Analysis for buckling and vibrations of composite ... - ResearchGate

Composite Structures 51 (2001) 361±370 

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Analysis for buckling and vibrations of composite sti€ened shells and 

plates 

R. Rikards * , A. Chate, O. Ozolinsh 

Institute of Computer Analysis of Structures, Faculty of Civil Engineering, Riga Technical University, Kalku St. 1, LV-1658 Riga, Latvia 

Abstract 

This paper deals with development of triangular ®nite element for buckling and vibration analysis of laminated composite 

sti€ened shells. For the laminated shell, an equivalent layer shell theory is employed. The ®rst-order shear deformation theory 

including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear 

stresses a correction of transverse shear sti€ness is employed. Based on the equivalent layer theory with six degrees of freedom (three 

displacements and three rotations), a ®nite element that ensures C 0 continuity of the displacement and rotation ®elds across interelement 

boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of 

the element. Results of vibration and buckling analysis of sti€ened plates and shells are discussed. Ó 2001 Elsevier Science Ltd. All 

rights reserved. 

Keywords: Finite element; Sti€ened shells; Vibrations; Buckling 

1. Introduction 

Due to their high speci®c modulus compared to 

metallic material, laminated composite materials o€er 

more ecient structures for application in aerospace 

engineering. Aerospace structures such as wings or fuselages 

are mostly assemblies of shell structures. In 

general, composite shell ®nite elements are based on the 

same shell theories as for conventional materials. 

However, there are some di€erences, which will be 

outlined below. 

Many curved shell ®nite element formulations have 

been developed and reported in the literature. A selected 

number of textbooks are given in [1±4]. Rather complete 

bibliography can be found in reviews [5,6]. 

Since laminated composites have a low transverse 

sti€ness shell theories accounting transverse shear deformations 

should be used. These are the so-called re- 

®ned shell theories [7,8]. For analysis of laminated 

structures, three main approaches can be employed. 

1. Approach based on a single layer theory. 

2. Approach based on equivalent layer theory. 

3. Approach based on layerwise theory. 

* Corresponding author. Tel.: +371-708-9264; fax: +371-782-0094. 

E-mail address: rikards@latnet.lv (R. Rikards). 

The ®rst approach based on single layer theory is 

applied for the homogeneous in transverse directions 

structures consisting of one layer. In this case, the approximation 

of displacements are performed throughout 

the thickness of shell. The second approach (equivalent 

layer theories) is employed for multilayered shells. In 

this case, the approximations are also performed for the 

whole stack of layers, but in addition it is necessary to 

perform some corrections of the transverse sti€ness. 

However, ®nally the laminated structure is considered as 

one equivalent layer. In the third approach (layerwise 

theories), the approximations are performed for each 

layer of the laminated structure. In this case, an interlayer 

continuity condition as well as top and bottom 

surface boundary conditions for the displacements and 

stresses are used. 

Re®ned theories are based on higher order approximations 

of the displacements. Higher order approximations 

for beams, plates and shells were introduced 

starting with investigations of Timoshenko [9], Reissner 

[10], Hencky [11], Mindlin [12], Naghdi [13] and other 

researchers. A general ®rst-order shear deformation 

theory for small de¯ections and small rotations of the 

shell was outlined by Naghdi [14]. A general non-linear 

theory of shells for large de¯ections and large rotations 

including transverse shear deformations and extension 

of the normal line was developed by Galimov [15] and 

later by other authors. 

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. 

PII: S 0 2 6 3 - 8 2 2 3 ( 0 0 ) 0 0 151-3

362 R. Rikards et al. / Composite Structures 51 (2001) 361±370 

There is extensive literature for analysis of sti€ened 

plates and shells. For example, review and analysis of 

buckling problems of sti€ened plates are presented in 

[16]. The eciency of the ®nite element method in 

comparison with other methods for analysis of sti€ened 

structures was shown. 

In the present paper, the equivalent layer theory 

based on the ®rst-order shear deformation theory including 

extension of normal line (FOSDT-6) is employed 

in order to develop the ®nite element for analysis 

of laminated sti€ened shells. Previously, similar ®nite 

element was developed and applied for analysis of 

smooth shells [17,18]. 

2. Kinematics of shell 

Kinematic equations for the ®rst-order shear deformation 

theory (FOSDT) including extension of the 

normal line can be obtained from the 3D equations of 

the theory of elasticity by using a well-known ®rst-order 

approximation of a vector function with respect to the 

coordinate x 3 . Further a well-known geometric relations 

for the shell in normal coordinates (see, for example, 

[14]) are used. 

Let us assume that vector r characterizes the position 

of an arbitrary point of the shell in the initial reference 

state (see point B in Fig. 1) and vector R is the position 

of the same point (point B 0 ) in the deformed state. The 

position of a point at the midsurface of the shell (point 

A) in the initial state is characterized by vector r, and 

position of the same point in the deformed state (point 

A 0 ) is characterized by vector R. Normal curvilinear 

coordinates x i ˆ‰x a ; x 3 Š at the midsurface of the shell in 

the initial state are de®ned by the right-handed triad of 

the base vectors ‰a a ; a 3 Š. Here, a unit vector a 3 is normal 

to the midsurface of shell. Therefore (see Fig. 1) 

r…x i †ˆr…x a †‡x 3 a 3 : 

…1† 

Similarly, curvilinear coordinates X i ˆ‰X a ; X 3 Š in the 

deformed state is de®ned by the triad of vectors ‰A a ; A 3 Š. 

Fig. 1. Kinematics of the ®rst-order shear deformation theory. 

In the deformed state, the vector A 3 may be not perpendicular 

to the midsurface of shell. 

Vector function R can be expanded in a Taylor's series 

with respect to coordinate x 3 normal to the midsurface 

of shell 

R…x i †ˆR…x a †‡x 3 rR a 3 ‡ 

…2† 

Here rR ˆ A a a a ˆ G is a second-order tensor 

(see, for example, [19]), which characterizes the gradient 

of strains with respect to coordinate x 3 , r is a Hamiltonian 

operator, and is a dyadic product of the tensors. 

Further, the notations for displacement at the midsurface 

v and displacement of an arbitrary point of shell 

u are introduced. The following expressions can be 

written (see Fig. 1) 

R ˆ r ‡ u; R ˆ r ‡ v: …3† 

From expressions (1)±(3) the representation of the displacement 

u of an arbitrary point of the shell for the 

®rst-order approximation can be obtained 

u…x a †ˆv…x a †‡x 3 c…x a †: 

…4† 

Here c is vector of rotation at the midsurface of shell 

c…x a †ˆA 3 …x a †a 3 …x a †; 

…5† 

where 

A 3 ˆ G a 3 : 

…6† 

In expression (4), the vectors v and c each have three 

components 

v ˆ v a a a ‡ wa 3 ; c ˆ c a a a ‡ ca 3 : …7† 

The motion of the shell is completely described by the 

two vectors, the displacement vector v and the rotation 

vector c. Therefore, the displacement ®eld of the shell in 

the FOSDT including extension of the normal line is 

represented by six unknown functions ± three displacements 

(v a ; w) and three rotations (c a ; c). These quantities 

are only functions of the shell midsurface coordinates x a . 

From displacements and rotations, the two-dimensional 

strains can be obtained. 

A three-dimensional Green's strain tensor in the linear 

case (in®nitesimal strain theory) is given by (see, for 

example, [20]) 

2e ij ˆ u ;i g j ‡ u ;j g i : 

…8† 

Here, partial di€erentiation is denoted by a comma 

…† ;i 

o=ox i and g i is a triad of base vectors for the 

spatial coordinates x i at the surfaces (x 3 ˆ const:) parallel 

to the midsurface (reference surface) of shell. The 

metric tensor of the spatial system is given by g ij ˆ g i g j 

and the metric properties of the midsurface are characterized 

by a ab ˆ a a a b . Substituting the approximation 

(4) into Eq. (8) the strain displacement relations can be 

obtained in a routine way (see, for example, [14]). From 

these strain displacement relations, kinematic equations

in di€erent curvilinear coordinates can be obtained and 

written through the physical components in the matrix 

form 

ˆ Lw: 

…9† 

Here w is physical components of vectors of displacement 

v and rotation c 

w T ˆ‰u; v; w; c …1† ; c …2† ; cŠ: 

R. Rikards et al. / Composite Structures 51 (2001) 361±370 363 

…10† 

The shell theory outlined above is the so-called ®rstorder 

shear deformation theory with six degrees of 

freedom (FOSDT-6). Note, that in the present theory 

the normal line undergoes a small rotation relative to 

the reference surface and also a small, uniform extension. 

The third component of the vector c, i.e., the 

function c ˆ c…x a † characterizes the deformation in the 

direction of normal line. Often this deformation is neglected 

c ˆ 0. Thus, the ®rst-order shear deformation 

theory with ®ve degrees of freedom (FOSDT-5) can be 

obtained. In the case of ¯exural deformation of plates 

with symmetric layer stacking sequence the in-plane 

displacements u ˆ 0 and v ˆ 0. Therefore, for analysis 

of such plates the ®rst-order shear deformation theory 

with three degrees of freedom (FOSDT-3) can be employed. 

3. Strain energy and sti€ness of laminated shell 

The strain energy U of the shell represented as a 

three-dimensional body is given by the expression, 

where in curvilinear coordinates the contravariant stress 

tensor r ij is contracted with the covariant strain tensor 

e ij 

U ˆ 1 Z 

r ij e ij dV ; 

2 V 

where dV is a volume element of the shell 

dV ˆ 

dA ˆ 

p g dx 1 dx 2 dx 3 ˆ 

p a dx 1 dx 2 : 

r 

g 

dx 3 dA; 

a 

…11† 

…12† 

Here dA is the shell midsurface element and a determinants 

of the metric tensors are given by a ˆ det…a ab † and 

g ˆ det…g ij †. 

In the laminated structures, the sti€ness properties 

are function of the normal coordinate. In Fig. 2, a crosssection 

of laminated shell composed of K layers is presented. 

The thickness of the kth layer is h k , the inner 

coordinate of the ®rst layer is x …0† 

3 

, the coordinate at the 

interface of the ®rst and the second layer is denoted by 

x …1† 

3 

, etc. A linear elastic properties of the anisotropic 

layers are characterized by the tensor of elasticity A. 

This tensor is function of the coordinate x 3 , i.e., 

A ˆ A…x 3 †. 

Fig. 2. Cross-section of laminated shell composed of K layers. 

In many applications, it can be assumed that calculations 

of shell sti€ness properties can be performed 

neglecting the di€erences in spatial and shell midsurface 

metrics, i.e., a ab g ab . In this case, the Hooke's law for 

each layer can be written by 

r ij ˆ A ijkl e kl : 

…13† 

Here A ijkl is the component of the elasticity tensor in the 

midsurface metrics. Substituting Eq. (13) into (11) and 

taking into account the relations (12) the strain energy U 

can be expressed by 

Z Z h=2 

U ˆ A ijkl e ij e kl dx 3 dA: 

…14† 

A 

h=2 

After integration, throughout the thickness, the strain 

energy can be obtained in terms of shell quantities: stress 

resultants and couples and laminated shell sti€ness 

characteristics 

…Q abcd ; B abcd ; D abcd †ˆ 

Z h=2 

h=2 

A abcd …1; x 3 ; x 2 3 † dx 3; 

Z h=2 

Q a3b3 ˆ c …a3† A a3b3 dx 3 : 

h=2 

…15† 

Here c …a3† are shear correction coecients. These shear 

correction coecients must be used for laminated shells 

for more correct representation of the transverse shear 

sti€ness properties. Calculation of the shear correction 

coecients can be performed by using various methods. 

Some approaches have been discussed in [21±23]. Further 

in numerical examples the matrix notations for the 

shear correction coecients are used. 

k 2 4 ˆ c …23†; k 2 5 ˆ c …13†: 

Since the elastic properties are to be constant throughout 

the layer thickness the expressions (15) can be easily 

integrated


Q abcd ˆ XK 

B abcd ˆ 1 

2 

kˆ1 

X K 

kˆ1 

A abcd 

…k† 

‰x 3…k† x 3…k1† Š; 

A abcd 

…k† 

‰x 2 3…k† x2 3…k1† Š; 

D abcd ˆ 1 X K 

A abcd 

…k† 

‰x 3 3…k† 

3 

x3 3…k1† Š; 

kˆ1 

Q a3b3 ˆ c …a3† 

X K 

Here A ijkl 

…k† 

kth layer. 

kˆ1 

4. Finite elements 

A a3b3 

…k† 

‰x 3…k† x 3…k1† Š: 

…16† 

are components of the elasticity tensor of the 

In [17] for the shell analysis, it was proposed to use 

two ®nite elements based on FOSDT-5 (see Fig. 3). The 

®rst element with six nodes called SH30 is based on the 

second-order approximation of displacements. The second 

element with 10 nodes called SH50 is based on the 

third-order approximation of displacements. Since the 

element SH50 has internal node and is too complicated, 

further only the quadratic element was developed [18]. 

However, the cubic element has higher convergence rate 

[17]. 

For sti€ened shells, two di€erent shell surfaces should 

be assembled. In this case, in order to obtain a continuous 

displacement and rotation ®elds the FOSDT-6 

should be used. The displacements (10) are approximated 

by the second-order polynomial shape functions 

(see [1]) 

w T ˆ‰N 1 I; N 2 I; ...; N 6 IŠv e ˆ Nv e : 

…17† 

Here I is a 6 6 unity matrix for the FOSDT-6, a 5 5 

unity matrix for the FOSDT-5 and a 3 3 unity matrix 

for the FOSDT-3. For the FOSDT-6, the element displacement 

vector v e is given by 

v T e ˆ‰ve 1 ; ve 2 ; ...; ve 6 Š; 

v eT 

i 

ˆ‰u i ; v i ; w i ; c i …1† ; ci …2† ; c iŠ: 

…18† 

Here v e i 

…i ˆ 1; 2; ...; 6† are displacements for the ith 

node. For the FOSDT-5, the element displacement 

vector in Eq. (17) is given by 

v T e ˆ ve 1 ; ve 2 ; ...; h 

i 

ve 6 ; v 

eT 

i 

ˆ u i ; v i ; w i ; c i …1† ; ci …2† 

: …19† 

For the FOSDT-3, the element displacement vector in 

Eq. (17) is given by 

v T e ˆ ve 1 ; ve 2 ; ...; h i 

ve 6 ; v 

eT 

i 

ˆ w i ; c i …1† ; ci …2† 

: …20† 

Therefore, at all three triangular ®nite elements for 

analysis of shells and plates are considered. 

1. The ®nite element SH36 based on second-order approximation 

of displacements according the FOS- 

DT-6. 

2. The ®nite element SH30 based on second-order approximation 


DT-5. 

3. The ®nite element PL18 based on second-order approximation 


DT-3. 

Here, the plate element PL18 is considered only for 

the convergence studies. 

For discretization of the sti€ened structures, several 

idealizations can be used in which the assemblies can be 

treated as collections of the shell, plate and beam elements. 

The sti€eners depending on dimensions can be 

idealized as a shell or beam elements. For light sti€eners, 

the overall (global) buckling or vibration modes occur 

and in this case the beam elements can be used. If the 

sti€ener is ¯exurally and rotationally sti€ and skin of the 

shell is thin, then the local mode of the skin is dominant. 

In this case for sti€eners also the beam elements can be 

used. In the case of torsionally weak sti€ener, a local 

buckling occurs in the sti€ener. In this case, the sti€ener 

should be idealized by using the shell (plate) elements. 

Therefore, the idealization depends upon the geometric 

characteristics of the attached sti€ener. 

In order to reduce the degrees of freedom of assembled 

structure for some geometric dimensions of sti€ener 

a spatial beam elements can be used. Let us consider the 

spatial beam element compatible with the shell ®nite 

element SH36. So, for the beam a second-order approximation 

of displacements and rotations should be 

used. A beam element with three nodes is presented in 

Fig. 4. For the present element a Timoshenko's beam 

theory is used to derive the element matrices. In addition 

a torsional deformation of the beam is taken into 

Fig. 3. Isoparametric triangular ®nite elements with six (a) and 10 (b) 

nodes. Fig. 4. Spatial Timoshenko's beam ®nite element B18.


account. So, the beam displacement vector is the same as 

for shell Eq. (10) and the element displacement vector is 

given by 

v T e ˆ‰ve 1 ; ve 2 ; ve 3 Š; 

h 

veT i 

ˆ u i ; v i ; w i ; c i …1† ; ci …2† ; c i 

i 

: …21† 

At all for the present beam element there are 18 DOF 

and the element is designated as B18. Similar Timoshenko's 

beam ®nite element employing the third-order 

approximations was developed in [24]. 

The ®nite elements SH36 and B18 can be used to 

assemble the sti€ened shells of an arbitrary shape since 

for both isoparametric elements the mapping technique 

is applied. The locking phenomenon for these elements 

can be avoided by using a selective integration technique 

[25]. 

5. Numerical examples 

Further the vibration and buckling problems of 

di€erent structures are analyzed. In order to evaluate 

the natural frequencies a linear eigenvalue problem is 

solved 

Ku kMu ˆ 0: 

…22† 

Here K and M is a global sti€ness and mass matrices, u 

is global displacement vector and k ˆ x 2 , where x ˆ 

2pf is a circular frequency and f is a frequency. The 

subspace iteration method [26], which is widely used for 

large scale ®nite element calculations, is employed to 

solve Eq. (22). 

Di€erent methods are employed to estimate buckling 

loads. The simplest is linearized buckling analysis, where 

Fig. 5. Geometry of laminated plate. 

the critical load parameter k can be estimated by solving 

the equation 

Ku kGu ˆ 0: 

…23† 

Here G is a geometric sti€ness matrix. 

5.1. Convergence study of ®nite element PL18 

Let us study the convergence properties of the element 

PL18 for the vibration problem of square plate. 

The vibration problem is solved employing Eq. (22). For 

this a consistent mass matrix M should be derived, i.e., 

the same shape functions should be used to develop the 

sti€ness and mass matrices. 

The error estimates for the linear ®nite element 

problems are given in [1,27]. If within an element of size 

h a polynomial approximation of degree k 1 is employed 

and the higher derivative of the displacements in 

the functional to be minimized is m, the error of the 

strain energy is O…h 2…km† †. Therefore, for the quadratic 

elements (SH36, PL18 or B18) these numbers are as 

follows …k ˆ 3; m ˆ 1† and the error estimate for the 

strain energy is O…h 4 †. The same error estimate is for the 

eigenvalues of the Eq. (22). Note that element size is 

inversely proportional to the number of ®nite elements 

per side of the plate h N 1 . 

Let us consider a square cross-ply plate with layer 

stacking sequence [0/90/0] (see Fig. 5). The ®bers of the 

top and bottom layers is in direction of x-axis. This 

example was examined using the re®ned plate theory in 

[28] and by employing a 3D elasticity solution in [29]. 

The plate parameters are as follows: 

a ˆ b ˆ 5; h ˆ 1; E 1 ˆ 40; E 2 ˆ 1; 

G 12 ˆ G 13 ˆ 0:6; G 23 ˆ 0:5; m 12 ˆ 0:25; q ˆ 1: 

The unidirectionally reinforced layers are to be transverselly 

isotropic, where 1 is ®ber direction. The plate is 

simply supported and can be analyzed employing the 

following boundary conditions: 

v; w; c y j xˆ0;a ˆ 0; u; w; c x j yˆ0;b ˆ 0: 

Results for the ®rst natural frequency are presented in 

Table 1. The actual values of shear correction coecients 

of the present plate calculated by the approach 

outlined in [22] are k4 2 ˆ 0:827; k2 5 ˆ 0:521. In [28], 

the value k4 2 ˆ k2 5 ˆ 5=6 as for homogeneous plate was 

used. 

In Table 1 results are compared with a 3D elasticity 

solution, higher order shear deformation plate theory 

(HSDPT) and classical plate theory (CPT). It should be 

Table 1 

p 

Non-dimensional ®rst frequency x ˆ x 

 

qh 2 =E 2 of the cross-ply plate 

3D elasticity [29] HSDPT [28] FOSDT [28] Present CPT 

0.4300 0.4315 0.4342 0.4229 0.7319


Fig. 7. Geometry of the single ribbed plate. 

Fig. 6. Convergence study for the eigenvalue of vibrating plate. 

noted that result employing the present element PL18 

with k4 2 ˆ k2 5 ˆ 5=6 is x ˆ 0:4342, i.e., the same as 

reference solution by Reddy (FOSDT). Present results 

were obtained using the mesh 32 32 (N ˆ 32), i.e., the 

plate is divided in 2048 elements. This is a rather ®ne 

mesh and the result can be treated as exact. 

The numerical (FEM) and theoretical convergence 

graphs for the fourth eigenvalue k 4 are presented in 

Fig. 6. Results are presented in logarithmic scale and 

numerical error is calculated by expression 

e ˆ j^k k exact j 

k exact 

: 

Here ^k is approximate (numerical) solution and k exact is 

exact solution (for the mesh 32 32). In Fig. 6, a slope 

of the theoretical error estimate O…N 4 † for the quadratic 

element are also presented in logarithmic scale. It 

is seen that for the element PL18 a convergence rate of 

numerical solution is the same as theoretical error estimate. 

The element matrices are evaluated by using numerical 

integration. For plates, when the elements are ¯at 

triangles, it is easy to determine the order of numerical 

integration formula, by which the strain energy can be 

exactly integrated. In this case, the order of polynomial 

which should be integrated is known. More complicated 

is the case of using the isoparametric ®nite elements. In 

this case, after discretization of the strain energy the 

rational functions should be integrated. Thus, the required 

order of integration formula should be chosen 

empirically. 

5.2. Vibration of the isotropic sti€ened plate 

The vibrations of clamped isotropic plate with a 

single rib is examined (see Fig. 7). For the present single 

ribbed square plate, the experimental natural frequencies 

were measured in [30]. The plate parameters are as 

follows: 

a ˆ b ˆ 203 mm; h ˆ 1:37 mm; t s ˆ 6:35 mm; 

h s ˆ 12:7 mm; E ˆ 68:7 GPa; m ˆ 0:3; 

q ˆ 2820 kg=m 3 : 

Here a and b are the side lengths of the plate, h the plate 

thickness, t s the rib thickness, h s the rib depth, E the 

Young's modulus, m the Poisson's ratio and q is a density. 

The plate was assembled in two di€erent ways. The 

®rst assembly was created by using for the skin and rib 

the shell element SH36 (designated as shell model) and 

the second by using the shell element for the skin and 

beam element B18 for the rib (designated as beam 

model). Results for the 24 ®rst natural frequencies are 

presented in Table 2. Present results for both assemblies 

was obtained using the mesh 24 24. For the shell 

model, a lumped (diagonal) mass matrix was used. For 

the beam model, a consistent mass matrix was employed 

in Eq. (22). Experimental frequencies are taken from the 

paper [30]. For comparison numerical frequencies have 

also been calculated employing the code ANSYS. In this 

case, the mesh 24 24 was used and the assembly of the 

plate (shell) elements for the skin and rib was employed. 

From analysis of the results presented in Table 2, it is 

seen that the best agreement with the experiment is for 

numerical frequencies calculated by ANSYS. At all 18 

frequencies are closer to the experiment than other 

predictions. Present analysis employing the beam model 

gives the best prediction for ®ve frequencies including 

the ®rst one. Present analysis using the shell model gives 

the best prediction for the frequency f 10 . 

Selected vibration mode shapes are presented in 

Fig. 8. Mode 1 is overall vibration mode, modes 12 and 

24 are a local skin vibration modes and mode 20 is a 

coupled vibration mode with skin±rib interaction. In 

general, it can be concluded that for present geometric 

dimensions (rib of low aspect ratio h s =t s ) it is sucient to 

assemble the sti€ened plate by a compatible combination 

of beam and plate (shell) elements since such assembly 

can accurately predict the natural frequencies 

and mode shapes. Advantage of the beam model is also 

less DOF in comparison with assembly of shell elements 

only.


Table 2 

Experimental and numerical frequencies [Hz] of single ribbed clamped plate 

Mode Ref. [30] ANSYS Present 

Theory Experiment Shell model Beam model Shell model 

1 718.1 689 712.6 693.2 722.8 

2 751.4 725 742.9 751 754.6 

3 997.4 961 983.8 984.8 997.9 

4 1007.1 986 993.6 1007 1009 

5 1419.8 1376 1398.5 1417 1425 

6 1424.3 1413 1402.5 1427 1430 

7 1631.5 1512 1599.5 1651 1690 

8 1853.9 1770 1831 1829 1866 

9 2022.8 1995 1983.6 2019 2027 

10 2025 2069 1985.8 2024 2029 

11 2224.9 2158 2175.6 2191 2227 

12 2234.9 2200 2185 2231 2240 

13 2400.9 2347 2344.6 2393 2423 

14 2653.9 2597 2585.9 2645 2671 

15 2670.2 2614 2597.5 2674 2683 

16 2802.4 2784 2733.9 2789 2800 

17 2804.6 2784 2735.1 2793 2802 

18 3259 3174 3143.7 3254 3276 

19 3265.9 3174 3148.8 3271 3283 

20 3414.2 3332 3316.8 3583 3403 

21 3754 3660 3644.3 3724 3741 

22 3754.8 3730 3644.8 3726 3741 

23 3985.5 3780 3798.8 3909 3950 

24 4045.9 3913 3862.9 4019 4054 

obtained for the clamped sti€ened plate made of a carbon/epoxy 

composite material (AS1/3501-6). The properties 

of that material are listed in Table 3. The layer 

stacking sequence of the skin plate is ‰0= 45=90Š s 

. The 

play thickness is 0.13 mm and at all in the skin there are 

eight layers. The rib is made from the cross-ply laminate 

with 28 layers. The layer stacking sequence throughout 

the rib thickness is ‰0 7 =90 7 Š s 

. The plate parameters are as 

follows (see Fig. 7). 

a ˆ 250 mm; b ˆ 500 mm; t p ˆ 1:04 mm; 

h s ˆ 10:5 mm; t s ˆ 3:64 mm: 

Fig. 8. Vibration modes of sti€ened plate. 

5.3. Vibration of the laminated sti€ened plate 

The vibrations of laminated composite plate with a 

single rib is examined. The reference solution is taken 

from [31] in order to verify the shell element model for a 

plate with composite sti€ener. Numerical results are 

The composite sti€ened plate was calculated by the shell 

model, i.e., the rib also was assembled by using the shell 

elements SH36. The mesh was 24 24 elements. Present 

solution was obtained employing the diagonal mass 

matrix in Eq. (22). Natural frequencies were calculated 

also by the code ANSYS employing the shell model. In 

this case, the mesh was 16 16. Results are compared 

with the reference solution [31], which was obtained 

employing the beam model. Numerical results for the 

®rst ®ve frequencies are presented in Table 4. For 

comparison, the same plate was calculated without 

sti€ener. 

Table 3 

Material properties of the composite plate with cross-ply sti€ener 

Material E 1 (GPa) E 2 (GPa) G 12 ˆ G 13 (GPa) G 23 (GPa) m 12 q (kg/m 3 ) 

Carbon/epoxy 128 11 4.48 1.53 0.25 1500


Table 4 

Natural frequencies [Hz] of laminated composite plate 

Mode Ref. [31] ANSYS Present 

With rib No rib With rib No rib With rib No rib 

1 213.8 85.1 213.1 85 215 85.6 

2 229.4 134 220.8 133.9 235.5 135.7 

3 270.2 207.4 270.3 206.3 274.5 208.1 

4 313.8 216.1 308.8 215.5 315.4 219.9 

5 354 252.5 353.5 251.4 361.4 256.3 

Exploring the mode shapes of the sti€ened plate it 

should be noted that, for example, for the ®rst frequency 

there is a coupled skin±rib antisymmetric vibration 

mode. This is due to low torsional sti€ness of the rib 

since the shear modulus G 23 of the composite is very low 

(see Table 3). 

5.4. Buckling of sti€ened isotropic plate under axial 

compression 

In order to verify the present element SH36 for solution 

of buckling problems a clamped sti€ened plate 

under axial compression is considered (see Fig. 9). The 

reference solution of the present numerical example was 

obtained in [32]. 

The critical stress of the plate is given by 

 

r cr ˆ k 

 

; D ˆ 

p 2 D 

Eh 3 

a 2 h 

12…1 m 2 † : …24† 

Here critical parameter k is function of non-dimensional 

quantities. 

b ˆ b 

a ; 

j ˆ EI 

aD ; 

e ˆ A 

ah ; 

…25† 

where A is area, I the second moment of area of the 

sti€ener and h is a skin-plate thickness. The boundary 

conditions of the clamped plate are as follows: 

u; v; w; c x ; c y ; cj xˆ0;a; yˆ0 ˆ 0; u; w; c x ; c y ; cj yˆb ˆ 0: 

…26† 

The shell model was used in order to calculate the critical 

stress by linearized buckling approach. In analysis, a 

18 18 mesh was employed assuming the values of nondimensional 

parameters: b ˆ 1; e ˆ 0:2; j ˆ 20; h s ˆ 

10:483h; h s =t s ˆ 2:75; a=h ˆ 200. The present analysis 

gives the value of critical parameter k ˆ 24:85 in Eq. 

(24). The reference solution is k ˆ 25:46 [32]. Note that 

reference solution was obtained employing the beam 

model for the sti€ener. The result obtained by the ®nite 

element code ANSYS is k ˆ 23:44. In this case, the mesh 

20 20 was employed. The buckling mode is antisymmetric 

in x-direction, however the sti€ener's aspect ratio 

h s =t s ˆ 2:75 is rather low and it has enough torsional 

rigidity. 

5.5. Buckling of composite cylindrical panel under axial 

compression 

Buckling load of composite cylindrical panel under 

axial compression (see Fig. 10) is compared with the 

reference solution [33], where the ®rst-order shear deformation 

theory is also used for buckling analysis. The 

laminate of 12 plies is considered. The ply thickness is 

0.018 in and the laminate ply stacking sequence is 

‰45=0=90= 45Š s 

. The nominal ply mechanical properties 

for the composite material used are given by (see 

[33]) 

E 1 ˆ 13:75 Msi; E 2 ˆ 1:03 Msi; G 12 ˆ G 13 ˆ G 23 

ˆ 0:420 Msi; m 12 ˆ 0:25: 

Geometric parameters of the shell are as follows: 

R ˆ 40 in; L ˆ 22 in; h ˆ 0:216 in; a ˆ 180°: 

Fig. 9. Clamped sti€ened plate under axial compression. 

Fig. 10. Cylindrical composite panel under axial compression.


Table 5 

Buckling loads [lbs/in] of composite cylindrical panel 

Ref. [33] ANSYS Present 

STAGS Segment approach 

3328 3278 3285 3313 

Acknowledgements 

The investigations concerning the development of ®- 

nite elements for analysis of laminated shells are sponsored 

through Contract No. G4RD-CT-1999-00103. 

The author are pleased to acknowledge the ®nancial 

support by the Commission of European Union. 

Thanks are also due to Latvian Council of Science for 

®nancial support through Grant No. 6430.2. 

References 

Fig. 11. Buckling mode of cylindrical panel under axial compression. 

The boundary conditions for a simply supported cylindrical 

panel are given by 

u ˆ wj aˆ0°;180° ˆ 0; v; wj xˆ0 ˆ 0; wj xˆL ˆ 0: 

Table 5 shows the results for cylindrical panel. The 

present analysis (see Fig. 11) is performed by a mesh 

14 30 elements in the axial and circumferential direction, 

respectively, while the STAGS and ANSYS ®nite 

element model consists of a mesh 20 40 elements. The 

results obtained by the present element SH36 are found 

to be in good agreement with the reference solutions. 

6. Conclusions 

The ®nite element for vibration and buckling analysis 

of laminated composite sti€ened shells has been developed. 

The element is based on ®rst-order shear deformation 

theory with six degrees of freedom of normal 

line. For analysis of sti€ened structures two models are 

considered: the beam and the shell model. It was shown 

that for torsionally rigid sti€eners the beam model can 

be employed. Natural frequencies and buckling loads 

for sti€ened structures obtained from the present analysis 

are found to be in good agreement with the reference 

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