Composite Structures 104 (2013) 94–117

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Composite Structures

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General higher-order equivalent single layer theory for free vibrationsof doubly-curved laminated composite shells and panels

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.04.009

⇑ Corresponding author. Tel.: +39 0512093500; fax: +39 0512093496.E-mail address: francesco.tornabene@unibo.it (F. Tornabene).URL: http://software.dicam.unibo.it/diqumaspab-project.

Francesco Tornabene ⇑, Erasmo Viola, Nicholas FantuzziDICAM – Department, Faculty of Engineering, University of Bologna, Italy

a r t i c l e i n f o

Article history:Available online 18 April 2013

Keywords:Doubly-curved shells and panelsLaminated compositesHigher-order shear deformation theoryEquivalent single layer approachGeneralized differential quadrature method

a b s t r a c t

The present paper provides a general formulation of a 2D higher-order equivalent single layer theory forfree vibrations of thin and thick doubly-curved laminated composite shells and panels with different cur-vatures. The theoretical framework covers the dynamic analysis of shell structures by using a general dis-placement field based on the Carrera’s Unified Formulation (CUF), including the stretching and zig-zageffects. The order of the expansion along the thickness direction is taken as a free parameter. The startingpoint of the present general higher-order formulation is the proposal of a kinematic assumption, with anarbitrary number of degrees of freedom, which is suitable to represent most of the displacement field pre-sented in literature. The main aim of this work is to determine the explicit fundamental operators thatcan be used not only for the Equivalent Single Layer (ESL) approach, but also for the Layer Wise (LW)approach. Such fundamental operators, expressed in the orthogonal curvilinear co-ordinate system, areobtained for the first time by the authors. The 2D free vibration shell problems are numerically solvedusing the Generalized Differential Quadrature (GDQ) and Generalized Integral Quadrature (GIQ) tech-niques. GDQ results are compared with recent papers in the literature and commercial codes.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The study of doubly-curved laminated shell structures hasbeen recently growing in many branches of structural engineer-ing. Many researchers have been deeply studying these struc-tures, over the years, mainly for their great capacity of carryingexternal loads. This phenomenon is due to shell curvature thatcouples the membrane and flexural behaviors of doubly and sin-gly curved structures. The historical framework of shell structuresis known since the 1940s [1–52]. Over the years the theoreticalapproaches have been following the engineering applicationrequirements, due to the increasing usage of laminated shellstructures. An extremely wide group of shell theories has beenimplemented so far [1–52]. They represent the basis of all thepresent theories on shells. Furthermore, not only linear solutionsbut also nonlinear shell problems have been investigated and ex-posed in the cited books [1–52].

In general, there are three different ways to study anisotropicshell structures: the 3D Elasticity [1,4,5,18,21,32], Equivalent Sin-gle Layer (ESL) [2,3,6–17,19,20,22–31,33–52] and Layer Wise(LW) [40,50] theories. The mechanical model used in the presentpaper is based on a unified approach named Carrera’s Unified

Formulation (CUF). This theoretical model, valid for all kinds ofengineering structures from plates to doubly-curved shells, is re-ported in the book by Carrera et al. [50], where basic principlesof classic and advanced structures are described. Furthermore, fi-nite element applications are reported in [50] for the evaluationof the static and dynamic behavior of plates and shells.

Composite plates and shell structures have been studied withina significant number of Higher-order Shear Deformation Theories(HSDTs) by many researchers. In fact, among them, several numer-ical solutions were proposed [53–58], which include first-orderand higher-order theories. The interested reader can refer to Carre-ra [53–55] where a unified description of several models based ondisplacements and transverse stress assumptions was given. Inparticular, the degree of freedom expansion along the thicknessdirection has been assumed as a free parameter. Furthermore, in-depth description about LW and ESL was pointed out includingzig-zag effect and inter-laminar continuity. Various shell geome-tries and mechanical shell complexities were conducted in the re-view article by Qatu [56], where the dynamic behavior oflaminated composite shells for different geometries has beenexamined in the first decade of the new millennium.

Nowadays, in order to carry out the static and dynamic analysesof anisotropic shell structures, the most popular numerical tool isthe finite element method [23,40,50,51]. The generalized colloca-tion method based on the ring elements has also been applied. Inthis method, it is necessary to transform, into an infinite Fourier

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 95

series of harmonic components, the static and kinematic variables[23,40]. The 2D problem can be simplified using standard Fourierdecomposition when a revolution shell is considered. On the con-trary, the reduction operation, illustrated above, cannot be usedfor panel structures. A two dimensional field has to be considered,as it is done in the present study. One dimensional formulation ofthe equilibrium of the shell is not considered in the following, be-cause completely revolution shells are special cases of panels.Kinematic and physical compatibility conditions have to beenforced.

As far as the numerical methods for solving differential equa-tions are concerned, the Generalized Differential Quadrature(GDQ) method is used in this paper to discretize the derivativesin the governing equations. The mathematical fundamentals andrecent developments of the GDQ method as well as its major appli-cations in engineering are discussed in detail in the book by Shu[59]. The interest of researches in this procedure is increasingdue to its great simplicity and versatility, as shown in the literature[60]. A lot of papers have been appeared over the years [61–94],and it is nearly impossible to cite all of them. The GDQ methodhas been applied extensively, since it gives very accurate resultsby using merely a few grid points.

Regarding the classification of plate and shell deformation the-ories reported above, in the class of the Equivalent Single LayerTheories, there are mainly three major theories, which are usuallyknown as: the Classical Plate Theory (CPT) [1,6–22,25], the First-order Shear Deformation Theory (FSDT) [2,3,23,26–28,31–46] andthe Higher-order Shear Deformation Theory (HSDT) [40,50,53–55]. The latter approach [40,50] has brought a significant contrib-ute for the study on the static and dynamic behavior of platesand shell structures under mechanical loading, as well as underthermal actions. Principally, the accuracy of the transverse shearstresses is improved due to the neglecting of the shear correctionfactor when these kinds of theories are used. Generally, the shearcorrection factor is overcome considering an expansion of the dis-placements upon the middle surface. In fact, the kinematic modelswhich are presented in this paper have been developed improvingthe analysis of shell responses and they involve a lot of researchers[95–139].

Recently, Viola et al. [91,92] conducted numerical dynamic andstatic investigations on doubly-curved shells and panels using theGeneralized Higher-order Shear Deformation Theory (GHSDT) cou-pled with the stress recovery via GDQ method. The suggested the-oretical model was derived from a 2-D third-order sheardeformation theory. The displacement field, having a fixed nine de-grees of freedom (9DOF), was considered and different shear func-tions proposed in literature [95–139] have been used. Surveys ofvarious shear deformation theories for plates and shells is pre-sented in the papers by Viola et al. [91,92], where the shear func-tions developed by a number of researchers are also reported. Inthese studies the effect of the shear function choice has been dee-ply investigated and comparisons with 3D elastic solutions are alsoreported in detail, not only for the dynamic case [91], but also forthe static case [92]. The main novelty proposed by Viola et al.[91,92] is a generalization of the third order theories presentedin literature without considering the stretching and the zig-zag ef-fects. Thus, in order to improve the previous higher-order theory,including the two mentioned effects, in this study a General Equiv-alent Single Layer Theory is presented. Furthermore, due to thelack of studies regarding the application of higher-order theoriesto completely doubly-curved shells, in this paper an extension ofthe Carrera’s Unified Formulation is developed for the first timeby the authors.

It should be remembered among all the researchers cited above[95–139], that shear deformation advanced theories have beendeeply investigated by Ferreira et al. [113,123] and Neves et al.

[129,138], where global multiquadratic radial basis functions(RBFs) were used for the first time by the authors [113] for model-ing symmetric composite plates using a trigonometric shear defor-mation theory. It should be noted that the trigonometric theorywas applied along the thickness, too. RBF collocation was appliedalso to laminated orthotropic elastic shells, using a sinusoidal sheardeformation theory [123] accounting through-the-thickness shelldeformation. The CUF approach [50] has been employed to obtainthe equations of motion and the boundary conditions. In the articleby Neves et al. [129] a different expansion for the out-of-plane andin-plane displacements has been proposed using an hyperbolicsine shear deformation theory. The stretching effect has beenadded studying the flexural and free vibration analysis of function-ally graded plates. Finally, in [138] RBFs have been used for study-ing the dynamic behavior of functionally graded shells withconstant curvature using an higher-order shear deformationtheory.

Unfortunately, the dynamic behavior of moderately thick lami-nated doubly-curved shells and panels is not fully detailed. In fact,it can be found in literature that a lot of papers present shells withconstant curvatures, while the theoretical proposed approach isvalid for doubly-curved shells. In this work, instead, the CUF ap-proach is exploited to the free vibration of deep multilayered shellsand panels with different curvatures. It is underlined that for thefirst time the explicit fundamental operators are presented. Theseoperators can be employed either in the ESL and the LW approach.Furthermore, each operator has the initial curvature effect embed-ded in its definition. It should be added that the description of themiddle surface of the shell has been done with Differential Geom-etry [13]. Hence, the GDQ method has been applied to differentialgeometry formulae for having the problem discretization. SinceGDQ method needs a set of discrete points for the fundamentalequation resolution, its application to differential geometry is aconvenient starting point for the computation.

Summarizing, the paper is arranged as follows. In the secondsection, the theoretical framework for the general higher-ordershear deformation theory concerning doubly-curved thin and thickshell structures is presented. The differential geometry is used todescribe the middle surface of shell structures by means of the po-sition vector written in the global reference system. All the deriv-atives of the geometric quantities and the integral through thethickness are numerically evaluated using the Generalized Differ-ential Quadrature method [59] and the Generalized Integral Quad-rature method [59]. In the third section, the GDQ implementationis shown and the discretization of the shell domain is presented. Inthe fourth section a lot of numerical results are reported. They con-cern with different types of structures: degenerate shells such asplates, singly-curved and doubly-curved shells and panels are con-sidered. For each structure different higher-order theories areworked out and compared with each other and with 3D FEM elas-ticity solutions. Different lamination schemes are also taken intoaccount. Finally, in the fifth section some remarks and conclusionsare drawn.

2. Geometry description and shell fundamental system ofequations

A 2D Equivalent Single Layer model is proposed to study genericdoubly-curved shells and panels. The position of an arbitrary pointwithin the shell medium is defined by curvilinear principal co-ordinates a1 a0

1 6 a1 6 a11

� �, a2 a0

2 6 a2 6 a12

� �upon the middle

surface or reference surface r(a1, a2), and f directed along the out-ward normal n(a1, a2), and measured from the reference surface(�h/2 6 f 6 h/2), where h(a1, a2) is the total thickness of the shell(Fig. 1). The differential geometry [10–13,18,38,43,65,89,91,92] is

W

ES

N

Fig. 1. Geometry description and co-ordinate system of a doubly-curved shell.

96 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

used to describe the shell structure. The position vector written inthe global reference system is the starting point:

Rða1;a2; fÞ ¼ rða1;a2Þ þhða1;a2Þ

2znða1;a2Þ ð1Þ

where z = 2f/h(a1, a2) and z 2 [�1, 1]. The basic configuration of theproblem considered is a laminated composite doubly-curved shell,as shown in Fig. 1. For this type of structure made of l laminae orplies, the total thickness h is defined as:

h ¼Xl

k¼1

hk ð2Þ

in which hk = fk+1 � fk is the thickness of the kth lamina or ply. Inthis study, doubly-curved shells and degenerate shells such asplates are investigated. It should be noted from Eq. (1) that the loca-tion of each point of the 3D shell is a function of the location of thepoint on the reference surface r(a1, a2) and the normal vector n(a1,a2) to the reference surface itself at the given point (Fig. 1). More-over, the position of the generic point of the shell volume is also afunction of the shell thickness h(a1, a2). In writing the position vec-tor of the reference surface, it is possible to define the three compo-nents along the three global axes Ox1x2x3 as:

rða1;a2Þ ¼ r1ða1;a2Þe1 þ r2ða1;a2Þe2 þ r3ða1;a2Þe3 ð3Þ

where e1, e2, e3 are the unit vectors of the global reference systemOx1x2x3. From the definition of the first fundamental form [10–13,18,38,43,65,89,91,92] of the reference surface r(a1, a2), theLamè parameters can be derived:

A1ða1;a2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir;1 � r;1

pA2ða1;a2Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir;2 � r;2

p ð4Þ

where the scalar product symbol � represents the scalar product andthe comma stands for the partial derivative with respect to a1, a2

co-ordinates, respectively. Moreover, by considering an orthogonalcurvilinear co-ordinate system O0a1a2f [10–13,18,38,43,65,89,91,92], from the position vector (3) the normal vector n(a1, a2)can be deduced and written as:

nða1;a2Þ ¼r;1 � r;2

A1A2ð5Þ

where the cross-product symbol� denotes the vector product. Final-ly, due to the fact that an orthogonal curvilinear co-ordinate systemO0a1a2f is considered and following the definition of the second fun-damental form [10–13,18,38,43,65,89,91,92] of the reference surfacer(a1, a2), the principal radii of curvature can be evaluated as:

R1ða1;a2Þ ¼ �r;1 � r;1r;11 � n

R2ða1;a2Þ ¼ �r;2 � r;2r;22 � n

ð6Þ

As far as the shell theory is concerned, the present study is based on thefollowing assumptions: (1) the transverse normal is extensible, so thatthe stretching effect is considered. In short, the normal strain is notequal to zero: en = en(a1, a2, f, t) – 0; (2) the higher-order transverseshear deformations are considered to influence the governing equa-tions, so that normal lines to the reference surface of the shell beforedeformation do not remain straight and are not necessarily normal afterdeformation; (3) the shell deflections are small and the strains are infin-itesimal; (4) the shell is moderately thick, therefore it is not possible toassume that the thickness-direction normal stress is negligible, so thatthe plane stress assumption cannot be invoked: rn = rn(a1, a2, f, t) – 0;(5) the linear elastic behavior of anisotropic materials is assumed; (6)the rotary inertias and the initial curvatures are taken into account;(7) the zig-zag effect is also considered [50].

The development of the theory under consideration involvesmoderately thick and thick shells, in which the following ratiosof thickness to curvature and thickness to length are valid:

1100

6maxh

Rmin;

hLmin

� �6

15

ð7Þ

It is noticed that the above presented shell theory is consistent withthe Higher-order Shear Deformation Theory (HSDT). Using the Car-rera’s Unified Formulation [50,53–55] the displacement field con-sidered in the present study can be put in the following form:

U ¼XNþ1

s¼0

FsuðsÞ ¼ FsuðsÞ ð8Þ

where U ¼ U1ða1;a2; f; tÞ U2ða1;a2; f; tÞ U3ða1;a2; f; tÞ½ �T is thedisplacement component vector for the three-dimensional shell,

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 97

uðsÞ ¼ uðsÞ1 ða1;a2; tÞ uðsÞ2 ða1;a2; tÞ uðsÞ3 ða1;a2; tÞh iT

is the sth ordergeneralized displacement component vector of points lying on themiddle surface (f = 0) of the shell, whereas t represents the timevariable. Fs is the thickness function matrix defined as follows:

Fs ¼Fs 0 00 Fs 00 0 Fs

264375 ð9Þ

In an expanded explicit form the kinematic model (8) can be put,considering s = N order of expansion, in the following way:

U1 ¼ F0uð0Þ1 þ F1uð1Þ1 þ F2uð2Þ1 þ F3uð3Þ1 þ � � � þ FNuðNÞ1 þ FNþ1uðNþ1Þ1

U2 ¼ F0uð0Þ2 þ F1uð1Þ2 þ F2uð2Þ2 þ F3uð3Þ2 þ � � � þ FNuðNÞ2 þ FNþ1uðNþ1Þ2

U3 ¼ F0uð0Þ3 þ F1uð1Þ3 þ F2uð2Þ3 þ F3uð3Þ3 þ � � � þ FNuðNÞ3 þ FNþ1uðNþ1Þ3

ð10Þ

The thickness functions Fs(f) can be assumed in the classical man-ner [50] as:

Fs ¼fs for s ¼ 0;1; . . . ;Nð�1Þkzk for s ¼ N þ 1

�ð11Þ

The dimensionless thickness co-ordinate zk(f) 2 [�1, 1] at layer le-vel [50] is assumed as:

zk ¼2

fkþ1 � fkf� fkþ1 þ fk

fkþ1 � fkð12Þ

As it can be seen from the definitions (11), the thickness functionFN+1(f) represents the Murakami function [50] that allows to con-

Table 1Several thickness functions Fs(f) proposed in literature.

½Cs�½50;55�FsðfÞ ¼ fs for s ¼ 0;1; . . . ;N

½CNþ1� ¼ ½Z�½50;55�FNþ1ðfÞ ¼ ð�1Þk 2

fkþ1�fkf� fkþ1þfk

fkþ1�fk

� for s ¼ N þ 1

½Ls�½50;55�FsðfÞ ¼ Ls for s ¼ 0;1; . . . ;NL0 ¼ C0 ¼ 1; L1 ¼ C1 ¼ f;L2 ¼ 1

2 ð3f2 � 1Þ; L3 ¼ 12 fð5f2 � 3Þ; L4 ¼ 1

8 ð35f4 � 30f2 þ 3Þ

½LSTF�½95;104;105;113�FsðfÞ ¼ h

p sin ph f� �

½KI�½108�FsðfÞ ¼ h

p cos ph f� �

½MIm�½132�FsðfÞ ¼ h

mp tan mph f

� �� mp

h f� �

½MIIm;k� for k ¼ 0;1½130�FsðfÞ ¼ ð�1Þk h

mp tan mph f

� �� mp

h sec2 mp2

� �f

� �½MIIIm�½131�FsðfÞ ¼ h

p sin ph f� �

em cos phfð Þ þ mp

h f�

½Xm;k� for k ¼ 0;1½125�FsðfÞ ¼ ð�1Þk 1

m2h

� �m�1fm

½KIIk� for k ¼ 0;1½111�FsðfÞ ¼ feð�1Þk2 f

hð Þ2

½Am;k� for k ¼ 0;1½118�FsðfÞ ¼ fmð�1Þk 2

ln mðfhÞ

2

½MIVm;k� for k ¼ 0;1½126�FsðfÞ ¼ fmð�1Þk2 f

hð Þ2

½N�½129�FsðfÞ ¼ h

p sinh ph f� �

½Sk� for k ¼ 0;1½109�FsðfÞ ¼ ð�1Þk h

p sinh ph f� �

� f cosh p2

� �� �½ATI�½116�FsðfÞ ¼ 3p

2 h tanh fh

� �� 3p

2 f sec h2 12

� �

sider the zig-zag effect through the thickness [53–55]. Another clas-sical thickness function choice is represented by the Legendrepolynomials Fs(f) = Ls [50]:

F0ðfÞ ¼ L0 ¼ 1; F1ðfÞ ¼ L1 ¼ f; F2ðfÞ ¼ L2 ¼12ð3f2 � 1Þ;

F3ðfÞ ¼ L3 ¼12

fð5f2 � 3Þ; F4ðfÞ ¼ L4 ¼18ð35f4 � 30f2 þ 3Þ

ð13Þ

The kinematic hypothesis expressed by relations (8) should be con-strained by the statement that the shell deflections are small and strainsare infinitesimal, that is uðsÞ3 � h. It should be noted that the displace-ments U1, U2 and U3 do not vary linearly through the thickness.

Due to the generality of the proposed theory, each sth orderthickness function Fs(f) can also assume different expressions.For the sake of completeness, various thickness functions Fs(f), alsonamed shear functions [50,53–55,91,92,95–139], are reported inTable 1. By choosing the thickness functions Fs(f) between thoseproposed in Table 1, different kinds of kinematical models can bedirectly derived and studied. For example, by considering thefourth order of expansion (s = 4), the classical higher-order kine-matical model assumes the following form:

U1 ¼ uð0Þ1 þ fuð1Þ1 þ f2uð2Þ1 þ f3uð3Þ1 þ f4uð4Þ1 þ ð�1Þkzkuð5Þ1

U2 ¼ uð0Þ2 þ fuð1Þ2 þ f2uð2Þ2 þ f3uð3Þ2 þ f4uð4Þ2 þ ð�1Þkzkuð5Þ2

U3 ¼ uð0Þ3 þ fuð1Þ3 þ f2uð2Þ3 þ f3uð3Þ3 þ f4uð4Þ3 þ ð�1Þkzkuð5Þ3

ð14Þ

where the thickness functions Fs(f) are chosen as:

ðF0; F1; F2; F3; F4; F5Þ ¼ ð1; f; f2; f3; f4; ð�1ÞkzkÞ ð15Þ

½ATII �½116�FsðfÞ ¼ f sec h p f

h

� �2�

� f sec h p4

� �1� p

2 tanh p4

� �� �½E�½124�FsðfÞ ¼ h

p sinh ph f� �

� f� �

= cosh p2

� �� 1

� �½Gm�½139�FsðfÞ ¼ h

m arcsin h mh f� �

� 2fffiffiffiffiffiffiffiffiffiffim2þ4p

½PTd1 d2 d3�½91�

FsðfÞ ¼ d1fþ d2h f2 þ d3

h2 f3

½PA� ¼ ½PT1=8;0;�1=6�½96�d1 ¼ 1

8 ; d2 ¼ 0;d3 ¼ � 16

½PKR� ¼ ½PT3=2;0;�2�½103�d1 ¼ 3

2 ; d2 ¼ 0;d3 ¼ �2

½PKPRS� ¼ ½PT5=4;0;�5=3�½97—99;115�d1 ¼ 5

4 ; d2 ¼ 0;d3 ¼ � 53

½PLMR� ¼ ½PT1;0;�4=3�½100—102�d1 ¼ 1; d2 ¼ 0;d3 ¼ � 4

3

½PSTh� ¼ ½PT1=4;0;�5=3�½136�d1 ¼ 1

4 ; d2 ¼ 0;d3 ¼ � 53

½PRMTVSI� ¼ ½PT5=4;�5;0�½2;3;44;110�d1 ¼ 5

4 ; d2 ¼ �5; d3 ¼ 0

½PRMTVSII� ¼ ½PT5=4;0;�5�½2;3;44;110�d1 ¼ 5

4 ; d2 ¼ 0;d3 ¼ �5

½TI�½91�FsðfÞ ¼ 2h

p tan p2h f� �

½TII�½91�FsðfÞ ¼ h

p sinh ph f� �

� f

½TIII�½91�FsðfÞ ¼ h

p cosh ph f� �

� 1� �

½TIVk� for k ¼ 0;1½91�FsðfÞ ¼ feð�1Þk2

hf

98 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

in which the last one allows to consider the zig-zag effect [50,53–55].Otherwise, if different thickness functions are assumed, such as thethickness function by Neves et al. [129] and the other one by Violaet al. [91], the following kinematical model can be considered:

U1 ¼ uð0Þ1 þhp

sinhph

f�

uð1Þ1 þhp

coshph

f�

� 1�

uð2Þ1 þ ð�1Þkzkuð3Þ1

U2 ¼ uð0Þ2 þhp

sinhph

f�

uð1Þ2 þhp

coshph

f�

� 1�

uð2Þ2 þ ð�1Þkzkuð3Þ2

U3 ¼ uð0Þ3 þhp

sinhph

f�

uð1Þ3 þhp

coshph

f�

� 1�

uð2Þ3 þ ð�1Þkzkuð3Þ3

ð16Þ

where the thickness functions Fs(f) are chosen as:

ðF0; F1; F2; F3Þ ¼ 1;hp

sinhph

f�

;hp

coshðph

fÞ � 1�

; ð�1Þkzk

� �ð17Þ

As can be seen form the two proposed kinematical models (14) and(16), the main differences are in the choice of the shear functions Fsand in the different number of degrees of freedom u(s) of the as-sumed displacement model (8). As a matter of fact, the first model(14) has 18 independent variables, whereas the second one (16)only has 12 independent variables.

In order to simplify the notation of the general model (10), it ispossible to use the following representation:

ED�ða1Þ ½F0�½F1�½F2�½F3� . . . ½FN �½FNþ1�ða2Þ ½F0�½F1�½F2�½F3� . . . ½FN �½FNþ1�ðfÞ ½F0�½F1�½F2�½F3� . . . ½FN �½FNþ1�

ð18Þ

where E indicates that an Equivalent Single Layer theory is consid-ered; D specifies that the governing equations are only expressed interms of generalized displacements; a1, a2, f denote the principaldirections of the variable expansion in the kinematical model,respectively; [Fs] stands for the type of thickness function Fs(f) cho-sen for the sth order of expansion (see Table 1) in each principaldirection; finally, [FN+1] = [CN+1] = [Z] represents the zig-zag func-tion or Murakami function (11), (12). The introduced symbology(18) considers the general displacement field in which the degreesof freedom are multiplied by different shear functions. The symbol-ogy (18) can be further simplified when the same thickness func-tions Fs (f) are chosen for each displacement of the kinematicalmodel assumed (8), as shown in the following:

ED�ða1Þ ½F0�½F1�½F2�½F3�½F4�½Z�ða2Þ ½F0�½F1�½F2�½F3�½F4�½Z�ðfÞ ½F0�½F1�½F2�½F3�½F4�½Z�

¼ ED� ½F0�½F1�½F2�½F3�½F4�½Z� ¼ EDZ4

ED�ða1Þ ½F0�½F1�½F2�½F3�½F4�ða2Þ ½F0�½F1�½F2�½F3�½F4�ðfÞ ½F0�½F1�½F2�½F3�½F4�

¼ ED� ½F0�½F1�½F2�½F3�½F4� ¼ ED4

ED�ða1Þ ½F0�½F1�½F2�½F3�½Z�ða2Þ ½F0�½F1�½F2�½F3�½Z�ðfÞ ½F0�½F1�½F2�½F3�½Z�

¼ ED� ½F0�½F1�½F2�½F3�½Z� ¼ EDZ3

ED�ða1Þ ½F0�½F1�½F2�½F3�ða2Þ ½F0�½F1�½F2�½F3�ðfÞ ½F0�½F1�½F2�½F3�

¼ ED� ½F0�½F1�½F2�½F3� ¼ ED3

ED�ða1Þ ½F0�½F1�½F2�½Z�ða2Þ ½F0�½F1�½F2�½Z�ðfÞ ½F0�½F1�½F2�½Z�

¼ ED� ½F0�½F1�½F2�½Z� ¼ EDZ2

ED�ða1Þ ½F0�½F1�½F2�ða2Þ ½F0�½F1�½F2�ðfÞ ½F0�½F1�½F2�

¼ ED� ½F0�½F1�½F2� ¼ ED2

ED�ða1Þ ½F0�½F1�½Z�ða2Þ ½F0�½F1�½Z�ðfÞ ½F0�½F1�½Z�

¼ ED� ½F0�½F1�½Z� ¼ EDZ1

ED�ða1Þ ½F0�½F1�ða2Þ ½F0�½F1�ðfÞ ½F0�½F1�

¼ ED� ½F0�½F1� ¼ ED1

The reader can refer to the book by Carrera et al. [50] for further de-tails about the last symbols on the right side of expressions (19). Thus,by using the nomenclature reported in Table 1, the previous kinemat-ical models (14) and (16) can be now indicated, respectively, as:

ED�ða1Þ ½C0�½C1�½C2�½C3�½C4�½Z�ða2Þ ½C0�½C1�½C2�½C3�½C4�½Z�ðfÞ ½C0�½C1�½C2�½C3�½C4�½Z�

¼ ED� ½C0�½C1�½C2�½C3�½C4�½Z� ¼ EDZ4

ED�ða1Þ ½C0�½N�½TIII�½Z�ða2Þ ½C0�½N�½TIII�½Z�ðfÞ ½C0�½N�½TIII�½Z�

¼ ED� ½C0�½N�½TIII�½Z�

ð20Þ

Furthermore, the previous HSDT theories, presented by the authorsin the works [91,92], can be easily represented by the general rep-resentation (18) as:

ED�ða1Þ ½C0�½F1�½F2�½F3�ða2Þ ½C0�½F1�½F2�½F3�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �½g�½e�ða2Þ ½1�½f �½g�½e�ðfÞ ½1�

¼ TSDT� ½f �½g�½e�

ED�ða1Þ ½C0�½F1�½F2�ða2Þ ½C0�½F1�½F2�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �½g�ða2Þ ½1�½f �½g�ðfÞ ½1�

¼ SSDT� ½f �½g�

ED�ða1Þ ½C0�½F1�ða2Þ ½C0�½F1�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �ða2Þ ½1�½f �ðfÞ ½1�

¼ FSDT� ½f �

ð21Þ

By considering the zig-zag function (11) the previous kinematicmodels (21) assume the following aspect:

ED�ða1Þ ½C0�½F1�½F2�½F3�½Z�ða2Þ ½C0�½F1�½F2�½F3�½Z�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �½g�½e�½Z�ða2Þ ½1�½f �½g�½e�½Z�ðfÞ ½1�

¼ TSDT � ½f �½g�½e�½Z�

ED�ða1Þ ½C0�½F1�½F2�½Z�ða2Þ ½C0�½F1�½F2�½Z�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �½g�½Z�ða2Þ ½1�½f �½g�½Z�ðfÞ ½1�

¼ SSDT� ½f �½g�½Z�

ED�ða1Þ ½C0�½F1�½Z�ða2Þ ½C0�½F1�½Z�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f �½Z�ða2Þ ½1�½f �½Z�ðfÞ ½1�

¼ FSDT� ½f �½Z�

ð22Þ

The well-known Third-order Shear Deformation Theory (TSDT) andFirst-order Shear Deformation Theory (FSDT) with or without thezig-zag function (11) can now be indicated as:

ED�ða1Þ ½C0�½C1�½C2�½C3�½Z�ða2Þ ½C0�½C1�½C2�½C3�½Z�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f�½f2�½f3�½Z�ða2Þ ½1�½f�½f2�½f3�½Z�ðfÞ ½1�

¼ TSDTZ

ED�ða1Þ ½C0�½C1�½Z�ða2Þ ½C0�½C1�½Z�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f�½Z�ða2Þ ½1�½f�½Z�ðfÞ ½1�

¼ FSDTZ

ED�ða1Þ ½C0�½C1�½C2�½C3�ða2Þ ½C0�½C1�½C2�½C3�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f�½f2�½f3�ða2Þ ½1�½f�½f2�½f3�ðfÞ ½1�

¼ TSDT

ED�ða1Þ ½C0�½C1�ða2Þ ½C0�½C1�ðfÞ ½C0�

¼ ED�ða1Þ ½1�½f�ða2Þ ½1�½f�ðfÞ ½1�

¼ FSDT ð23Þ

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 99

As it appears from expressions (18)–(23), most of the theories pre-sented in literature are included as special cases in the proposed Gen-eral Higher-order Equivalent Single Layer Theory (GHESLT) (8) and(18). Moreover, it should be noted that the GHESLT (8) and (18) isan extension of the Carrera’s Unified Formulation (CUF) [50,53–55].

The strain–displacement relations for a 3D solid shell in orthog-onal co-ordinate system [4,5,8–13,17,18,40,42,43,65,89,91,92] arethe well-known following relationships:e ¼ DU ¼ DfDXU ð24Þ

where

Df ¼

1H1

0 0 0 0 0 0 0 0

0 1H2

0 0 0 0 0 0 0

0 0 1H1

1H2

0 0 0 0 0

0 0 0 0 1H1

0 @@f 0 0

0 0 0 0 0 1H2

0 @@f 0

0 0 0 0 0 0 0 0 @@f

266666666664

377777777775ð25Þ

and

DX ¼

1A1

@@a1

1A1A2

@A1@a2

1R1

1A1A2

@A2@a1

1A2

@@a2

1R2

� 1A1A2

@A1@a2

1A1

@@a1

01

A2

@@a2

� 1A1A2

@A2@a1

0

� 1R1

0 1A1

@@a1

0 � 1R2

1A2

@@a2

1 0 00 1 00 0 1

266666666666666666664

377777777777777777775

ð26Þ

The strain component vector in Eq. (24) is defined as eða1;a2; f; tÞ ¼e1 e2 c12 c13 c23 e3½ �T , whereas the quantities H1, H2 intro-

duced in the matrix (25) are equal to H1 = 1 + f/R1, H2 = 1 + f/R2,respectively. As shown in (6), R1, R2 denote the curvature radii ofthe a1, a2 co-ordinate curves at the generic point of the middle sur-face of the shell, respectively. It is noted that the Df matrix consistsof all terms that depend on the shell thickness. On the other hand,the DX matrix includes only middle surface quantities.

Substituting the displacement field (8) into the strain–displace-ment relations (24), valid for the doubly-curved shells under con-sideration, the following equations are derived:

e ¼XNþ1

s¼0

ZðsÞeðsÞ ð27Þ

where eðsÞða1;a2; tÞ ¼ eðsÞ1 eðsÞ2 cðsÞ1 cðsÞ2 cðsÞ13 cðsÞ23 xðsÞ13 xðsÞ23 eðsÞ3

h iT

is the sth order generalized strain component vector and the matrix

Z(s) is defined as:

bFðsÞf ¼

FsH2 0 0 0 0 0 0 00 FsH1 0 0 0 0 0 00 0 FsH2 0 0 0 0 00 0 0 FsH1 0 0 0 00 0 0 0 FsH2 0 0 00 0 0 0 0 FsH1 0 0

0 0 0 0 0 0@Fs

@fH1H2 0

0 0 0 0 0 0 0@Fs

@fH1H2

0 0 0 0 0 0 0 0

26666666666666666666664

ZðsÞ ¼ DfFs ¼

FsH1

0 0 0 0 0 0 0 0

0 FsH2

0 0 0 0 0 0 0

0 0 FsH1

FsH2

0 0 0 0 0

0 0 0 0 FsH1

0 @Fs@f 0 0

0 0 0 0 0 FsH2

0 @Fs@f 0

0 0 0 0 0 0 0 0 @Fs@f

2666666666666664

3777777777777775ð28Þ

Due to the chosen kinematical model (8), the relationships betweengeneralized strains e(s) and generalized displacements u(s) for adoubly-curved shell are the following:eðsÞ ¼ DXuðsÞ for s ¼ 0;1;2; . . . ;N;N þ 1 ð29ÞIt is worth noting that in (29) different generalized strain vectorsare involved e(s) for s = 0, 1, 2, . . ., N, N + 1, where the index s refersto the order of the strain, e.g. s = 1 represents the first order defor-mation. Since an higher order theory has been introduced, only e(0)

and e(1) have a physical meaning, whereas e(s) for s = 2, . . ., N repre-sent the fictitious parts of the deformation. Finally, e(N+1) representsfictitious part of the deformation associated to the zig-zag effect. Inthe following, the shell material is a laminated composite linearelastic one. In order to write the constitutive equations in termsof stress resultants, the relations between stresses and strains mustbe written for the kth lamina [4,5,11,40,42,43,65,89]:rðkÞ ¼ CðkÞeðkÞ ð30Þ

where the constitutive matrix CðkÞ for the kth lamina assumes the aspect:

CðkÞ ¼

CðkÞ11 CðkÞ12 CðkÞ16 0 0 CðkÞ13

CðkÞ12 CðkÞ22 CðkÞ26 0 0 CðkÞ23

CðkÞ16 CðkÞ26 CðkÞ66 0 0 CðkÞ36

0 0 0 CðkÞ44 CðkÞ45 0

0 0 0 CðkÞ45 CðkÞ55 0

CðkÞ13 CðkÞ23 CðkÞ36 0 0 CðkÞ33

266666666664

377777777775ð31Þ

In Eq. (30) rðkÞða1;a2; f; tÞ ¼ rðkÞ1 rðkÞ2 sðkÞ12 sðkÞ13 sðkÞ23 rðkÞ3

h iTis

the stress component vector and eðkÞða1;a2; f; tÞ ¼

eðkÞ1 eðkÞ2 cðkÞ12 cðkÞ13 cðkÞ23 eðkÞ3

h iTis the strain component vector,

whereas in matrix (31) CðkÞnm represent the material constants writtenwith respect to the curvilinear reference system O0a1a2f after theapplication of the equations of transformation [12,15,34,53,54].

By using the Hamilton’s principle for the 3D doubly-curved so-lid, it is possible to define the generalized internal actions or resul-tants of sth order as:

SðsÞ ¼Xl

k¼1

Z fkþ1

fk

bFðsÞf r̂ðkÞdf for s ¼ 0;1;2; . . . ;N;N þ 1 ð32Þ

where

000000

0

0

@Fs

@fH1H2

37777777777777777777775

ð33Þ

100 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

In (32) SðsÞða1;a2; tÞ ¼ NðsÞ1 NðsÞ2 NðsÞ12 NðsÞ21 TðsÞ1 TðsÞ2 PðsÞ1 PðsÞ2 SðsÞ3

h iT

is the sth order resultant or internal action vector and

r̂ðkÞða1;a2;f; tÞ ¼ rðkÞ1 rðkÞ2 sðkÞ12 sðkÞ12 sðkÞ13 sðkÞ23 sðkÞ13 sðkÞ23 rðkÞ3

h iTis the

generalized stress component vector for the kth lamina. Therefore,it is possible to write the sth order resultants in terms of the general-ized sth order strains e(s)(a1, a2, t) by using the definitions (32), (30),(29) and (27):

SðsÞ ¼XNþ1

s¼0

AðssÞeðsÞ for s ¼ 0;1;2; . . . ;N;N þ 1 ð34Þ

where

AðssÞ ¼Xl

k¼1

Z fkþ1

fk

bFðsÞfbCðkÞbZðsÞdf

!¼

AðssÞ11ð20Þ AðssÞ

12ð11Þ AðssÞ16ð20Þ AðssÞ

16ð11Þ 0 0 0 0 Aðs~sÞ13ð10Þ

AðssÞ12ð11Þ AðssÞ

22ð02Þ AðssÞ26ð11Þ AðssÞ

26ð02Þ 0 0 0 0 Aðs~sÞ23ð01Þ

AðssÞ16ð20Þ AðssÞ

26ð11Þ AðssÞ66ð20Þ AðssÞ

66ð11Þ 0 0 0 0 Aðs~sÞ36ð10Þ

AðssÞ16ð11Þ AðssÞ

26ð02Þ AðssÞ66ð11Þ AðssÞ

66ð02Þ 0 0 0 0 Aðs~sÞ36ð01Þ

0 0 0 0 AðssÞ44ð20Þ AðssÞ

45ð11Þ Aðs~sÞ44ð10Þ Aðs~sÞ

45ð10Þ 0

0 0 0 0 AðssÞ45ð11Þ AðssÞ

55ð02Þ Aðs~sÞ45ð01Þ Aðs~sÞ

55ð01Þ 0

0 0 0 0 Að~ssÞ44ð10Þ Að~ssÞ

45ð01Þ Að~s~sÞ44ð00Þ Að~s~sÞ

45ð00Þ 0

0 0 0 0 Að~ssÞ45ð10Þ Að~ssÞ

55ð01Þ Að~s~sÞ45ð00Þ Að~s~sÞ

55ð00Þ 0

Að~ssÞ13ð10Þ Að~ssÞ

23ð01Þ Að~ssÞ36ð10Þ Að~ssÞ

36ð01Þ 0 0 0 0 Að~s~sÞ33ð00Þ

266666666666666666666664

377777777777777777777775

ð35Þ

The bCðkÞ and bZðsÞ matrices are shown in the following:

bCðkÞ ¼

CðkÞ11 CðkÞ12 CðkÞ16 CðkÞ16 0 0 0 0 CðkÞ13

CðkÞ12 CðkÞ22 CðkÞ26 CðkÞ26 0 0 0 0 CðkÞ23

CðkÞ16 CðkÞ26 CðkÞ66 CðkÞ66 0 0 0 0 CðkÞ36

CðkÞ16 CðkÞ26 CðkÞ66 CðkÞ66 0 0 0 0 CðkÞ36

0 0 0 0 CðkÞ44 CðkÞ45 CðkÞ44 CðkÞ45 0

0 0 0 0 CðkÞ45 CðkÞ55 CðkÞ45 CðkÞ55 0

0 0 0 0 CðkÞ44 CðkÞ45 CðkÞ44 CðkÞ45 0

0 0 0 0 CðkÞ45 CðkÞ55 CðkÞ45 CðkÞ55 0

CðkÞ13 CðkÞ23 CðkÞ36 CðkÞ36 0 0 0 0 CðkÞ33

266666666666666666664

377777777777777777775

ð36Þ

bZðsÞ ¼

FsH1

0 0 0 0 0 0 0 0

0 FsH2

0 0 0 0 0 0 0

0 0 FsH1

0 0 0 0 0 0

0 0 0 FsH2

0 0 0 0 0

0 0 0 0 FsH1

0 0 0 0

0 0 0 0 0 FsH2

0 0 0

0 0 0 0 0 0 @Fs@f 0 0

0 0 0 0 0 0 0 @Fs@f 0

0 0 0 0 0 0 0 0 @Fs@f

2666666666666666666664

3777777777777777777775

ð37Þ

The elastic coefficients of the constitutive matrix (35) take theform:

AðssÞnmðpqÞ ¼

Xl

k¼1

Z fkþ1

fk

CðkÞnmFsFsH1H2

Hp1Hq

2

df

Að~ssÞnmðpqÞ ¼

Xl

k¼1

Z fkþ1

fk

CðkÞnmFs@Fs

@fH1H2

Hp1Hq

2

df

Aðs~sÞnmðpqÞ ¼

Xl

k¼1

Z fkþ1

fk

CðkÞnm@Fs

@fFs

H1H2

Hp1Hq

2

df

Að~s~sÞnmðpqÞ ¼

Xl

k¼1

Z fkþ1

fk

CðkÞnm@Fs

@f@Fs

@fH1H2

Hp1Hq

2

df

for s;s¼0;1;2; . . . ;N;Nþ1for n;m¼1;2;3;4;5;6for p;q¼0;1;2

ð38Þ

where s, s are the superscripts indicating the corresponding thick-ness function Fs, Fs, respectively. The superscripts ~s;~s denote the

derivatives of the corresponding thickness function Fs, Fs with re-spect to f (@Fs/@f or @Fs/@f), respectively. The subscripts p, q onthe left-hand side terms stand for the exponents of the quantities,in the right terms H1, H2, whereas n, m are the indexes of the mate-rial constants CðkÞnm defined in Eq. (30) for the kth lamina.

Different approaches can be found in literature to evaluate the

elastic engineering constants AðssÞnmðpqÞ; Að~ssÞ

nmðpqÞ; Aðs~sÞnmðpqÞ; Að~s~sÞ

nmðpqÞ

[2,3,11,13,38,40,42,87,90,93]. In the present paper, in order toavoid numerical instabilities, the relations of the elastic engineer-

ing stiffnesses AðssÞnmðpqÞ; Að~ssÞ

nmðpqÞ; Aðs~sÞnmðpqÞ; Að~s~sÞ

nmðpqÞ are numerically eval-uated using the Generalized Integral Quadrature (GIQ) rule [59]:

Z xj

xi

f ðxÞdx ¼XT

k¼1

1Ijk � 1I

ik

� f ðxkÞ; k ¼ 1;2; . . . ; T ð39Þ

where the coefficients 1Iik; 1I

jk are explicitly defined in the book by

Shu [59]. Since the elastic engineering stiffnesses AðssÞnmðpqÞ; Að~ssÞ

nmðpqÞ;

Aðs~sÞnmðpqÞ; Að~s~sÞ

nmðpqÞ depend on the shell curvatures, the correspondingderivatives with respect to the co-ordinates along a1 and a2 direc-tions of the reference surface have to be evaluated. The derivatives

of the elastic engineering constants AðssÞnmðpqÞ; Að~ssÞ

nmðpqÞ; Aðs~sÞnmðpqÞ; Að~s~sÞ

nmðpqÞ

are numerically evaluated:

@nf ðxÞ@xn

x¼xm

¼XT

k¼1

1ðnÞmkf ðxkÞ; k ¼ 1;2; . . . ; T ð40Þ

using the Generalized Differential Quadrature rule exposed in [59],where the weighting coefficients 1ðnÞmk are entirely defined. More

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 101

details about GDQ applications and numerical accuracy can befound in literature [60–94].

The governing equations of motion and the correspondingboundary conditions can be derived from the Hamilton’s principle[4,5,7–13,39–43,50,65,89]. For a shell structure the following set ofthree motion equations for each sth order of displacement expan-sion in terms of internal actions assumes the aspect:

D�XSðsÞ ¼XNþ1

s¼0

MðssÞ€uðsÞ for s ¼ 0;1;2; . . . ;N;N þ 1 ð41Þ

Lð00Þ Lð01Þ Lð02Þ Lð03Þ Lð04Þ Lð05Þ

Lð10Þ Lð11Þ Lð12Þ Lð13Þ Lð14Þ Lð15Þ

Lð20Þ Lð21Þ Lð22Þ Lð23Þ Lð24Þ Lð25Þ

Lð30Þ Lð31Þ Lð32Þ Lð33Þ Lð34Þ Lð35Þ

Lð40Þ Lð41Þ Lð42Þ Lð43Þ Lð44Þ Lð45Þ

Lð50Þ Lð51Þ Lð52Þ Lð53Þ Lð54Þ Lð55Þ

26666666664

37777777775

uð0Þ

uð1Þ

uð2Þ

uð3Þ

uð4Þ

uð5Þ

2666666664

3777777775¼

Mð00Þ Mð01Þ Mð02Þ Mð03Þ Mð04Þ Mð05Þ

Mð10Þ Mð11Þ Mð12Þ Mð13Þ Mð14Þ Mð15Þ

Mð20Þ Mð21Þ Mð22Þ Mð23Þ Mð24Þ Mð25Þ

Mð30Þ Mð31Þ Mð32Þ Mð33Þ Mð34Þ Mð35Þ

Mð40Þ Mð41Þ Mð42Þ Mð43Þ Mð44Þ Mð45Þ

Mð50Þ Mð51Þ Mð52Þ Mð53Þ Mð54Þ Mð55Þ

26666666664

37777777775

€uð0Þ

€uð1Þ

€uð2Þ

€uð3Þ

€uð4Þ

€uð5Þ

2666666664

3777777775ð47Þ

The motion or equilibrium operator D�X is defined as:

D�X ¼

1A1

@@a1þ 1

A1A2

@A2@a1

� 1A1A2

@A2@a1

1A1A2

@A1@a2

1A2

@@a2þ 1

A1A2

@A1@a2

1R1

0 �1 0 0

� 1A1A2

@A1@a2

1A2

@@a2þ 1

A1A2

@A1@a2

1A1

@@a1þ 1

A1A2

@A2@a1

1A1A2

@A2@a1

0 1R2

0 �1 0

� 1R1

� 1R2

0 0 1A1

@@a1þ 1

A1A2

@A2@a1

1A2

@@a2þ 1

A1A2

@A1@a2

0 0 �1

26643775 ð42Þ

and the inertia matrix M(ss) has the form:

MðssÞ ¼IðssÞ

0 0 0

0 IðssÞ0 0

0 0 IðssÞ0

26643775 for s; s ¼ 0;1;2; . . . ;N;N þ 1 ð43Þ

in which the mass inertia terms IðssÞ0 are defined by:

IðssÞ0 ¼

Xl

k¼1

Z fkþ1

fk

qðkÞFsFsH1H2df for s; s ¼ 0;1;2; . . . ;N;Nþ 1 ð44Þ

In expression (44) q(k) represents the mass density of the materialper unit of volume of the kth ply.

The kinematic (29), constitutive (35) and motion (41) equationscan be combined to give the fundamental system of equations, alsoknown as the governing system of equations. By replacing thekinematic Eqs. (29) into the constitutive Eq. (35) and the resultof this substitution into the motion Eq. (41), the governing equa-tions in terms of displacement components are deduced:XNþ1

s¼0

LðssÞuðsÞ ¼XNþ1

s¼0

MðssÞ€uðsÞ for s ¼ 0;1;2; . . . ;N;N þ 1 ð45Þ

where

LðssÞ ¼ D�XAðssÞDX ¼LðssÞ

11 LðssÞ12 LðssÞ

13

LðssÞ21 LðssÞ

22 LðssÞ23

LðssÞ31 LðssÞ

32 LðssÞ33

26643775 ð46Þ

The equilibrium operators in (46), LðssÞij ; i; j ¼ 1;2;3, s,s = 0, 1, 2,

. . ., N, N + 1, which are derived for the first time by the authors,

are reported in the Appendix in their explicit form. It should benoted that the total number of motion equations depends onthe order of expansion s = 0, 1, 2, . . ., N, N + 1. In fact, the totalnumber of motion equations are equal to 3 � (N + 2) and can beobtained by using the fundamental nuclei defined in Eq. (45).The complete system of equations for the fourth order N = 4 ofexpansion (s,s = 0, 1, 2, 3, 4, 4 + 1) can be written in matrix formas follows:

In order to solve the differential system (47), the boundary con-ditions must be imposed. Two kinds of boundary conditions areconsidered in the following, namely the fully clamped edgeboundary condition (C) and the free edge boundary condition(F). The equations describing the boundary conditions can bewritten as follows:

Clamped edge boundary conditions (C)

uðsÞ1 ¼ uðsÞ2 ¼ uðsÞ3 ¼ 0 for s ¼ 0;1;2; . . . ;N;N þ 1;

at a1 ¼ a01 or a1 ¼ a1

1;a02 6 a2 6 a1

2 ð48ÞuðsÞ1 ¼ uðsÞ2 ¼ uðsÞ3 ¼ 0 for s ¼ 0;1;2; . . . ;N;N þ 1;

at a2 ¼ a02 or a2 ¼ a1

2;a01 6 a1 6 a1

1 ð49Þ

Free edge boundary conditions (F)

NðsÞ1 ¼ 0;NðsÞ12 ¼ 0; TðsÞ1 ¼ 0 for s ¼ 0;1;2; . . . ;N;N þ 1;

at a1 ¼ a01 or a1 ¼ a1

1;a02 6 a2 6 a1

2 ð50ÞNðsÞ21 ¼ 0;NðsÞ2 ¼ 0; TðsÞ2 ¼ 0 for s ¼ 0;1;2; . . . ;N;N þ 1;

at a2 ¼ a02 or a2 ¼ a1

2;a01 6 a1 6 a1

1 ð51Þ

In addition to the external boundary conditions (48)–(51), the kine-matic and physical compatibility conditions should be satisfied atthe common closing meridians defined by a2 = 0, 2p, when a com-plete shell of revolution is considered. The kinematic compatibilityconditions include the continuity of displacements. The physicalcompatibility conditions can only be represented by the continuousconditions for the generalized stress resultants. To consider com-plete shells of revolution characterized by a1

2 ¼ 2p, it is necessary

102 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

to implement the kinematic and physical compatibility conditionsbetween the two computational meridians with a0

2 ¼ 0 and witha1

2 ¼ 2p:Kinematic compatibility conditions along the closing meridian

(a2 = 0, 2p)

uðsÞ1 ða1;0; tÞ ¼ uðsÞ1 ða1;2p; tÞuðsÞ2 ða1;0; tÞ ¼ uðsÞ2 ða1;2p; tÞ for s ¼ 0;1;2; . . . ;N;N þ 1; a0

1 6 a1 6 a11

uðsÞ3 ða1;0; tÞ ¼ uðsÞ3 ða1;2p; tÞð52Þ

Physical compatibility conditions along the closing meridian (a2 = 0,2p)

NðsÞ21 ða1;0; tÞ ¼ NðsÞ21 ða1;2p; tÞNðsÞ2 ða1;0; tÞ ¼ NðsÞ2 ða1;2p; tÞ for s ¼ 0;1;2; . . . ;N;N þ 1;a0

1 6 a1 6 a11

TðsÞ2 ða1;0; tÞ ¼ TðsÞ2 ða1;2p; tÞð53Þ

In analogous way, in order to consider a toroidal shell it is necessaryto implement the kinematic and physical compatibility conditionsbetween the two computational parallels with a0

1 ¼ 0 and witha1

1 ¼ 2p:Kinematic compatibility conditions along the closing parallel

(a1 = 0, 2p)

uðsÞ1 ð0;a2; tÞ ¼ uðsÞ1 ð2p;a2; tÞuðsÞ2 ð0;a2; tÞ ¼ uðsÞ2 ð2p;a2; tÞ for s ¼ 0;1;2; . . . ;N;N þ 1;

a02 6 a2 6 a1

2

uðsÞ3 ð0;a2; tÞ ¼ uðsÞ3 ð2p;a2; tÞ

ð54Þ

Physical compatibility conditions along the closing parallel (a1 = 0, 2p)

NðsÞ12 ð0;a2; tÞ ¼ NðsÞ12 ð2p;a2; tÞNðsÞ1 ð0;a2; tÞ ¼ NðsÞ1 ð2p;a2; tÞ for s ¼ 0;1;2; . . . ;N;N þ 1;a0

2 6 a2 6 a12

TðsÞ1 ð0;a2; tÞ ¼ TðsÞ1 ð2p;a2; tÞð55Þ

3. Discretized equations and numerical implementation

The GDQ method is used to discretize the derivatives in the gov-erning equations in terms of generalized displacements, as well asboundary conditions (see Tornabene [71] for a brief review). TheChebyshev–Gauss–Lobatto (C–G–L) grid distribution is assumedthroughout the paper, in which the co-ordinates of grid points(a1i, a2j) along the reference surface are in the discrete form:

a1i ¼ 1� cosi� 1

IN � 1p

� �� �a1

1 � a01

� �2

þ a01; i ¼ 1;2; . . . ; IN ;

for a1 2 a01;a

11

� �a2j ¼ 1� cos

j� 1IM � 1

p� �� �

a12 � a0

2

� �2

þ a02; j ¼ 1;2; . . . ; IM;

for a2 2 a02;a

12

� �ð56Þ

where IN, IM are the total number of sampling points used to discret-ize the domain in a1 and a2 directions, respectively, of the doubly-curved shell. It has been proven that, for the Lagrange interpolatingpolynomials, the C-G-L sampling point rule guarantees convergenceand efficiency to the GDQ technique [63,65–67,69,81,89]. TheGDQ procedure enables to write the governing equations (45) andthe boundary and compatibility conditions (48)–(55) in discreteform, transforming each space of derivatives into a weighted

sum of node values of independent variables using the GDQ rule(40) [59,71].

Using the method of separation of variables, it is possible toseek solutions that are harmonic in time and whose frequency isf = x/2p. The generalized displacements can be written as follows:

uðsÞða1;a2; tÞ ¼ UðsÞða1;a2Þeixt ð57Þ

where the vibration spatial amplitude values UðsÞ ¼

UðsÞ1 ða1;a2Þ UðsÞ2 ða1;a2Þ UðsÞ3 ða1;a2Þh iT

fulfil the fundamental dif-

ferential system (45). Each approximate equation is valid in a singlesampling point. Thus, the whole system of differential equationscan be discretized and the global assembling leads to a set of lineareigenvalue problem. When the kinematic condensation of nondo-main degrees of freedom is performed, one gets:

ðKdd � KdbðKbbÞ�1KbdÞdd ¼ x2Mdddd ð58Þ

The natural frequencies of the structure fr = xr/2p can be deter-mined by solving the standard eigenvalue problem (58). Thenumerical problem is partitioned, dividing the boundary algebraicequations b and the domain equations d. In this way it is possibleto avoid numerical instabilities and ill-conditioned matrices. Fur-thermore, the eigs function, embedded in MATLAB software, is usedto obtain the results in terms of natural frequencies of the struc-tures under consideration. It have to be added that GDQ techniqueis computational cost saving because no integration is needed forthe assembly of the linear algebraic global system, differently fromthe finite element approach.

4. Numerical results

In the present section, some results and considerations aboutthe free vibration problem of laminated composite doubly-curvedshells and panels are presented. The geometrical boundary condi-tions for a panel are identified by the following convention. Consid-ering the Fig. 1, the West edge (W) is defined by the relationsa2 ¼ a0

2; a01 6 a1 6 a1

1, whereas its opposite, the East edge (E), ischaracterized by the relations a2 ¼ a1

2; a01 6 a1 6 a1

1. Likewise,the North edge (N) is defined by the relations a1 ¼ a0

1; a02 6

a2 6 a12, whereas its opposite, the South edge (S), is characterized

by the relations a1 ¼ a11; a0

2 6 a2 6 a12. Thus, the boundary condi-

tion sequence for a panel structure can be represented with the fol-lowing symbology WSEN. In this way, the first side is the Westedge, the second one is the South edge, the third one is the Eastedge and finally, the last one is the North edge. For example, thesymbolism CFCF shows that the West and East edges are clamped,whereas the South and North edges are free. Differently, the CFsymbol denotes that the South and North edges are clamped andfree for a revolution shell, whereas the West and East edges areclamped and free for a toroidal shell. The missing boundary condi-tions are the kinematical and physical compatibility conditionsthat are applied at the same closing meridians, the West and Eastedges, for a revolution shell and at the same closing parallels, theSouth and North edges, for a toroidal shell, respectively.

In order to verify the theoretical formulation proposed in thesecond section, the solution procedure has been implemented ina MATLAB code [140]. With reference to the previous works [64–66,69,71–75,81–83,85–90], in this study the geometric parametersare calculated by using the differential geometry, as it is shown in[91,92]. Fig. 2 shows the eight investigated shell structures: anelliptic paraboloid, an hyperbolic paraboloid, a curved pipe panel,a curved shell pipe, a catenoidal panel, a conical shell, a parabolicalco-ordinate plate and a bi-polar co-ordinate plate. Tables 2–10present the first ten natural frequencies obtained by consideringvarious ESL higher-order theories proposed in the present work.Numerical comparisons with 3D FEM results for different lamina-

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 103

tion schemes and boundary condition combinations are alsoshown.

The elliptic paraboloid of Fig. 2a is obtained by moving aparabola along another parabola as reported in [91,92,140]. Forthe sake of simplicity, the elliptic paraboloid at issue is gener-ated by two parabolas having the same geometric parameters.The geometric description of a parabolic curve can be seen in[65,81,89,91,92,140] and the parameters describing the twoparabolas under consideration are shown in Table 2. The ortho-tropic materials and the laminae thickness, as well as the bound-ary conditions and lamination scheme are also reported inTable 2.

Regarding the hyperbolic paraboloid of Fig. 2b, it is obtained inthe same way as the elliptic paraboloid. On the contrary, for thisgeometry, the second parabola has an inverted concavity with re-spect to the first one. As in the previous case, the hyperbolic parab-oloid is generated by two parabolas having the same geometricparameters, but different concavities. The geometric descriptionof a parabolic curve can be found in [65,81,89,91,92,140] and theparameters describing the two parabolas under consideration areshown in Table 3. The orthotropic materials and the laminae thick-ness, as well as the boundary conditions and lamination schemeare also reported in Table 3. The elliptic and the hyperbolic parab-

(a) Elliptic Paraboloid (doubly-curved panel)

(c) Curved Pipe Panel (doubly-curved panel) (an elliptic curve slides on a catenary curve)

(e) Catenoidal Panel (doubly-curved panel of revolution)

(g) Parabolical Co-ordinate Plate (degenerate panel)

Fig. 2. Different shell and panel structures and relative

oloids of Tables 2 and 3, respectively, are made of three layers ori-ented as (30/45/70). The external laminae are made of differentorthotropic materials with respect to the middle lamina (see Tables2 and 3). The thickness of the three laminae are equal toh1 = h2 = h3 = 0.2 m.

Tables 4 and 5 present results for two different isotropic curvedcylinders or pipes. The curved pipe panel of Table 4 (see Fig. 2c) isobtained by moving an elliptic curve along a catenary curve, whilethe curved shell pipe of Table 5 (see Fig. 2d) is drawn by sliding anellipse along a Bézier curve. All the geometric details regarding theelliptic curve, the catenary curve and the Bézier curve are reportedin [65,81,83,89,91,92,140]. The position vector of a curved pipe canbe defined in the following general form:

rða1;a2Þ ¼ xa21 � B cos a1 sin a2

� �e1

þ xa23 þ B cos a1 cos a2

� �e2 þ b sin a1e3 ð59Þ

where xa21 ; xa2

3 are the plane co-ordinates of the directrix curve anda2 is the angle that the normal to the directrix curve forms with thevertical axis xa2

3 in the plane xa21 ; xa2

3

� �containing the curve itself. B

and b are the semi-major and the semi-minor diameters of the ellip-tic curve and a1 is the angle, that the line joining the generic point

(b) Hyperbolic Paraboloid (doubly-curved panel)

(d) Curved Shell Pipe (doubly-curved shell) (an ellipse slides on an Bézier curve)

(f) Conical Shell (singly-curved shell)

(h) Bi-polar Co-ordinate Plate with variable thickness (degenerate panel)

GDQ discretization for computer implementations.

Table 2First ten frequencies for a (30/45/70) CCCC elliptic paraboloid.

Geometric characteristics of the two parabolas: a = 3 m, c = �3 m, d = 0 m, b = 0.8 m, j ¼ ða2 � d2Þ=b ¼ 11:25 mProperties of the laminae 1 and 3: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h3 = 0.2 mProperties of the lamina 2: E1 = 53.78 GPa, E2 = E3 = 17.93 GPa, G12 = G13 = 8.96 GPa, G23 = 3.45 GPa, m12 = m13 = 0.25, m23 = 0.34, q = 1900 kg/m3, h2 = 0.2 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 209.448 210.470 212.985 208.779 210.680 210.383f2 214.619 214.950 216.120 214.149 214.717 214.630f3 233.005 233.349 236.163 231.589 232.933 232.532f4 276.777 277.170 284.668 275.750 277.702 277.289f5 305.705 307.227 312.888 304.395 308.321 307.574f6 333.035 333.838 338.257 330.964 334.573 334.183f7 334.430 334.752 341.616 333.306 335.824 334.682f8 354.381 354.647 361.956 353.504 355.305 354.865f9 377.360 379.231 386.715 375.430 380.836 379.835f10 412.940 411.159 430.736 410.466 413.532 412.643

FSDTZ TSDTZ EDZ1 EDZ2 EDZ3 EDZ4

f1 212.177 209.790 215.789 211.664 209.959 209.653f2 215.449 214.779 216.862 214.961 214.487 214.399f3 235.083 232.808 238.129 233.808 232.326 231.903f4 279.109 276.044 286.968 278.594 276.546 276.094f5 311.236 305.586 318.430 310.306 306.613 305.843f6 335.655 332.475 342.283 335.238 333.226 332.752f7 339.905 333.265 344.948 338.202 334.220 333.048f8 356.778 353.601 364.290 356.273 354.110 353.652f9 384.008 376.982 393.635 382.951 378.500 377.435f10 414.670 407.986 431.973 414.288 410.364 409.299

[C0][N][TIII] [C0][N][TIII][Z] [C0][C1][TII][TIII][Z] [L0][L1][L2][L3] [L0][L1][L2][L3][Z] 3D FEM Abaqus

f1 214.580 213.781 209.961 210.680 209.959 209.56f2 216.538 216.288 214.495 214.717 214.487 214.34f3 237.043 236.344 232.331 232.933 232.326 231.79f4 285.707 284.301 276.519 277.702 276.546 275.95f5 315.964 314.050 306.609 308.321 306.613 305.63f6 341.523 340.150 333.182 334.573 333.226 332.59f7 341.931 340.457 334.236 335.824 334.220 332.80f8 362.785 360.821 354.071 355.305 354.110 353.49f9 390.624 387.942 378.436 380.836 378.500 377.17f10 430.333 426.392 410.168 413.532 410.364 409.02

Table 3First ten frequencies for a (30/45/70) CCCC hyperbolic paraboloid.

Geometric characteristics of the two parabolas: a = 3 m, c = �3 m, d = 0 m, b = 1 m, j ¼ ða2 � d2Þ=b ¼ 9 mProperties of the laminae 1 and 3: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h3 = 0.2 mProperties of the lamina 2: E1 = 53.78 GPa, E2 = E3 = 17.93 GPa, G12 = G13 = 8.96 GPa, G23 = 3.45 GPa, m12 = m13 = 0.25, m23 = 0.34, q = 1900 kg/m3, h2 = 0.2 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 193.013 193.188 194.925 193.073 193.428 193.365f2 193.882 194.801 198.446 194.047 195.943 195.522f3 215.131 215.540 220.158 214.988 216.263 215.953f4 238.325 239.183 246.485 237.926 240.336 239.846f5 292.198 293.236 300.136 291.121 294.849 293.910f6 299.547 299.729 310.776 298.719 301.776 301.013f7 304.307 306.535 312.134 303.497 308.507 307.444f8 339.960 340.083 341.774 339.304 339.908 339.684f9 351.414 354.398 359.401 350.766 356.366 355.418f10 367.137 366.691 383.977 365.870 369.736 368.735

FSDTZ TSDTZ EDZ1 EDZ2 EDZ3 EDZ4

f1 193.474 193.072 195.427 193.595 193.311 193.249f2 196.625 194.071 201.255 196.960 195.189 194.763f3 216.841 215.020 221.815 216.946 215.732 215.422f4 241.352 237.955 249.524 241.540 239.097 238.586f5 297.062 291.811 305.014 296.503 293.400 292.461f6 302.661 297.696 315.041 302.997 299.741 298.916f7 311.337 304.822 317.773 310.959 306.755 305.702f8 341.180 339.595 342.396 340.556 339.419 339.144f9 359.109 352.415 367.741 359.063 354.269 353.354f10 370.330 363.635 386.778 370.994 366.691 365.559

(continued on next page)

104 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

Table 4First ten frequencies for an isotropic CCCC curved pipe panel (an elliptic curve slides on a catenary curve).

Geometric characteristics of the catenary curve: a = 2 m, c = 2 m, d = 2 mGeometric characteristics of the elliptic curve: B = 1 m, b = 0.5 m, a2 2 [�4p/9, 4p/9] (B, b are the semi-major and semi-minor diameters of the elliptic curve)Mechanical properties: E = 70 GPa, m = 0.3, q = 2707 kg/m3, h = 0.2 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 379.959 380.497 404.015 381.653 383.088 382.611f2 450.249 450.761 473.005 451.141 452.595 452.161f3 556.957 557.473 579.700 556.997 558.600 558.194f4 686.143 686.715 711.394 685.476 687.440 687.031f5 741.170 741.183 741.360 741.230 741.258 741.243f6 826.788 827.468 857.086 825.570 828.157 827.714f7 919.278 921.646 963.447 920.771 926.102 924.551f8 952.795 955.226 999.446 953.506 959.186 957.614f9 974.345 975.190 1004.182 972.715 976.228 975.725f10 998.190 998.651 1011.730 999.186 999.869 999.702

[C0][C1][PKPRS] [C0][C1][G3] [C0][C1][S0] [C0][N][TIII] [C0][C1][C2][KII1] [C0][C1][C2][TI]

f1 404.755 405.186 404.745 404.745 383.211 383.078f2 473.751 474.174 473.741 473.741 452.712 452.588f3 580.514 580.957 580.503 580.503 558.718 558.595f4 712.365 712.876 712.353 712.353 687.572 687.438f5 741.384 741.392 741.383 741.383 741.261 741.256f6 858.311 858.947 858.296 858.296 828.315 828.159f7 966.034 967.353 965.976 965.976 926.556 926.051f8 1002.464 1004.019 1002.396 1002.396 959.651 959.135f9 1004.699 1005.008 1004.686 1004.686 976.425 976.237f10 1013.306 1014.128 1013.287 1013.287 999.916 999.861

[C0][C1][LSTF][KI] [C0][C1][TII][TIII] [L0][L1][L2] [L0][L1][L2][L3] [L0][L1][L2][L3][L4] 3D FEM Abaqus

f1 383.839 382.993 388.042 383.088 382.611 382.76f2 453.331 452.501 457.384 452.595 452.161 452.31f3 559.365 558.502 563.516 558.600 558.194 558.36f4 688.311 687.331 693.005 687.440 687.031 687.23f5 741.264 741.255 741.337 741.258 741.243 741.28f6 829.217 828.026 834.957 828.157 827.714 827.95f7 928.200 925.458 943.194 926.102 924.551 924.90f8 961.373 958.517 976.957 959.186 957.614 957.96f9 977.552 976.057 984.870 976.228 975.725 976.02f10 1000.073 999.805 1001.446 999.869 999.702 999.79

Table 5First ten frequencies for an isotropic CC curved shell pipe (an ellipse slides on an Bézier curve).

Control points and weights of the Bézier curve: �x1 ¼ ½1 7 3 9 �; �x03 ¼ ½1 1 9 9 �; w ¼ ½1 1 1 1 �Geometric characteristics of the elliptic curve: B = 1.5 m, b = 1 m, a2 2 [0, 2p] (B, b are the semi-major and semi-minor diameters of the elliptic curve)Mechanical properties: E = 210 GPa, m = 0.3, q = 7800 kg/m3, h = 0.3 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 63.042 63.118 63.251 63.110 63.176 63.184f2 83.941 84.085 84.500 84.458 84.617 84.618f3 88.797 88.848 89.165 88.912 88.929 88.948f4 130.922 131.016 138.350 131.018 131.147 131.128f5 141.721 141.725 152.176 141.890 141.804 141.786f6 157.765 157.837 161.967 157.672 157.756 157.756f7 163.836 163.896 170.326 163.624 163.668 163.657f8 187.143 187.230 191.255 187.726 187.800 187.796f9 193.437 193.528 198.563 193.363 193.514 193.498f10 193.866 193.993 203.954 193.426 193.551 193.544

(continued on next page)

Table 3 (continued)

[C0][N][TIII] [C0][N][TIII][Z] [C0][C1][TII][TIII][Z] [L0][L1][L2][L3] [L0][L1][L2][L3][Z] 3D FEM Abaqus

f1 195.181 195.045 193.322 193.428 193.311 193.18f2 199.977 199.131 195.186 195.943 195.189 194.64f3 220.942 220.312 215.746 216.263 215.732 215.31f4 247.902 246.426 239.036 240.336 239.097 238.41f5 302.829 301.130 293.447 294.849 293.400 292.24f6 313.218 310.726 299.625 301.776 299.741 298.68f7 314.788 312.870 306.784 308.507 306.755 305.41f8 341.814 341.464 339.428 339.908 339.419 339.02f9 364.084 361.566 354.203 356.366 354.269 353.03f10 384.627 380.855 366.469 369.736 366.691 365.26

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 105

Table 6First ten frequencies for a (30/45) CCCC catenoidal panel.

Geometric characteristics of the catenary: a = 2 m, c = 2 m, d = 1 m, Rb = 0 m, a2 2 [0, p/2]Properties of the laminae 1 and 2: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h2 = 0.25 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 380.900 382.166 388.308 383.713 385.770 385.534f2 431.133 434.525 437.697 433.287 438.192 437.786f3 478.345 480.778 494.159 481.418 486.224 485.746f4 563.458 568.854 581.848 566.153 575.034 574.379f5 628.233 633.991 648.045 630.748 641.128 640.149f6 650.281 652.393 661.839 654.556 658.609 658.062f7 688.947 697.446 700.492 691.837 704.102 703.282f8 725.591 732.862 755.727 728.553 742.392 741.281f9 774.192 786.871 793.074 776.913 795.469 794.331f10 822.250 832.161 850.082 824.862 842.327 840.867

FSDTZ TSDTZ EDZ1 EDZ2 EDZ3 EDZ4

f1 387.621 382.156 392.361 390.922 385.665 385.524f2 443.065 434.507 446.998 445.592 438.017 437.769f3 491.877 480.758 499.700 495.702 486.010 485.730f4 582.892 568.835 591.576 586.500 574.753 574.359f5 651.413 633.944 660.187 654.858 640.697 640.123f6 664.059 652.370 671.567 669.250 658.365 658.039f7 715.086 697.406 721.674 718.860 703.743 703.255f8 752.579 732.827 765.333 756.730 741.911 741.252f9 808.153 786.832 817.548 812.270 794.990 794.298f10 855.825 832.095 866.769 859.492 841.692 840.830

[C0][N][TIII] [C0][N][TIII][Z] [C0][C1][TII][TIII][Z] [L0][L1][L2][L3] [L0][L1][L2][L3][Z] 3D FEM Abaqus [91]

f1 389.802 387.026 385.566 385.770 385.665 385.32f2 441.645 439.308 437.806 438.192 438.017 437.28f3 497.172 489.823 485.808 486.224 486.010 485.43f4 588.246 579.570 574.400 575.034 574.753 573.56f5 654.914 645.875 640.270 641.128 640.697 639.50f6 664.420 660.592 658.182 658.609 658.365 657.58f7 710.524 706.228 703.264 704.102 703.743 701.88f8 764.583 750.204 741.372 742.392 741.911 740.32f9 807.953 799.922 794.290 795.469 794.990 792.20f10 860.684 849.093 841.025 842.327 841.692 839.68

Table 5 (continued)

[C0][C1][PKPRS] [C0][C1][G3] [C0][C1][S0] [C0][N][TIII] [C0][C1][C2][KII1] [C0][C1][C2][TI]

f1 63.334 63.335 63.334 63.334 63.176 63.176f2 84.662 84.662 84.662 84.662 84.617 84.617f3 89.218 89.219 89.218 89.218 88.930 88.929f4 138.437 138.464 138.438 138.438 131.153 131.152f5 152.130 152.173 152.131 152.131 141.813 141.812f6 162.042 162.058 162.042 162.042 157.760 157.759f7 170.397 170.422 170.397 170.397 163.675 163.674f8 191.309 191.330 191.310 191.310 187.802 187.802f9 198.627 198.648 198.627 198.627 193.524 193.522f10 204.076 204.128 204.078 204.078 193.556 193.555

[C0][C1][LSTF][KI] [C0][C1][TII][TIII] [L0][L1][L2] [L0][L1][L2][L3] [L0][L1][L2][L3][L4] 3D FEM Abaqus

f1 63.180 63.176 63.205 63.176 63.184 63.28f2 84.617 84.618 84.608 84.617 84.618 83.65f3 88.936 88.930 88.978 88.929 88.948 88.37f4 131.308 131.192 131.541 131.147 131.128 133.95f5 142.039 141.847 142.485 141.804 141.786 137.96f6 157.855 157.783 158.021 157.756 157.756 157.01f7 163.839 163.701 164.149 163.668 163.657 158.88f8 187.860 187.812 187.988 187.800 187.796 185.47f9 193.665 193.561 193.862 193.514 193.498 190.05f10 193.755 193.582 194.230 193.551 193.544 192.39

106 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

of the elliptic curve and the origin, defines with the horizontal axisin the plane containing the elliptic curve itself. The parametersdescribing the two geometries under consideration are shown inTables 4 and 5, respectively. In particular, for the first structure

(see Fig. 2c) a catenary curve is considered as directrix curve,whereas for the second structure (see Fig. 2d) the directrix curveis a free-form curve (Bézier curve) [83]. The curved pipe panel of Ta-ble 4 and the curved shell pipe of Table 5 are made of isotropic

Table 7First ten frequencies for a (30/45) catenoidal panel with different boundary conditions.

Geometric characteristics of the catenary: a = 2 m, c = 2 m, d = 1 m, Rb = 0 m, a2 2 [0, p/2]Properties of the laminae 1 and 2: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h2 = 0.25 m

Mode (Hz) FCCF CCFF CFCF

EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus

f1 56.360 56.55 96.059 96.01 162.214 162.15f2 153.173 153.20 198.937 198.75 233.132 233.10f3 186.795 186.74 253.941 253.72 259.554 259.44f4 270.665 270.49 327.148 326.88 314.170 313.95f5 315.672 315.77 399.055 398.89 320.415 320.25f6 348.782 348.88 415.905 415.71 388.284 388.14f7 395.806 395.63 444.126 443.59 398.870 398.89f8 437.949 437.74 470.467 470.05 452.516 452.53f9 472.906 472.71 96.059 96.01 162.214 162.15f10 512.310 511.64 198.937 198.75 233.132 233.10

CFFC FFCC FCFC

EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus

f1 36.637 36.63 72.282 72.24 298.129 297.96f2 103.225 103.31 141.435 141.32 305.578 305.40f3 154.429 154.37 181.889 181.76 347.426 347.19f4 221.351 221.39 264.153 264.06 366.191 365.90f5 249.707 249.65 307.296 307.11 416.689 416.32f6 270.764 270.80 371.741 371.54 476.825 476.56f7 307.438 307.32 413.239 413.13 500.864 500.57f8 394.619 394.60 441.891 441.41 529.458 529.15f9 36.637 36.63 72.282 72.24 298.129 297.96f10 103.225 103.31 141.435 141.32 305.578 305.40

FCCC CCFC CCCF

EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus EDZ4 3D FEM Abaqus

f1 313.667 313.53 302.900 302.72 264.839 264.65f2 341.751 341.53 347.899 347.53 303.618 303.41f3 394.275 394.03 403.663 403.43 370.188 370.12f4 465.173 464.67 477.809 477.30 445.804 445.46f5 488.827 488.50 493.660 493.34 495.686 495.08f6 550.731 550.65 540.894 540.65 522.480 522.23f7 595.445 594.76 601.576 600.99 577.579 577.16f8 651.825 650.76 661.871 660.69 607.371 606.83f9 668.851 668.28 680.145 679.42 652.699 651.67f10 708.024 706.62 711.543 710.01 707.931 707.30

Table 8First ten frequencies for a (�45/45) FC conical shell.

Geometric characteristics of the conical shell: Rb = 1 m, L = 3 m, a = p/6, a2 2 [0, 2p]Properties of the laminae 1 and 2: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h2 = 0.15 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 61.601 61.545 62.262 61.343 61.550 61.408f2 61.601 61.545 62.262 61.343 61.550 61.408f3 95.800 95.832 95.989 95.651 95.762 95.689f4 95.800 95.832 95.989 95.651 95.762 95.689f5 99.059 98.560 101.520 98.286 98.640 98.277f6 99.059 98.560 101.520 98.286 98.640 98.277f7 167.652 166.584 172.139 165.873 166.691 165.941f8 167.652 166.584 172.139 165.873 166.691 165.941f9 181.369 181.064 182.026 181.462 181.381 181.289f10 221.296 221.123 220.673 220.558 220.237 220.219

FSDTZ TSDTZ EDZ1 EDZ2 EDZ3 EDZ4

f1 61.726 61.489 61.907 61.716 61.481 61.396f2 61.726 61.489 61.907 61.716 61.481 61.396f3 96.134 95.809 96.156 96.071 95.724 95.677f4 96.134 95.809 96.156 96.071 95.724 95.677f5 98.707 98.376 99.441 98.781 98.447 98.267f6 98.707 98.376 99.441 98.781 98.447 98.267f7 166.978 166.197 168.296 167.038 166.291 165.926f8 166.978 166.197 168.296 167.038 166.291 165.926f9 181.333 180.977 181.769 181.680 181.300 181.270f10 221.733 221.112 221.022 220.997 220.217 220.212

(continued on next page)

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 107

Table 9First ten frequencies for an isotropic CFFC parabolical co-ordinate plate.

Geometric characteristics: a1 2 [0.2 m, 2 m], a2 2 [0.2 m, 2 m]Mechanical properties: E = 70 GPa, m = 0.3, q = 2707 kg/m3, h = 0.3 m

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 30.301 30.304 32.535 30.360 30.375 30.297f2 101.242 101.265 104.654 101.420 101.519 101.524f3 136.393 136.427 144.278 136.644 136.812 136.815f4 236.081 236.081 236.552 236.512 236.526 236.500f5 276.653 276.784 290.177 276.996 277.614 277.515f6 280.528 280.659 298.141 280.956 281.559 281.414f7 323.481 323.660 341.481 324.003 324.873 324.725f8 453.890 453.890 454.784 454.715 454.713 454.678f9 503.526 503.903 506.351 504.062 505.886 505.712f10 506.232 506.232 527.488 506.276 506.272 506.271

[C0][C1][PKPRS] [C0][C1][G3] [C0][C1][S0] [C0][N][TIII] [C0][C1][C2][KII1] [C0][C1][C2][TI]

f1 32.540 32.544 32.540 32.540 30.376 30.375f2 104.684 104.716 104.684 104.684 101.528 101.520f3 144.325 144.379 144.325 144.325 136.826 136.814f4 236.583 236.581 236.584 236.584 236.523 236.531f5 290.348 290.521 290.347 290.347 277.653 277.622f6 298.315 298.486 298.315 298.315 281.603 281.569f7 341.723 341.969 341.722 341.722 324.931 324.886f8 454.831 454.832 454.831 454.831 454.714 454.714f9 506.363 506.365 506.363 506.363 505.992 505.912f10 528.005 528.489 528.000 528.000 506.273 506.271

[C0][C1][LSTF][KI] [C0][C1][TII][TIII] [L0][L1][L2] [L0][L1][L2][L3] [L0][L1][L2][L3][L4] 3D FEM Abaqus

f1 30.414 30.393 30.417 30.375 30.297 30.37f2 101.595 101.533 101.944 101.519 101.524 101.50f3 136.980 136.863 137.492 136.812 136.815 136.79f4 236.527 236.526 236.541 236.526 236.500 236.51f5 278.051 277.739 279.582 277.614 277.515 277.53f6 281.823 281.573 283.751 281.559 281.414 281.47f7 325.320 324.936 327.791 324.873 324.725 324.80f8 454.720 454.709 454.778 454.713 454.678 454.70f9 506.274 505.950 506.281 505.886 505.712 505.71f10 506.642 506.272 511.289 506.272 506.271 506.32

Table 10First ten frequencies for a isotropic FFCF bi-polar co-ordinate plate with a linear variable thickness.

Geometric characteristics: a1 2 [�1 m, 1 m], a2 2 [p/4, p], a = 2 m

Mechanical properties: E = 70 GPa, m = 0.3, q = 2707 kg/m3, h0 = 0.3 m, hða1; a2Þ ¼ h0 1þ da2�a0

2a1

2�a02

� , d = 2

Mode (Hz) FSDT TSDT ED1 ED2 ED3 ED4

f1 25.856 25.853 27.627 25.958 25.970 25.967f2 41.960 42.142 42.870 42.070 42.285 42.279f3 62.141 61.901 62.662 62.218 62.283 62.275f4 78.943 79.242 80.677 79.098 79.413 79.412f5 121.222 121.582 122.415 121.326 121.733 121.732f6 146.605 147.083 161.425 147.072 147.782 147.755f7 185.738 186.119 190.097 185.788 186.238 186.250f8 233.742 234.300 242.678 233.982 234.930 234.900

Table 8 (continued on next page)

Geometric characteristics of the conical shell: Rb = 1 m, L = 3 m, a = p/6, a2 2 [0, 2p]Properties of the laminae 1 and 2: E1 = 137.9 GPa, E2 = E3 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = m13 = 0.3, m23 = 0.49, q = 1450 kg/m3, h1 = h2 = 0.15 m

[C0][N][TIII] [C0][N][TIII][Z] [C0][C1][TII][TIII][Z] [L0][L1][L2][L3] [L0][L1][L2][L3][Z] 3D FEM Abaqus

f1 62.441 61.674 61.476 61.550 61.481 61.35f2 62.441 61.674 61.476 61.550 61.481 61.35f3 96.088 95.817 95.721 95.762 95.724 95.60f4 96.088 95.817 95.721 95.762 95.724 95.60f5 101.766 99.104 98.448 98.640 98.447 98.15f6 101.766 99.104 98.448 98.640 98.447 98.15f7 172.714 167.533 166.295 166.691 166.291 165.68f8 172.714 167.533 166.295 166.691 166.291 165.68f9 181.859 181.407 181.308 181.381 181.300 181.15f10 220.370 220.267 220.233 220.237 220.217 219.91

108 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 3. First six mode shapes of a (30/45/70) CCCC elliptic paraboloid.

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 4. First six mode shapes of a (30/45/70) CCCC hyperbolic paraboloid.

Table 10 (continued)

Geometric characteristics: a1 2 [�1 m, 1 m], a2 2 [p/4, p], a = 2 m

Mechanical properties: E = 70 GPa, m = 0.3, q = 2707 kg/m3, h0 = 0.3 m, hða1; a2Þ ¼ h0 1þ da2�a0

2a1

2�a02

� , d = 2

f9 243.922 243.087 244.847 243.841 244.044 244.032f10 265.028 264.299 265.402 264.591 264.730 264.716

[C0][C1][PKPRS] [C0][C1][G3] [C0][C1][S0] [C0][N][TIII] [C0][C1][C2][KII1] [C0][C1][C2][TI]

f1 27.640 27.648 27.640 27.640 25.960 25.957f2 42.923 42.947 42.922 42.922 42.121 42.111f3 62.678 62.681 62.678 62.678 62.280 62.278f4 80.714 80.746 80.714 80.714 79.131 79.123f5 122.528 122.596 122.525 122.525 121.474 121.455f6 161.576 161.684 161.574 161.574 147.388 147.363f7 190.169 190.252 190.169 190.169 185.971 185.957f8 242.987 243.171 242.981 242.981 234.631 234.581f9 244.888 244.892 244.887 244.887 244.040 244.036f10 265.427 265.429 265.427 265.427 264.730 264.729

[C0][C1][LSTF][KI] [C0][C1][TII][TIII] [L0][L1][L2] [L0][L1][L2][L3] [L0][L1][L2][L3][L4] 3D FEM Abaqus

f1 25.913 25.954 26.067 25.970 25.967 25.970f2 41.994 42.095 42.434 42.285 42.279 42.274f3 62.374 62.500 62.294 62.283 62.275 62.276f4 78.976 79.112 79.613 79.413 79.412 79.398f5 121.352 121.459 122.424 121.733 121.732 121.700f6 147.308 147.371 148.473 147.782 147.755 147.780f7 185.973 186.002 187.263 186.238 186.250 186.210f8 234.583 234.594 236.806 234.930 234.900 234.880f9 243.760 244.637 244.075 244.044 244.032 244.030f10 264.251 265.182 264.737 264.730 264.716 264.730

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 109

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 5. First six mode shapes of an isotropic CCCC curved pipe panel (an elliptic curve slides on a catenary curve).

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 6. First six mode shapes of an isotropic CC curved shell pipe (an ellipse slides on an Bézier curve).

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 7. First six mode shapes of a (30/45) CCFF catenoidal panel.

110 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

material. The mechanical properties, the lamina thickness and theboundary conditions are reported in Tables 4 and 5.

The catenoidal panel depicted in Fig. 2e has the same parame-ters reported in [91], as it can be seen in Tables 6 and 7. Thenumerical results obtained in this paper about the catenoidal panel(Table 6) are compared to the ones presented in the work by Violaet al. [91]. In Table 7, the GDQ results for the same catenoidal panelare compared with the 3D FEM solutions considering variousboundary condition combinations. The orthotropic materials, the

laminae thickness, the boundary conditions and laminationschemes are reported in Tables 6 and 7. The catenoidal panel of Ta-ble 6 and 7 has the (30/45) stacking sequence withh1 = h2 = 0.25 m.

The conical shell of Fig. 2f has the same parameters reported in[91]. Since it is a revolution shell, it has a0

2 ¼ 0 and a12 ¼ 2p. The

meridian abscissa is defined as a1 in [0,L], where L is the lengthof the straight meridian. The presented parameters are graphicallyshown in [64,65,71,74,89,140]. All mechanical and geometrical

1st - 2nd modes 3rd - 4th modes 5th - 6th modes

7th - 8th modes 9th mode 10th mode

Fig. 8. First six mode shapes of a (�45/45) FC conical shell.

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 9. First six mode shapes of an isotropic CFFC parabolical co-ordinate plate.

1st mode 2nd mode 3rd mode

4th mode 5th mode 6th mode

Fig. 10. First six mode shapes of an isotropic FFCF bi-polar co-ordinate plate with a linear variable thickness.

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 111

112 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

parameters, as well as the boundary conditions and laminationscheme are also reported in Table 8. In particular, the conical shellof Table 8 presents a (�45/45) stacking sequence withh1 = h2 = 0.15 m.

As far as degenerate shells are concerned, the position vector ofa parabolical co-ordinate degenerate plate (see Fig. 2g) r(a1, a2) canbe described as follows:

rða1;a2Þ ¼12

a22 � a2

1

� �e1 � a1a2e2 ð60Þ

The curvilinear co-ordinate lines are orthogonal and are defined inthe following ranges a1 = [0.2 m, 2 m] and a2 = [0.2 m, 2 m]. Theisotropic material, the plate thickness, the boundary conditionsare reported in Table 9. Finally, the position vector of a bi-polarco-ordinate degenerate plate (see Fig. 2h) r(a1, a2) can be writtenin the form:

rða1;a2Þ ¼a sinh a1

cosh a1 � cos a2e1 þ

a sin a2

cosh a1 � cos a2e2 ð61Þ

where a is a geometric parameter which defines the semi-distancebetween the two poles of the bi-polar co-ordinates. The curvilinearco-ordinate lines are orthogonal and are defined in the followingranges a1 = [�1 m, 1 m] and a2 = [p/4, p]. It is worth noting thatfor the bi-polar co-ordinate plate a linear variation of the thicknessof the plate is considered along the second co-ordinate line a2. Thelinear thickness variation is defined as:

hða1; a2Þ ¼ h0 1þ da2 � a0

2

a12 � a0

2

� �ð62Þ

where the parameter d is equal to 2. The geometric parameter a, theisotropic mechanical properties, the thickness variation, as well asthe boundary conditions are reported in Table 10. The parabolicalco-ordinate plate of Table 9 and the bi-polar co-ordinate plate of Ta-ble 10 are made of isotropic material.

From Tables 2–10, it appears that the numerical results ob-tained from the GDQ methodology are very close to those obtainedby the 3D FEM elasticity solutions: an excellent agreement isshown. For the present GDQ results, a Chebyshev–Gauss–Lobattogrid distribution (56) with IN = IM = 31 is considered for all the shellstructures, except for the curved pipe of Fig. 2d (an ellipse slidingon a Bézier curve) in which a IN = 52, IM = 32 grid is used. The ref-erence 3D FEM meshes are taken as 40 � 40 � 12, where 12 repre-sents the number of elements along the thickness direction.

For the curved pipe of Fig. 2d (an ellipse sliding on a Béziercurve) and for the conical shell, the FEM meshes have more ele-ments 60 � 100 � 12. Brick elements with 20 nodes are used forall the eight structures under discussion.

From Tables 2–10 it appears that the FSDT theory producesquite accurate results in agreement with the 3D FEM referencesolution. In its formulation the FSDT theory still needs the shearcorrection factor that is taken equal to 5/6. In order to neglectthe shear correction factor itself and obtain more accurate results,it might be useful to use shear functions to the displacement fieldand higher-order theories.

Increasing the number of independent parameters, passing from aFSDT to a TSDT, the solution in terms of natural frequencies improves.The shear functions’ choice slightly affects the solution, as it can be seenin Tables 2–10. In other words, when the number of independentparameters increases, most of the investigated cases show similar solu-tions and converge to the reference 3D elasticity solution.

In Tables 2, 3, 6, 8, in addition to the FSDT and TSDT classical the-ories, higher-order theories with different order of expansions areshown, such as ED1, ED2, ED3, ED4. The solutions connected withthe aforementioned theories are reported below the first row of eachtable. Furthermore, in order to study the zig-zag effect the Muraka-

mi’s function (11), (12) is introduced in the previous theories involv-ing the following higher-order theories FSDTZ, TSDTZ, EDZ1, EDZ2,EDZ3 and EDZ4. The results obtained with this complicating effectare shown below the second row of each table. Finally, results ob-tained from 3D FEM elasticity and from other kinematic modelsinvolving different thickness functions chosen in Table 1, are re-ported below the third row. Otherwise, Tables 4, 5, 9 and 10 show re-sults obtained from various kinematic models without consideringthe Murakami’s function (11), (12), because this function is notapplicable for an isotropic single lamina structure.

In Tables 2–10 it is shown that most of the thickness functionsused in literature produce good results and a better agreement isobtained with HSDT theories, with the zig-zag effect in laminatedstructures and without the zig-zag effect for isotropic structures.Moreover, the use of full Taylor’s expansions for all the f powerlaws can produce good results with a less number of independentparameters, with respect to the HSDT theories that are not compu-tationally low cost. In fact, they combine the odd powers and theeven powers of the f function.

The first six mode shapes of the aforementioned structures aregraphically reported in Figs. 3–10. In particular, due to the symme-try of the problem for shells of revolution (see the conical shell ofFig. 8), some symmetrical mode shapes are present and summa-rized in one picture. The depiction of the mode shapes for all theeight structures under consideration is generated by the authors’MATLAB code [140]. The three-dimensional view of the modalforms have been reconstructed by means of the displacement field(8) from the solution of the eigenvalue problem (58).

5. Conclusions

This paper shows comparisons among various plates and shellstheories in order to emphasize the differences between the well-known First-order Shear Deformation Theory (FSDT) and severalHigher-order Shear Deformation Theories (HSDTs). All computa-tions are worked out through the CUF approach, which leads to ageneralized formulation for any kind of displacement field. Fur-thermore, the differential geometry is used for the geometricdescription of doubly-curved shells. The GDQ method is used toperform the numerical evaluation of the quantities involved inthe differential geometry of doubly-curved shells, as well as tocompute the derivatives of the fundamental equations of motion.The frequency parameters evaluated by the ESL higher-order for-mulation are in good agreement with the results obtained by the3D finite element method. The first six mode shapes of eight differ-ent shell structures are also graphically presented. Regarding thetheoretical development, a general extension of the CUF approachto completely doubly-curved shells is proposed in this paper. Theusage of different thickness functions is coupled within a generaltheoretical framework to study and mix together all the thicknessfunctions presented in literature. In fact, any displacement fieldcan be enriched to obtain various higher order theories. The appli-cation of different thickness functions is shown in this study. Inparticular, the enforcement effect of these functions to the high-er-order terms of various kinematical models on the solution interms of frequencies is pointed out. A designer can choose properfunctions for tailoring a specific theory to be composed of lessindependent parameters. So, the computational cost of the analysiscan be reduced without losing the accuracy.

Acknowledgments

This research was supported by the Italian Ministry for Univer-sity and Scientific, Technological Research MIUR (40% and 60%).The research topic is one of the subjects of the Centre of Study

Appendix A

The equilibrium operators LðssÞij ; i; j ¼ 1;2;3, s,s = 0, 1, 2, . . ., N, N + 1, of the fundamental nuclei (45) for a general doubly-curved shell

described in orthogonal curvilinear co-ordinates (a1, a2), are derived and presented in their explicit form for the first time:

LðssÞ11 ¼

AðssÞ11ð20Þ

A21

@2

@a21

þ �AðssÞ

11ð20Þ

A31

@A1

@a1þ

AðssÞ11ð20Þ

A21A2

@A2

@a1þ 1

A21

@AðssÞ11ð20Þ

@a1þ 1

A1A2

@AðssÞ16ð11Þ

@a2

!@

@a1þ

2AðssÞ16ð11Þ

A1A2

@2

@a1@a2

þAðssÞ

66ð02Þ

A22

@2

@a22

þ �AðssÞ

66ð02Þ

A32

@A2

@a2þ

AðssÞ66ð02Þ

A1A22

@A1

@a2þ 1

A22

@AðssÞ66ð02Þ

@a2þ 1

A1A2

@AðssÞ16ð11Þ

@a1

!@

@a2

þAðssÞ12ð11Þ

1

A21A2

@2A2

@a21

� 1

A31A2

@A1

@a1

@A2

@a1

!þ AðssÞ

16ð20Þ1

A31A2

@A1

@a1

@A1

@a2� 1

A21A2

@2A1

@a1@a2

!þ AðssÞ

66ð11Þ1

A1A32

@A1@a2

@A2@a2� 1

A1A22

@2A1@a2

2

!

þAðssÞ26ð02Þ

1

A1A22

@2A2

@a1@a2� 1

A1A32

@A2

@a1

@A2

@a2

!þ 1

A21A2

@AðssÞ12ð11Þ

@a1þ 1

A1A22

@AðssÞ26ð02Þ

@a2

!@A2

@a1� 1

A21A2

@AðssÞ16ð20Þ

@a1þ 1

A1A22

@AðssÞ66ð11Þ

@a2

!@A1

@a2

þ2AðssÞ

26ð11Þ

A21A2

2

@A1

@a2

@A2

@a1�

AðssÞ22ð02Þ

A21A2

2

@A2

@a1

� �2

�AðssÞ

66ð20Þ

A21A2

2

@A1

@a2

� �2

�AðssÞ

44ð20Þ

R21

þAðs~sÞ

44ð10Þ þ Að~ssÞ44ð10Þ

R1� Að~s~sÞ

44ð00Þ

LðssÞ12 ¼

AðssÞ16ð20Þ

A21

@2

@a21

þ �AðssÞ

16ð20Þ

A31

@A1

@a1þ

AðssÞ11ð20Þ þ AðssÞ

66ð20Þ

A21A2

@A1

@a2þ

AðssÞ16ð20Þ � AðssÞ

16ð11Þ � AðssÞ26ð11Þ

A21A2

@A2

@a1þ 1

A21

@AðssÞ16ð20Þ

@a1þ 1

A1A2

@AðssÞ66ð11Þ

@a2

!@

@a1

þAðssÞ

12ð11Þ þ AðssÞ66ð11Þ

A1A2

@2

@a1@a2þ

AðssÞ26ð02Þ

A22

@2

@a22

þ �AðssÞ

26ð02Þ

A32

@A2

@a2�

AðssÞ22ð02Þ þ AðssÞ

66ð02Þ

A1A22

@A2

@a1þ

AðssÞ26ð02Þ þ AðssÞ

16ð11Þ þ AðssÞ26ð11Þ

A1A22

@A1

@a2þ 1

A22

@AðssÞ26ð02Þ

@a2þ 1

A1A2

@AðssÞ12ð11Þ

@a1

!@

@a2

þAðssÞ11ð20Þ

1

A21A2

@2A1

@a1@a2� 1

A31A2

@A1

@a1

@A1

@a2

!þ AðssÞ

16ð11Þ1

A1A22

@2A1

@a22

� 1

A1A32

@A1

@a2

@A2

@a2þ 1

A31A2

@A1

@a1

@A2

@a1� 1

A21A2

@2A2

@a21

!

þAðssÞ66ð02Þ

1

A1A32

@A2

@a1

@A2

@a2� 1

A1A22

@2A2

@a1@a2

!þ 1

A21A2

@AðssÞ11ð20Þ

@a1þ 1

A1A22

@AðssÞ16ð11Þ

@a2

!@A1

@a2� 1

A21A2

@AðssÞ16ð11Þ

@a1þ 1

A1A22

@AðssÞ66ð02Þ

@a2

!@A2

@a1

�AðssÞ

12ð11Þ þ AðssÞ66ð11Þ

A21A2

2

@A1

@a2

@A2

@a1þ

AðssÞ26ð02Þ

A21A2

2

@A2

@a1

� �2

þAðssÞ

16ð20Þ

A21A2

2

@A1

@a2

� �2

�AðssÞ

45ð11Þ

R1R2þ

Aðs~sÞ45ð10Þ

R1þ

Að~ssÞ45ð01Þ

R2� Að~s~sÞ

45ð00Þ

LðssÞ13 ¼

AðssÞ11ð20Þ þ AðssÞ

44ð20Þ

A1R1þ

AðssÞ12ð11Þ

A1R2þ

Aðs~sÞ13ð10Þ � Að~ssÞ

44ð10Þ

A1

!@

@a1þ

AðssÞ16ð11Þ þ AðssÞ

45ð11Þ

A2R1þ

AðssÞ26ð02Þ

A2R2þ

Aðs~sÞ36ð01Þ � Að~ssÞ

45ð01Þ

A2

!@

@a2

þAðssÞ

11ð20Þ � AðssÞ12ð11Þ

A1A2R1þ

AðssÞ12ð11Þ � AðssÞ

22ð02Þ

A1A2R2þ

Aðs~sÞ13ð10Þ � Aðs~sÞ

23ð01Þ

A1A2

!@A2

@a1þ

AðssÞ16ð11Þ þ AðssÞ

16ð20Þ

A1A2R1þ

AðssÞ26ð02Þ þ AðssÞ

26ð11Þ

A1A2R2þ

Aðs~sÞ36ð01Þ þ Aðs~sÞ

36ð10Þ

A1A2

!@A1

@a2

þ 1A1R1

@AðssÞ11ð20Þ

@a1þ 1

A1R2

@AðssÞ12ð11Þ

@a1þ 1

A2R1

@AðssÞ16ð11Þ

@a2þ 1

A2R2

@AðssÞ26ð02Þ

@a2þ 1

A1

@Aðs~sÞ13ð10Þ

@a1þ 1

A2

@Aðs~sÞ36ð01Þ

@a2

�AðssÞ

11ð20Þ

A1R21

@R1

@a1�

AðssÞ12ð11Þ

A1R22

@R2

@a1�

AðssÞ16ð11Þ

A2R21

@R1

@a2�

AðssÞ26ð02Þ

A2R22

@R2

@a2

LðssÞ21 ¼

AðssÞ16ð20Þ

A21

@2

@a21

þ �AðssÞ

16ð20Þ

A31

@A1

@a1�

AðssÞ11ð20Þ þ AðssÞ

66ð20Þ

A21A2

@A1

@a2þ

AðssÞ16ð20Þ þ AðssÞ

16ð11Þ þ AðssÞ26ð11Þ

A21A2

@A2

@a1þ 1

A21

@AðssÞ16ð20Þ

@a1þ 1

A1A2

@AðssÞ12ð11Þ

@a2

!@

@a1

þAðssÞ

12ð11Þ þ AðssÞ66ð11Þ

A1A2

@2

@a1@a2þ

AðssÞ26ð02Þ

A22

@2

@a22

þ �AðssÞ

26ð02Þ

A32

@A2

@a2þ

AðssÞ22ð02Þ þ AðssÞ

66ð02Þ

A1A22

@A2

@a1þ

AðssÞ26ð02Þ � AðssÞ

16ð11Þ � AðssÞ26ð11Þ

A1A22

@A1

@a2þ 1

A22

@AðssÞ26ð02Þ

@a2þ 1

A1A2

@AðssÞ66ð11Þ

@a1

!@

@a2

�AðssÞ66ð20Þ

1

A21A2

@2A1

@a1@a2� 1

A31A2

@A1

@a1

@A1

@a2

!� AðssÞ

26ð11Þ1

A1A22

@2A1

@a22

� 1

A1A32

@A1

@a2

@A2

@a2þ 1

A31A2

@A1

@a1

@A2

@a1� 1

A21A2

@2A2

@a21

!

�AðssÞ22ð02Þ

1

A1A32

@A2

@a1

@A2

@a2� 1

A1A22

@2A2

@a1@a2

!� 1

A21A2

@AðssÞ66ð20Þ

@a1þ 1

A1A22

@AðssÞ26ð11Þ

@a2

!@A1

@a2þ 1

A21A2

@AðssÞ26ð11Þ

@a1þ 1

A1A22

@AðssÞ22ð02Þ

@a2

!@A2

@a1

�AðssÞ

12ð11Þ þ AðssÞ66ð11Þ

A21A2

2

@A1

@a2

@A2

@a1þ

AðssÞ26ð02Þ

A21A2

2

ð@A2

@a1Þ2 þ

AðssÞ16ð20Þ

A21A2

2

ð@A1

@a2Þ2 �

AðssÞ45ð11Þ

R1R2þ

Að~ssÞ45ð10Þ

R1þ

Aðs~sÞ45ð01Þ

R2� Að~s~sÞ

45ð00Þ

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 113

,

LðssÞ22 ¼

AðssÞ66ð20Þ

A21

@2

@a21

þ �AðssÞ

66ð20Þ

A31

@A1

@a1þ

AðssÞ66ð20Þ

A21A2

@A2

@a1þ 1

A21

@AðssÞ66ð20Þ

@a1þ 1

A1A2

@AðssÞ26ð11Þ

@a2

!@

@a1þ

2AðssÞ26ð11Þ

A1A2

@2

@a1@a2

þAðssÞ

22ð02Þ

A22

@2

@a22

þ �AðssÞ

22ð02Þ

A32

@A2

@a2þ

AðssÞ22ð02Þ

A1A22

@A1

@a2þ 1

A22

@AðssÞ22ð02Þ

@a2þ 1

A1A2

@AðssÞ26ð11Þ

@a1

!@

@a2

�AðssÞ66ð11Þ

�1

A21A2

@2A2

@a21

� 1

A31A2

@A1

@a1

@A2

@a1

�� AðssÞ

16ð20Þ

�1

A31A2

@A1

@a1

@A1

@a2� 1

A21A2

@2A1

@a1@a2

�� AðssÞ

12ð11Þ

�1

A1A32

@A1

@a2

@A2

@a2� 1

A1A22

@2A1

@a22

�

�AðssÞ26ð02Þ

1

A1A22

@2A2

@a1@a2� 1

A1A32

@A2

@a1

@A2

@a2

!� 1

A21A2

@AðssÞ66ð11Þ

@a1þ 1

A1A22

@AðssÞ26ð02Þ

@a2

!@A2

@a1þ 1

A21A2

@AðssÞ16ð20Þ

@a1þ 1

A1A22

@AðssÞ12ð11Þ

@a2

!@A1

@a2

þ2AðssÞ

16ð11Þ

A21A2

2

@A1

@a2

@A2

@a1�

AðssÞ66ð02Þ

A21A2

2

@A2

@a1

� �2

�AðssÞ

11ð20Þ

A21A2

2

@A1

@a2

� �2

�AðssÞ

55ð02Þ

R22

þAðs~sÞ

55ð01Þ þ Að~ssÞ55ð01Þ

R2� Að~s~sÞ

55ð00Þ

LðssÞ23 ¼

AðssÞ16ð20Þ

A1R1þ

AðssÞ26ð11Þ þ AðssÞ

45ð11Þ

A1R2þ

Aðs~sÞ36ð10Þ � Að~ssÞ

45ð10Þ

A1

!@

@a1þ

AðssÞ12ð11Þ

A2R1þ

AðssÞ22ð02Þ þ AðssÞ

55ð02Þ

A2R2þ

Aðs~sÞ23ð01Þ � Að~ssÞ

55ð01Þ

A2

!@

@a2

þAðssÞ

16ð20Þ þ AðssÞ16ð11Þ

A1A2R1þ

AðssÞ26ð11Þ þ AðssÞ

26ð02Þ

A1A2R2þ

Aðs~sÞ36ð10Þ þ Aðs~sÞ

36ð01Þ

A1A2

!@A2

@a1þ

AðssÞ12ð11Þ � AðssÞ

11ð02Þ

A1A2R1þ

AðssÞ22ð02Þ � AðssÞ

12ð11Þ

A1A2R2þ

Aðs~sÞ23ð01Þ � Aðs~sÞ

13ð10Þ

A1A2

!@A1

@a2

þ 1A1R1

@AðssÞ16ð20Þ

@a1þ 1

A1R2

@AðssÞ26ð11Þ

@a1þ 1

A2R1

@AðssÞ12ð11Þ

@a2þ 1

A2R2

@AðssÞ22ð02Þ

@a2þ 1

A1

@Aðs~sÞ36ð10Þ

@a1þ 1

A2

@Aðs~sÞ23ð01Þ

@a2

�AðssÞ

16ð20Þ

A1R21

@R1

@a1�

AðssÞ26ð11Þ

A1R22

@R2

@a1�

AðssÞ12ð11Þ

A2R21

@R1

@a2�

AðssÞ22ð02Þ

A2R22

@R2

@a2

LðssÞ31 ¼ �

AðssÞ11ð20Þ þ AðssÞ

44ð20Þ

A1R1þ

AðssÞ12ð11Þ

A1R2þ

Að~ssÞ13ð10Þ � Aðs~sÞ

44ð10Þ

A1

!@

@a1�

AðssÞ16ð11Þ þ AðssÞ

45ð11Þ

A2R1þ

AðssÞ26ð02Þ

A2R2þ

Að~ssÞ36ð01Þ � Aðs~sÞ

45ð01Þ

A2

!@

@a2

þ �AðssÞ

44ð20Þ þ AðssÞ12ð11Þ

A1A2R1�

AðssÞ22ð02Þ

A1A2R2þ

Aðs~sÞ44ð10Þ � Að~ssÞ

23ð01Þ

A1A2

!@A2

@a1þ

AðssÞ16ð20Þ � AðssÞ

45ð11Þ

A1A2R1þ

AðssÞ26ð11Þ

A1A2R2þ

Aðs~sÞ45ð01Þ þ Að~ssÞ

36ð10Þ

A1A2

!@A1

@a2

� 1A1R1

@AðssÞ44ð20Þ

@a1þ 1

A1

@Aðs~sÞ44ð10Þ

@a1� 1

A2R1

@AðssÞ45ð11Þ

@a2þ 1

A2

@Aðs~sÞ45ð01Þ

@a2þ

AðssÞ44ð20Þ

A1R21

@R1

@a1þ

AðssÞ45ð11Þ

A2R21

@R1

@a2

LðssÞ32 ¼ �

AðssÞ16ð20Þ

A1R1þ

AðssÞ26ð11Þ þ AðssÞ

45ð11Þ

A1R2þ

Að~ssÞ36ð10Þ � Aðs~sÞ

45ð10Þ

A1

!@

@a1�

AðssÞ12ð20Þ

A2R1þ

AðssÞ22ð02Þ þ AðssÞ

55ð02Þ

A2R2þ

Að~ssÞ23ð01Þ � Aðs~sÞ

55ð01Þ

A2

!@

@a2

þAðssÞ

16ð11Þ

A1A2R1þ

AðssÞ26ð02Þ � AðssÞ

45ð11Þ

A1A2R2þ

Aðs~sÞ45ð10Þ þ Að~ssÞ

36ð01Þ

A1A2

!@A2

@a1þ �

AðssÞ11ð20Þ

A1A2R1�

AðssÞ12ð11Þ þ AðssÞ

55ð02Þ

A1A2R2þ

Aðs~sÞ55ð01Þ � Að~ssÞ

13ð10Þ

A1A2

!@A1

@a2

� 1A1R2

@AðssÞ45ð11Þ

@a1þ 1

A1

@Aðs~sÞ45ð10Þ

@a1� 1

A2R2

@AðssÞ55ð02Þ

@a2þ 1

A2

@Aðs~sÞ55ð01Þ

@a2þ

AðssÞ45ð11Þ

A1R22

@R2

@a1þ

AðssÞ55ð02Þ

A2R22

@R2

@a2

LðssÞ33 ¼

AðssÞ44ð20Þ

A21

@2

@a21

þ �AðssÞ

44ð20Þ

A31

@A1

@a1þ

AðssÞ44ð20Þ

A21A2

@A2

@a1þ 1

A21

@AðssÞ44ð20Þ

@a1þ 1

A1A2

@AðssÞ45ð11Þ

@a2

!@

@a1þ

2AðssÞ45ð11Þ

A1A2

@2

@a1@a2

þAðssÞ

55ð02Þ

A22

@2

@a22

þ �AðssÞ

55ð02Þ

A32

@A2

@a2þ

AðssÞ55ð02Þ

A1A22

@A1

@a2þ 1

A22

@AðssÞ55ð02Þ

@a2þ 1

A1A2

@AðssÞ45ð11Þ

@a1

!@

@a2

�AðssÞ

11ð20Þ

R21

�AðssÞ

22ð02Þ

R22

�2AðssÞ

12ð11Þ

R1R2�

Aðs~sÞ13ð10Þ þ Að~ssÞ

13ð10Þ

R1�

Aðs~sÞ23ð01Þ þ Að~ssÞ

23ð01Þ

R2� Að~s~sÞ

33ð00Þ

Finally, the explicit form of the sth order resultants S(s)(a1, a2, t) (34) in terms of the generalized displacements u(s), useful for the impositionof the boundary conditions Eqs. (50), (51), (53) and (55), are reported in the following:

114 F. Tornabene et al. / Composite Structures 104 (2013) 94–117

NðsÞ1

NðsÞ2

NðsÞ12

NðsÞ21

TðsÞ1

TðsÞ2

PðsÞ1

PðsÞ2

SðsÞ3

266666666666666666664

377777777777777777775

¼XNþ1

s¼0

AðssÞ11ð20Þ

A1

@

@a1þ

AðssÞ12ð11Þ

A1A2

@A2

@a1�

AðssÞ16ð20Þ

A1A2

@A1

@a2þ

AðssÞ16ð11Þ

A2

@

@a2

AðssÞ11ð20Þ

A1A2

@A1

@a2þ

AðssÞ12ð11Þ

A2

@

@a2þ

AðssÞ16ð20Þ

A1

@

@a1�

AðssÞ16ð11Þ

A1A2

@A2

@a1

AðssÞ11ð20Þ

R1þ

AðssÞ12ð11Þ

R2þ Aðs~sÞ

13ð10Þ

AðssÞ12ð11Þ

A1

@

@a1þ

AðssÞ22ð02Þ

A1A2

@A2

@a1�

AðssÞ26ð11Þ

A1A2

@A1

@a2þ

AðssÞ26ð02Þ

A2

@

@a2

AðssÞ12ð11Þ

A1A2

@A1

@a2þ

AðssÞ22ð02Þ

A2

@

@a2þ

AðssÞ26ð11Þ

A1

@

@a1�

AðssÞ26ð02Þ

A1A2

@A2

@a1

AðssÞ12ð11Þ

R1þ

AðssÞ22ð02Þ

R2þ Aðs~sÞ

23ð01Þ

AðssÞ16ð20Þ

A1

@

@a1þ

AðssÞ26ð11Þ

A1A2

@A2

@a1�

AðssÞ66ð20Þ

A1A2

@A1

@a2þ

AðssÞ66ð11Þ

A2

@

@a2

AðssÞ16ð20Þ

A1A2

@A1

@a2þ

AðssÞ26ð11Þ

A2

@

@a2þ

AðssÞ66ð20Þ

A1

@

@a1�

AðssÞ66ð11Þ

A1A2

@A2

@a1

AðssÞ16ð20Þ

R1þ

AðssÞ26ð11Þ

R2þ Aðs~sÞ

36ð10Þ

AðssÞ16ð11Þ

A1

@

@a1þ

AðssÞ26ð02Þ

A1A2

@A2

@a1�

AðssÞ66ð11Þ

A1A2

@A1

@a2þ

AðssÞ66ð02Þ

A2

@

@a2

AðssÞ16ð11ÞA1A2

@A1@a2þ

AðssÞ26ð02ÞA2

@@a2þ

AðssÞ66ð11ÞA1

@@a1�

AðssÞ66ð02ÞA1A2

@A2@a1

AðssÞ16ð11ÞR1þ

AðssÞ26ð02ÞR2þ Aðs~sÞ

36ð01Þ

�AðssÞ

44ð20Þ

R1þ Aðs~sÞ

44ð10Þ �AðssÞ

45ð11Þ

R2þ Aðs~sÞ

45ð10ÞAðssÞ

44ð20Þ

A1

@

@a1þ

AðssÞ45ð11Þ

A2

@

@a2

�AðssÞ

45ð11Þ

R1þ Aðs~sÞ

45ð01Þ �AðssÞ

55ð02Þ

R2þ Aðs~sÞ

55ð01ÞAðssÞ

45ð11Þ

A1

@

@a1þ

AðssÞ55ð02Þ

A2

@

@a2

�Að~ssÞ

44ð10Þ

R1þ Að~s~sÞ

44ð00Þ �Að~ssÞ

45ð01Þ

R2þ Að~s~sÞaia2

45ð00ÞAð~ssÞ

44ð10Þ

A1

@

@a1þ

Að~ssÞ45ð01Þ

A2

@

@a2

�Að

~ssÞ45ð10ÞR1þ Að~s~sÞ

45ð00Þ �Að

~ssÞ55ð01ÞR2þ Að~s~sÞ

55ð00ÞAð~ssÞ

45ð10Þ

A1

@

@a1þ

Að~ssÞ55ð01Þ

A2

@

@a2

Að~ssÞ13ð10Þ

A1

@

@a1þ

Að~ssÞ23ð01Þ

A1A2

@A2

@a1�

Að~ssÞ36ð10Þ

A1A2

@A1

@a2þ

Að~ssÞ36ð01Þ

A2

@

@a2

Að~ssÞ13ð10Þ

A1A2

@A1

@a2þ

Að~ssÞ23ð01Þ

A2

@

@a2þ

Að~ssÞ36ð10Þ

A1

@

@a1�

Að~ssÞ36ð01Þ

A1A2

@A2

@a1

Að~ssÞ13ð10Þ

R1þ

Að~ssÞ23ð01Þ

R2þ Að~s~sÞ

33ð00Þ

266666666666666666666666666666666666666664

377777777777777777777777777777777777777775

uðsÞ1

uðsÞ2

uðsÞ3

26643775

The compact form of the sth order resultant vector S(s)(a1, a2, t) is obtained by combining Eqs. (34) and (29):

SðsÞ ¼XNþ1

s¼0

AðssÞDXuðsÞ

F. Tornabene et al. / Composite Structures 104 (2013) 94–117 115

and Research for the Identification of Materials and Structures(CIMEST)-‘‘M. Capurso’’ of the University of Bologna (Italy).

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