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ORIGINAL ARTICLE

Received: March 01, 2019. In Revised Form: August 19, 2019. Accepted: September 02, 2019. Available online: September 06, 2019

http://dx.doi.org/10.1590/1679-78255535

Latin American Journal of Solids and Structures. ISSN 1679-7825. Copyright © 2019. This is an Open Access article distributed under the terms of the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 1/26

A simple spline finite strip for buckling analysis of composite

cylindrical panel with cutout

Davood Poorveisa*

Amin Khajehdezfulya

Shapour Moradib

Elahe Shirshekana

a Department of Civil Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran. E-mail: dpoorveis@scu.ac.ir,

amin_dezfuly@scu.ac.ir, elahe_scu89@yahoo.com

b Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Email: moradis@scu.ac.ir

* Corresponding author

https://doi.org/10.1590/1679-78255535

Abstract

In this study, a new sub-parametric strip element is developed to simulate the axially loaded composite

cylindrical panel with arbitrary cutout. For this purpose, a code called SSFSM is developed in FORTRAN to

analyze the buckling of panels. The first order shear deformation theory is used to form the strain-

displacement relations. Spline and Lagrangian functions are used to derive element shape functions in the

longitudinal and transverse directions, respectively. The computational cost of the SSFSM is decreased

dramatically, as mapping functions of the strip element are very simple. The results obtained from the

SSFSM are compared with those of the literature and the results obtained by ABAQUS to show the validity

of the proposed approach. A parametric study is performed to show the capability of the SSFSM in

calculating the panel buckling load. Results indicate that increasing the panel thickness and panel central

angles cause an increase in panel buckling load. The cutout shape is an important factor influencing the

panel buckling load. For instance, when the angle between the direction of big chord of the elliptical cutout

and compressive load direction are 0 and 90 degrees, the panel buckling load reaches its minimum and

maximum magnitude, respectively.

Keywords

Composite, perforated cylindrical panel, spline finite strip, sub-parametric

1 INTRODUCTION

In the recent decades, plates and cylindrical composite panels with and without holes are used widely in the

aerospace, civil and mechanical industries (Shuohui et al. 2016a-b, Nejati et al. 2017, Asadi and Qatu 2013, Khalfi et al.

2014, Khetir et al. 2017, Asadi et al. 2012). In this matter, buckling analysis of these structures under different load

cases has been investigated by several researchers using different methods (Tiantang et al. 2017, Tiantang et al. 2016,

Patel et al. 2006, Vuong and Dung 2017, Soltani et al. 2019, Asadi and Qatu 2013). Magnucki and Mackiewicz (2006)

investigated the effects of panel boundary conditions on the elastic buckling of composite cylindrical panel. In their

research, an analytical approach was used to calculate the buckling load of a panel without any cutout. Effects of

rectangular and circular cutouts on the buckling load of laminated plate were investigated analytically by Masia et al

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 2/26

(2005) and Ashrafi et al (2013), respectively. Although analytical approaches are simple to calculate the buckling load,

but they have some limitations such as cutout shape, panel boundary conditions and the structure shape. To overcome

these limitations, the Finite Element Method (FEM) was used extensively to calculate the buckling load of composite

cylindrical panels. Anil et al (2007) evaluated the effect of rectangular cutout on the critical load of composite plate

using FEM. The influence of circular cutout on the buckling load of laminated plate was investigated by Komur and

Sonmez (Komur and Sonmez 2008). The effects of cutouts with various shapes (such as circular and elliptical shapes) on

the buckling load of composite plates were also evaluated by FEM in the several researches (Arbelo et al. 2015,

Shuohui et al. 2015, Thinh et al. 2016, Rajanna et al. 2016, Soni et al. 2013, Pascan et al. 2012).

Use of the Finite Strip Method (FSM) is simpler and more economical compared to the FEM, in order to analyze

long thin-walled structures such as cylindrical panels (Mirzaei et al. 2015, Fazilati 2017, Balogh and Logo 2015, Rondal

1998). Moreover, while in the FEM the structure is discretized in transverse and longitudinal directions, it is only

discretized in transverse direction in FSM (Rondal 1998). FSM can be divided into two categories. In the first one, called

semi-analytical FSM, the longitudinal shape function of strip element is considered as harmonic function. Also, the

transverse shape function of strip element is considered as polynomial function. Buckling load of laminated plates

without any cutout subjected to different load cases was evaluated using semi-analytical FSM (Yao et al. 2015,

Ovesy et al. 2015, Dawe et al. 1995, Ovesy and Fazilati 2012b). Semi- analytical FSM is very useful and efficient for

buckling analysis of laminated plate when the boundary conditions of both ends of strip elements are simply supported

(Wang et al. 1998, Dawe and Wang 1994, Cheung and Tham 1997). The longitudinal harmonic shape functions of strip

element are complicated in case of more complex plate/panel boundary conditions (Wang et al. 1998, Dawe and Wang

1994, Cheung and Tham 1997). Spline FSM (SFSM) is used in the analysis of buckling load of plate/panel with different

boundary conditions (Second category of the FSM) (Wang et al. 1998, Dawe and Wang 1994). The spline function is

used as the shape function of strip element in the longitudinal direction. Au and Cheung (1996) investigated the free

vibration and stability of shell without any cutout utilizing iso-parametric SFSM. In their work, First Order Shear

Deformation plate Theory (FSDT) was used to model the shell. Fazilati and Ovesy (2013) investigated the effects of

structure boundary conditions and material on the buckling load of laminated stiffened cylindrical panel with

rectangular cutout. They developed a 3D model of cylindrical panel by using SFSM based on Higher order Shear

Deformation Theory (HSDT). Finally, Eccher et al. (2008) evaluated the elastic buckling of perforated thin-walled

structures such as cylindrical panels by iso-parametric SFSM. In their research, an iso-parametric spline finite strip

element was developed based on the FSDT. The strip element shape functions of longitudinal and transverse directions

have been considered as polynomial and B3 spline respectively (Eccher et al. 2008). Based on the literature, the order

of mapping function of sub-parametric element is lower than that of iso-parametric element (Akin 2005) and therefore,

equilibrium equations obtained in case of sub-parametric element have lower order than iso-parametric ones and

could cause lower computational cost (Akin 2005).

A review of the literature reveals that there are two major limitations in buckling analysis of perforated laminated

cylindrical panel. While the computational cost of sub-parametric element is lower than that of the iso-parametric

element, there is no sign of considering sub-parametric SFSM in buckling analysis of perforated laminated cylindrical

panel. Moreover, although the effects of single cutout on the buckling load of cylindrical panel are investigated in the

literature, the effects of cutout with different shapes on the buckling load of composite laminated cylindrical panel are

not compared with each other. Therefore, this paper is a response to these needs. In this regard, at first the shape and

mapping functions of sub-parametric spline finite strip element are developed. Next, the displacement functions are

derived. Then, the components of Jacobian matrix are calculated and based on the FSDT, the strain-displacement and

stress-strain relationships are extended. According to the shape functions, the load vector of strip element is evaluated.

Then, components of elastic stiffness and stability stiffness matrices of sub-parametric element are derived. Finally,

according to the developed element, the elastic buckling analysis of laminated perforated cylindrical panel is carried

out. To validate the results obtained in this paper, comparison are made between the results obtained in this study and

those published in the literature and those obtained from commercial finite element software, namely ABAQUS

(ABAQUS 2006).

2 Methodology

The cylindrical panel with arbitrary cutout is shown in Figure 1. As it is shown in Figure 1, R, B, and L represent

the panel radius, width, central angle and length, respectively. The x, y and z axes are considered in the longitudinal,

circumferential and radial directions of the panel, respectively. A local coordinate system is established for the panel

using x, y and z axes. Based on the first order shear deformation theory and local coordinate system considered in

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 3/26

Figure 1, five degrees of freedom (u, v, w, ,) are considered for each node of the panel (Moradi et al. 2011). These

degrees of freedom are shown in Figure 1. u and v are membrane (in-plane) displacement in x and y directions, w is the

panel deflection in radial direction (out of plane), and and are the strip edge rotations about the x and y

directions, respectively. According to the load vector and linear stiffness matrix of the cylindrical panel, the pre-

buckling displacement vector of the panel is established. Based on the panel pre-buckling displacement vector,

membrane force vector of the panel is calculated. Also, using the panel membrane force vector, the geometrical

stiffness matrix is calculated. Finally, the buckling load of the cylindrical panel through the solution of linearized

eigenvalue problem is calculated. A finite strip model of perforated cylindrical panel is shown in Figure 2. The process

described above was carried out in a FORTAN code called Sub-parametric Spline Finite Strip Method (SSFSM)

developed in this paper. All stages performed in this code are shown in Figure 3. The details of the process are

described in the following sections. In this study, a sub-parametric strip element is developed to calculate the

cylindrical panel axial buckling load. In this regard, some features of strip element such as shape functions, mapping

functions, Jacobian matrix and stress-strain relationships are established. Based on these features, the stiffness matrix

and load vector of the strip element are developed. The stiffness matrices and load vectors of strip elements are used

in assemblage of global stiffness matrix and load vector of the cylindrical panel, respectively. In the tenth step of the

algorithm, the calculation of [] for large number of degrees of freedom is very time-consuming and in the current

analysis, [] is decomposed by the modified Cholsky method as []=[][][], in which [U] is an upper triangular

matrix and [D] is a diagonal matrix (Rakotonirina 2013). After that, the pre-buckling deformations [d0] can be extracted.

Also, in the step 13th, the linear eigenvalue problem is solved by the inverse iteration method (Van Ness 1969) which

taking into account [] and [] as symmetric banded matrices. Several eigenvalues and their eigenvectors can be

extracted using the aforementioned method.

Figure 1. View of cylindrical panel.

Figure 2. Finite strip model of cylindrical panel with a circular cutout.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 4/26

Figure 3. SSFSM algorithm.

2.1 Shape functions of strip element

The B spline (Moradi et al. 2011) and four-noded Lagrangian functions (Moradi et al. 2011) are used to estimate

the strip element displacement functions in the longitudinal and transverse directions, respectively. As it is indicated in

Figure 4, the coordinate is considered in the transverse direction (y direction) of the strip element. The shape

functions of strip element in the transverse direction are presented in equation 1. In this equation, L() is the

Lagrangian function for jth nodal line in coordinate. is varied between -1 and +1. As it is presented in equation 1,

the shape functions of strip element in the transverse direction are third degree polynomial function.

()=9

16 (1

9+1

9)

()=27

16 (1

3+1

3)

()=27

16 (+1

31

3)

()=

+

(1)

The C0 continuity is implied in the strip transverse direction (Eccher et al. 2008). Due to the use of this shape

function, it is possible to impose any arbitrary boundary condition on the longitudinal strip edges. Four nodal lines are

considered in the longitudinal direction of the strip element and m knots (node) are considered in each nodal line. In

other word, each nodal line is divided into m sections. The B spline function is used to derive the strip element

displacement function in the longitudinal direction as shown in equation 2.

()=

()

()()()

()()()

()

(2)

where, () is the spline function for the ith knot in coordinate, while is varied between -1 to m+1. In Figure 5a, a

B spline function is drawn. For each knot (node) considered on each nodal line, a spline function is assigned. In Figure

5b, all of the spline functions considered on a nodal line are drawn. Two virtual nodes are considered outside the panel

(nodes -1 and m+1 in Figure 5b) for each nodal line to apply any arbitrary boundary condition on the transverse edges

of the strip element (Moradi et al. 2011). The displacement functions of the strip element are derived based on the

shape functions considered in the transverse and longitudinal directions of the element (see equation 3). The

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 5/26

displacement function presented in the equation 3, is derived based on the FSDT (Khayat et al. 2016a). In equation 3,

Z(,) is the displacement of strip element at degree of freedom Z in coordinates and , j is the number of nodal line

and i is the number of knot (node) considered on the jth nodal line, L() is the Lagrangian function for the jth nodal

line, () is the spline function for ith knot, and Z is the unknown multiplier for the jth nodal line, ith knot and degree

of freedom Z. Z could be considered as u, v, w, or .

(,)=()×()×

(,)=()×()×

(,)=()×()×

(,)=()×()×

(,)= ()×()×

(3)

Figure 4. Lagrangian function in transverse direction.

The displacement function of the strip element is defined based on the 4 × (m + 3) knots. equation 3 can be

expressed as equation 4 in which {Δ} is strip element displacement vector, Δ is the unknown multiplier for ith knot of

jth nodal line, is the ith knot spline function and L is the jth nodal line Lagrangian function. {Δ} is comprised of {u},

{v}, {w}, {} and (see equation 5).

Figure 5. Spline function in longitudinal direction (a) spline function (b) Spline functions for a nodal line.

{Δ}= Δ

(4)

{Δ}={ } (5)

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 6/26

2.2 Mapping functions

The sub-parametric strip element in the local Cartesian coordinate X and Y is shown in Figure 6a. This element is

mapped into the parent element as it is shown in Figure 6b. The mapping functions are used to map the element from

(X,Y) coordinate to the (,) coordinate system. The strip element considered in this paper has two advantages. The

longitudinal edges of the strip element are linear and parallel to each other and therefore, the width of the strip (b) is

constant in the longitudinal direction of the strip element. On the other hand, the transverse edges of strip element

(start and end edges of the element) are defined as curved function and consequently any arbitrary cutout could be

considered in the cylindrical panel. According to these advantages, the mapping functions of the strip element are

estimated using eight nodes on the start and end edges of the strip element. These nodes are shown in Figure 6. In

fact, the mapping functions of the element are derived based on the eight nodes while the displacement functions of

the element are estimated based on the 4 × (m + 3) nodes. In other word, this element is a sub-parametric element

(Eccher et al. 2008). x is estimated with the x-coordinate of eight nodes as equation 6. In this equation, N is the

mapping function of the ith node and x is the x-coordinate of the ith node.

=

(6)

Figure 6. Strip element (a) actual element (b) parent element.

The mapping functions N to N are written as equation 7. The magnitude of N is 1 at the ith node and 0 at the

other nodes. In equation 7, m is the number of knots considered on the nodal line.

=9

16× () × (1) × (1

9)

=9

16×()×(+ 1)×1

9

=9

16×× (+ 1) × (1

9)

=9

16×× (1) × (1

9)

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 7/26

=27

16×()×(1)×1

3

=27

16×()×(1)×+1

3

=27

16××(1)×+1

3

=

××(1)×

(7)

N is simplified as the multiplication of two functions, N and N where N and N are functions of and ,

respectively (see equation 8). The constant multiplier related to the m is considered as a part of N.

=× (8)

shape functions are cubic Lagrangian fuctions and are used to approximate boundary curve of each strip. So,

each closed boundary of cutout is estimated by the several cubic polynomials. As it is illustrated in equation 6, x is

proportional to ξ, therefore

and

can be written as the following relations.

x

=×

=××

(9)

Same as x, the y-coordinate is estimated by eight nodes laid down on the first and last strip edges. Therefore, y

can be calculated by equation 10. In this equation, i is the node number, N is mapping function of the ith node and y is

the y-coordinate of the ith node.

=

(10)

Based on the advantages of the strip elements mentioned in this section of the paper, y is simplified and

calculated based on the node numbers 1, 2, 3 and 4 (four corner nodes of the element). Therefore, equation 10 can be

rewritten as the following equation.

=

=×+×+×+×++

×++

()×+

+

()×++

× (11)

where, y-coordinate of nodes 5, 6, 7 and 8 is calculated based on the y-coordinate of node number 1, 2, 3 and 4.

Inserting equations 7 and 8 into equation 11, it can be simplified to get the following relation.

=×=×

×()×(1)×

×()×(+ 1)+×

××(+ 1)

×

××(1) (12)

where, y-coordinate of strip element is estimated using four corner nodes of the element. In equation 12, N is the

modified mapping function of ith node. Moreover, as the width of the strip element is constant in the longitudinal

direction (y= y , y= y), therefore equation 12 can be more simplified and rewritten as equation 13.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 8/26

=×=

×+×=

+

(13)

In equation 13, N is the simplified mapping function of strip element in y direction. As it is illustrated in this

equation, y-coordinate of the element is estimated just with nodes number 1 and 2, and consequently y is only a

function of η. In other word, the simplified mapping function (N) in the y direction is only a function of η. Thus,

and

of the strip element can be calculated by the following equations (see equation 14).

y

= 0

=

(14)

Therefore, the computational cost of calculation of Jacobian matrix is decreased dramatically due to the simplicity

in evaluation of

,

,

and

.

2.3 Jacobian matrix

The determinant of the Jacobian matrix (|J|) is used to transform the integration from actual element to the

parent element. It is calculated by equation 15 (Khayat et al. 2016a).

||=

×

×

(15)

As it is shown in equation 15, the determinant of Jacobian matrix is related to the

,

,

and

terms which

calculated in the previous section. Inserting equations 9 and 14 into equation 15, the determinant of the Jacobian

matrix can be reached as equation 16.

||=()=

×

(16)

where, b is the width of the strip element, i is node number, x is the x-coordinate of the ith node and N is the

mapping function of ith node (see equation 9). As it is shown in equation 16, the determinant of Jacobian matrix is only

a function of . Therefore, ξ and η coordinates can be separated in the integration process to calculate the load vector,

linear stiffness matrix and geometrical stiffness matrix of strip element and result in decreasing the computational cost

dramatically (Khayat et al. 2016a). Moreover, the partial derivatives with respect to x and y coordinates are presented

in equation 17.

=

||×

(17)

For iso-parametric strip element used in the literature, approximation of geometry (x, y) follows the same rule as

for displacement shape functions (Eccher et al. 2008). So, the geometry of iso-parametric strip element is

approximated by following equations (Eccher et al. 2008).

=()()

= ()()

(18)

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 9/26

in which coefficients and are calculated based on the geometry of each strip element. A full Jacobian matrix is

derived using equation 18. Therefore, the Jacobian matrix determinant and derivatives of Jacobian matrix with respect

to x and y are become dependent to and . It leads to increase the computational cost of iso-parametric strip

element (available element developed in the literature) compared to that of the sub-parametric element (developed in

this study).

2.4 Strain- displacement relationships

In this paper, the strain- displacement relationship is assumed to be Sanders-type (Moradi et al. 2011,

Khayat et al. 2016a, Khayat et al. 2016b). As it is illustrated in equation 19, the cylindrical panel strain vector {} is

comprised of linear part ({}) and nonlinear part ({}).

{}={}+{} (19)

{}=

=

+

+

+

+

+

(20)

The components of linear part of strain vector ({}) are presented in equation 20 where, in this equation,

,

,

, k

, k

, k

,

and

are the linear in-plane strain in x direction, in-plane linear strain in y direction, linear

shear strain in xy plane, linear curvature strain about the x axis, linear curvature strain about the y axis, linear curvature

strain in xy plane, linear shear strain in zx plane and linear shear strain in zy plane respectively. The components of

nonlinear part of strain vector ({}) are presented in equation 21 (Wang et al. 1998), where

,

and

are the

nonlinear in-plane strain in x direction, nonlinear in-plane strain in y direction and nonlinear shear strain in xy plane,

respectively.

{}=

0

0

0

0

0

=

(

)

(

)

(

)

0

0

0

0

0

(21)

The relationship between generalized linear stress vector () and linear strain vector ({}) is presented in the

following equation.

{}=[]{} (22)

where the components of generalized linear stress vector (i.e. ) is shown by the equation 23.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 10/26

{}= (23)

In this equation, N, N, N, M, M, M, Q and Q are the membrane stress in x direction, membrane

stress in y direction, shear membrane stress in xy plane, moment stress about y axis, moment stress about x axis,

torsional stress in xy plane, transverse shear stress in xz plane and transverse shear stress in yz plane, respectively

(Moradi et al. 2011).

=

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

(24)

In equation 22, [M] is the material constitutive matrix which is expanded by equation 24. where, A, B and D

are the extensional, bending, and bending-extensional coupling rigidities respectively and can be calculated from the

following relations (Moradi et al. 2011).

,,=(1, ,) (,= 1,2,6)

(25)

Note that, A, A and A are the shear stiffness constants (Moradi et al. 2011). In equation 25, N is the number

of layers, and t and t are the depth of lower and upper surfaces of the kth layer, respectively. Moreover, Q

are

the transformed elastic constants which are obtained from equation 26 (Eccher et al. 2008, Moradi et al. 2011,

Khayat et al. 2016a).

=

2

+

4

4

2

4

2()

2

2()

()

(26)

where, c and s stand for the cos and sin , in which θ is the lamination angle for each layer. Q are rigidities of an

orthotropic layer in its material directions (1 and 2) and are calculated by the following equation.

Q=E

(1 vv)

Q=E

(1 vv)

Q=vE

(1 vv)

Q= (27)

In equation 27, E, E, , v and v are material properties which are well-known in the literature

(Eccher et al. 2008, Moradi et al. 2011, Khayat et al. 2016a).

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 11/26

2.5 Strip element linear stiffness matrix

The linear stiffness matrix components are calculated based on the internal virtual work (W

) done by

components of the stress vector ({}) on the strip element volume. The internal virtual work (W

) is written as

equation 28.

=

+

+

+

+

+

+

+

|| (28)

where m is the number of ith element knots on a nodal line. The integrations on and are separated from each

other because of the simplicity of Jacobian matrix determinant (see equation 16). So, the one-dimensional Gauss

quadrature integration method is applied separately in each direction to compute the equation 28 (Eccher et al. 2008).

In the iso-parametric element presented in the literature (Eccher et al. 2008), the determinant of Jacobian is

complicated and the integrations on and are dependent to each other. This prevents of decomposition of surface

integrals encountered in the calculation of stiffness matrix. It proves that the computational cost of sub-parametric

strip element developed in this study is lower than that of the iso-parametric strip element due to the simplicity of its

Jacobian matrix. According to the internal virtual work, the linear stiffness matrix of the strip element ([K]) is obtained

from equation 29.

={δ}[]{Δ} (29)

in which, {Δ} is the strip element displacement vector (see equations 4 and 5) and {δ}is transpose of the virtual

displacement vector of the panel.

2.6 Strip element force vector

There are two types of strip element based on the panel loading condition and location of the cutout on the panel.

In Figure 7, a circular cutout with free edges is considered in the center of the panel. The figure shows two types of

strip elements, extended strip element (i.e. ith element) and truncated strip element (i.e. jth element). The length of the

extended strip element is equal to the panel length. The end edges of extended strip element are under compressive

distributed load with magnitude of q. The truncated strip element is between the panel transverse edge and cutout

edge; therefore, its length is less than the length of the panel. Also as can be seen from Figure 7, for the truncated strip

elements only one transverse edge is under compression load of q in u direction. The external virtual work (W

)

done by compressive load applied on transverse edges of the cylindrical panel is written to obtain the load vector of

the panel by the following relation (Khayat et al. 2016a):

=()×()()×()

(30)

In equation 30, N is the total number of strip elements, b is the width of ith strip element, l is the length of ith

strip element, q is the distributed compressive load on the element edges in direction u and u is the virtual

displacement. For the truncated strip elements, one of the q() or q() is set to zero. Also due to the costraint on

u in =, equation 30 can be summerized as equation 31.

=()×()

(31)

According to Figure 1, y-coordinate is along circumferential direction of the strip and can be related to the natural

coordinate as:

dy =

(32)

Virtual displacement u() can be rewritten as equation 33 based on equation 3.

()=

× (33)

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 12/26

It should be noted that, three spline functions (, and ) are contributed to the external virtual work

(

) (Eccher et al. 2008). Substituting equations 32 and 33 into equation 31 leads to following equation.

=

(++)××

(34)

Finally, equation 30 is simplified as equation 35 to obtain the load vector of cylindrical panel.

={}{} (35)

where, {F} and {} are the load vector and transpose of the virtual displacement of the cylindrical panel,

respectively.

Figure 7. Cylindrical panel loading.

2.7 Strip geometrical stiffness matrix

An initial deflection is induced in the cylindrical panel due to the distributed compressive load applied on the

edges of the panel. This pre-buckling displacement vector is calculated by the following equation.

{}={}[] (36)

where, {} is the pre-buckling displacement vector, {F} is the panel load vector and [K] is the linear stiffness matrix

of the cylindrical panel. Based on the initial displacement vector of the panel, the pre-buckling generalized linear stress

vector {} is obtained by equation 22. In the linearized buckling analysis it is assumed that the load-deflection relation

is linear up to the onset of buckling perturbation. The stresses induced in this stage will remain constant and are

independent of buckling deformations. In the geometrical nonlinear analysis which is another way to obtain buckling

load of structures, the stress vector {} depend on the linear and nonlinear strains. This makes the problem to be

nonlinear in terms of displacements. This pre-buckling bending deformations have significant effects on the calculation

of the critical axial load in shells. The components of the initial generalized linear stress vector {} are shown in

equation 37.

{}=

(37)

From the internal virtual work done by the initial linear in-plane stresses (W

) (Khayat et al. 2016a), the

geometrical stiffness matrix of the panel is obtained (see equation 38).

= (

)+

+

(38)

In equation 38, N

is the initial membrane stress in x direction, N

is the initial membrane stress in y direction,

N

is the initial shear membrane stress in xy plane,

is the nonlinear in-plane strain in x direction,

is the

nonlinear in-plane strain in y direction and

is the nonlinear shear strain in xy plane. equation 38 is rewritten and

simplified as equation 39.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 13/26

={δ}[]{Δ} (39)

where, [K] is the geometrical stiffness matrix, {Δ} is the displacement vector and {} is the virtual displacement

vector of the strip element.

2.8 Buckling analysis

The linear stiffness matrix ([K]) and geometrical stiffness matrix ([K]) of the cylindrical panel are obtained by

assembling linear and geometrical stiffness matrices of all strip elements. To calculate the buckling load of the panel,

the eigenvalue problem presented in equation 40 is solved by the inverse iteration method (Khayat et al. 2016b). In this

equation, is the eigenvalue and is its associated eigenvector.

(){}= 0 (40)

3 Verification

Based on the formulation presented in the previous sections, a computer program namely SSFSM, was written in

FORTRAN to compute the buckling load of a cylindrical panel having cutouts with different shapes. In order to check

the validity of the presented formulation, several case studies are carried out and the results obtained from SSFSM are

compared with the results available in the literature and the results obtained from the ABAQUS commercial finite

element software (ABAQUS 2006). The details are as follows.

3.1 Buckling load of cylindrical panel without cutout

Au and Cheung (1996) calculated the buckling load of a square cylindrical panel without any cutout using spline

finite strip method. In their research, the length of the panel, panel thickness, modulus of elasticity and poison ratio

were considered as 600 mm, 5 mm, 70 kN/mm and 0.3, respectively. Moreover, all edges of the panel were simply

supported. A view of the cylindrical panel considered in the Au and Cheung study is shown in Figure 8. In their research,

a uniform compressive load was applied at the transverse edges of the panel. The cylindrical panel was simulated in

ABAQUS using 3600 S8R5 elements and roughly 11000 degrees of freedom (ABAQUS 2006). S8R5 is an eight nodded

iso-parametric shell element. In this element, the stiffness matrix is obtained using reduced integraton. A cylindrical

coordinate system is defined in the ABAQUS to define/apply the boundary conditions on the panel edges. A unique

uniform load is applied on the curved edge of the panel using the option named as "shell edges load" in the ABAUQS.

"Buckle analysis" is performed to calculate the panel buckling load in the ABAQUS. Their panel was modeled in SSFSM

with 6 strip elements, 20 sections and roughly 1900 degrees of freedom. A view of the cylindrical panel modeled in the

SSFSM and ABAQUS is shown in Figure 8. The cylindrical panel buckling load for different central angle

(0, 0.1, 0.2 and 0.3 radian) is calculated by SSFSM and ABAQUS. In Figure 9, the buckling load obtained from the SSFSM

is compared with those of Au and Cheung (1996) and ABAQUS. As it is illustrated in the figure, as the central angle of

the panel is increased, the buckling load is increased. The figure also shows that there is a good agreement between

results obtained from the SSFSM and those of the Au and Cheung (1996) and ABAQUS.

Moreover, in order to verify the convergence ability of the SSFSM in calculating the panel buckling load, the

buckling load of the panel for different number of strip elements and knots are calculated and presented in Table 1. As

it is indicated in the table, 6 strip elements and 15 knots are sufficient for convergence of the panel buckling load.

Figure 8. SSFSM and ABAQUS models of cylindrical panel (a) ABAQUS (b) SSFSM.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 14/26

Figure 9. Cylindrical panel buckling load obtained from Au and Cheung, SSFSM and ABAQUS

Table 1 Convergence of SSFSM buckling loads

Number of strips

2

3

4

5

6

7

8

Number of knots

5

183.2950

183.0030

182.7109

182.7028

182.6947

182.7677

182.8407

10

183.2098

182.9117

182.6136

182.5649

182.5162

182.5933

182.6704

15

183.1247 182.8204 182.5162 182.5081 182.5000 182.5000 182.5000

20 183.0395 182.7779 182.5162 182.5081 182.5000 182.5000 182.5000

25 182.9543 182.7353 182.5162 182.5081 182.5000 182.5000 182.5000

In order to evaluate the computational cost of SSFSM, a comparsion is performed between execution time of

SSFSM (sub-parametric element) and the same code with iso-parametric element. In this regrad, the panel described in

this section (Figure 8) is discretized by six sub-parametric strip elements and the buckling load of the panel is

calculated. Different numbers of konts are considered to simulate the panel. The execution time for both codes is

measured for different degrees of freedom. The solution time of code is dependent on the number of degrees of

freedom. For instance, when degrees of freedom is 2040, the solution time of SSFSM is about 148 seconds for a

2.8 GHz CPU. Figure 10 shows the computational cost of SSFSM against the iso-parametric code. In this figure, the

precentage of reduction in execution time for sub-parametric element is drawn against the number of degrees of

freedom. As shown in Figure 10, the execution time of SSFSM is significantly lower than that of the iso-parametric one.

Two factors are responsible for drastical reduction of SSFSM computational cost. The simplicity of Jacobian

determinant in the SSFSM causes decomposition of surface integral encountered in the linear elastic and geometric

stiffness matrices. The number of Gauss points requried in the SSFSM to accuratly integrate the stiffness matrices is

lower than those of the iso-parametric. As the degrees of freedom is increased from 675 to 2040, the precentage of

reduction in the execution time is reduced from 62 to 57.5 percent. This reduction is related to the calculation of

eigenvalue problem and as the degrees of freedom increases, the time to solve the eigenvalue problem increases.

Figure 10. Computational cost of sub-parametric strip element developed in this study.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 15/26

3.2 Buckling load of cylindrical panel with cutout

A composite cylindrical panel with layup configurations of (/) and (/) is simulated in ABAQUS and

SSFSM. The angle layer is considered as 0, 15, 30, 45, 60, 75 and 90 degrees. The material properties of panel are

presented in Table 2. The panel length, width and radius are considered 180 mm, 180 mm and 1800 mm, respectively.

The ratio of thickness to length of the panel is 1/45. The panel comprises of eight layers and all edges are simply

supported. The panel is under uniform compressive load on the transverse edges. A cutout with a square shape and

area of 1296 mm was considered in the center of the panel.

Figure 11. View of cylindrical panel with square cutout (a) ABAQUS (b) SSFSM.

Figure 12. Panel buckling load obtained from ABAQUS and SSFSM for layup (/).

The cylindrical panel was modeled in the ABAQUS with 700 of S8R5 elements type. Moreover, the panel was

simulated in the SSFSM with 10 strip elements and 20 knots. ABAQUS and SSFSM models are shown in Figure 11. The

buckling loads of the panel calculated from the SSFSM and ABAQUS are shown in Figures 12 and 13. As seen from

Figures 12 and 13, the maximum and minimum buckling load of the panel is about 490 and 221 N/mm for layup angle

45 and 90 degree, respectively. The maximum difference between results obtained from the SSFSM and those of the

ABAQUS is about 4 percent, which shows that there is a good agreement between the result obtained from the SSFSM

and the ABAQUS.

Figure 13. Panel buckling load obtained from ABAQUS and SSFSM for layup (/).

4 Evaluation of SSFSM capability in calculation of panel axial buckling load

A parametric study was carried out to clarify the capability of the SSFSM in calculating the buckling load of the

composite cylindrical panel with cutout. The effects of panel boundary condition, cutout shape, cutout area, panel

thickness, central angle of the panel and layup angle configuration on buckling load of panel were investigated. The

details are as follows.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 16/26

4.1 Effect of cutout shape and panel thickness on panel buckling load

In this section, the effects of cutout shape and panel thickness are investigated on the buckling load of the panel.

To this end, the effect of several cutout shapes such as square, diamond, circular and elliptical on the buckling load of

the panel are investigated. All cutouts are considered in the center of the panel and their cutout area are considered to

be 1296 mm (the ratio of cutout area to panel area is 4%). The ratio of big chord to small chord of the elliptical cutout

is kept at 0.5, 2 3

, 1, 1.5 and 2. The panel is simply supported on its edges and the in-plane displacements are

considered to be free on all of its edges. The edges of the cutout are considered to be free of any mechanical

constraint. The boundary of cutout coincides with boundaries of small strips and is approximated by eight cubic

polynomials, and subsequently, 24 nodes are contributed in approximating the boundary of circular/elliptical cutout.

The panel width, length and central angle are 180 mm, 180 mm and 0.3 radians, respectively. The properties of the

panel material are presented in Table 2.

Table 2 Mechanical properties of panel material

181

kN

mm

10.3

kN

mm

7.17

kN

mm

7.17

kN

mm

6.21

kN

mm

0.28

The parametric study is performed for two cross-ply layups configuration of (0/90) and (0/90). Three types of

panel thickness to length ratio of 1/25, 1/50 and 1/100 are used in the parametric study. The cylindrical panel is

modeled in the SSFSM using 12 four-nodded strip elements and 20 knots (roughly 3700 degrees of freedom). Figure 14

shows the configuration of the panel with different cutout shapes as modeled in SSFSM. The buckling loads of the

panels computed from SSFSM are tabulated in Table 3.

The results presented in Table 3 indicate that the buckling load of the panel increases as its thickness is increased.

In case of the panel without any cutout, when its thickness is getting larger by 2 and 4 times, the buckling load of the

panel is increased by 3.7 and 26 times, respectively. It means that the buckling load of the panel is changed nonlinearly

in terms of its thickness. In all cases, buckling load of the panel with cutout is less than that of the panel without any

cutout. Although all cutouts areas are the same, the cutout shape has an important effect on the buckling load.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 17/26

Figure 14 Cylindrical panel simulated by SSFSM (a) Without cutout (b) Square cutout (c) Diamond cutout (d) Elliptical cutout with ratio of 0.5

(e) Elliptical cutout with ratio of 2/3 (f) Circular cutout (g) Elliptical cutout with ratio of 1.5 (h) Elliptical cutout with ratio of 2.

When the elliptical cutout rotates about the normal axis to the center of the panel, the buckling load varies,

dramatically. In all cases, when big chord of the elliptical cutout is normal to the compressive load direction (elliptical

cutout with ratio 2), the cutout has negligible effect on the buckling load of the panel compared with other cutout

shapes. For these kinds of cutouts, the panel buckling load decreases by 17%, 11% and 3% for thick, medium and thin

panels respectively. It indicates that when the panel thickness is decreased, the effect of cutout on changes of the

panel buckling load is decreased. In most cases, when the big chord of the elliptical cutout is parallel to compressive

load direction (elliptical cutout with ratio 0.5), the maximum reduction is seen in the panel buckling load. The buckling

load of the panel with diamond cutout is always larger than that of the panel with square cutout. Moreover, the

buckling load of the panel with square cutout is always less than that of the panel with circular cutout. As it is

illustrated in Table 3, the layup configuration has an interesting influence on the buckling load of the panel. In the most

cases, the buckling load of the panel with layup configuration of [0/90] is more than that of [0/90] configuration.

But as an interesting point, when the elliptical cutouts with ratio 1.5 and 2 are considered in the center of the panel,

the buckling load for [0/90] layup configuration is more than that of the [0/90].

4.2 Effects of cutout size, layup, panel central angle and boundary conditions on the buckling load

In this section, the effects of cutout area, layup configuration and panel central angle are investigated on the

buckling load of the panel. A circular cutout is considered at the center of the panel (see Figure 14(f)). The panel length,

width and the ratio of panel thickness to its length are considered as 180 mm, 180 mm and 1/50, respectively. The

panel is composed of eight layers and its material properties are presented in Table 2. The cutout area is considered as

5%, 10%, 15%, 20% and 25% of panel area. Furthermore, the central angle of the panel is considered as

0.1, 0.3 and 0.5 radian. The layup arrangements of [0], [90], [0/90], [0/90], [45], [45/45], [30/30] and

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 18/26

[0/45/90/45] are used in the study. At the first step, the boundary conditions of the panel and cutout are the same

as those used in previous section. The buckling loads of the simply supported panel calculated from the SSFSM are

presented in Table 4.

Table 3 Panel buckling load for different cutout shapes.

Panel thickness

Thick (1/25) Cutout shape

Buckling load (N/mm)

(/)

(/)

Thick (1/25)

Without cutout

2201.668

2244.298

Square

1792.830

1822.665

Diamond

1712.729

1730.092

Elliptical (a/b=0.5)

1640.561

1733.972

Elliptical (a/b=2/3)

1697.209

1762.393

Circle

1772.675

1798.089

Elliptical (a/b=1.5)

1847.753

1829.905

Elliptical (a/b=2)

1912.064

1859.781

Medium (1/50)

Without cutout

363.776

368.892

Square

313.892

317.070

Diamond

304.338

305.599

Elliptical (a/b=0.5)

286.228

297.150

Elliptical (a/b=2/3)

297.237

304.590

Circle

310.934

313.378

Elliptical (a/b=1.5)

323.922

321.119

Elliptical (a/b=2)

334.446

327.491

Thin (1/100)

Without cutout

84.535

86.0832

Square

79.550

80.271

Diamond

78.240

78.357

Elliptical (a/b=0.5)

72.711

74.263

Elliptical (a/b=2/3)

75.844

76.921

Circle

79.298

79.802

Elliptical (a/b=1.5)

82.169

82.062

Elliptical (a/b=2)

84.198

83.566

The results presented in Table 4 indicate that the panel buckling load is decreased as the cutout area is increased

or central angle of the panel is decreased. For instance, in case of [0] layup and central angle of 0.1 radian, when the

area contribution is changed from 5% to 25% (cutout area is increased 5 times), the buckling load of the panel is

decreased from 191 to 110.7 N/m (the panel buckling load is decreased 42%). On the other hand, for the same layup

and cutout area ratio of 25%, when the central angle is changed from 0.5 to 0.1 radian (central angle is decreased

5 times), the buckling load of the panel is decreased from 151 to 110.7 N/m (the panel buckling load is decreased 27%).

This shows that the variation of cutout area has more effect on the buckling load than the variation of the panel central

angle. The results obtained in this section indicate that the panel layup configuration is one of the most important

factors on its buckling load. The panel buckling load is reached at the maximum magnitude of 707.5 N/m for cutout

area ratio of 5%, central panel angle of 0.5 radian and [0/45/90/45] layup configuration. While, it is reached at

639.6 and 513.3 N/m in [45/45] and [30/30] layup configuration, respectively, the minimum buckling load is

reached at 110.6 N/m for [0] layup configuration. As an interesting point, in case of [90] layup and cutout area ratio

of 25%, when the central angle is varied from 0.1 to 0.3, the panel buckling load is changed from 127.5 to 127.6 N/m

which shows only a slight variation.

Figure 15 shows the buckling modes of laminated composite cylindrical panel having 15% central circular cutout

with different layups and central angles. Panel stacking sequence is considered as [90]8, [45]8, [45/-45]4 and [0/45/90/-45]s. The

central angle of the panel is 0.1 and 0.5 radians. For uni-directional layups [90]8 and [45]8 wavelength in the fiber

direction is more than that of fiber transverse direction. This is pronounced with increasing in the panel central angle.

The local shape of modes in these cases justifies reduction of buckling loads. The buckling mode shapes of multi-

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 19/26

directional laminates such as [45/-45]4 and [0/45/90/-45]s behave globally and therefore endure more axial buckling

loads.

To show the capability of SSFSM to consider the effects of different cylindrical boundary conditions on the

buckling load, the buckling load of the panel was calculated using SSFSM when the loading edges of the panel were

fixed. The panel properties and geometry were considered as mentioned at the beginning of this section. All layups

presented in Table 4 were considered in the analysis. The central angle of the panel was considered as

0.1 and 0.5 radian. The area ratio of circular cutout was considered as 5%, 10%, 15%, 20% and 25%. The buckling load

of the simply supported cylindrical panel (simple panel) is compared with that of fix supported cylindrical panel (fixed

panel) in Figures 16 to 23.

Table 4 Buckling load of cylindrical panel with circular cutout.

Layup Panel central

angle (Radian)

buckling load (N/mm)

Cutout area to panel area ratio

%

%

15%

20%

25%

[]

0.1

191.565

157.045

136.396

121.755

110.602

0.3

224.302

185.779

159.564

140.111

124.960

0.5

291.893 240.619

202.801

173.774

150.949

[]

0.1

152.005

139.993

133.221

129.046

127.449

0.3 200.351 164.023

144.246

133.015 127.633

0.5

269.667

193.089

157.496

138.601

128.390

[/]

0.1

211.942

184.828

169.256

158.427

149.793

0.3

307.965

275.370

250.331

228.820

209.289

0.5

493.606

441.490

390.148

343.196

301.987

[/]

0.1

207.756

181.447

166.355

155.683

146.882

0.3

305.682

273.351

248.185

226.400

206.613

0.5

491.235

438.266

384.229

335.748

294.745

[]

0.1

184.912

171.255

164.816

157.359

148.421

0.3

237.307

207.731

188.777

172.980

159.064

0.5

334.709

263.502

219.285

190.414

170.618

[/]

0.1

386.753

370.087

368.392

366.912

361.698

0.3

474.016

439.697

419.616

405.847

392.931

0.5

639.636

563.598

503.161

465.010

439.207

[/]

0.1

339.907

317.775

307.730

298.518

287.405

0.3

397.086

366.716

345.626

327.320

309.259

0.5

513.295

460.208

416.627

380.318

348.782

[///]

0.1

271.319

247.538

235.994

227.448

218.481

0.3

423.066

378.692

342.118

310.317

281.378

0.5

707.511

603.327

502.818

420.998

358.210

The buckling load of the panel with layup [0] is shown in Figure 16. As shown in Figure 16, the buckling load of a

fixed panel with layup [0] is greater than that of simple panel. As the cutout area ratio is increased or central angle of

the panel is decreased, the buckling load is decreased. The maximum buckling load of fixed panel is about 840 N/mm.

In the fixed panel, the effect of central angle on the buckling load of panel decreases dramatically when cutout area

ratio is increased. In the simple panel with central angle of 0.1 radians, the area of the cutout has a negligible effect on

the buckling load. The buckling load of the panel with layup [90] is presented in Figure 17. As an interesting point, the

boundary condition of the panel has not considerable effect on the variation of buckling load of the panel. For instance,

the maximum buckling load of the simple panel is 1% lower than that of the fixed panel. This is due to the fact that the

lamination angle is perpendicular to the loading direction. The central angle of the panel has very significant effect on

the buckling load. As the central angle decreases, the buckling load decreases. As the cutout area is increased, the

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

Davood Poorveiset al.

Latin American Journal of Solids and Structures, 2019, 16(8), e227 20/26

buckling load of the panels (fixed and simple) with central angle of 0.5 radians is decreased. On the other hand, the

cutout area has a slightly effect on the variation of panel buckling load with central angle of 0.1 radians. For instance,

when the cutout area ratio is changed from 5 to 25%, the buckling load of the simple and fixed panels with central

angle of 0.1 radians is changed about 15%. Moreover, when the cutout area ratio is reached to 25%, the effect of

central angle of the panel on the buckling load of the panel is negligible. For instance, the buckling loads of fixed panel

with central angle of 0.5 radian and simple panel with central angle of 0.1 radian are about 133 and 127 N/mm,

respectively (in presence of a circular hole with cutout area ratio of 25%).

Figure 15 Cylindrical panel mode shape (a) layup [90]8 and central angle of 0.1 (b) layup [90]8 and central angle of 0.5 (c) layup [45]8

and central angle of 0.1 (d) layup [45]8 and central angle of 0.5 (e) layup [45/-45]4 and central angle of 0.1 (f) layup [45/-45]4 and

central angle of 0.5 (g) layup [0/45/90/-45]s and central angle of 0.1 (h) layup [0/45/90/-45]s and central angle of 0.5.

The effect of boundary condition on the buckling load the panel with layup [0/90] is shown in Figure 18. As

illustrated in this figure, as the cutout ratio increases, the buckling load of the panel decreases. However, when the

central angle of the panel is considered as 0.1 radians, the reduction is negligible for any boundary condition. Figure 19

shows the effect of boundary condition on the buckling load for layup [0/90]. The trend of variation of buckling load

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 21/26

in this figure is same as Figure 18. The buckling loads of the fixed and simple panels with layups [45] and [45/45]

are presented in Figures 20 and 21, respectively. As an interesting point depicted in Figure 20, when the central angle

of the fixed and simple panels is low (0.1 radians), the cutout area has not significant effect on the buckling load of the

panel. In both panels (fixed and simple), considerable difference is seen between the buckling loads of the panel with

central angles of 0.1 and 0.5 radian. However, as the cutout area increases, the difference is decreased, dramatically.

As shown in Figure 19 (panel with layup [45/45]), the cutout area has not a significant effect on the buckling load of

the panel with central angle of 0.1 radians (fixed and simple panels). The buckling load of the panel is decreased when

the central angle of the panel is 0.5 radians. When the panel layup is [30/30] (Figure 22), the trend of variation of

the buckling load is same as Figure 20 (layup [45]). The panel with Layup [0/45/90/45] is reached to the

maximum and minimum in comparison with other layups (Figure 23). The trend of variation of the buckling load of the

panel is same as Figure 20.

Figure 16. Effect of boundary conditions on the buckling load for layup [0].

Figure 17. Effect of boundary conditions on the buckling load for layup [90].

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 22/26

Figure 18. Effect of boundary conditions on the buckling load for layup [0/90].

Figure 19. Effect of boundary conditions on the buckling load for layup [0/90].

Figure 20. Effect of boundary conditions on the buckling load for layup [45].

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 23/26

Figure 21. Effect of boundary conditions on the buckling load for layup [45/45].

Figure 22. Effect of boundary conditions on the buckling load for layup [30/30].

Figure 23. Effect of boundary conditions on the buckling load for layup [0/45/90/45].

5 CONCLUSION

In this paper a sub-parametric spline finite strip element is developed to study the buckling analysis of perforated

composite cylindrical panel. To this end, shape, mapping and displacement functions, Jacobian matrix, strain-

displacement relations based on first order shear deformation theory, load vector, and linear and geometrical stiffness

matrices of the strip element were derived. The obtained eigenvalue problem was solved by inverse subspace iteration

to calculate the buckling load of the cylindrical panel. All of these steps were carried out in a FORTRAN code called

SSFSM. The results obtained by the SSFSM were compared with those of the published literature and those obtained

from ABAQUS finite element software to check the validity of the results.

A simple spline finite strip for buckling analysis of composite cylindrical panel with cutout

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Latin American Journal of Solids and Structures, 2019, 16(8), e227 24/26

A parametric study was performed to show the influences of cutout shape and area, panel thickness, central angle

of the panel, boundary conditions of the panel and layup scheme on the axial buckling load of the composite cylindrical

panel. The cutouts with the shape of circle, elliptic, square and diamond were considered in the center of the panel.

The ratio of big to small chords of the elliptical cutout and the ratio of circular cutout area to panel area were varied. In

addition eight layup configurations (angle-ply and cross-ply), three panel thicknesses (thick, medium and thin) and

three panel central angles were considered in the parametric study. The following conclusions can be extracted based

on the current study.

- The simplicity of the Jacobian matrix in the current sub-parametric study causes the surface integrals encountered

in the derivation of stiffness matrices turn into multiplication of two single variable integrals.

- Another advantage of current sub-parametric formulation over iso-parametric one returns to the reduction in the

required number of Gauss points to accurately estimate the domain integrals.

- The reduction in execution time of current sub-parametric code against the standard iso-parametric formulation is

about 60 percent. With increasing in panel degrees of freedom due to increasing in eigenvalue solvers

contribution, this reduction is decreased slightly.

- Increasing in the cutout chord prependicular to the loading direction while unaltering cutout area increases the

axial buckling load of cylindrical panel.

- Among the layups considered in the parametric study, quasi-isotropic laminate [0/45/90/-45]s and uni-directional

[0]8 laminate carry the maximum and minimum axial buckling loads, respectively.

- Fixed or simply supported curved loaded edges of the panel have not any effect on the axial buckling load of the

panel for [90]8 layup while they have the most effect in the case of [0]8 arrangement.

- Axial buckling mode of uni-directional laminates such as [90]8 and [45]8 has local expansion in the fibre direction

and this is pronounced with increasing in the panel curvature. Unlike the uni-directional, in the multi-directional

laminates like [45/-45]4 and [0/45/90/-45]s global expansion is illustrated.

Author's Contributions: Conceptualization, D Poorveis and A Khajehdezfuly; Methodology, D Poorveis,

A Khajehdezfuly, S Moradi and E Shirshekan; Investigation, D Poorveis, A Khajehdezfuly and E Shirshekan; Writing -

original draft, D Pooveis and Amin Khajehdezfuly; Writing - review & editing, D Poorveis, A Khajehdezfuly and S Moradi;

Supervision, D Poorveis.

Editor: Marco L. Bittencourt.

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