 2 5 .. 24 7 2 5 3 .. 2 5 42 2 3 242 4 1 6 2 1 6 2 2 2 2 2 9 16 3 4 9 16 3 8 3 44 3 4 3 4 x w H I H I H I H I I v R I I HR I IK H Q x S Hx M x S Hx M                                                                 ...(7)         5,4,1, 6,3,2,1,,1,, 2 3     idzzKiQi idzzzSiMiNi i i       7,6,5,4,3,2,1 ,,,,,,1 65432    idz zzzzzzI ii  Substituting the above resultant forces in the developed equations and then the assumed displacement components according to Navier׳s Solution Ref. [8] (for simply supported boundary conditions), the stiffness and mass matrices are obtained. Solution of the Developed Equations: The solution is assumed as follows:     mni m n mn mni m n mn exBtzxv exAtzxu                 1 1 0 1 1 0 cossin,,, sincos,,,     mni m n mn mni m n mn exDtzx exCtzxw                 1 1 1 1 1 0 sincos,,, sinsin,,,   mni m n mn exEtzx         1 1 2 cossin,,, … (8) where α=       L m , β=n The mass and stiffness matrices are obtained from the solution of the eigenvalue equation as follows:      02  AMK  …(9) From which the natural frequencies and mode shapes are obtained. Finite element modeling Finite element modeling for the laminated shells is done using ANSYS (5.4), following the major steps [9]: 1. Building the structure using quad shell element 99 as shown in Fig. (1). 2. Applying boundary conditions. 3. Solve the natural frequency problem and getting the results. Fig. 1. 100-layer Shell-99 element. Results and Discussions The developed analytical solution using the general third order shear deformation theory (HSDT) will be employed to investigate its applicability in investigating the dynamic analysis of symmetric and non-symmetric cross- ply laminated cylindrical and spherical shells. The results are composite plates used by other researchers. (a) Spherical shell In order to obtain the fundamental natural frequencies from the developed analytical solution, different ratios of radius-to-side length 5,10 and 20 for both type of shells thin(a/ H) =100 and thick (a/H)=10 for the composites (0,90,0) and (0,90,90,0).The results are shown in table 1. It is seen that the maximum percentages of discrepancy is 8.97%. The results indicate that the thicker shells have lower frequency parameter than the thinner shells, and for shells with smaller (R/a) the frequency parameter is greater than that for larger ratios .In the above calculations the material properties are as followings [10]. E1=2e6, E2=E3=1e6, G12=G13=0.5e6, G23=0.2e6, ν12 = ν13 = 0.24, ν23 = 0 Another interesting result is that the fundemenral frequency for symmetric shells is Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 69-75 (2008) 72 greater than that for antisymmetric one. However, Fig. 2 shows the first three modes shapes for the spherical shell of composite type (0,90). The shapes are consistent with the predicted modes. Fig. 3 shows the frequency parameter change 2 2 EH a          with the ratio (a/H) which shows that the fundamental natural frequency is increased with the increasing (a/H) ,for both types of composites investigated ( 0,90,0) and (0,90) (b) Cylindrical Shell General Third Order Theory (HSDT) is employed to investigate its capability level for dynamic analysis of the symmetric and non- symmetric cross-ply laminated cylindrical shells, and compared with other theories used by other researchers such as FSDT. In this respect, the fundamental natural frequency was obtained for the ratios of radius to side length (5, 10, and 20) and (a/H)=100 thick shell and (a/H)=10, thin shell using the composite, (0,90,0) and ( 0, 90, 90, 0). The results of using higher shear deformation theory are compared with those obtained from using first order shear deformation and the Ansys Package with a percentage of discrepancy 9 %. The results indicates a decrease in the frequency parameter, for example for composite (0,90) a decrease from 16.69 for (R/a)= 5 and 10.27 for (R/a)= 20 while in thin cylinder (a/H)= 10, No noticeable change , the frequency parameter decreases from 9.023 for (R/a)= 5 to 8.972 for (R/a)= 2. Similar trends were noticed for the other composites. However Fig. 4 shows theses trends for both composies (0, 90) and (0, 90, 0). Table 1 Nondimensionalized fundamental frequencies versus Radius-to-side length ratios of spherical shell. (R/a) Theory [0-90] [0-90-0] [0-90-90-0] (a/H)=100 (a/H)=10 (a/H)=100 (a/H)=10 (a/H)=100 (a/H)=10 5 FST 28.825 9.230 30.993 12.372 31.079 12.437 Present Work 28.829 9.307 30.999 12.018 31.083 12.007 FEM 27.563 8.872 29.253 11.563 30.146 11.683 Discrepancy% 4.3 4.7 5.63 3.67 3 2.6 10 FST 16.706 8.984 20.347 12.215 20.380 12.280 Present Work 16.710 9.064 20.353 11.853 20.385 11.840 FEM 16.001 8.254 19.754 11.102 19.831 11.024 Discrepancy% 4.24 9 2.9 6.1 2.71 6.64 20 FST 11.841 8.921 16.627 12.176 16.638 12.240 Present Work 11.847 9.002 16.634 11.811 16.643 11.798 FEM 11.011 8.201 16.001 11.310 15.885 11.023 Discrepancy% 7.06 8.97 3.807 4.11 4.55 6.33 Table 2 Nondimensionalized fundamental frequencies versus Radius-to-side length ratios of cylindrical shell. (R/a) Theory [0-90] [0-90-0] [0-90-90-0] (a/H)=100 (a/H)=10 (a/H)=100 (a/H)=10 (a/H)=100 (a/H)=10 5 FST 16.668 8.9082 20.332 12.207 20.361 12.267 Present Work 16.690 9.0230 20.330 11.850 20.360 11.830 FEM 16.001 8.731 19.705 11.203 19.871 11.211 Discrepancy% 4.13 4.99 3.54 2.86 3.09 4.55 10 FST 11.831 8.887 16.625 12.173 16.634 12.236 Present Work 11.840 8.979 16.620 11.8 16.630 11.79 FEM 11.151 8.535 16.031 11.451 16.115 11.233 Discrepancy% 5.82 4.99 3.54 2.86 3.09 4.55 20 FST 10.265 8.89 15.556 12.166 15.559 12.23 Present Work 10.27 8.972 15.55 11.79 15.55 11.78 FEM 9.891 8.420 15.067 11.247 15.004 11.136 Discrepancy% 3.69 6.2 3.1 4.48 3.509 5.265 Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-75 (2008) 73 (a)- first mode (b)-second mode (c)-third mode Fig. 2 Mode shapes of [0-90] spherical shell. 0 20 40 60 80 100 (a/H) 8 12 16 20 24 F re q u e n c y P a ra m e te r [0-90-0] [0-90] Fig. 3 Frequency parameter 2 2 EH a           change with (a/H) for cylindrical shell. 0 20 40 60 80 100 (a/H) 0 10 20 30 40 F re q u e n c y P a ra m e te r [0-90-0] [0-90] Fig. 4 Frequency parameter 2 2 EH a           change with (a/H) for spherical shell Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-75 (2008) 74 Conclusions The following points may be summarized from the current work: 1. The developed analytical solution may be used for the dynamic analysis of thin composite shell, spherical or cylindrical and may be extended to the conical shells. The validity obtained between the analytical and numerical results were in good agreement with a maximum discrepancy of 6.2%. 2. The natural frequency tends to increase with the increasing of radius to side length for all the composite shells (0, 90, 0, 90, 0) and (0, 90, 90, 0). 3. The natural frequency is decreased with increasing the radius/side ratio for all the types of composites. (0, 90), (0, 90, 90, 0). 4. The frequency parameter 2 2 EH a           for spherical shell is more than that of the cylindrical shell for the same radius/side ratio. Nomenclature a : radius mm a : Side length (mm) E : Modulus of elasticity (N/m 2 ) G : Modulus of rigidity (N/m 2 ) H : Thickness (mm) Ii : Integrations Mi : Resultant moments per unit length N.m/mm Ni : Resultant forces per unit length N/m Ρ : Density (kg/m 3 ) R : Radius (mm) t : Time (sec) u, v, w : Displacement components in x, y, and z directions respectively (mm) δU : Change in strain energy (N.m) δK : Change in kinetic energy (N.m) εi : Strain component in the principal direction (I, i=1,….6) ζi : Stress component in the principal direction (I, i=1,….6) Φi, θi : Rotations ω : Frequency (rad/s) ν : Poisons ratio References [1] Timoshenko "Theory of plates and shells" [2] Humayun R H. Kabir,"Application of linear shallow shell theory of reissner to frequency response of thin cylindrical panel with arbitrary lamination" comp.stru. vol. 56, 2002. [3] H.M Wang "Dynamic solution of multilayered orthotropic piezoelectric hollow cylinder for axisymmetric plane strain problems" Int.J of solid and strctures,vol.42,2004 [4] Rong-Tyai wanga and Zung-Xian Lin "Vibration analysis of ring-stiffened cross-ply laminated cylindrical shell" J.of sound and vibration, vol.295, 2006. [5] Penzes L.and Buring G "Free vibration of thin isotropic oblate spheroidas shells"Generas Dynamic Report No GD/C-BTD 65-113, 1956 [6] Al-Najim , F.A"An Investigation into the free ax. Symmetric Vibration charachterstks shell" M.Sc thesis, university of Baghdad, 1990 [7] Al-Azzawy W. I. “Fatique and Vibration Characteristics of Laminated Composite Shelles of Revolution” Ph. D thesis, University of Baghdad, 2007 [8] Reddy, J. N. and Lium C. F. “A Higher–Order Shear Deformation Theory of Laminated Elastic Shells”, International Journal of Eng. Since, Vol. 23, No.3, 1985 [9] B.M Pandya and T Kant "Finite Element of Laminated Composite Plates using a Higher Order Displacement Model", Composite Science and Technology 1988. [10] George Lubin "Hand Book of Composites", Van Nostrand Reimhold Compony, Inc, 1982. Qusai H. Jebur Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-75 (2008) 75 التحليل الذيناميكي للقشريات االسطوانية والكروية الخفيفة المصنوعة من مواد مركبة قصي حاتم جبر .هيئت انخعهيى انخقُي بغذاد/انكهيت انخقُيت الخالصة اسخُذ انحم انخحهيهي عهى َظشيت حشىِ انقض يٍ . في هزا انبحث حى دساست انخحهيم انذيُبييكي نهقششيبث االسطىاَيت وانكشويت انًشكبت اٌ رنك يشًم ايجبد انخشدداث . انشحبت انعهيب نهقششيبث انًشَت وانخي يُهب َسخطيع اشخقبق انًعبدالث انخبطت ببنطبقبث انغيش سىيت انخىاص . انطبيعيت واَسبق االهخضاصاث نهقششيبث االسطىاَيت وانكشويت انًشكبت وانًسُذة ببسبطت . ANSYحى اثببث انُخبئج انخحهيهيت وانخي حى انحظىل عهيهب ببسخخذاو انًعبدالث انًطىسة بىاسطت حقُيت انعُبطش انًحذدة يٍ خالل بشَبيج . في انخحهيم انذيُبييي نهقششيبث االسطىاَيت وانكشويت%9بيُج انُخبئج حىافقب جيذا وببعظى َسبت حفبوث