Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

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International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells M.H. Toorani ∗ Institute for Aerospace Research, National Research Council Canada, Montreal Road, Ottawa, Ont., Canada K1A 0R6

Abstract A general approach, based on shearable shell theory, to predict the in)uence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The e+ects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid -nite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear sti+ness matrices are derived by exact analytical integration. In part two, the modal coe1cients are obtained, using Green’s exact strain–displacement relations, for these displacement functions. The secondand third-order non-linear sti+ness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.

R esum e Dans cet article, une approche g5en5erale est present5ee a-n de pr5edire l’in)uence de la non-lin5eairit5e g5eom5etrique sur les fr5equences naturelles des coques cylindriques e5 lastiques et anisotropes lamin5es bas5ee sur la th5eorie ra1n5ee (la th5eorie de d5eformations de cisaillement du premier ordre) des coques, en incorporant les grands d5eplacements et rotations. Les e+ets de d5eformations de cisaillement et de l’inertie rotative sont pris en compte dans les e5 quations du mouvement. La m5ethode utilis5ee est celle des e5 l5ements -nis hybrides qui est la combinaison de la m5ethode des e5 l5ements -nis et de la th5eorie des d5eformations de cisaillement. L5a, o5u les e5 quations des coques sont utilis5ees int5egralement ce qui permet d’utiliser les e5 quations enti6eres d’5equilibre pour d5eriver les fonctions du d5eplacement. La solution analytique est divis5ee en deux parties. En premi6ere partie, les fonctions de d5eplacements sont obtenues de la th5eorie ra1n5ee des coques et par la suite, les matrices de masse et de rigidit5e lin5eaire sont obtenues par la proc5edure des e5 l5ements -nis hybrides et l’integration analytique exacte. En deuxi6eme partie, les coe1cients modaux sont obtenus, en utilisant les relations exactes de d5eformation-d5eplacements de Green, pour les fonctions des d5eplacements nomm5ees au-dessus. Les expressions d5ecrivant les matrices de rigidit5es non-lin5eaires de deuxi5eme et troisi6eme ordre sont obtenues par l’int5egration analytique pr5ecise et sont int5egr5ees dans le syst6eme lin5eaire pour obtenir l’5equation non-lin5eaire de mouvement. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear shells; Dynamics; Cylindrical shells; Anisotropic; Shear deformations

∗ Nuclear Engineering Department, Babcock and Wilcox Canada, 581 Coronation Boulevard, P.O. Box 310, Cambridge, Ont., Canada N1R 5V3. Tel.: +1-519-621-2120 ext. 2878; fax: +1-519-622-7222. E-mail address: toorani [email protected] (M.H. Toorani).

0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 2 ) 0 0 0 7 3 - 2

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Nomenclature A; B; C; D; E AAjk ; BBjk ; : : : ; TTjk AAijk ; BBijk ; : : : ; TTijk AAijks ; BBijks ; : : : ; TTijks

de-ned by Eq. (5) modal coe1cients determined by Eq. (17) modal coe1cients determined by Eq. (B.1) modal coe1cients determined by Eq. (B.2)

(1) (58) AUXijk ; : : : ; AUXijk

modal coe1cients determined by Eq. (B.1)

fi (i = 1–10) GG(p; q) hi (i = 1; 2) kijL NL2 kijk

modal coe1cients determined by Eq. (B.2) coe1cients of determinant of the matrix [H ] Eq. (6) de-ned by Eq. (22a) Lam5e’s parameters Eq. (1) general element of linear sti+ness matrix Eq. (20) general element of second-order non-linear sti+ness matrix Eq. (20)

NL3 kijks L Li m mL mij Mx ; M ; Mx ; Mx n Nx ; N ; Nx ; Nx Pij Qxx ; Q R SA(p; q) t u; v; w Um ; Vm ; Wm ; (xm ; (m x )i , (i , *i and +i (x and ( ,i -ij -pq -x0 and -0 *0x and *0 .x and . /x and / 0x0 and 00  1T

general element of third-order non-linear sti+ness matrix Eq. (20) length of shell equations of motion Eq. (4) axial mode number de-ned by m=L general element of mass matrix Eq. (14) the moment resultants circumferential wave number the in-plane force resultants terms of elasticity matrix (i = 1; : : : ; 10; j = 1; : : : ; 10) Eq. (4) the transverse force resultants Eq. (3) mean radius of the shell de-ned by Eq. (22b) thickness of the shell the axial, circumferential and radial displacement respectively amplitudes of u; v; w; (x , and ( associated with mth axial mode number axial coordinate de-ned by Eq. (7) the rotations of the normal about the coordinates of the reference surface complex roots of the characteristic Eq. (6) deformation vector components Eq. (A.1) the term (p; q) of matrix [A−1 ][A−1 ]T Eq. (22b) normal strains of the reference surface in-plane shearing strains of the reference surface change in the curvature of the reference surface torsion of the reference surface the shearing strains circumferential coordinate angle for the whole open shell

(1) (58) AUXijks ; : : : ; AUXijks

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

2 3i i

List of matrices [A](10×10) [B](10×10) [EA](10×10) = [A−1 ][A−1 ]T [G](10×10) [H ](5×5) [k (L) ] [K (NL2) ] [K (NL3) ] [k (NL2) ] [k (NL3) ] [M ] [m] [N ] [P] [QQ] [R] [S] [T1 ] {C} {q} {+i } {+T } {-L } and {-NL }

1317

density of the shell material vibration amplitude function determined Eq. (29)

de-ned by Eq. (9) de-ned by Eq. (11) de-ned by Eq. (22b) de-ned by Eq. (14) de-ned by Eq. (6) linear sti+ness matrix Eq. (14) second-order global sti+ness matrix third-order global sti+ness matrix second-order sti+ness matrix Eq. (21a) third-order sti+ness matrix Eq. (21b) global mass matrix local mass matrix Eq. (14) shape function matrix (10) elasticity matrix Eq. (3) de-ned by Eq. (11) de-ned by Eq. (10) de-ned by Eq. (14) transformation matrix Eq. (8) vector for arbitrary constants Eq. (8) time-related vector Eq. (25) degrees of freedom at node i degrees of freedom for total shell linear and non-linear deformation vector Eq. (A.1)

1. Introduction Multilayered composite shells are being used extensively as structural elements in modern construction engineering, ship building, nuclear, space and aeronautical industries as well as the petroleum and petrochemical industries (pressure vessel, pipeline). It is very important to know the static and dynamic behavior of these structures subjected to di+erent loads in order to avoid destructive e+ects during their industrial usage. There are many situations where these structures are subjected to severe environment conditions, impact, collision or other intensive transient loads, which can cause large transient structural deformations and damage, or catastrophic failure. These phenomena may be attributed to changes in the equilibrium state characterizing the load-response mode. In addition, due to high ratio of tangential Young’s modulus to transverse shear modulus in composite materials such as graphite-epoxy and boron-epoxy (i.e., of order 25 – 40 instead of 2.6 for isotropic materials), the shear deformation e+ect on the non-linear behavior of anisotropic composite shells is more signi-cant than that of isotropic ones. This e+ect plays a very important role in reducing the e+ective )exural sti+ness of anisotropic composite plates and shells.

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Therefore, response of these types of structures may be predicted only when one accounts for their geometric non-linear behavior. Accordingly, the need for accurate and e1cient methods for structural analysis and design, especially for this category of large-de)ection (geometrically non-linearity) and elastic-plastic (material non-linearity) dynamic response problems, has increased. Several articles are available dealing with geometrical non-linearities, based on the classical shell theory, in isotropic shells of arbitrary shapes [1–3]. In case of anisotropic composite shells, the reader is referred to the Refs. [4 –7] where geometrically non-linear behavior of shearable anisotropic composite plates and shells is analyzed. A re-ned theory accounting the small strains and moderate rotation for anisotropic shell has been developed by Librescu and Schmidt [6]. Successive approximations, as steps towards an estimate of exact shell strain–displacement relations where displacements, large strains and rotations, were initially allowed, are presented for isotropic shells by Sanders [2] and for anisotropic shells by Librescu et al. [4 –7]. Also, a number of theories for layered anisotropic shells exist in literature [8], which are developed for thin shells and are based on the Kirchho+–Love hypotheses. The -rst paper to deal with non-linear vibrations of shells was the pioneering work of Reissner [9], who used the method of assumed mode shapes to investigate the vibration of curved cylindrical panels. Nowinski [10] investigated the non-linear vibration of orthotropic cylindrical shells. A more rigorous study of non-linear free )exural vibrations of circular cylindrical shells was conducted by Atluri [11] who compared his results with the available data and concluded the possibility of the softening type of non-linearity. Reddy and Chandrashekhara [12] solved the laminated shell, both cylindrical and spherical, assuming Reissner–Mindlin (RM) theory and an intermediate non-linearity. A non-linear two-dimensional theory for laminated thick plates and shells, which can predict the in-plane stresses as well as transverse direct stresses and transverse shearing stresses, was made by Stein [13]. The equations derived were similar to those of [9] except that non-linear strain–displacement relations were used and expanded into a series that contained all -rst and second degree terms by retaining only the -rst few terms. Rotter and Jumiski [14] have presented a set of non-linear strain–displacement relations for axisymmetric thin shells, based on the Kirchho+ assumptions, subject to large displacements with moderate rotations by retaining more terms. Their work is based on the Kirchho+ assumptions. They have shown that non-linear strains arising from products of in-plane strain terms, which were omitted in previous theories, may be important in certain buckling problems. The new relations are particularly important when branched shells are being studied and when the buckling mode may involve a translation of the branching joint. Most of these approaches can include various degrees of non-linearity in the strain–displacement relations in representing the displacements and rotations. Considerable simpli-cation was achieved in the Donnell equations by assuming that the non-linear membrane strains derived only from out-of-plane rotations. For example, Donnell theory is not suitable for the analysis of shells in which the buckling mode involves fewer than three full waves around the circumference [14]. More accurate non-linear shell equations are given by Sanders [2], but these are somewhat more complex than the Donnell equations. More terms are retained because fewer assumptions are made about the relative magnitude of various terms in the non-linear strain–displacement relations. The in)uence of large amplitudes on the free vibrations of -nite circular cylindrical shells with linearly varying wall thickness, embedded in an incompressible )uid is made by Ramachandran [15]. Most of mentioned works have been based on Galerkin’s approach [10], the small perturbation procedure [11,16], the Rayleigh–Ritz method [15], the modal expansion method [17,18], the -nite element method [19] and the hybrid -nite element method [20 –22]. All of these methods have their advantages and disadvantages. The best of any method is probably its general content and the capacity to predict, with precision, both the high and low frequencies of vibration. These criteria were not met in Galerkin’s and small perturbation methods. Adopting a perturbation technique, Chen and Babcock [16] also considered the large-amplitude vibration of a thin-walled cylindrical shell. Ramachandran [15] studied the non-linear vibration of cylindrical shells with varying thickness. The small

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

Z

V

W

U

R r

1319

X

t

L

(A) 1 1

φΤ (B)

2 m

i j

N

Wmi β1mi Vmi i β2mi U mi

m j

φ

(e)

(C)

Fig. 1. (A) Circular cylindrical shell geometry; (B) -nite element discretization; (C) nodal displacement at node i for the m’th element. N: number of elements.

perturbation method was used [16] to transform the non-linear equations to a linear system by expanding the unknown variables in a power series with respect to small parameters. The major advantage of this technique, compared to other methods requiring an initial hypothesis regarding the form of the vibration mode, is that the results are not preconceived. By incorporating the modal expansion (assumed mode shapes) technique, Radwan and Genin [17] eliminated some weakness and serious drawbacks of other theories, by using Sanders–Koiter [2,3] general non-linear theory. Their work is limited to a -nite simply supported isotropic cylindrical shell. After adopting the -nite element method, Raju and Rao [19] obtained the frequency variation of thin shells of revolution in conjunction with the maximum normal displacement for various boundary conditions. However, the modal expansion and -nite element methods appear to be ideally suited to the analysis of complex shell structures from this viewpoint. Numerous general computer programs, based on the -nite element method are available for industrial use for the linear and non-linear analysis where the displacement functions of the -nite element used are assumed to be polynomial but precise prediction of both the high and the low frequencies requires the use of a great many elements in the classical -nite element method. The present study presents a general approach to the dynamics non-linear analysis, incorporating the large displacements and rotations, of anisotropic laminated open or closed cylindrical shells based on the re-ned shell theory in which the shear deformation and rotary inertia e+ects are taken into account. The method used, in this work, is a combination of -nite element analysis and shearable shell theory. In this method the equations of cylindrical shells are used in full to obtain the pertinent displacement functions, instead of using the more common arbitrary polynomial forms. The -nite element method employed is a cylindrical panel-segment -nite element, Fig. 1, rather than the more commonly used triangular or rectangular elements. The displacement functions over an element are derived by exact solution of the equilibrium equations of a cylindrical shell (in case of shearable shell theory), and the mass and sti+ness matrices of each element are derived by exact analytical integration. In doing so, the accuracy of the formulation will be less a+ected as the number of elements used is decreased (thus reducing computation time) and as the dynamic characteristics of the shell are required at higher beam-mode (m) or higher shell-mode (n), a signi-cant advantage over polynomial interpolation. Therefore, this method is more accurate than the more usual -nite element methods.

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The present method o+ers many advantages, some of which are: (a) simple inclusion of thickness discontinuities, material property variations and di+erences in materials comprising the shell, (b) arbitrary boundary conditions without changing the displacement functions in each case, (c) high and low frequencies may be obtained with high accuracy as shown in [20 –22], (d) this approach has also been applied, with satisfactory results, to the dynamic analysis of shells containing a )owing )uid or partially -lled with liquid [20,22]. The analytical solution involves two steps: (i) Using the strain–displacement relations expressed in an arbitrary orthogonal curvilinear coordinate system and Green’s exact stress–strain relationships for anisotropic laminated materials. These relations are then inserted into equilibrium equations given in [21,22] including the transverse shear deformations and rotary inertia e+ects. The displacement functions are analytically determined by solving the linear equation system. The mass and linear sti+ness matrices are then determined by exact analytical integration, for each element and are assembled for the complete shell [21,22]. The displacement functions presented in this work allow dynamic behavior analysis of open or closed cylindrical shells with arbitrary boundary conditions. (ii) Using the non-linear part of the general strain–displacement relations as the linear part [23] and then applying the approach proposed by Radwan and Genin [17] to obtain the coe1cients of the modal equations from the displacement functions. The non-linear sti+ness matrices of the second- and third-order are then calculated by precise analytical integration with respect to modal coe1cients. These matrices, once calculated, are superimposed on the linear part of equations to establish the non-linear modal equations. There are several reasons for undertaking the development of the present theory. The -rst is to develop a theory for either dynamic or stress analysis of anisotropic laminated open or closed cylindrical shells. The accurate prediction of the dynamic response or failure characteristics of these structures, made up from advanced composite materials, requires the use of non-linear re-ned shell theory where the shear deformation and geometrical non-linear e+ects as well as rotary inertia e+ect are taken into account. The second deals with the numerical simulation of non-linear dynamic behavior of anisotropic laminated cylindrical shells based on the present theory. At the same time, the )owing )uid e+ect on the natural frequencies will be studied. One of the criteria of success of a method may be considered to be its capability of yielding the high and low natural frequencies and modal shapes with comparable high accuracy. The numerical method is based on a combination of hybrid -nite element analysis [21,22] and non-linear re-ned shell theory (shearable shell theory). This allows us to use the shell equations in full for the determination of the displacement functions, and hence the mass, sti+ness and stress-resultant matrices, instead of the more usual polynomial displacement functions.

2. Hypotheses The -rst-order transverse shear deformation theory of shells and modal expansion approaches have been used to develop the non-linear dynamic equations of motion. This theory is based on the following hypotheses: (a) It is assumed that the normal stress is negligible compared with stress tangential to the shell surface. (b) Linear elastic behavior of laminated anisotropic materials. (c) Use the Green’s exact strain–displacement relations expressed in arbitrary orthogonal curvilinear systems.

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

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(d) The -rst-order transverse shear deformation theory of the shells is adopted so the transverse shear deformations and also the rotary inertia e+ects are taken into account to develop the governing equations. (e) The large displacements and rotations are incorporated into the theory.

3. Linear matrix construction 3.1. Strain–displacement and stress–strain relations The physical strains, -ij , in the orthogonal curvilinear coordinates, are de-ned as below [23] -ij =

*ij ; hi hj

(1)

where, hi are called the Lam1e’s parameters and de-ned by Gij = h2i (no sum), Gij is a metric tensor which links the initial undeformed con-guration to the actual deformed one. The scale factors hi are de-ned as below for cylindrical shell geometry h1 =

√

E ∗ (1 − 6=R1 );

h2 =

√

G ∗ (1 − 6=R2 );

h3 = 1;

(2)

where E ∗ and G ∗ are the -rst fundamental magnitudes which are related to the elements of the surface metric [23]. For rigid body motion, the elongation -ii (no sum) and the shear -ij (i = j) are identically zero, then there are no theoretical limitations. Based on these de-nitions, the deformation vector {-} for the cylindrical shells (Fig. 1) is obtained and given in Appendix A. Unlike classical shell theory, the transverse shear strains do not vanish in the present theory and therefore (i cannot be expressed in terms of displacement components. The -ve degrees of freedom at each node, U; V; W; (x and ( are function of the in-plane coordinates in which U; V and W are the axial, circumferential and radial displacements, respectively. The (x and ( are, respectively, rotations of the normal about the coordinates of the reference surface oriented along the parametric lines of middle surface. The constitutive relations between the stress and deformation vectors of an anisotropic laminated cylindrical shell can be written as [21] {Nxx ; Nx ; Qxx ; N ; Nx ; Q ; Mxx ; Mx ; M ; Mx }T = [P](10×10) {-};

(3)

where [P] is the anisotropic matrix elasticity. The elements Pij ’s in [P] depend on the mechanical characteristics of the structure’s materials and shell geometrical parameters. This matrix is given in [21]. The formulations of governing equations will be developed hereafter in terms of displacement measures only. There are other formulations in terms of stress resultants and stress couples, in terms of strain measures, as well, mixed formulation in terms of stress resultants (or stress potential functions like the Airy functions) and the displacement quantities. In order to obtain such equations, the reader may follow the procedures described by Sanders [24] and Brull and Librescu [25]. In composite laminated plates and shells, the transverse shear stresses vary through layer thickness. This discrepancy is often corrected in computing the transverse shear force resultants (Qxx and Q ) by considering shear correction factor. This factor is computed such that the strain energy due to transverse shear stresses equals the strain energy due to the true transverse stresses predicted by the three-dimensional elasticity theory. This factor depends, in general, on the lamination parameters such as number of layers, stacking sequence, degree of orthotropy and -ber orientation in each individual layer.

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The equations of motion of cylindrical shells in terms of U; V and W; (x and ( and in terms of Pij ’s elements are written as follows: Lm (U; V; W; (x ; ( ; Pij ) = 0;

m = 1; : : : ; 5

and

i; j = 1; 2; : : : ; 10;

(4)

where Lm (m = 1; 2; : : : ; 5) are -ve linear di+erential operators fully given in [21]. It is noted that the sixth equation of equilibrium is identically satis-ed by the integral de-nitions of the shearing-stress resultants in terms of the shearing-stress components. Two basic approaches that enable us to eliminate exactly this sixth equation of equilibrium have been developed: (i) Derivation of constitutive equations which ful-ll identically this extra equation of equilibrium. For the classical theory of isotropic, such constitutive equations are known as FlSugge–Lure–Byrne constitutive equations. In the case of composite shells, the reader is referred to the following monograph [7,26,27] by Librescu. (ii) Formulation of modi-ed stress resultants and stress couple measures, satisfying identically this sixth extra equation of equilibrium. For the classical theory of isotropic shells, such modi-ed stress resultants and stress couples have been de-ned by Sanders [2,24]. For anisotropic shells, a similar stress resultants and stress couples were considered by Librescu [7,26,27]. Neither 6th equation of equilibrium nor the -ve -rst ones are given in this paper, and the interested reader is referred to Ref. [21] where all six equations are de-ned. 3.2. Displacement functions The -ve equations of motion (4) are expressed in terms of displacement measures. Therefore, the -ve boundary conditions must be speci-ed at each edge of the shell. The shell is subdivided into several -nite elements de-ned by line-nodes, i and j, and by components U; V; W; (x and ( representing axial, tangential and radial displacements and two rotations, respectively (Fig. 1). The displacement functions associated with the axial wave number are assumed to be m , m , U (x; ) = A cos xe ; (x (x; ) = D cos xe ; L L V (x; ) = B sin

m , xe ; L

W (x; ) = C sin

( (x; ) = E sin

m , xe ; L

m , xe ; L

(5)

where m is the axial mode number and , is a complex number. Substituting de-nition (5) into equations of motion (4) and setting equal to zero the determinant of the coe1cients in order to have a non-trivial solution, leads to a tenth-order polynomial characteristic equation in terms of ,: Det([H ]) = f10 ,10 + f8 ,8 + f6 ,6 + f4 ,4 + f2 ,2 + f0 ;

(6)

where fi (i = 0–10) are the coe1cients of the determinant of the matrix [H ] given in [22]. Each root of the characteristic equation yields a solution of equation (4). The complete solution of equations of motion can be obtained by adding the ten independent solutions, obtaining for each root of the characteristic equation involving the constants Ai ; Bi ; Ci ; Di and Ei (i = 1–10). As these constants are not independents, Ai ; Bi ; Di and Ei can be expressed as a function of Ci Ai = ) i C i ;

Bi = (i Ci ;

Di = *i Ci ;

and

Ei = +i Ci (i = 1; 2; : : : ; 10):

(7)

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The values of )i ; (i ; *i and +i can be obtained from the linear system (4) by introducing relations (7). After carrying out the some manipulations, the displacements U (x; ); V (x; ) and W (x; ) as well as (x (x; ) and ( (x; ) can be expressed in conjunction with ten Ci constants only.   U (x; )               V (x; )       W (x; ) = [T1 ](5×5) [R](5×10) {C}(10×1) ;           ( (x; ) x           ( (x; )

(8)

where matrices [T1 ](5×5) and [R](5×10) are given in [22], and {C} is a tenth-order vector of the Ci constants. The Ci constants can be determined using ten boundary conditions for each -nite element. The axial, tangential and radial displacements (U; V , and W ) as well as the rotations ((x ; ( ) have to be speci-ed for each node. Thus, the element displacement vector at the boundaries can be given by following relation: 

+i

= {Ui Vi Wi (xi (i Uj Vj Wj (xj (j }T = [A](10×10) {C}(10×1) ;

+j

(9)

where the terms of matrix [A], given in [22], are obtained from those of [R] matrix by successively setting  = 0 and  = ’. Multiplying Eq. (9) by [A−1 ] and substituting that into Eq. (8), we obtain     U (x; )            V (x; )          +i +i −1 W (x; ) = [T1 ][R][A] = [N ] ;     +j +j      (x (x; )            ( (x; )

(10)

where the matrices [T1 ], [R] and [A] are previously de-ned and [N ] matrix represents the displacement function matrix.

3.3. Mass and linear sti3ness matrices for an element From the expressions given in the linear part of deformation vector (given in Appendix A) and the expressions of the displacement functions (10), we obtain

{-} =

[T1 ] 0 0 [T1 ]



 [QQ][A]−1

+i +j



 = [B]

+i +j

;

(11)

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where [QQ] is a (10 × 10) matrix given in [22]. The constitutive relations between the stress and deformation vectors of the anisotropic laminated cylindrical shells are given as  +i {Nxx ; Nx ; Qxx ; N ; Nx ; Q ; Mxx ; Mx ; M ; Mx }T = [P](10×10) {-} = [P][B] : (12) +j The Pij ’s elements describe the elasticity matrix of the anisotropic cylindrical shells, which depends on the mechanical properties of the material of the structure. Some coupling as in-plane extensional-shear, extensional-bending and bending-twisting can be present in anisotropic laminated composite shells due to anti-symmetry of scheme lamination or -ber orientation. The mass and linear sti+ness matrices for one -nite element can be expressed as L ’ L ’ T [m] = 2t [N ] [N ] dA [KL ] = [B]T [P][B] dA; (13) 0

0

0

0

where dA = R d x d and 2 is the density of the shell, [P] the elasticity matrix and the matrices [N ] and [B] are de-ned before in Eqs. (10) and (11), respectively. Substituting them in (13) and analytically integrating with respect to x and , we obtain the matrices [m] and [k]. [m] = 2t[A−1 ]T [S][A−1 ]

[k] = [A−1 ]T [G][A−1 ];

(14)

where the Sij ’s and Gij ’s general elements are given in [22].

4. Non-linear part 4.1. Non-linear sti3ness matrix construction for an element The exact Green strain relations are used in order to describe the linear and non-linear, including large displacements and rotations, behavior of anisotropic open cylindrical shells. In common with linear theory, it is based on re-ned shell theory in which the shear deformations and rotary inertia e+ects are taken into account. The approach developed by Radwan and Genin [17] is used with particular attention to geometric non-linearities. The coe1cients of the modal equations are obtained through the Lagrange method. Thus, the non-linear sti+ness matrices of second- and third-order, once calculated, are superimposed on the linear part of equations to establish the non-linear modal equations. This section is limited to the relevant details of the method used to -nd the non-linear sti+ness matrices. The main steps of this method are as follow: (a) Shell displacements are expressed as generalized product of coordinate sums and spatial functions:

u= qi (t)Ui (x; ); (x = qi (t)(xi (x; ); i

v=

i

qi (t)Vi (x; );

i

w=

( =

qi (t)(i (x; );

i

i

qi (t)Wi (x; );

(15)

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

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where the qi (t)’s functions are the generalized coordinates and the spatial functions U; V; W; (x and ( are given by Eq. (5). (b) The deformation vector is written as a function of the generalized coordinates by separating the linear part from non-linear one given in Appendix A: {-} = {-L } + {-NL } = {-x0 ; *0x ; 0x0 ; -0 ; *0 ; 00 ; .x ; /x ; . ; / }T ;

(16)

where subscripts “L” and “NL” mean “linear” and “non-linear”, respectively. The -i0 ; *0i ; .i ; /i and 0i0 are de-ned in Appendix A. In general, these terms can be expressed in the following form:



-x0 = aj qj + AAjk qj qk ; 00 = gj qj + GGjk qj qk ; *0x =

j

j



bj qj +

j

0x0 =

j

c j qj +

j

-0 = *0 =

dj qj +

j

ej qj +

j

n j qj +

CCjk qj qk ;

j

/x =

p j qj +

. =

j

sj q j +

/ =

j

k

j

tj qj +

SSjk qj qk ;

k

j

PPjk qj qk ;

k



j

EEjk qj qk ;

NNjk qj qk ;

k



j

DDjk qj qk ;

k



j

k

j

.x =

k

j

BBjk qj qk ;

j

k

j

j

k

TTjk qj qk :

(17)

k

Note: AAij = AAji ; BBij = BBji and etc. The aj ; bj ; : : : ; tj and AAij ; BBij ; : : : ; TTij are given in Appendix B. (c) Using Eq. (15) and Hamilton’s principle leads to Lagrange’s equations of motion in the generalized coordinates qi (t):   @T d @T @V − + = Qi ; (18) dt @q˙i @qi @qi where T is the total kinetic energy, V the total elastic strain energy of deformation and the Qi ’s are the generalized forces. Assuming {-NL } = {-1 ; -2 ; : : : ; -10 }T , the strain energy V can be de-ned as follows: a V= 2

0

L

0

1

(Pij -i -j + Pkl -k -l )R d x d;

(19)

where a=1

if i = j or k = l (i; j = 1; 2; : : : ; 10);

(k; l = 3; 6)

a=2

if i = j or k = l

(k; l = 3; 6):

(i; j = 1; 2; : : : ; 10)

and i; j = 3; 6;

(d) After developing the total kinetic and strain energy, using de-nitions (17), and then substituting into the Lagrange Eq. (18) and carrying out a large number of the intermediate manipulations, that are not displayed here, the following non-linear modal equations are obtained. These non-linear modal equations

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M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

are used to study the dynamic behavior of anisotropic cylindrical shells.

mij +Sj +

j

j

kij(L) +j +

j

k

(NL2) kijk +j +k +



j

k

s

(NL3) kijks +j +k +s = Q i

i = 1; 2; : : : ;

(20)

(NL2) where mij ; kij(L) are the terms of mass and linear sti+ness matrices given by Eq. (14). The terms kijk

(NL3) and kijks , represent the second- and third-order non-linear sti+ness matrices. These terms, in the case of anisotropic laminated cylindrical shell, are given in Appendix B. Performing some intermediate manipulations, explained in Appendix B, the following expression are obtained for the second- and third-order non-linear sti+ness matrices:

[kijk ] = [k (NL2) ] = [A−1 ]T [J (NL2) ][A−1 ];

(21a)

[kijks ] = [k (NL3) ] = [A−1 ]T [J (NL3) ][A−1 ];

(21b)

where the (i; j) term in matrices [J (NL2) ] and [J (NL3) ] are written as  10

RGG(i; j)    [e(,i +,j +,k )1 − 1] if (,i + ,j + ,k )1 = 0;    (,i + ,j + ,k ) k=1 J(i;(NL2) j) =  10

   RGG(i; j)1 if (,i + ,j + ,k )1 = 0;  

(22a)

k=1

J(i;(NL3) s) =

 10 10

RL  -pq SA(i; s)   [e(,i +,s +,p +,q )1 − 1]   4 (,i + ,s + ,p + ,q ) 

if (,i + ,s + ,p + ,q )1 = 0;

q=1 p=1

10 10 

 RL   -pq SA(i; s)1   4

(22b) if (,i + ,s + ,p + ,q )1 = 0;

q=1 p=1

where the GG(i; j) and SA(i; s) are coe1cients in conjunction with ); (; *; +; , and Pij ’s elements in matrix [P]. The -pq is the term (p; q) of matrix [EA], where [EA] represents a matrix of constants de-ned by [EA] = [A−1 ][A−1 ]T .

5. The in"uence of geometric non-linearities on the natural frequencies of cylindrical shells The mass and sti+ness matrices, either linear or non-linear, given in Eqs. (14) and (21) are only determined for one element. After subdividing the shell into several elements (see Fig. 1), the global mass and sti+ness matrices are obtained by assembling the matrices for each element. Assembling is done in such way that all the equations of motion and the continuity of displacements at each node are satis-ed. These matrices are designated as [M ]; [K L ]; [K NL2 ] and [K NL3 ], respectively. S + [K (L) ]{+} + [K (NL2) ]{+2 } + [K (NL3) ]{+3 } = {0}; [M ]{+}

(23)

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

1327

where {+} is the displacement vector and [M ]; [K L ]; [K NL2 ] and [K NL3 ] are, respectively the mass, linear and second- and third-order non-linear sti+ness matrices of the system. They are square matrices of order NDF(N + 1), where NDF represents the number of degrees of freedom, (=5 in the present theory), at each nodal line and N represents the number of -nite elements. These matrices are then reduced to square matrices of order NREDUC = NDF(N + 1)-NC, where NC represents the number of constraints applied. The system of Eq. (23) then becomes 2 3 (r) (r) (r) [M (r) ]{+S } + [KL(r) ]{+(r) } + [KNL2 ]{+(r) } + [KNL3 ]{+(r) } = {0};

(24)

where the superscript “r” means “reduced”. The matrices contained in the linear part of Eqs. (24) can be reduced to diagonal matrices. Setting: {+(r) } = {B}{q};

(25)

where [B] represents the square matrix for eigenvectors of the linear system and {q} is a time related vector. Substituting relation (25) into Eq. (24) and multiplying by [B]T , we obtain [B]T [M (r) ][B]{q} S + [B]T [KL(r) ][B]{q} (r) (r) + [B]T [KNL2 ]([B]{q})2 + [B]T [KNL3 ]([B]{q})3 = 0:

(26)

The matrix product of ([B]T [ ][B]) represents diagonal matrices, written as [M (D) ] and [KL(D) ]. By neglecting the cross-product terms in ([B]{q})2 and ([B]{q})3 of Eq. (26), we obtain mjj qSj + kjj(L) qj +

NREDUC

kji(NL2) qi2 +

i=1

NREDUC

kji(NL3) qi3 = 0;

(27)

i=1

where coe1cients mjj and kjj(L) represent the “jth” diagonal terms of linear matrices. The kji(NL2) and kji(NL3) (r) (r) are the (j; i) term of the product ([B]T [KNL2 ][B2 ]) and ([B]T [KNL3 ][B3 ]), respectively. There are “NREDUC” simultaneous equations of the form of (27). Turn now to solution of Eq. (27) associated with the parametric function q(t). For this purpose, a solution q(t) of the Eq. (27) has to be found. It is expedient to introduce the following representation: qj (t) = 3j j (t)

(28)

which permits the use of the normalized initial conditions: j (0)

=1

and

˙ (0) = 0: j

(29)

Substituting de-nition (28) into Eq. (27) and dividing by mjj , we obtain S + !2 j jj

j

+ D(NL2) (3j =t) j

2 j

+ D(NL3) (3j =t)2 j

3 j

= 0;

(30)

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M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

where “t” and !j2 = kjjL =mjj represent shell thickness and linear vibration frequency of the shell, respectively, and D(NL2) = j

kjj(NL2) t mjj

D(NL3) = j

and

kjj(NL3) 2 t : mjj

(31)

The solution j (t) of the non-linear di+erential Eq. (30), which satis-es condition (29), is calculated by a fourth-order Runge–Kutta numerical method. The linear and non-linear natural frequencies are evaluated by a symmetric search for the j (t) roots as a function of time. The !NL =!L ratio of non-linear and linear frequency is expressed as a function of non-dimensional ratio (3j =t) where 3j is the vibration amplitude. 6. Numerical results and discussion This paper is focused on the application of the hybrid -nite element method, based on shearable shell theory and modal expansion approach, to anisotropic laminated cylindrical shells to evaluate both transverse shear deformation and geometrically non-linear e+ects. Non-linear e+ects produce either hardening or softening behavior in circular cylindrical shells. The method has been developed to demonstrate the in)uence of the shear deformation e+ect and that of geometrical non-linearities on the free vibration of open or closed cylindrical shells. It is a hybrid -nite element based on a combination of shearable shell theory and modal expansion approach such that the displacement functions could be derived directly from shearable shell theory. This method is capable of obtaining the high as well as low frequencies with high accuracy. The values of the shear correction factors used in calculations have been taken 2 =12. As the -rst numerical example, Fig. 2 shows the natural frequencies computed for a closed simply supported, circular cylindrical shell for m = 1 and 2 and compared with the experimental results given in [28]. As can be 1800

1600 Present Experiment [28]

Natural frequencies (Hz)

1400

1200

1000

800 m=2 600

400

m=1

200

0 1

2

3

4

5

6

7

8

9

Circumferential wave number (n)

Fig. 2. Natural frequencies of simply supported cylindrical shell.

10

11

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

1329

45 x

40

Present Ref. [29]

Non-dimensional fundamental frequencies 2 1/2 Ω=ω oR (ρ / E 2) /t

y 35

m=1 0˚/90˚/90˚/0˚

30 R/t=100 25

20

R/t=50

15

10

5

R/t=20

0 0

2

4

6

8

10

The length radius ratio L/R ◦

◦

◦

◦

Fig. 3. Non-dimensional fundamental frequencies of simply supported cylindrical shell with symmetric cross-ply 0 =90 =90 =0 .

Table 1 ◦ ◦ ◦ Non-dimensional fundamental frequencies, E=!0 L2 =(2=E2 )1=2 =t of simply-supported cylindrical shell with symmetric cross-ply 0 =90 =0

R=L

L=t = 100

1 2 3 4 5 6 7 8 9 10 a These

L=t = 10

Sanders’ theorya

Reddy [30]

Present

Sanders’ theorya

Reddy [30]

Present

73.05 38.77 28.36 23.73 22.15 21.05 20.28 19.69 19.23 18.58

66.583 36.770 27.116 22.709 20.232 — — — — 16.625

63.50 33.51 24.52 20.96 19.18 18.17 17.25 16.96 16.79 16.60

15.811 14.794 14.726 14.688 14.675 14.638 14.612 14.575 14.568 14.537

13.172 12.438 12.287 12.233 12.207 — — — — 12.173

13.135 12.129 11.716 11.542 11.461 11.367 11.356 11.350 11.341 11.332

results have been obtained by author for Sanders’ theory, not considering shear deformation e+ect.

seen, there is good agreement between the present theoretical results and those of experimental. Dimensions and material properties are given as follows: R = 0:175 (m)

L = 0:664 (m) t = 1 (mm)

E = 206 (GPa) F = 0:3 2s = 7680 (kg=m3 ): The non-dimensional fundamental frequencies obtained from the present theory are shown in Fig. 3 along with corresponding values given in Ref. [29], for a four layer cross-ply cylindrical shell to demonstrate the accuracy and range of applicability of the present theory. All layers, for Fig. 3 and Table 1, are assumed to

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M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335 2.5

Present 2 1/2 Ω =ω o R ( ρ (1- ν ) / E )

Non-Dimensional Frequency

2

Sanders' Theory

1.5

R/t=10 n=2 m=5

φΤ = π

1

0.5

m=3

0 2

3

4

5

6

7

8

9

10

The Length Radius Ratio L/ R

Fig. 4. Frequency distribution of an open cylindrical shell in conjunction with L=R and m variations.

have the same geometric and material parameters and the individual layer is assumed to be orthotropic with the following material properties: E1 = 25E2

G23 = 0:2E2 G13 = G12 = 0:5E2

F12 = 0:25 and 2 = 1: The e+ect of variation of the length-to-radius ratio on the frequency parameters of an open cylindrical shell having its straight edges clamped and the curved edges simply supported is shown in Fig. 4 and compared with corresponding results based on Sanders’ theory, obtained by author. Fig. 5 shows the in)uence of geometrical non-linear e+ects on the free vibrations of a simply supported cylindrical shell, along with corresponding results given in Refs. [10,19]. The given results in Ref. [10] were obtained based on Donnell’s simpli-ed non-linear method where only lateral displacement was considered. Raju and Rao used the -nite element method based on an energy formulation. It is observed that the variation ratio between the non-linear and liner frequency !Nl: =!L increases as the ratio 3j =t increases although these variations are small for values 3j =t below 1.0. It can also be seen that the non-linearity has a hardening e+ect. The di+erence between results obtained by the present theory and those of [10,19] might be due to the fact that Nowinski [10] neglected in-plane inertia and took account only lateral displacement. Raju and Rao [19] expressed the displacement components along the shell generator in polynomial form. The next step of calculations is the non-linear dynamics characteristics of an orthotropic open cylindrical shell, with clamped boundary conditions along its straight edges and simply supported along its curved edges, as a function of circumferential and axial mode numbers. The results are shown in Fig. 6. As can be seen,

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335 1.3 Present Raju and Rao [19] Nowinski [10] E=200GPa ν=0.3

Relative frequncy ω NL / ω L

1.2

m=1 n=4

ρ=7800kg / m3 R=2.54 cm L=40 cm t=0.0254 cm 1.1

1

0.9 0

0.5

1

1.5

2

2.5

3

Amplitude to Thickness Ratio Γj /t

Fig. 5. Relative frequency versus relative amplitude for non-linear vibration of a simply supported cylindrical shell. 1.5

n=1 n=2 n=3 n=4

1.4

m=2

E x =1.0x1011 N / m2 E θ = 0.05 Ex

νxθ = 0.2 G xθ = 0.05 E x

Relative frequency ωNL/ωL

1.3

ρ = 7800kg / m3 R / L = 0.1 and R / t=50 φT = π

1.2

1.1

1

0.9 0

0.5

1

1.5

2

2.5

3

Amplitude to Thickness Ratio Γ j / t

Fig. 6. In)uence of large amplitude on frequency of clamped–clamped orthotropic open cylindrical shell.

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M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

the non-linearity is of the hardening type for these modes. It is also seen that the non-linear e+ect is more pronounced for the circumferential mode (n = 1). Table 1 presents a parametric study based on shell thickness to determine the e+ect of transverse shear deformation on the natural frequencies of cross-ply cylindrical shells, for various di+erent ratios R=L and L=t. The natural frequencies for L=t = 10 are less sensitive to L=R variations than those of thin shells, L=t = 100. Classical shell theory over-predicts the frequency even for thin shells. 7. Conclusions This paper deals with some of the problems that arise when considering geometric non-linearities (incorporating the large displacements and rotations), shear deformation and rotary inertia e+ects in the study of static and dynamic behavior of elastic, anisotropic open or closed cylindrical shells. An e1cient hybrid -nite element method, modal expansion approach and shearable shell theory including shear deformation e+ects have been used to develop the non-linear dynamic equations. The shape functions are derived by analytical integration, from exact solution of shearable shell equations. It is believed that the re-ned shear deformation theory and the non-linear e+ects presented here are essential for predicting an accurate response for anisotropic shell structures. The shear deformation and rotary inertia e+ects are taken into account to develop the equations of motion. In case of rotary inertia, some preliminary results indicate that the e+ects of the rotary inertia are practically limited. On the other hand, Librescu [7] found that the inertia e+ect, in the case of anisotropic plate, was practically inexistant at the lowest branch of the frequency spectrum. Consequently, the rotary inertia e+ect is discarded in the remaining equations (after obtaining the equations of motion), so the present study focuses only on the shear deformation and geometrically non-linear e+ects. From the results obtained based on this theory, the following can be concluded: (1) The hybrid -nite element model is con-rmed as a very e1cient tools to analyses the anisotropic laminated shells. (2) Re-ned shell theory, taking into account the e+ects of shear deformation, plays a fundamental role in vibration analysis of anisotropic composite structures. (3) The assumed mode shapes (modal expansion) appears to be the ideal tool for the non-linear analysis of complex shell structures, which could eliminate some weaknesses and serious drawbacks of other theories such as Galerkin’s and perturbation’s methods. (4) A good description of geometrical non-linear e+ects on the vibration behavior of shells could be presented when the coupling between di+erent modes is taken into account by considering the cross-product terms of non-linear sti+ness matrices. The non-linear strain terms arising from products of in-plane strain terms may be important in buckling problems. The equations obtained in this work are shown to be suitable for practical applications involving large static or dynamic deformations of anisotropic open or closed cylindrical shells for di+erent parameter variations such as arbitrary boundary conditions and e+ect of materials properties. The non-linear modal equations derived here can render possible complete post-buckling investigations for cylindrical shells. The next work, under preparation, deals with the stability analysis of anisotropic cylindrical shells conveying )uid to show the reliability and e+ectiveness of the present formulations. Acknowledgements The author would like to acknowledge Dr. D.G. Zimcik, Institute for Aerospace Research, NRC, for his valuable suggestions and reviewing the manuscript.

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

1333

Appendix These appendices contain the various equations for non-linear dynamic analysis of anisotropic laminated cylindrical shells in which the shear deformation and rotary inertia e+ects are taken into account in the formulations.

Appendix A The strain vector for an anisotropic cylindrical shell (Eq. (1)) {-} = {-L }+{-NL }                                        1     2  2R           

         @U              @x            1 @V @V @V       +W    @V     0    R @x @ @x           @x x                      @U @V           0 @W ( +(      x  *     x +( @x @x      x           @x          

            0 2 2 2      0     @U @W @V x      @V W 1      + V − + W+      +           @ @ @ R @ R      0                                 @U 1      1 @U @U @W @W @W         + −V   *0      R @ R @x @ @x @ @x = = + (A.1);      0         @W 1 V       0      1 @V @U − +(              ( +W(+(x R @ R                   R @ @             .x             @( x                   @U @( @( @V x        @x       +       /x       @x @x @x @x                         @V @( 1                +       .   @x 2R @x  @V @( @V 1 @( @(             + +W                   R @x @ @x @ @x               / 1 @(                   1 @( @( @U @W @( @V R @     x       +W + +V( −(           2     R @ @ @ @ @ @         1 @(x − 1 @U           2  R @ 2R @ L(10×1)    1 @U @( @U @W @(   x x   + −(    R @x @ @x @ @x NL(10×1) 1 2



@U @x

2  2  2  @V @W + + @x @x

where -i0 ; *0i ; .i ; /i and 0i0 are normal and in-plane shearing strain, change in the curvature and torsion of the reference surface and the shearing strain components, respectively, and subscripts ‘L’ and ‘NL’ mean ‘linear’ and ‘non-linear’, respectively.

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M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

Appendix B Non-linear sti+ness matrices (Eq. (20))   AA + P BB + P CC + P DD + P EE ) (P 11 ijk 22 ijk 33 ijk 44 ijk 55 ijk   (NL2)  +(P66 GGijk + P77 NNijk + P88 PPijk + P99 SSijk + P1010 TTijk )  dA = kijk   I +(Pmn (AUXijk

and (NL3) = kijks



   

+

J AUXijk ))

+

57 (P36 (AUXijk

+

(P11 AAijks + P22 BBijks + P33 CCijks + P44 DDijks + P55 EEijks )



 +(P66 GGijks + P77 NNijks + P88 PPijks + P99 SSijks + P1010 TTijks )   dA; I +(Pmn (AUXijks

+

J AUXijks ))

+

57 (P36 (AUXijks

(B.1)

58 AUXijk ))

+

(B.2)

58 AUXijks ))

where dA = R d x d and m = 1; 2; : : : ; 9; m; n = 3; 6;

I = 1; 3; 5; : : : ; 55;

n = m + 1 to 10;

and

J = I + 1:

58 The Pij ’s are the terms of the elasticity matrix [P], given in [21], and the terms AAijk ; BBijk ; : : : ; AUXijk and 58 AAijks ; BBijks ; : : : ; AUXijks represent the coe1cients of the modal equations. The expressions for the coe1cients 1 1 ; AUXijks and ai are given here. The other coe1cients are obtained in the same way. AAij ; AAijk ; AAijks ; AUXijk

AAijk = ai AAjk + aj AAki + ak AAij ; AAijks = 2AAis AAjk ;   @Vi @Vj @Wi @Wj 1 @Ui @Uj + + ; AAij = 2 @x @x @x @x @x @x 1 AUXijk = aj BBki + ak BBij + bi AAjk ; 1 AUXijks = 2AAjk BBis ;

ai =

@Ui ; @x

(B.3)

where U; V , and W are the spatial functions determined by Eq. (5) of the text. The subscripts “i; j” and “i; j; k; s” represent the coupling between two and four distinct modes, respectively. Substituting Eq. (5), of the text, into Eq. (B.3), we obtain (,i +,j ) AAij = Ci ()i )j mL 2 sin2 mx L + (1 + (i + (j )mL 2 cos2 mx)C L ; je

ai = −Ci )i mL sin mxe L ,i  :

(B.4)

The ,i (i = 1; : : : ; 10) are the roots of characteristic Eq. (6), of the text, and mL = (m=L)x in which m is the axial mode number and )i ; (i are de-ned by Eq. (7). The constants Ci (i = 1; : : : ; 10) can be obtained using ten boundary conditions applied on the nodal lines i and j.

M.H. Toorani / International Journal of Non-Linear Mechanics 38 (2003) 1315 – 1335

1335

1 to 58 Using the relation (B.3) for AAijk ; BBijk ; : : : ; TTijk ; AUXijk and replacing these terms by their expressions (Eq. (B.4)) into Eq. (B.1) and then integrating over x and , we obtain the expression (21a) for the 1 to 58 second-order non-linear matrix for an element. Substituting the terms of AAijks ; BBijks ; : : : ; TTijks and AUXijks by their expressions, (B.3) and (B.4), into Eq. (B.2) and then integrating over x and , we obtain the expression (21b) that de-nes the third-order non-linear matrix for an element of anisotropic cylindrical shell.

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