Vibration analysis of laminated anisotropic shells of revolution

COMPUTER METHODS NORTH-HOLLAND IN APPLIED VIBRATION MECHANICS AND ENGINEERING 61 (1987) 277-301 ANALYSIS OF LAMINATED ANISOTROPIC SHELLS OF REVOLUTION Ahmed K. NOOR and Jeanne M. PETERS zyxwvutsrqponmlkjihgfedcbaZYX George W ashington University Center, NASA Langley Research Center, Hampton, VA 23665, U.S.A. Received 14 March 1986 Revised manuscript received 16 October 1986 An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the SandersBudiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction. The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meri~onal direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain ~om~nents on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique. The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness. Nomenclature a radial distance for parameters for a toroidal shell (see shell element (see Fig. 51, (3) and (4% cij, A?, d, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA shell stiffness GLT, Gl-l- shear moduli in plane coefficients, (i, j = 1,2, . . . ,6) of fibers and cd4, cd5, c55 transverse shear normal to it, stiffness coefficients vectors of stress {4), W} of the shell, resultant EL, E, elastic moduli in parameters for a direction of shell element (see fibers (3) and (411, and normal to it, h total thickness of the vectors of strain shell, w7 09 004% 7825/87/$3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland) 278 A. K. Noor, J. M . Peters, Vibration analysis of laminated shells meridional and elemental matrices circumferential (see (3) and (4)) directions, linear stiffness respectively (see matrices of the Fig. l), reduced system Y normal distance from (see (W-(12)), L total length of the shell axis to the reference cylindrical shell, surface (see Fig. m,, m2, ml translational, rotational, and 1) number of global coupling inertia approximation terms, vectors, (and MS,MO,MS0 bending zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA elemental matrices twisting) stress resultants (see Fig. (see (3) and (4)) linear strain2) displacement WJCWI consistent mass matrices of the matrices of the shell element (see shell element (see (3) and (4)), (3) and (4)) s meridional coordinate mass matrix of the of the shell (see reduced system Fig. l), (see (13)) Fourier harmonic in u, u, w displacement components of the the circumferential middle surface of direction, the shell in the extensional stress meridional, resultants (see Fig. 2), circumferential, shape functions used in approximating and normal directions, the generalized respectively, displacements, vectors of generalized x-9 shape functions used nodal displacement in approximating coefficients for a the stress resultants shell element (see and strain components, (3) and (4))Y coordinate normal to transverse shear the shell middle stress resultants surface, (see Fig. 2), matrices of global radius of curvature of approximation cylindrical shell, vectors defined in principal radii of curvature of the (5)-P), A. K. Noor, J.M . K,, Ko, 279 Peters, Vibration analysis of laminated shells the sensitivity of extensional strains of the global response the middle surface of the shell to of the shell, nonorthotropic transverse shear material strains of the shell, coefficients (see circumferential (hoop) coordinate (14)) > 5 dimensionless of the shell (see coordinate along Fig. l), the meridian (see bending strains of the 2Kse Fig. 5), shell, rotation components A tracing parameter ‘p,, cpe of the middle identifying all the surface of the shell nonorthotropic (see Fig. 2), (anisotropic) terms { $} vector of amplitudes in the governing of global finite element approximation equations (see (3) vectors, and (4)), 0 circular frequency of major Poisson’s ratio ‘LT zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA vibration of the of the individual shell, layers, mass density of the 0 eigenvalue parameter, P 1 length of element in material of the the meridional shell, direction. parameters measuring PI a, = alas. Superscript t denotes transposition. Subscript II refers to the nth Fourier harmonic. Shell variables with a bar are the coefficients of the sine terms of the Fourier series. Ranges of superscripts i, j: 1 to number of displacement nodes in the element. 4: 1 to number of parameters used in approximating each of the stress resultants strain components. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and 1. Introduction In the past two decades a substantial capability has been developed for the numerical analysis of arbitrary shells of revolution. However, most of this capability is for orthotropic and transversely isotropic shells for which the principal material directions coincide with the coordinate lines. Interest in the use of filament-winding techniques for manufacturing composite shells of revolution has recently been expanded in aircraft, shipbuilding, and other industries. Also, 280 A. K. Noor, J. M . Peters, Vibration analysis of laminated shells modern tires are made of flexible cord-rubber composites. Therefore, an understanding of the vibration characteristics of laminated anisotropic shells of revolution is desirable. The most commonly used approach for the free vibration analysis of shells of revolution is based on the representation of the shell variables by a Fourier series in the circumferential coordinate 0, combined with the use of a numerical discretization technique (such as finite elements, finite differences, or numerical integration in the meridional direction-see, for example, [l-5]). For shells with uniform circumferential properties, the Fourier-series representation permits separation of variables, and the equations uncouple in harmonics. Moreover, for orthotropic and isotropic shells the governing equations describing the symmetric and antisymmetric vibrations (with respect to 0 = 0), associated with each Fourier harmonic, uncouple. In the axisymmetric case (corresponding to the zeroth Fourier harmonic), the symmetric and antisymmetric vibrations correspond to the bending and torsional vibrations, respectively. In the first case (symmetric vibrations), only the generalized displacements U, W, spsand the stress resultants N,, N8, M,, M,, and Q, can be nonzero; and in the second case (antisymmetric vibrations), only the generalized displacements u, q* and the stress resultants fV$,, ~~~, and Q, can be nonzero (see Figs. 1 and 2). For anisotropic shells of revolution, the two sets of symmetric and antisymmetric stress resultants and displacements, associated with each Fourier harmonic, are coupled. To account for this coupling, a number of studies used complex series representation of the shell variables and applied complex numerical integration techniques to the vibration analysis of anisotropic shells (see [6, 71). Based on these techniques, the computational time required for the vibration analysis of anisotropic shells of revolution is significantly higher than that of the corresponding orthotropic shells. The present study aims at making the computational cost of predicting the free vibration response of anisotropic shells of revolution comparable with that of the corresponding orthotropic shells. Specifically, the objectives of this paper are: (a) to present a simple and efficient procedure for predicting the free vibration response of laminated anisotropic shells of revolution; and, (b) to study the sensitivity of the vibration response of the shell to nonorthotropic (anisotropic) material parameters. shell theory [8-lo] The analytical formulation is based on a form of the Sanders-Budiansky including the effects of both the transverse shear deformation and the laminated anisotropic h Fig. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED 1. Shell geometry. 281 Fig. 2. Sign convention for stress resultants and generalized displacements. consist of the eight stress resultants IV,, A&, E,, se, 2~~~~KzyxwvutsrqponmlkjihgfedcbaZY ~- zyxwvutsrqponmlkjihgfedcba K @, i?rc ,,, 2 .~~~~ 2 ~~~~ and the five generalized displacements U, U, W, rpS, and ‘p9of the shell (see Fig. 2 for sign convention). The three key elements of the proposed computational procedure are: (a) use of three-field mixed finite element models in the meridional direction, with discontinuous stress resultants and strain components at the element interfaces; (b) operator splitting or decomposition of the material ~~rnplian~e matrix of the shell into an orthotropic and nono~hotropic (anis~tropic) parts; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique. material response. The fundamental unknowns Uses Mz, i&, MS@,&,, and e,, the eight strain components 2. I. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ~~~~~a~ ~~s~~e~~~~~~o~ of the shdt For asymmetric vibrations, each of the generalized displacements, the stress resultants, and the strain components can be expressed in terms of trigonometric functions of the circumferential coordinate 6. The discretization in the meridional direction is performed by using a three-field mixed finite element model. The following expressions are used for approximating the generalized displacements, stress resultants, and strain components within each element: zyxwvutsr -i u?l ) -i u, -i wn -i rp -i pPe,n s,n sin n6 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON , ! 282 A. K. Noor, J.M . Peters, Vibration analysis of laminated shells and sin n0 1 , (2) where X’ and &” are the polynomial interpolation functions used in approximating the generalized displacements, and the stress resultants and strain components in the meridional direction; the generalized displacements with superscript i and subscript n represent the nodal displacement coefficients associated with the Fourier harmonic n; the stress resultants and strain components with superscript 4 and subscript II represent the parameters associated with the Fourier harmonic n. Note that the degree of the interpolation functions p” is lower than that of Xi. Moreover, the continuity of the stress resultants and strain components is not imposed at the interelement boundaries and, therefore, the stress resultant and strain parameters can be eliminated on the element level. In 11) and (2) the range of the superscript i is 1 to the number of dispIacement nodes in the element; the range of the superscript 9 is 1 to the number of parameters used in approximating each of the stress resultants and strain components; the shell variables without a bar are the coefficients of the cosine terms, and the barred shell variables are the coefficients of the sine terms; and a repeated superscript denotes summation over its entire range. 2.2. Governing finite element equations The governing finite element equations are obtained by applying the three-field HuWashizu mixed variational principle. For the purpose of simplifying the analysis, the material A.K. Noor, J.M . Peters, Vibration analysis of laminated shells 283 stiffness matrix is decomposed into the sum of orthotropic and nonorthotropic parts. Moreover, the vectors of nodal displacements, stress resultant and strain parameters are each partitioned into two subvectors: {X},, , {Z?}, ; {@}, , {X}, ; and {e}, , { %‘}, , respectively. Each of these subvectors is associated with a symmetric or antisymmetric set of fundamental unknowns corresponding to the Fourier harmonic II. The two sets are listed in Table 1. For shells with uniform circumferential properties the finite element equations uncouple in harmonics. The governing finite element equations for each individual element are partitioned into two sets of coupled equations associated with the symmetric and antisymmetric sets of family of fundamental unknowns. These equations are embedded in a single-parameter equations of the form: (3) and where [RI,, [Xl,, [El,; [I%],, [.%!I,; and [G],, [Y],, are “generalized” stiffness matrices; [%R],, [.&I, are consistent mass matrices; o is the circular frequency of vibration of the shell; A is a tracing parameter which identifies all the nonorthotropic (anisotropic) contributions in the governing equations; and superscript t denotes transposition. The explicit forms of the arrays appearing in (3) and (4) are given in Appendix A. Because of the use of the three-field mixed model, no coupling submatrices exist between (3) and (4) other than the matrix [xl,. Note that for it > 0, [R] = [JX], [‘$I]= [.?%!I,[G] = [P’], [ZR]= [Ju]; and the vibration frequencies obtained from (3) and (4) are of multiplicity two (i.e., each pair of eigenvalues is equal). For A = 0 (orthotropic and isotropic shells), (3) and (4) are uncoupled; and for IZ> 0, the two sets of equations are identical. Therefore, the vibration frequencies, predicted by using (3) and (4), are equal. 3. Basic idea of proposed computational procedure The proposed Table 1 computational procedure can be conveniently divided into the following two 284 A. K. Noor, J. M . Peters, Vibration analysis of laminated shells distinct steps: (1) generation of global approximation vectors using the discretized shell model; and (2) use of the classical Bubnov-Galerkin technique to substantially reduce the size of the eigenvalue problem. The procedure is described subsequently. 3.1. Generation of the global approximation vectors For each Fourier harmonic II, the global approximation vectors are selected to be the eigenvectors {E}, {@}, and {X}; {%}, {%‘}, and {Z?} associated with A = 0; and its various-order derivatives with respect to A (evaluated at A = 0). Henceforth, the derivatives with respect to A will be referred to as path derivatives. The path derivatives are obtained by successive differentiation of the governing finite element equations (3) and (4). The equations used in generating these vectors are listed in Appendix B. Note that for A = 0, (3) and (4) are uncoupled, and for n > 0, the vibration frequencies o obtained by using (3) and (4) are identical. The procedure for generating these vectors is as follows: (a) The eigenvalue problem is solved at A = 0 for each value of n, using either (3) or (4). (b) The derivatives of {E}, {Q}, {X} and { %}, {X}, {%} with respect to A are obtained by using the uncoupled sets of equations given in Appendix B. Since the matrices on the left-hand sides of these equations are singular, the solution of each of these equations is expressed as the sum of a particular solution obtained by fixing one of the components to be zero (to remove the singularity of the matrix) and a multiple of the eigenvector (see [ll] and Appendix B). Note that if (3) are used to evaluate o at A = 0, the vectors { %}, {E}, and { %} and all their even-order derivatives with respect to A will be zero. Also, the odd-order derivatives with respect to A of {E}, {a}, and {X} will vanish. On the other hand, if (4) are used, the situation is reversed with respect to { %}, {X}, { %‘} and {E}, {@}, {X}. 3.2. Basis reduction and extraction of eigenvalues The eigenvectors of the anisotropic shell are expressed global approximation vectors as follows: as linear combinations of a few and where the matrices [r], are transformation matrices; and {I/J}, are unknown parameters representing the amplitudes of the global approximation vectors (eigenvectors of the reduced system). The columns of the matrices [r], consist of the eigenvector and its various-order path derivatives, evaluated at A = 0, i.e., A. K. Nmw, J.M . Peters, Vibration analysis of lminnted shells 2x5 with similar expressions for fr,],, [I-++],, and [&I, in terms of {8>, (X), and f%?). A Bubnov-Galerkin technique is used to replace the original equations (3) and (4) by the The partieuIar choice of the globaf approximation vecturs used herein provides a direct quantitative measure uf the sensitivity of the response quantities to nonorthotrop~~ (anisotropic) material coefficients of the shell. The derivatives of the eigenvalues o* with respect to A can be used as quantitative measures for the sensitivity of the global vibrational response of the shell to nonorthotropic material coefficients. Specifically, dimensionless parameters I;I are introduced such that 2 where bath , 1 not summed, to2and its der~va~ves with respect to h are e~a~~at~d a%n = 0. The srna~l~~ the vafues of Gfthe less sensitive tke response is to the nonorthotr~~i~ material coefficients of the shell and vice versa. Also, small values of & (less than 0.10) indicate that only a few global approximation vectors are needed for approximating the response of the anisotropic panel; and that Taylor-series expansion can provide an acceptable approximation of the vibration frequencies of the anisotropic shell. On the other hand, large values of & (greater than l*O) indicate that more global approximation vectors are needed for approximating the vibration frequencies of the anisotropic shell, Note that the presence of anisotropy results in reducing the stiffness of the shell. Therefore, the vibration frequencies evaluated at i\ = 0 are higher than those of the original anisotropic shell (~orrespon~ng to A = 1). 286 A.K. Noor, J.M . Peters, Vibration analysis of laminated shells 5. Numerical studies To evaluate the effectiveness of the foregoing computational procedure, a number of free vibration problems of anisotropic shells of revolution have been solved by this procedure. For each problem, the solution obtained by the foregoing procedure was compared with the direct solution of the anisotropic shell and with previously published solutions (whenever available). Herein, the results of two typical free vibration problems are discussed. The two problems are: (1) simply supported two-layered cylind~cal shell; and (2) clamped two-layered anisotropic toroidal shell. The cylindrical shell is made of boron-epoxy composite material and the toroidal shell is made of cord-rubber composite material. For the two shells both axisymmetric (n = 0) as well as nonaxisymmetric motions are considered. A classical Rayleigh-Ritz solution for the cylindrical shell is given in [12]. Also, a numerical solution for the same problem based on complex numerical integration is given in [6]. Both solutions are based on the classical shell theory (with transverse shear deformation and rotatory inertia neglected). For each structure the three-field mixed finite element models are used for the discretization in the meridional direction. Because of the symmetry of the shell meridian, only one-half of the meridian is analyzed using eight elements. The boundary conditions at the centerline are taken to be the symmetric or antisymmetric conditions. Quadratic Lagrangian interpolation functions are used for approximating each of the stress resultants and strain components. Cubic Lagrangian interpolation functions are used for approximating the generalized displacements. The integrals in the governing equations are evaluated using a three-point GaussLegendre numerical quadrature formula. The odd-order derivatives of the eigenvalues evaluated at h = 0 are zero, therefore &, &, vectors was PST.. .? are zero (see (24)). The particular solution for the global approximation obtained by setting to zero the maximum absolute value of the generalized displacement at the centerline. 5. I, Simply supported awn-lay ered cylindrical shell The first problem considered is that of a two-layered anisotropic circular cylindrical shell with simply supported edges. The fibers of the outer layer are in the circumferential direction and those of the inner layer are making 45” with the meridian. The material and geometric characteristics of the shell are shown in Fig. 3. The shell is modeled by using eight elements in half the meridian (192 stress-resultant parameters and 120 nonzero displacement degrees of freedom). Typical results are given in Table 2 and in Fig. 4. The foregoing procedure was applied to this problem and eight global approximation vectors were generated at h = 0 for each Fourier harmonic II. The minimum eigenvalues for the anisotropic shell associated with y1= 0, . . . , 8 are listed in Table 2 along with the corresponding values for h = 0 and the derivatives of the eigenvalues with respect to A. As can be seen from Table 2, the fundamental frequency of the shell corresponds to II = 2. The ratio of the minimum eigenvalue for A = 0 to that for h = 1 is 1.68 when y1= 1. The ratio reduces to only 1.03 when II = 2 and increases when n > 2. The ratio is 1.44 when n = 8. This is also reflected in the values of p,. The smallest i, correspond to 12= 2, & = -2.77 X lo-’ and &id= -4.09 x 10-3. Note that for n = 0, 1, 2, and 4 the minimum eigenvalues correspond to symmetric boundary conditions (see Fig. 3), and for II = 3, 5, 6, 7, and 8 the minimum eigenvalue corresponds to antisymmetric boundary conditions. An indication of the accuracy A. K. Noor, J. M . Peters, Vibration analysk of laminated shells L = 0.8001 R = 6.302 Pr oper t ies m x IO‘* m E = 21 L h=5.08x 10dm Boundar y AT s = ET condit ions 0, L : v=w=Q) lJ=F= AT s = L/ 2 = 1.862 x x lOlo lOlo Pa Pa 5.171 x 9 lo9 Pa Gy [. = 4 .137 x 10 Pa = 0.28 2051.88 Number s Fiber Symmet r y 374 layer s = P = : . individual GLT vLT =fj B 5 z(-J 6 of 287 zyxwvutsr of k g/ m3 layer s or ient at ion: = 2 90/ 4 5 condit ion: u=* s=v=~ =@ p Ant isymm~ t r y condit ion: Y=w =~ * =~ =;s=O Fig. 3. Two-layered anisotropic cylindrical shell used in the present study. and convergence of the minimum eigenvalues obtained by the foregoing procedure, with increasing number of vectors, is given in Fig. 4. For the sake of comparison, the accuracy of the eigenvalues obtained by using the Taylor series are also given in Fig. 4. As can be seen from Fig. 4, the eigenvalues obtained by the foregoing procedure are considerably more accurate than those obtained by using the Taylor-series expansion. For zyxwvutsr n 2 5 the Taylor-series expansions did not converge and the eigenvalues obtained by using five or more terms were grossly in error and could not be shown in Fig. 4. Table 2 Derivatives of minimum eigenvalues D = 100 pR2w21E,, evaluated at A = 0. Simply-supported two-layered anisotropic cylindrical shell shown in Fig. 3 OS 1s 2s 3A 4s 5A 6A 7A 8A 11.76 0.3316 0.0269 0.1447 0.4823 1.298 2.834 5.500 9.745 -20.85 -2.370 -0.0149 -0.1380 -0.9049 -7.731 -18.56 -62.15 -188.4 -14.41 -0.3454 -0.0026 -0.8255 -0.4048 48.66 92.43 1.348 x lo3 1.513 x lo4 as and A refer to symmet~c SJL = 0.5. -233.1 -1.207 -0.01402 -116.3 -13.77 -2.582 x -3.829 x -1.915 x -7.237 x and antisymmet~c lo4 lo4 lo6 10’ 9.5326 0.1969 0.0260 0.1078 0.3852 0.9823 2.0817 3.9129 6.7537 modes with respect to 288 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A. K. Noor, .I. M . Peters, Vibration analysis of laminated shells 0.28 0.21 OvFg 0.14 0.07 0 1 2 3 4 Circumferential we 5 6 7 8 number, n 0.28 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0.21 A LI@g : :’ / 0.14 0.07 0 a 1 2 3 4 5 6 Circum~re~tial wave number, n I I 7 8 Fig. 4. Accuracy of minimum frequencies obtained by proposed computational expansion. Two-layered anisotropic cylindrical shell shown in Fig. 3. 5.2. Tao-iayered procedure and Taylor-series a~isotro~ic e~Zi~tica~toroid The second problem considered is that of a two-layered anisotropic elliptical toroid. The shell properties are given in Fig. 5 and are chosen to simulate those of chord-rubber composites used in tires. Typical results are presented in Tables 3, 4, and 5, and in Figs. 6, 7 and 8. As in the previous problem, eight global approximation vectors were generated at A = 0, for each Fourier harmonic n. A. K. Noor, .I. M . Peters, Vibration analysis of laminated shells Table 3 Derivatives of minimum eigenvalues fl = 100 ph202/E,, evaluated A = 0. Two-layered anisotropic toroidal shell shown in Fig. 5 OS 1A 2s 3s 4s 5s 6s 7s 8s 1.094 0.855 1.731 2.588 3.453 3.869 4.170 4.596 5.192 -1.836 -0.6394 -2.290 -2.559 -1.864 -1.820 -2.971 -4.689 -6.826 -5.151 -2.194 -5.642 -5.613 -4.587 -5.496 -8.158 -11.39 -14.87 - 16.58 -6.555 -24.50 -33.14 -24.15 -17.18 -28.63 -49.38 -78.99 a s and A refer to symmetric and antisymmetric .$ = 0.5. 289 at 0.7268 0.7006 1.297 2.096 3.057 3.480 3.576 3.723 3.983 modes with respect to The minimum eigenvalues associated with different circumferential wave numbers II are listed in Table 3, along with the corresponding values for A = 0 and the derivatives of the eigenvalues with respect to A. As can be seen from Table 3, the eigenvalues for A = 0 are 11.2% (n = 5) to 50.5% (n = 0) higher than those for A = 1. The fundamental frequency corresponds to n = 1. The vibration mode shape associated with the fundamental frequency is shown in Fig. 6 and the distribution of the first three nonzero global approximation vectors along the meridian for the case 12= 1 is shown in Fig. 7. An indication of the accuracy and convergence of the minimum eigenvalues obtained by the foregoing procedure and the Taylor-series expansion is given in Tables 4 and 5 and in Fig. 8. As can be seen from Fig. 8 and Tables 4 and 5, the eigenvalues obtained by the proposed zyxwvutsrq EL = ET =8.27x 106Pa 517 x 106 Pa = 3.10 x lo6 Pa = 1.86 x lo6 Pa = 0.4 GIT Gl-r V Boundar y IT h : condit ions b. At 5 = = 1.067 = 5.08~ x zyxwvutsrqponmlkjihgfedcb 10eL low 2 m m 0: bl = b2 = 6.223 x 10m2 m u=~ =w =@ s=QO, ij=“ =i=6 c = x 10m2 m =lpO s At 5.385 a = 19.558 x 10S2m 0.5: Symmet r y : condit ion u=@s=“=w=@ Ant isymmet r y P = e Fig. 5. Two-layered k g/ m3 =fJ Number of layer s condit ion: Fiber ii= 1190 WGjB = r= anisotropic i$s = or ient at ion = 2 : + 55/ -55 0 toroidal shell used in the present study. 290 A. K. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Nom, J.M. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Peters, Vibration analysis of laminated she& “r h zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IQ.0 2 -2.0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE -‘r h h 4.0 6.0 -I*oou6 . . . . 0-6 , . . . 5 E lU.Of- 20.0 j-- 6.0 *I -ii 0.0 -6.0 40.0 0 0.1 0.2 0.9 0.4 0.5 0 0.1 0.2 0.3 0.4 0.6 5 Fig. 6. vibration mode shapes associated with the fundamental shown in Fig. 5, n = 1. frequency. Two-layered anisotro~ic toroidal shell ml 1&l had 50 0 0.1 0.2 0.3 6 O*# 0.5 0 0.1 I t t f 0.2 0.3 O” 4 0.5 6 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0.1 O” 2 0.3 0.4 0.5 0 0.1 0.2 0.3 a.4 c c 0.5 292 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A. K. Noor, J.M . Peters, Vibration analysis of laminated shells Table 4 Accuracy of minimum proposed computational shell shown in Fig. 5 eigenvalues procedure. R = 100 ph2021E,, obtained by Two-layered anisotropic toroidal n rl” r=3 r=4 r=5 OS 0.7285 1A 2s 3s 4s 0.7110 1.307 2.141 3.108 3.490 3.585 3.733 3.994 0.7271 0.7053 1.302 2.126 3.089 3.484 3.580 3.727 3.988 0.7269 0.7016 1.299 2.112 3.075 3.481 3.577 3.724 3.985 SS 6S 7s 8s r-6 0.7268 0.7007 1.297 2.103 3.068 3.480 3.576 3.724 3.984 a s and A refer to symmetric and antisymmetric to 5 = 0.5. procedure are considerably Taylor-series expansion. fl full system zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO 0.7268 0.7006 1.297 2.096 3.057 3.480 3.576 3.723 3.983 modes with respect more accurate than the corresponding eigenvalues obtained by the 6. Concluding remarks An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to Table 5 Accuracy of minimum eigenvalues a = 100 ph2u21E,, obtained by Taylor-series expansion.’ Two-layered anisotropic toroidal shell shown in Fig. 5 n nb r=3 r=5 r=7 OS 0.8362 0.7454 1.449 2.308 3.224 3.594 3.762 4.027 4.448 0.7597 0.7188 1.354 2.201 3.146 3.518 3.638 3.831 4.164 0.7367 0.7097 1.320 2.156 3.112 3.494 3.598 3.763 4.054 1A 2s 3s 4s 5s 6s 7s 8s R full system 0.7268 0.7006 1.297 2.096 3.057 3.480 3.576 3.723 3.983 Notes : ’ The Taylor series expansion is &, 1 = f&=, + -I . . ., h(m/aA) + ( A2/2!)(ab/aSJ’) + (A3/3!)(a3lNah3) and the odd-order derivatives of 0 at A = 0 are all zero. ’ s and A refer to symmetric and antisymmetri~ modes with respect to 5 = OS. A. K. Noor. 1. M . Peters, Vibration analysis of laminated shells - A=1 .., . . . . A = 293 Full syst zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP em 0 > 0 4 vect or s Reduct ion + 6 Vect or s met hod I I 1 1 2 3 I 4 Cir cumf er ent ial 0 I.. I I 1 1 2 3 J I J 1 5 6 7 8 number , n w ave I 4 ~ ~ r ~ ur n~ r an~ ia~ Fig. 8. Accuracy of minimum frequencies obtained by proposed expansion. Two-layered anisotropic toroidal shell shown in Fig. 5. I I 5 b w ave number , I 7 I 8 n computational procedure and Taylor-series anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiausky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response, The fundamental unknowns consist of the eight strain components, eight stress resultants, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate, and a three-field mixed finite element modei is used for the discreti~a~on in the meridional direction The element characteristic arrays are obtained by using the three-field mu-Washi~u mixed variational principle. The polynomial 294 A. I(. Noor, J.M . Peters, Vibration analysis of laminated shells interpolation (or shape) functions for approximating the strain components and stress resultants are the same and are of a lower degree than those used for approximating the generalized displacements. The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric parameters (with respect to 8 = 0) for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the classical BubnovGalerkin technique is used to substantially reduce the size of the eigenvalue problem. For each Fourier harmonic, the global approximation vectors are selected to be the eigenvector corresponding to zero nonorthotropic material stiffness matrix, and its variousorder path derivatives (derivatives of the eigenvector with respect to an anisotropic tracing parameter h, identifying all the nonorthotropic material coefficients, evaluated at A = 0). The size of the eigenvalue problem used in generating the global approximation vectors is the same as that of the corresponding orthotropic shell. Two numerical examples of free vibration problems of laminated anisotropic cylindrical and toroidal shells are used to demonstrate the effectiveness of the foregoing computational procedure. The results of the present study suggest the following conclusions relative to the selection of the global approximation vectors and to the potential of the proposed computational procedure. These conclusions are as follows: (1) The use of the eigenvectors at A = 0 and their path derivatives as global approximation vectors leads to accurate predictions of the vibration frequencies with a small number of vectors. Therefore, the time required to solve the reduced eigenvalue problem is relatively small, and the total analysis time, to a first approximation, equals the time required to evaluate the eigenvectors and their derivatives at A = 0 and to generate the reduced equations. (2) The global approximation vectors provide a direct measure of the sensitivity of the different response quantities (strain components, stress resultants, and generalized displacements) to the nonorthotropic (anisotropic) material coefficients of the shell. The sensitivity of the global measure is assessed by using the derivatives of the eigenvalues. (3) The foregoing computational procedure exploits the best elements of the operator the finite element method, and the Bubnov-Galerkin technique as splitting technique, follows: (a) the operator splitting technique is first applied to reduce the size of the eigenvalue problem of the anisotropic shell to that of the corresponding o~hotropic shell; (b) the finite element method is then used as a general approach for generating global approximation vectors; (c) the Bubnov-Galerkin technique is used as an efficient procedure for minimizing and distributing the error throughout the shell, and for substantially reducing the size of the e~genvalue problem. (4) The foregoing procedure extends the range of applicability of the classical perturbation A. K. Noor, J.M . 295 Peters, Vibration analysis zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ of laminated shells technique (or Taylor-series expansion) by relaxing the requirement A in the expansion. of using a small parameter Appendix A. Formulas for coefficients in the finite element equations for individual elements The explicit forms of the elemental arrays [S],, [X]-],, [I?],,, [%I,,, [%I,, [G],, [Y’],,, [9X],, and [A], associated with the harmonic n are given in this appendix. For convenience, each of these arrays is partitioned into blocks corresponding to contributions from individual nodes, stress resultants, or strain approximation functions. The expressions of the typical partitions (or blocks) are given subsequently. Note that the order of the strain parameters in these partitions is Ed, E@, 2~,,, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ‘c ,, K ~, 21csB, 2~~,,, and 2~~~; the order of the stress resultant parameters is MS, Iv@, N,, , M,, M,, MT,, Q,, and Q, ; and the order of the generalized displacement coefficients is U, U, W, ‘p,, and q*. A typical $9 partition of the elemental array [R], is given by: d 11 d,, - ; q d22 - 1 e zyxwvutsrqponmlkjihgfedcbaZYXWVUT . cos2 n0 * -/-- - ____-. c55 * Symmet~c . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG . f zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG __-- 66 n0 1 rdOds. (A.1) d66 The $2 partition of the elemental array [X], is given by (A.l) after interchanging and sin’ n& The $9 partition of the elemental array I?], is given by: c . . . * . . - --_____ ‘16 ‘26 . _--- .I. . _____T___~,l____--;_ . . : . . ---_____ . . P26 0 _--. * . - d,, i - . co? rzfI A. K. Noor, J.M . Peters, Vibration analysis of laminated shells 297 (A.4) The 9j partition of the elemental array [Y’],, is given by (A.4) after interchanging and sin2 ~3. The ij partition of the elemental array [%X1,,is given by: 1 m2 * Symmetric ... m, 1 Symmetric - cos2 no -1 ml * * ml . . 1 sin2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK ne rdeds. (A-5) m2 The ij partition of the elemental array [JR], is given by (A.5) after interchanging cos2 no and sin2 n& 1 is the length of the element; Jhr” are the interpolation functions for the In (A.l)-(AS) strain components and stress resultants; JY’ are the interpolation functions for the generalized displacements; the c, f, and d are shell stiffnesses; r is the normal distance from the shell axis to the reference surface (see Fig. 1); R, and R, are principal radii of curvature in the meridional and circumferential directions, respectively; s and 8 are the meridional and circumferential coordinates of the shell; a, = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE a/as; m,, m2, and m, are translational, rotational, and coupling inertia terms. 298 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A. K. Noor, J. M . Peters, Vibration analysis of laminated shells Appendix B. Ev~uation of global approximation vectors For each Fourier harmonic 12, the global approximation vectors are chosen to be the eigenvector associated with A = 0, and its various-order path derivatives (derivatives of eigenvector with respect to A) evaluated at A = 0. For A = 0, (3) and (4) are uncoupled. The path derivatives are obtained by successive differentiation of the governing finite element equations (3) and (4) with respect to A, and solving the resulting sets of equations. For an individual element, the recursion formulas for the path derivatives can be written in the following compact form: and where 9 3 1. The vectors {@@‘} and {QC9)} on the right-hand sides of (B.l) are listed in Table B.l for 2 up to 5, The vectors {S!?@)} and { 9 (‘)} are obtained from the corresponding expressions of {‘@‘<‘) [Xrj *[I#, and {CC’) } given in Table B.l after making the following interchanges: [W+[JQ @‘>++{@), and {Q+W}. The derivatives of w2 with respect to A in Table B. 1. are given by: (B-4) A. K. Now, J. M . Peters, Vibration analy sis of laminated shells In (B.3)-(B,7) superscript t denotes transposition normalized with respect to the mass matrix, i.e., and the eigenvectors 299 are assumed to be The global matrices on the left-hand sides of (B.l)-(B.2) (obtained by assembling the elemental contributions) are singular, and, therefore, the solution of each of these equations is expressed as the sum of a particular solution, obtained by fixing one of the components to zero (to remove the singularity of the matrix) and a multiple of the eigenvector. This is described by the following set of equations: 300 A.K. Noor, J.M. Peters, Vibration analysis zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON qf laminated shells where the barred quantities represent the particular solutions of the assembled (B.1) and (B.2), and cd, are multipliers. The expressions of the first five ca are: equations (B.lO) (B.ll) (B.12) (B-13) Equations (B.lO)-(B.14) are obtained by successive differentiation the derivatives of the eigenvectors by their expressions from (B.9). of (B.8) and replacing References [l] J.E. Goldberg, Computer analysis of shells, in: D. Muster, ed., Proceedings of the Symposium on the Theory of Shells to Honor L.H. Donnell, University of Houston, Houston, TX (1967) 3-22. [2] A. Kalnins, Static, free vibration and stability analysis of thin, elastic shells of revolution, AFFDL-TR-68-144, U.S. Air Force, 1969. [3] D. Bushnell, Analysis of buckling and vibration of ring-stiffened segmented shells of revolution, Internat. J. Solids and Structures 6 (1970) 157-181. [4] A.K. Noor and W.B. Stephens, Comparison of finite difference schemes for analysis of shells of revolution, NASA TN D-7337, 1973. [5] S.K. Sen and P.L. Gould, Free vibration of shells of revolution using FEM, ASCE J. Engrg. Mech. Div. 100 (1974) 283-303. A. K. Noor, J.M . [6] [7] [8] [9] [lo] [ll] [12] Peters, Vibration analysis of laminated shells 301 J. Padovan, Quasi-analytical finite element procedures for axisymmetric anisotropic shells and solids, Comput. & Structures 4 (1974) 467-483. G.A. Cohen, FASOR-a second generation shell of revolution code, Comput. & Structures 10 (1979) 301-309. J.L. Sanders, Nonlinear theories for thin shells, Quart. Appl. Math. 21 (1963) 21-36. B. Budiansky, Notes on nonlinear shell theory, J. Appl. Mech. 35 (1968) 393-401. A.K. Noor and J.M. Peters, Analysis of laminated anisotropic shells of revolution, ASCE J. Engrg. Mech. 113 (1) (1987) 49-65. R.B. Nelson, Simplified calculation of eigenvector derivatives, AIAA J. 14 (9) (1976) 1201-1205. C.W. Bert, J.L. Baker and D.M. Egle, Free vibrations of multilayer anisotropic cylindrical shells, J. Composite Materials 3 (1969) 480-499.