Journal of Advanced Research 8 (2017) 635–648

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Journal of Advanced Research

journal homepage: www.elsevier.com/locate/jare

Original Article

A comparison between different finite elements for elastic and

aero-elastic analyses

Mohamed Mahran a,⇑, Adel ELsabbagh b, Hani Negm a

a

b

Aerospace Engineering Department, Cairo University, Giza 12613, Egypt

Asu Sound and Vibration Lab, Design and Production Engineering Department, Ain Shams University, Abbaseya, Cairo 11517, Egypt

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history:

Received 8 April 2017

Revised 23 June 2017

Accepted 28 June 2017

Available online 1 July 2017

Keywords:

Finite element method

Triangular element

Quadrilateral element

Free vibration analysis

Stress analysis

Aero-elastic analysis

a b s t r a c t

In the present paper, a comparison between five different shell finite elements, including the Linear

Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation

modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented. The shape functions and the element equations related to each element were presented through a detailed mathematical formulation. Additionally, the Jacobian matrix for the second order derivatives was simplified and

used to derive each element’s strain-displacement matrix in bending. The elements were compared using

carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements

needed to reach convergence, the resulting accuracy, and the needed computation time. The best suitable

element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with

deformation-based shape functions, whereas the most suitable element for stress analysis was the 8Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node

Quadrilateral Element. Although the linear triangular element was the last choice for modal and stress

analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer

computation time. Additionally, the nine-node quadrilateral element was found to be the best choice

for laminated composite plates analysis.

Ó 2017 Production and hosting by Elsevier B.V. on behalf of Cairo University. This is an open access article

under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer review under responsibility of Cairo University.

⇑ Corresponding author.

E-mail address: abdu_aerospace@eng.cu.edu.eg (M. Mahran).

http://dx.doi.org/10.1016/j.jare.2017.06.009

2090-1232/Ó 2017 Production and hosting by Elsevier B.V. on behalf of Cairo University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

636

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Nomenclature

Symbol

A

As

Asd

Avlm

B

d

D

J

K

M

N

w

W

coefficient matrix for in-plane action

steady aerodynamic coefficient matrix in structural

coordinates

unsteady aerodynamic coefficient matrix in structural

coordinates

steady aerodynamic coefficient matrix

strain-displacement matrix

displacement in global coordinates

stress-strain matrix (isotropic material properties matrix)

Jacobian Matrix for first order derivatives

stiffness matrix

mass matrix

shape function matrix

structural bending displacement field

structural bending nodal displacements

Introduction

Numerical methods are usually the first choice for many

researchers and engineers to analyze complicated systems because

of their accessibility, flexibility and ability to solve complex systems. The Finite Element Method (FEM) as one of the powerful

numerical methods for structural analysis comes at the top of the

list of all numerical methods. As introduced in many Refs. [1–6],

the method is mainly based on dividing the whole structure into

a finite number of elements connected at nodes. The properties of

the whole structure such as mass and stiffness, which are continuous in nature, are discretized over the elements and approximate

solutions are obtained for the governing equations. The elements

equations are assembled together to reach a global system of algebraic equations, which can be solved for the unknown solution variables of the structure. The accuracy of the FEM solution depends on

many factors, such as the interpolation polynomials and subsequently the element shape functions, the number of degrees of freedoms selected for each element, the mesh size, and the type of

element used. The model accuracy is a result of the deep understanding of the effect of each factor on the final results.

The selection of the element interpolation functions is a key factor in the accuracy of the FEM solution. For this reason, intensive

researches have been made to develop new finite elements having

different shapes and interpolation functions. There are numerous

types of elements for different structural problems. In this paper,

the main focus is on two-dimensional shell elements. Finite shell

elements such as triangular elements [7–9], quadrilateral elements

[10,11], higher order elements [12–17], and improved elements

[18] are all tested and approved to achieve an acceptable level of

accuracy. Although a vast number of elements are available in literature, researchers cannot easily figure out which element is the

most suitable to select for their particular problem. The selection

problem is even more difficult for engineers who are mainly interested in the application rather than the theoretical background.

Additionally, the detailed mathematical formulation of some thin

shell bending elements, especially the higher order ones, cannot

be easily found in the literature.

Considering aero-elasticity in which the structural model is

coupled to an aerodynamic model adds more complications to

the problem, and makes the choice of the suitable element more

challenging. Aero-elasticity is crucial for structures such as aircraft,

wind turbines, and several other applications in which divergence

and flutter phenomena may occur leading to catastrophic failures

x, y, z

d

X, Y, Z

V

E

V inf

q1

r

br

JJ

k

t

1

x

e

n, g

q

element local coordinates

displacement vector in local coordinates

structural global coordinates

volume

elasticity modulus of the wing material

flow speed

dynamic pressure

stress

reference length (half the wing root chord)

Jacobian matrix for second order derivatives

reduced frequency

plate wing thickness

wing damping ratio

flutter frequency

strain vector

reference element coordinates

air density

of the structure. Therefore, designers of these structures are constrained by the design limits and definitely need accurate FEM

without being computationally expensive.

Therefore, the aim of the present work is to present a detailed

mathematical formulation for different thin shell finite elements

along with a complete comparison between them for specific problems in structures and aero-elasticity. The results of the selected

elements are compared based on (1) solution accuracy of each element, (2) number of elements needed to achieve convergence, and

(3) computational time. The comparison is for free vibration analysis, stress analysis, aero-elastic analysis, and laminated composite

analysis. Five different elements are selected for the present comparison with different nature. These finite elements are

(1) Three-node linear triangular element [1] denoted as LINTRI.

(2) Four-node linear quadrilateral element [1] denoted as

LINQUAD.

(3) Four-node linear quadrilateral element based on deformation modes (MKQ12 [18]).

(4) Eight-node quadrilateral element denoted as QUAD8NOD.

(5) Nine-node quadrilateral element denoted as QUAD9NOD.

These elements are selected with different nature ranging from

linear to higher order, triangular to quadrilateral, and improved to

regular elements to provide wide range of variety to the present

comparison. All these elements are tested using bench mark problems from the literature [19,20] for elastic and aero-elastic analyses with analytical results and/or experimental measurements.

The element shape functions are derived using MATHEMATICA

[21] software and then implemented into MATLAB [22] codes to

solve the selected problems.

The finite elements’ formulation

The present finite element model is based on either the classical

plate theory for metallic materials or laminated plate theory for composite materials. Both are based on the Kirchhoff assumptions which

neglect the transverse shear and transverse normal effects [2].

To formulate a finite shell element there is a standard procedure

that is usually followed.

(1) Start from the weak (integral) form of the governing

equation.

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M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

(2) Assume suitable interpolation polynomials for both the inplane Pp and bending Pb displacement fields.

(3) Calculate the coefficients of these polynomials by applying

the nodal movement conditions.

(4) Determine the shape functions for both the in-plane Np and

bending Nb actions.

(5) Derive the strain-displacement matrix B from the shape

functions’ derivatives.

(6) Integrate to obtain the element stiffness matrix K knowing

the material elasticity matrix and the strain displacement

relationships.

(7) Calculate the element mass matrix Me from the element

shape functions and the material density q, and finally,

(8) The structural matrices K and M can be obtained by assembling the element matrices obtained in steps 6 and 7.

All these steps were followed for each element considered in

the current study to derive the element shape functions, straindisplacement relationships, stiffness, and mass matrices for both

the in-plane and bending actions. The element shape functions,

presented in this section, are derived using MATHEMATICA software. All the strain-displacement matrices, stiffness, and mass

matrices are numerically integrated using MATLAB software.

General formulation

In the present section, general formulation of the element shape

functions and strain displacement matrices is developed. Based on

this formulation all the shell elements’ shape functions and subsequently the elements’ equations are derived

In-plane action

u ¼ Pp a

ð1Þ

where u is the in-plane displacement field at any point through the

element and a is a vector of constants to be determined from the

nodal in-plane displacements U.

U ¼ Aa

ð2Þ

8

9

u1 >

>

>

>

>

>

>

>

>

>

v

>

1 >

>

>

<

=

..

U¼

. >

>

>

>

>

>

>

> unnod >

>

>

>

>

>

:

;

ð3Þ

v nnod

nnod represents the total number of nodes in an element. The size

of U equals the total element in-plane degrees of freedoms.

Finally, the in-plane shape functions can be obtained,

À1

N p ¼ Pp A

ð4Þ

The in-plane strain displacement matrix can be obtained from the

shape functions’ derivatives by using the Jacobian matrix definition

9

8

9

8

9

ex >

Np;x

>

= >

< u;x >

= >

<

=

ey ¼

u;y

Np;y

¼

Bp ¼

>

>

>

:c >

; >

:

; >

:

;

u;y þ v ;x

Np;y þ Np;x

xy

2

¼

J1

J3

J2

J4

!

J4

1 6

4 ÀJ 3

J det

J4 À J3

ÀJ 2

J1

J1 À J2

3

&

'

7 Np;n

U

5

Np;g

ð5Þ

where J 1 ; J2 ; J3 ; J 4 are the elements of the Jacobian matrix and J det is

the Jacobian determinant

¼

x;n

y;n

x; g

y ;g

#

; J det ¼ J 1 J 4 À J 3 J 2

ð6Þ

Bending action

An interpolation function is chosen either from Pascal’s Triangle

or based on the displacements modes,

w ¼ Pb a

ð7Þ

where w is the bending displacement at any point on the element,

from which we can obtain the rotation around the x-axis (hx) and

the y-axis (hy) using the Jacobian matrix.

hx ¼

dw

dw

; and hy ¼ À

dy

dx

ð8Þ

Note that the Jacobian matrix elements are rearranged in JÃ so

that the displacement rotations are defined as,

&

hx

hy

'

¼

ÀJ 3

J1

J det ÀJ 4

J2

1

!&

w;n

'

w;g

¼ JÀ1

Ã

&

w;n

'

ð9Þ

w;g

Then, the three bending displacements can be calculated from

the equation,

8 9

8

9

>

! > Pb >

!

=

=

1 0 <

1 0

Pb;n a ¼

hx ¼

À1

À1 c a

>

>

0 JÃ >

0 JÃ

: >

:

;

;

Pb;g

hy

ð10Þ

a is the coefficients vector to be determined from the out of plane

nodal displacements W. The bending shape functions will have

the form

Nb ¼ Pb CÀ1

An interpolation function is chosen from Pascal’s Triangle [1,2],

8

>

<

J¼

"

1

0

!

ð11Þ

0 JÃ

where the C matrix is calculated from the c matrix after applying

the nodal boundary conditions, and

8

9

w1 >

>

>

>

>

>

>

>

>

hx1 >

>

>

>

>

>

>

8 9

>

>

>

h

>

>

y1 >

w

>

>

>

>

< =

<

=

..

hx ¼ Nb

¼ Nb W

.

>

>

>

: >

;

>

>

>

>

hy

>

>

>

> wnnod >

>

>

>

>

>

>

>

>

h

> xnnod >

>

>

>

:

;

hynnod

ð12Þ

Then the strain-displacement matrix can be derived and simplified from the Jacobian definition for second order derivatives. They

were derived and simplified by the authors to have the form

8

8

9

9

>

>

< wxx >

< wnn >

=

=

wyy

Bb ¼

¼ JJÀ1 wgg

>

>

>

>

:

:

;

;

2wng

2wxy

ð13Þ

where JJ is the Jacobian Matrix for the second order derivatives,

which can be calculated using the elements of the regular Jacobian

matrix to have the form,

8

9 2 2

J1

>

< wnn >

=

6

wgg

¼ 4 J 23

>

>

:

;

2wng

2J 1 J 3

J 22

J 24

2J 2 J 4

8

9

9

38

>

>

< wxx >

< wxx >

=

=

7

¼ JJ: wyy

5 wyy

J3 J4

>

>

>

>

:

:

;

;

2wxy

2wxy

J2 J3 þ J1 J4

J1 J2

ð14Þ

The inverse of the Jacobian Matrix for the second order derivatives is

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M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

2

À1

JJ

1 6

¼ 2 4

J det

J 24

J 22

J 23

J 21

ÀJ 1 J 3

À2J 3 J 4

À2J 1 J 2

J2 J3 þ J1 J4

ÀJ 2 J 4

3

7

5

Pp ¼ f 1; n; g; gn g

and subsequently the in-plane shape functions have the form

Based on this simple and detailed mathematical implementation, the considered elements’ equations can be derived. All the

shape functions for those elements are presented in the following

sections.

Notice that Pb and Pp are represented as row vectors all over the

present paper.

The LINTRI thin-shell element has three nodes. The element has

six degrees of freedom per node with a total of 18 degrees of freedom. Fig. 1a shows a schematic of the element with the element

global, local, and reference coordinates. The element interpolation

and shape functions are derived in the following.

For in-plane action

The interpolation polynomial for in-plane action has the form

ð16Þ

and subsequently the in-plane shape functions have the form

Np ¼ f 1 À g g À n n g

ð17Þ

For bending action

The interpolation polynomial for bending action based on the

element area coordinates has the form

(

Pb ¼

n; g; 1 À g À n; gn; ÀgðÀ1 þ g þ nÞ; ÀnðÀ1 þ g þ nÞ; gn2 ;

)

Àg2 ðÀ1 þ g þ nÞ; nðÀ1 þ g þ nÞ2

ð18Þ

and subsequently the bending shape functions are

N b1 ¼ ðÀ1 þ g þ nÞð2g2 þ gðÀ1 þ 2nÞ þ ðÀ1 þ nÞð1 þ 2nÞÞ

N b2 ¼ ðÀ1 þ g þ nÞðJ 4 ðÀ1 þ gÞg þ J 2 nðÀ1 þ g þ nÞÞ

N b3 ¼ ÀðÀ1 þ g þ nÞðJ 3 ðÀ1 þ gÞg þ J 1 nðÀ1 þ g þ nÞÞ

N b4 ¼ Àgð2g2 þ 2ðÀ1 þ nÞn þ gðÀ3 þ 2nÞÞ

N b5 ¼ gðÀJ 2 ðÀ1 þ nÞn þ J 4 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

1

Np ¼ f ð1 À gÞð1 À nÞ ð1 À gÞð1 þ nÞ ð1 þ gÞð1 þ nÞ ð1 þ gÞð1 À nÞ g

4

ð21Þ

For bending action

The interpolation basis functions for bending action selected

from Pascal’s Triangle has the form

The linear triangular element (LINTRI)

Pp ¼ f 1; n; g g

ð20Þ

ð15Þ

ð19Þ

Pb ¼ f1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; gn3 ; g3 ng

ð22Þ

1

Nb1 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðÀ2 þ g þ g2 þ n þ n2 Þ

8

1

Nb2 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðJ 4 ðÀ1 þ g2 Þ þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb3 ¼ ðÀ1 þ gÞðÀ1 þ nÞðJ 3 ðÀ1 þ g2 Þ þ J 1 ðÀ1 þ n2 ÞÞ

8

1

Nb4 ¼ ðÀ1 þ gÞð1 þ nÞðg þ g2 þ ðÀ2 þ nÞð1 þ nÞÞ

8

1

Nb5 ¼ ðÀ1 þ gÞð1 þ nÞðJ 2 þ J 4 ðÀ1 þ g2 Þ À J 2 n2 Þ

8

1

Nb6 ¼ ðÀ1 þ gÞð1 þ nÞðJ 3 À J 3 g2 þ J 1 ðÀ1 þ n2 ÞÞ

8

ð23Þ

and subsequently the bending shape functions are

1

Nb7 ¼ À ð1 þ gÞð1 þ nÞððÀ1 þ gÞg þ ðÀ2 þ nÞð1 þ nÞÞ

8

1

Nb8 ¼ ð1 þ gÞð1 þ nÞðJ 4 ðÀ1 þ g2 Þ þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb9 ¼ À ð1 þ gÞð1 þ nÞðJ 3 ðÀ1 þ g2 Þ þ J 1 ðÀ1 þ n2 ÞÞ

8

1

Nb10 ¼ ð1 þ gÞðÀ1 þ nÞðÀ2 þ ðÀ1 þ gÞg þ n þ n2 Þ

8

1

Nb11 ¼ ð1 þ gÞðÀ1 þ nÞðJ 4 À J 4 g2 þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb12 ¼ ð1 þ gÞðÀ1 þ nÞðJ 1 þ J 3 ðÀ1 þ g2 Þ À J 1 n2 Þ

8

ð24Þ

N b6 ¼ ÀgðÀJ 1 ðÀ1 þ nÞn þ J 3 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

N b7 ¼ ÀnðÀ3n þ 2ðg2 þ gðÀ1 þ nÞ þ n2 ÞÞ

N b8 ¼ nðJ 4 gn þ J 2 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

N b9 ¼ ÀnðJ 3 gn þ J 1 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

The linear quadrilateral element (LINQUAD)

The LINQUAD element consists of four nodes. It has six degrees

of freedom per node with a total of 24 degrees of freedom. Fig. 1b

shows a schematic of the element with the global, local, and reference coordinates. The element interpolation and shape functions

are derived to be as follows.

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

The linear quadrilateral element based on deformation modes

(MKQ12)

The MKQ12 element has four nodes. It has six degrees of freedom per node with a total of 24 degrees of freedom. It has the global, local, and reference coordinates shown in Fig. 1b. This element

was introduced by Karkon and Rezaiee-Pajand [18]. It has the same

in-plane shape functions of the LINQUAD element but with

improved bending shape functions based on the deformation

modes.

8

9

2

2

2

2

>

< 1;n; g; gn;0:5ðÀ1 þ n Þ; 0:5ðÀ1 þ g Þ; 0:5nðÀ1 þ n Þ; 0:5gðÀ1 þ g Þ; >

=

2

2

Pb ¼ 0:25ðÀ1 þ g2 Þnð3 À n Þ; 0:25gð3 À g2 ÞðÀ1 þ n Þ;

>

>

:

;

0:25gðÀ1 þ g2 Þnð3 À n2 Þ;0:25gð3 À g2 ÞnðÀ1 þ n2 Þ

ð25Þ

The shape functions are then

639

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Fig. 1. Finite elements local and reference coordinates.

1

N b1 ¼ ðÀ1 þ gÞðÀ1 þ nÞð2 À g À g2 À n þ gn þ g2 n À n2 þ gn2 þ g2 n2 Þ

8

1

N b2 ¼ ðÀ1 þ gÞ2 ðÀ1 þ nÞ2 ð2J1 þ 2J 3 þ J 1 g þ 2J 3 g þ 2J1 n þ J3 n þ J 1 gn þ J3 gnÞ

16

1

N b3 ¼ ðÀ1 þ gÞ2 ðÀ1 þ nÞ2 ð2J2 þ 2J 4 þ J 2 g þ 2J 4 g þ 2J2 n þ J4 n þ J 2 gn þ J4 gnÞ

16

1

N b4 ¼ ðÀ1 þ gÞð1 þ nÞðÀ2 þ g þ g2 À n þ gn þ g2 n þ n2 À gn2 À g2 n2 Þ

8

1

N b5 ¼ ðÀ1 þ gÞ2 ð1 þ nÞ2 ðÀ2J1 þ 2J 3 À J 1 g þ 2J 3 g þ 2J1 n À J3 n þ J 1 gn À J3 gnÞ

16

1

N b6 ¼ ðÀ1 þ gÞ2 ð1 þ nÞ2 ðÀ2J2 þ 2J 4 À J 2 g þ 2J 4 g þ 2J2 n À J4 n þ J 2 gn À J4 gnÞ

16

1

N b7 ¼ ð1 þ gÞð1 þ nÞð2 þ g À g2 þ n þ gn À g2 n À n2 À gn2 þ g2 n2 Þ

8

1

N b8 ¼ ð1 þ gÞ2 ð1 þ nÞ2 ðÀ2J1 À 2J3 þ J1 g þ 2J 3 g þ 2J 1 n þ J3 n À J1 gn À J 3 gnÞ

16

1

N b9 ¼ ð1 þ gÞ2 ð1 þ nÞ2 ðÀ2J2 À 2J4 þ J2 g þ 2J 4 g þ 2J 2 n þ J4 n À J2 gn À J 4 gnÞ

16

1

N b10 ¼ ð1 þ gÞðÀ1 þ nÞðÀ2 À g þ g2 þ n þ gn À g2 n þ n2 þ gn2 À g2 n2 Þ

8

1

N b11 ¼ ð1 þ gÞ2 ðÀ1 þ nÞ2 ð2J 1 À 2J 3 À J1 g þ 2J3 g þ 2J1 n À J 3 n À J1 gn þ J3 gnÞ

16

1

N b12 ¼ ð1 þ gÞ2 ðÀ1 þ nÞ2 ð2J 2 À 2J 4 À J2 g þ 2J4 g þ 2J2 n À J 4 n À J2 gn þ J4 gnÞ

16

ð26Þ

The eight-node quadrilateral element (QUAD8NOD)

The QUAD8NOD element has eight nodes. It has six degrees of

freedom per node with a total of 48 degrees of freedom. Fig. 1c

shows a schematic of the element with the global, local, and reference coordinates. The element interpolation and shape functions

were derived in the following.

Pp ¼

1; n; g; gn; n2 ; g2 ; n2 g; g2 n

É

ð27Þ

and subsequently the in-plane shape functions have the form

1

1

Np1 ¼ ð1 À gÞð1 À nÞðÀ1 À g À nÞ; Np2 ¼ ð1 À gÞð1 þ nÞðÀ1 À g þ nÞ

4

4

1

1

Np3 ¼ ð1 þ gÞð1 þ nÞðÀ1 þ g þ nÞ; Np4 ¼ ð1 þ gÞð1 À nÞðÀ1 þ g À nÞ

4

4

1

1

Np5 ¼ ð1 À gÞð1 À nÞð1 þ nÞ; Np6 ¼ ð1 À gÞð1 þ gÞð1 þ nÞ

2

2

1

1

Np7 ¼ ð1 þ gÞð1 À nÞð1 þ nÞ; Np8 ¼ ð1 À gÞð1 þ gÞð1 À nÞ

2

2

ð28Þ

For bending action

The interpolation polynomial for bending action was selected

carefully from the well-known Pascal Triangle. Initially, the following basis functions were selected;

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g2 n2

)

ð29Þ

g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; g2 n4 ; g3 n3 ; g4 n2

Using the above basis functions yields a singular C matrix. The

rank of the matrix turns out to be 22 instead of 24, which indicates

that two terms result in repeated equations. Different terms have

been replaced with higher order terms to detect the reason for

the singularity. The analysis revealed that the bilinear term gn,

the biquadratic term g2 n2 , and the bicubic term g3 n3 all yield similar equations. Therefore, the biquadratic and bicubic terms were

replaced with gn5 and g5 n. Finally, the bending interpolation function for the QUAD8NOD element is;

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

È

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n;

g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n

)

ð30Þ

640

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

This choice eliminates all the singularities, and subsequently

the bending shape functions are

bal, local, and reference coordinates shown in Fig. 1d. The element

interpolation and shape functions were derived in the following.

1

N b1 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðg3 þ 3g4 À gð1 þ nÞ2 À g2 ð5 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 ðÀ2 þ 3nÞÞ

8

!

1

J 4 ðÀ1 þ g2 ÞðÀ1 þ nÞðÀ3g3 þ g2 ð1 À 2nÞ þ 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

N b2 ¼

24 þJ 2 ðÀ1 þ gÞðÀ1 þ n2 ÞðÀ1 þ g2 þ g3 þ gðÀ1 þ ð3 À 2nÞnÞ þ nð3 þ n À 3n2 ÞÞ

N b3 ¼

1

J 3 ð1 þ gÞð1 þ 3g3 þ n À 3gð1 þ nÞ À n2 ð1 þ nÞ þ g2 ðÀ1 þ 2nÞÞ

ðÀ1 þ gÞðÀ1 þ nÞ

24

ÀJ 1 ð1 þ nÞðÀ1 þ g2 þ g3 þ gðÀ1 þ ð3 À 2nÞnÞ þ nð3 þ n À 3n2 ÞÞ

!

1

ðÀ1 þ gÞð1 þ nÞðg3 þ 3g4 þ g2 ðÀ5 þ nÞ À gðÀ1 þ nÞ2 þ ðÀ1 þ nÞ2 ð1 þ nÞð2 þ 3nÞÞ

8

!

1

J 4 ð1 þ gÞð3g3 þ 3gðÀ1 þ nÞ þ ðÀ1 þ nÞ2 ð1 þ nÞ À g2 ð1 þ 2nÞÞ

ðÀ1 þ gÞð1 þ nÞ

¼

24

ÀJ 2 ðÀ1 þ nÞðg2 þ g3 À gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

N b4 ¼

N b5

N b6

1

J 3 ð1 þ gÞð3g3 þ 3gðÀ1 þ nÞ þ ðÀ1 þ nÞ2 ð1 þ nÞ À g2 ð1 þ 2nÞÞ

¼ À ðÀ1 þ gÞð1 þ nÞ

24

ÀJ 1 ðÀ1 þ nÞðg2 þ g3 À gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

1

N b7 ¼ À ð1 þ gÞð1 þ nÞðÀg3 þ 3g4 þ g2 ðÀ5 þ nÞ þ gðÀ1 þ nÞ2 þ ðÀ1 þ nÞ2 ð1 þ nÞð2 þ 3nÞÞ

8

!

1

J 4 ðÀ1 þ gÞðÀ1 þ gðÀ3 þ g þ 3g2 Þ þ n þ gð3 þ 2gÞn þ n2 À n3 Þ

ð1 þ gÞð1 þ nÞ

N b8 ¼

24

þJ 2 ðÀ1 þ nÞðg2 À g3 þ gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

N b9 ¼ À

1

J 3 ðÀ1 þ gÞðÀ1 þ gðÀ3 þ g þ 3g2 Þ þ n þ gð3 þ 2gÞn þ n2 À n3 Þ

ð1 þ gÞð1 þ nÞ

24

þJ 1 ðÀ1 þ nÞðg2 À g3 þ gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

!

ð31Þ

!

1

ð1 þ gÞðÀ1 þ nÞððÀ1 þ gÞ2 ð1 þ gÞð2 þ 3gÞ À ðÀ1 þ gÞ2 n þ ðÀ5 þ gÞn2 þ n3 þ 3n4 Þ

8

!

J 2 ð1 þ gÞðÀ1 þ n2 ÞððÀ1 þ gÞ2 ð1 þ gÞ þ 3ðÀ1 þ gÞn À ð1 þ 2gÞn2 þ 3n3 Þ

1

¼

24 ÀJ 4 ðÀ1 þ g2 ÞðÀ1 þ nÞð3g3 þ g2 ð1 À 2nÞ À 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

!

2

1 ÀJ 1 ð1 þ gÞðÀ1 þ n2 ÞððÀ1 þ gÞ ð1 þ gÞ þ 3ðÀ1 þ gÞn À ð1 þ 2gÞn2 þ 3n3 Þ

¼

24 þJ 3 ðÀ1 þ g2 ÞðÀ1 þ nÞð3g3 þ g2 ð1 À 2nÞ À 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

N b10 ¼

N b11

N b12

1

N b13 ¼ À ðÀ1 þ gÞðÀ1 þ n2 ÞðÀ2 þ g þ g2 þ 2n2 Þ

4

1

ðÀ1 þ gÞðÀ1 þ n2 ÞðJ 4 ð1 þ gÞð1 þ gðÀ3 þ 2gÞ À n2 Þ À 6J 2 nðÀ1 þ n2 ÞÞ

N b14 ¼

12

1

N b15 ¼ À ðÀ1 þ gÞðÀ1 þ n2 ÞðJ 3 ð1 þ gÞð1 þ gðÀ3 þ 2gÞ À n2 Þ À 6J 1 nðÀ1 þ n2 ÞÞ

12

1

Nb16 ¼ ðÀ1 þ g2 Þð1 þ nÞð2g2 þ ðÀ2 þ nÞð1 þ nÞÞ

4

1

Nb17 ¼ ðÀ1 þ g2 Þð1 þ nÞð6J 4 gðÀ1 þ g2 Þ þ J 2 ðÀ1 þ nÞðg2 À ð1 þ nÞð1 þ 2nÞÞÞ

12

1

Nb18 ¼ À ðÀ1 þ g2 Þð1 þ nÞð6J 3 gðÀ1 þ g2 Þ þ J 1 ðÀ1 þ nÞðg2 À ð1 þ nÞð1 þ 2nÞÞÞ

12

1

Nb19 ¼ ð1 þ gÞðÀ1 þ n2 ÞðÀ2 À g þ g2 þ 2n2 Þ

4

1

Nb20 ¼ À ð1 þ gÞðÀ1 þ n2 ÞðJ4 ðÀ1 þ gÞð1 þ gð3 þ 2gÞ À n2 Þ À 6J 2 nðÀ1 þ n2 ÞÞ

12

1

N b21 ¼ ð1 þ gÞðÀ1 þ n2 ÞðJ3 ðÀ1 þ gÞð1 þ gð3 þ 2gÞ À n2 Þ À 6J1 nðÀ1 þ n2 ÞÞ

12

1

Nb22 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞðÀ2 þ 2g2 þ n þ n2 Þ

4

1

Nb23 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞð6J4 gðÀ1 þ g2 Þ þ J 2 ð1 þ nÞðÀ1 þ g2 þ ð3 À 2nÞnÞÞ

12

1

N b24 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞð6J3 gðÀ1 þ g2 Þ þ J1ð1 þ nÞðÀ1 þ g2 þ ð3 À 2nÞnÞÞ

12

ð32Þ

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

Pp ¼

È

1; n; g; gn; n2 ; g2 ; n2 g; g2 n; n2 g2

É

ð33Þ

and subsequently the in-plane shape functions have the form

1

1

ðÀg þ g2 ÞðÀn þ n2 Þ; Np2 ¼ ðÀg þ g2 Þðn þ n2 Þ

4

4

1

1

2

2

¼ ðg þ g Þðn þ n Þ; N p4 ¼ ðg þ g2 ÞðÀn þ n2 Þ

4

4

1

1

2

2

¼ ðÀg þ g Þð1 À n Þ; Np6 ¼ ð1 À g2 Þðn þ n2 Þ

2

2

1

1

2

2

¼ ðg þ g Þð1 À n Þ; Np8 ¼ ð1 À g2 ÞðÀn þ n2 Þ

2

2

¼ ð1 À g2 Þð1 À n2 Þ

Np1 ¼

Np3

Np5

Np7

Np9

ð34Þ

For bending action

The nine-node quadrilateral element (QUAD9NOD)

The QUAD9NOD element has nine nodes. It has six degrees of

freedom per node with a total of 54 degrees of freedom. It has the glo-

The interpolation polynomial for bending action was selected

carefully from the well-known Pascal Triangle. Initially, the following basis functions were selected;

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; g2 n2 ; n5

4

2 3

3 2

5

2 4

4 2

4 2

)

2 4

; gn ; g n ; g n ; g n; g ; gn ; g n ; g n ; g n; g n ; g n ;

4

5

5

ð35Þ

Using the above basis functions yielded singular C matrix. The

rank of the matrix turned out to be 23 instead of 24, which indicated that two terms result in repeated equations. Different terms

have been replaced with higher order terms to detect the origin of

the singularity. Finally, the bending interpolation function for the

QUAD9NOD element is;

(

Pb ¼

1; n; g; gn; n2 ; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; n5 ; gn4 ;

641

1

Nb23 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞnðÀ2J 4 gðÀ1 þ g2 Þ þ J 2 ðg þ ðÀ1 þ gÞn þ n3 ÞÞ

4

1

Nb24 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞnðÀ2J3 gðÀ1 þ g2 Þ þ J 1 ðg þ ðÀ1 þ gÞn þ n3 ÞÞ

4

Nb25 ¼ ÀðÀ1 þ g2 ÞðÀ1 þ n2 ÞðÀ1 þ g2 À 3gn þ n2 Þ

Nb26 ¼ ÀðÀ1 þ gÞð1 þ gÞðÀ1 þ nÞð1 þ nÞðJ 4 gðÀ1 þ g2 Þ þ J2 nðÀ1 þ n2 ÞÞ

Nb27 ¼ ðÀ1 þ gÞð1 þ gÞðÀ1 þ nÞð1 þ nÞðJ3 gðÀ1 þ g2 Þ þ J 1 nðÀ1 þ n2 ÞÞ

ð38Þ

)

The test problems

g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n; g2 n5 ; g5 n2 ; 12 ðg2 n2 þ g3 n3 Þ

ð36Þ

and subsequently the bending shape functions are

It has been mentioned earlier that the aim of the present work

was to compare between different shell finite elements with different behavior for elastic and aero-elastic analyses. This will enable

1

ðÀ1 þ gÞgðÀ1 þ nÞnð4 þ g2 þ 3g3 þ gð2 þ 6nÞ þ nð2 þ n þ 3n2 ÞÞ

8

1

Nb2 ¼ ðÀ1 þ gÞgðÀ1 þ nÞnðJ 4 ð1 þ g3 þ n þ gnÞ þ J 2 ð1 þ g þ gn þ n3 ÞÞ

8

1

Nb3 ¼ À ðÀ1 þ gÞgðÀ1 þ nÞnðJ 3 ð1 þ g3 þ n þ gnÞ þ J 1 ð1 þ g þ gn þ n3 ÞÞ

8

1

Nb4 ¼ ðÀ1 þ gÞgnð1 þ nÞðÀ8 þ gð2 þ gÞðÀ5 þ 3gÞ þ 10n þ 6gn þ n2 À 3n3 Þ

8

1

Nb5 ¼ ðÀ1 þ gÞgnð1 þ nÞðJ 4 ðÀ1 þ g3 þ gðÀ2 þ nÞ þ nÞ þ J 2 ð1 þ g À ð2 þ gÞn þ n3 ÞÞ

8

1

Nb6 ¼ À ðÀ1 þ gÞgnð1 þ nÞðJ 3 ðÀ1 þ g3 þ gðÀ2 þ nÞ þ nÞ þ J 1 ð1 þ g À ð2 þ gÞn þ n3 ÞÞ

8

1

Nb7 ¼ À gð1 þ gÞnð1 þ nÞðÀg2 þ 3g3 þ gð2 À 6nÞ þ ðÀ1 þ nÞð4 þ nð2 þ 3nÞÞÞ

8

1

Nb8 ¼ gð1 þ gÞnð1 þ nÞðJ 4 ðÀ1 þ g3 þ n À gnÞ þ J 2 ðÀ1 þ g À gn þ n3 ÞÞ

8

1

Nb9 ¼ À gð1 þ gÞnð1 þ nÞðJ 3 ðÀ1 þ g3 þ n À gnÞ þ J 1 ðÀ1 þ g À gn þ n3 ÞÞ

8

1

Nb10 ¼ À gð1 þ gÞðÀ1 þ nÞnð8 þ ðÀ2 þ gÞgð5 þ 3gÞ þ 10n À 6gn À n2 À 3n3 Þ

8

1

Nb11 ¼ gð1 þ gÞðÀ1 þ nÞnðJ 2 ðÀ1 þ g þ ðÀ2 þ gÞn þ n3 Þ þ J 4 ð1 þ g3 þ n À gð2 þ nÞÞÞ

8

1

Nb12 ¼ À gð1 þ gÞðÀ1 þ nÞnðJ 1 ðÀ1 þ g þ ðÀ2 þ gÞn þ n3 Þ þ J 3 ð1 þ g3 þ n À gð2 þ nÞÞÞ

8

1

Nb13 ¼ À ðÀ1 þ gÞgðgðÀ4 þ g þ 3g2 Þ þ 6ð1 þ gÞn À 2n2 ÞðÀ1 þ n2 Þ

4

1

Nb14 ¼ À ðÀ1 þ gÞgðÀ1 þ n2 ÞðJ 4 ðg3 þ gðÀ1 þ nÞ þ nÞ À 2J 2 nðÀ1 þ n2 ÞÞ

4

1

Nb15 ¼ ðÀ1 þ gÞgðÀ1 þ n2 ÞðJ 3 ðg3 þ gðÀ1 þ nÞ þ nÞ À 2J 1 nðÀ1 þ n2 ÞÞ

4

1

Nb16 ¼ ðÀ1 þ g2 Þnð1 þ nÞð2g2 À 6gðÀ1 þ nÞ þ nð1 þ nÞðÀ4 þ 3nÞÞ

4

Nb1 ¼

1

Nb17 ¼ ðÀ1 þ g2 Þnð1 þ nÞð2J 4 gðÀ1 þ g2 Þ þ J 2 ðgðÀ1 þ nÞ þ n À n3 ÞÞ

4

1

Nb18 ¼ À ðÀ1 þ g2 Þnð1 þ nÞð2J 3 gðÀ1 þ g2 Þ þ J1 ðgðÀ1 þ nÞ þ n À n3 ÞÞ

4

1

Nb19 ¼ gð1 þ gÞðÀ1 þ n2 Þðgð1 þ gÞðÀ4 þ 3gÞ À 6ðÀ1 þ gÞn þ 2n2 Þ

4

1

Nb20 ¼ À gð1 þ gÞðÀ1 þ n2 ÞðJ 4 ðÀ1 þ gÞðg þ g2 À nÞ À 2J 2 nðÀ1 þ n2 ÞÞ

4

1

Nb21 ¼ gð1 þ gÞðÀ1 þ n2 ÞðJ 3 ðÀ1 þ gÞðg þ g2 À nÞ À 2J 1 nðÀ1 þ n2 ÞÞ

4

1

Nb22 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞnð2g2 À 6gð1 þ nÞ À nðÀ4 þ n þ 3n2 ÞÞ

4

ð37Þ

any researcher to select the shell element which best suits his/

her specific application. Different test benchmark problems are

considered for elastic and aero-elastic analyses. These problems

are described in detail in this section in addition to their mathematical models. These models are implemented in the next section

into MATLAB codes, which are carefully constructed and validated.

For each problem, a suitable number of elements was selected

based on convergence analysis. The number of elements was

increased till the response converged to a certain value. Then the

results were compared with published experimental or analytical

solutions.

642

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Study of the natural frequencies of free vibration of an elastic

square plate

Problem formulation

This problem was presented by Safizadeh et al. [20] in which an

analytical solution was provided. It deals with a square plate

(1 m Â 1 m) with thickness t equal 0.003 m and the material properties, elastic modulus, mass density, and Poisson’s ratio

E ¼ 71 Gpa; q ¼ 2700 kg=m3 ; and

The mathematical model

m ¼ 0:3

The plate is fixed along all sides.

The mathematical model

The plate natural frequencies are calculated by solving the

eigenvalue problem

ðKb À x Mb ÞW ¼ 0

ð39Þ

2

where the subscript b refers to the bending action and the mass and

stiffness matrices are calculated from the strain displacement

matrix and the shape functions

Z

Keb ¼

Meb

¼

2

Z

V

V

t

BTb Ds Bb dV ¼

NTb Nb dV

0

0

3

t

12

ZZ

First the stiffness matrix for both the in-plane and bending

actions are derived. The stiffness matrix of the bending action Keb

is given in Eq. (40) and the in-plane stiffness matrix Kep has the

form

Z

Kep ¼

BTb Ds Bb J det dndg

NTb INb J det dnd

g

3

V

ZZ

BTp Ds Bp dV ¼ t

BTp Ds Bp J det dndg

8

9

>

>

< rx =

ry ¼ Ds ðBp Á dp À zBb Á db Þ

>

sxy ;

ð40Þ

t3

12

07

5

Wing aero-elastic analysis

0

0

t3

12

Problem formulation

The superscript e means that these matrices are calculated over

each element and then assembled in the global coordinates.

Ds is the isotropic material stiffness matrix

Ds ¼

1

m

E 6

4m 1

1 À m2

0 0

3

0

0 7

5

ð43Þ

where dp is the local in-plane displacements vector and db is the

local bending displacements vector.

6

I ¼ 40

2

ð42Þ

The elastic problem was solved to find the displacements, then

the stresses are calculated using the equation

>

:

ZZ

¼q

formed to select the suitable number of elements for both the aerodynamic and finite element analyses.

The material properties are

Young’s Modulus = 98E9 Pa, Poison’s ratio = 0.28, plate

thickness = 0.001 m

The flow properties are

Speed = 30 m/s, density = 1.225 kg/m3, AOA = 3°

AOA is the flow angle of attack.

ð41Þ

1Àm

2

Stress and deformation analysis of metallic plate wing

The problem formulation

The elastic stress and displacement analysis performance of the

considered elements was tested by analyzing a plate-like straightrectangular wing under aerodynamic load. The wing geometry was

shown in Fig. 2. The aerodynamic analysis was performed by using

the Doublet lattice method [23]. A convergence analysis was per-

The finite element selection in aero-elastic analysis is always a

problem. The point is to select the suitable element for accurate

aero-elastic calculations and load transformation between the

aerodynamic model and the structural model and vice versa. For

this purpose, a comparison was made between different finite elements for aero-elastic analysis.

The results are compared with published experimental results.

For all the finite elements, the shape functions are used in the

aero-elastic coupling, rather than the conventional spline interpolation, to make the model more accurate and consistent [24].

A straight-rectangular plate wing model made of laminated

composite materials was analyzed with different laminate configurations. The wing has the same plane form shown in Fig. 2. The

suitable number of elements was selected for both the aerodynamic and finite element analyses throughout convergence analyses. The lamina material properties were EL = 98 Gpa, ET = 7.9 Gpa,

GLT = 5.6 Gpa, tL = 0.28, q = 1520 kg/m3, t = 0.134eÀ3m

Mathematical model

Fig. 2. Plate wing plane form geometry.

The need to decrease the aircraft structural weight for economic

purposes leads to an increase in the aircraft flexibility, and subsequently the tendency for aero-elastic instability. The wing aeroelastic instability was categorized into divergence and flutter analyses [25–29]. In the former, analysts were interested in determining the minimum speed at which wing static torsional instability

takes place. In the latter, analysts were interested in determining

the minimum speed at which wing dynamic instability flutter

takes place. For both analyses, the doublet lattice method was used

for the steady and unsteady aerodynamic analyses [23] for all elements. An exception was the linear triangular element where the

vortex lattice method [30] was used in the steady aerodynamic

analysis because it produces more accurate results, although it

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

needs more elements and subsequently longer computation time.

For this reason, it was not considered for the rest of elements, as

the doublet lattice method was enough and smaller computational

time.

The problem was solved by developing two models; one for the

structural analysis and the other for the aerodynamic analysis.

Then, the aerodynamic coefficient or stiffness matrix was transformed into the structural nodes by means of either spline interpolation or by the same shape functions of the finite element. The use

of the shape functions of the finite element in the connection

between the finite element model and the aerodynamic model

was found to be more accurate and consistent than the spline

method [24]. The mathematical models for both the divergence

and flutter analyses are presented in this section.

The divergence analysis

Divergence can be regarded as a static torsional instability that

occurs for aircraft wings at a certain flight speed. The divergence

speed can be calculated by solving the eigenvalue problem

ðKb À q1 As ÞW ¼ 0

ð44Þ

where As is the aerodynamic stiffness transformed from the aerodynamic control points into the finite element nodes by using the element shape functions [31], and qdiv represents the dynamic

pressure at which divergence takes place.

qdiv ¼

1

qV 2div

2

ð45Þ

where V div is the velocity at which the divergence occurs.

The flutter analysis

Flutter can be regarded as a dynamic instability that occurs to

aircraft wings at a certain flight speed. The flutter speed was determined by solving the eigenvalue problem

KÀ1

Mb þ

b

qb2r

2k

A

2 sd

!

À

1 þ i1

x2

!

I W¼0

ð46Þ

where br is a reference length (chosen to be half the wing root

chord), Asd is the unsteady aerodynamic stiffness matrix transformed from the aerodynamic control points to the finite elements

nodes by using the element shape functions [31], while k is known

as the reduced frequency which is defined as

k¼

br x

Vf

ð47Þ

Eq. (46) comprises two unknowns; the speed V and the frequency x, and both can be obtained by iteration where the flutter

occurs at zero damping coefficient 1.

It is worth noting that Eq. (46) is nested and solved using the kmethod.

The aerodynamic coefficient matrix As for steady aerodynamic

analysis was calculated using either the Vortex Lattice Method

(VLM) [30] or the Doublet Lattice Method (DLM) [23]. Then, they

were transformed to the structural coordinates as following

ð48Þ

where Asteady is the steady aerodynamic coefficient matrix at the

aerodynamic control points. GN1 and GN2d are transformation

matrices calculated from the element bending shape functions.

ZZ

GN1 ¼ TT

NTb dxdyK

T and K are geometric transformation matrices that connect

between the global structural coordinates and the element local

coordinates.

GN2d ¼ TT

n

X

NT K

ð50Þ

i¼1

n represents the number of aerodynamic control points in each

element.

In flutter analysis, the unsteady aerodynamic coefficient matrix

Asd was calculated using the DLM.

Asd ¼ GN1 AÀ1

unsteady BcGN2f

ð51Þ

Aunsteady is the unsteady aerodynamic coefficient matrix at the aerodynamic control points. GN2f is calculated as GN2d, by considering

the lateral displacement w. Bc is a boundary conditions matrix calculated as

Bc ¼

iÃk

; À1

br

!

ð52Þ

Laminated plate elastic analysis

Problem formulation

To present a complete picture regarding the differences

between the considered finite elements, a comparison between

them for the analysis of a composite laminated plate was established. A square plate was considered with 25 cm side length and

1 cm total thickness [32]. The lamina material properties were

EL = 52.5 MPa, ET = 2.1 MPa, GLT = 1.05 MPa, tL = 0.25. Reddy [32]

presents an analytical solution for the maximum displacement of

the square plate subject to a distributed pressure of 1 N/cm2.

Two laminate configurations were considered as well as two different boundary conditions. The analytical results are represented in

the following section.

The mathematical model for this problem is exactly the same as

that of the elastic deformation problem, but with imposing the

effect of composite material on the stress-strain relation. More

details can be found in Reddy [32].

Results and discussion

The selected finite elements are tested by the four problems

described in the previous section, and the results are demonstrated

in this section. All the analyses have been performed on a personal

computer with an i7 processor CPU @ 3.6 GHz, intel core and 16 GB

RAM. The results for each analysis are shown in the next

subsection.

The dynamic elastic analysis

Aerodynamic analysis and aero-elastic coupling

As ¼ GN1 AÀ1

steady GN2d

643

ð49Þ

The problem of the square plate presented in the previous section was implemented on a computer code using MATLAB software.

Table 1 shows the predicted natural frequencies. The first column

contains the analytical solution [20] for the first five natural frequencies, while the finite element results are listed in the rest of

the columns. Results have shown that the natural frequencies predicted by the different types of finite elements are in general less

than those predicted by the analytical model. The frequency values

obtained by using the Linear Triangular Element are farthest from

the analytical ones and the number of elements (N_elem) needed

to reach convergence is a maximum. On the other hand, the frequencies obtained by using the Linear Quadrilateral Element based

on deformation modes are closest to the analytical solution. The

644

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 1

The natural frequencies of clamped square plate [Hz].

Mode

Analytical (20)

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

1

2

3

4

5

Nelem

28.47

58.06

58.06

85.66

104.09

–

26.2

53.35

53.35

78.35

95.54

800

26.54

54.06

54.06

79.09

96.91

225

26.81

54.78

54.78

81.49

98.29

100

26.68

54.39

54.39

80.3

97.42

100

26.61

54.27

54.25

80.01

97.24

100

Fig. 3. The average error and execution time of each finite element in the frequency analysis.

Table 2

The product of the average error percentage and processing time for various finite

elements.

Mode

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

avg Â Time

125.5

23.8

7.7

25.8

101

Table 3

Max. displacement and stress over the plate wing under aerodynamic load.

Element

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

dmax [mm]

rmax [MPa]

Nelem

Time [s]

19.1

28.9

192

20.4

20.3

32.9

120

5.2

20.3

33.2

120

8.1

20.4

34.4

60

6

19.9

34.1

60

7.9

MKQ12 performed even better than the higher order elements,

which were expected to produce accurate results in bending analysis. The number of elements needed to reach convergence in the

linear quadrilateral element based on deformation modes is also

the same as those of higher order elements.

The average error percentage and processing time of each element are demonstrated in Fig. 3. The product of the average error

percent and processing time can be used as a measure of excellence of the finite element, where the best element has the minimum value for the product. This product is given in Table 2,

which shows that the linear quadrilateral element based on deformation modes is the best element to use in this type of problems.

Plate wing stress and displacement analysis

The elastic performance of a plate wing under steady aerodynamic load was studied using the five finite elements under consideration. The values of the maximum Von Mises stresses and

maximum displacements are tabulated in Table 3 and plotted

in Fig. 4 together with the execution time. Fig. 5 shows the distribution of the Von Mises stress in the wing for each element

type. The stresses are calculated in the case of triangular elements over the mid-side points and then averaged over the element. In case of the quadrilateral elements, the stresses are

determined at the element integration points, and then averaged

over the element.

Since there was neither analytical nor experimental data available for this model, the error percentage cannot be computed for

this particular problem. However, it is clear from Fig. 5 that the

stress distribution resulting from using higher order elements

(QUAD8NOD and QUAD9NOD) is the smoothest and most realistic.

This can be attributed to the higher order interpolation functions

for displacements, which render the stress distribution (derived

from the displacement derivatives) continuous. Hence, if all the

results are considered together, the best performing element can

be considered to be the QUAD8NOD element, which results in

accurate displacement and stress distributions, together with a

reasonable computational time. Following this element comes

the QUAD9NOD in the second place. On the other hand, the LINTRI

element comes as the worst element for wing stress analysis from

the point of view of computation time and stress distribution as

seen in Fig. 5.

645

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Tme

32.9

LINTRI

LINQUAD

7.9

19.9

20.4

6

8.1

20.3

5.2

20.3

19.1

20.4

28.9

34.1

34.4

σ

33.2

d

MKQ12

QUAD8NOD

QUAD9NOD

Fig. 4. Max. displacement and stress of the plate wing in addition to the executing time.

Fig. 5. The Von Mises stresses for each element model.

646

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 4

Divergence speed of the plate wing related to different laminate configurations.

Laminate configuration

[0 0 90 90 0 0]

[45 À45 0 0 À45 45]

[45 45 0 0 45 45]

[À45 À45 0 0 À45 À45]

[30 30 0 0 30 30]

[À30 À30 0 0 À30 À30]

Nelem

FE mesh

DLM mesh

Divergence/Flutter Speed [m/s]

Exp [19]

LINTRI [31]

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

25F

>32

28F

12.5D

27F

11.7D

–

–

–

25.4/26.4

47.5F

27.8F

12.7/29.1

27.4F

12.8/48.1

48

3Â8

8 Â 12

25.4/24.47

43.8F

26.1F

11.4/26.9

26.1F

11.58/33.7

96

6 Â 16

6 Â 12

25.5/25.9

46.8F

27.6F

11.5/29

27.16F

11.67/35.09

96

6 Â 16

6 Â 12

25.4F

45.6F

29.2F

12.86/23.5

28.1F

13/31.4

12

2Â6

7 Â 12

52.6/25.3

46.6F

29F

12.88/32.4

27.8F

13/30.6

12

2Â6

6 Â 12

Table 5

The error % in each analysis and the computation time.

Laminate configuration

[0 0 90 90 0 0]

[45 À45 0 0 À45 45]

[45 45 0 0 45 45]

[À45 À45 0 0 À45 À45]

[30 30 0 0 30 30]

[À30 À30 0 0 À30 À30]

avg %

Time [s]

avg Â Time

Error %

LINTRI

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

5.6

–

0.7

1.6

1.5

5.6

2.3

120

276

2.1

–

6.8

8.8

3.3

2.1

5.2

20.3

106

3.7

–

1.4

7.7

0.6

3.7

3.4

22.8

77.5

1.6

–

4.4

2.9

4.2

1.6

3.3

17.6

58.1

1.3

–

3.6

3.1

3.2

1.3

2.8

18.6

52.1

choice for wing aero-elastic analysis, followed by the QUAD8NOD

element.

Composite plate wing aero-elastic analysis

The static and dynamic aero-elastic analysis of a composite

plate wing was carried out using the five elements. The results

are listed in Table 4 for the smaller of the divergence and Flutter

speeds. The wing aero-elastic analysis was performed for different

laminate configurations. The subscript D refers to the Divergence

speed while subscript F refers to the Flutter speed. The error percent, the average error, and the computation time are listed in

Table 5 and plotted in Fig. 6. Considering the minimum value of

the product of the average error and execution time to be the sign

of excellence, we find that the QUAD9NOD element was the best

Laminated plate elastic analysis

The square laminated plate was analyzed for [0, 90]o and

[À45, 45]° laminate configurations. Two boundary conditions were

considered; in the first all the plate sides were simply supported,

and in the second all the plate sides were clamped [32]. The analytical results are listed in Table 6, and the average error and computation time for each element are depicted in Fig. 7. The analyses are

obtained for the maximum normalized bending displacement,

Fig. 6. The average error and computation time for each element.

647

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 6

Laminated elastic plate maximum normalized displacement.

BC’s

Laminate config

Analytical

LINTRI

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

Simply supported

[0, 90]

[À45, 45]

[0, 90]

[À45, 45]

Nelem

1.6955

0.6773

0.3814

0.3891

–

1.719

0.6902

0.3952

0.3901

200

1.61

0.8554

0.3806

0.294

144

1.606

0.8721

0.371

0.2865

144

1.6958

0.7198

0.4096

0.4229

36

1.6996

0.6925

0.3968

0.4078

25

Fixed

Fig. 7. The average error and computational time of laminated plate elastic analysis.

¼

w

100wmax ET t3tot

L4 P

ð53Þ

where wmax is the plate maximum thickness, t tot is the total

laminate thickness, L is the side length, and P is the applied pressure

load.

The results have shown that the linear triangular and the

9-node quadrilateral elements are the best elements for laminated

composite analysis from the accuracy point of view. However, the

linear triangular element (LINTRI) needs higher number of elements, and subsequently, longer computational time. On the other

hand, the worst elements for laminated composite analysis were

the LINQUAD and the MKQ12 elements as they have the maximum

relative average error.

Conclusions

In the present paper, five different thin shell finite elements

were considered. The five elements were the Linear Triangular Element (LINTRI), the Linear Quadrilateral Element (LINQUAD), the

Linear Quadrilateral Element Based on Deformation Modes

(MKQ12), the 8-Node Quadrilateral Element (QUAD8NOD), and

the 9-Node Quadrilateral Element (QUAD9NOD). A simple and

detailed mathematical model to derive the interpolation functions

and the stiffness matrix of each element was presented. The basis

functions were selected from the well-known Pascal Triangle to

minimize the order of the interpolation functions. However, singularities existed and specific terms had to be removed and replaced

with other terms to eliminate the source of singularities. The five

elements were tested using several elastic and aero-elastic analyses through three carefully selected bench mark problems with

analytical or experimental results available in the literature. In

order to have a fair comparison, a convergence analysis was conducted for each element and the minimum number of elements

needed for convergence was used in the comparison.

From the present investigation, it was found that the MKQ12

element was the best choice for elastic free vibration analysis of

a plate, since it yields the most accurate results with the minimum

execution time. The second choice is the QUAD8NOD element, and

the worst results are produced by the LINTRI element. In case of

elastic thin shells, and if the stress analysis was sought, the most

accurate elements are naturally the higher order elements. From

the point of view of time and accuracy, the best element was found

to be the QUAD8NOD element, and the QUAD9NOD element comes

out second. The worst element for this kind of analysis was the

LINTRI element, which requires longer computational times and

produces discontinuous stress distributions.

For aero-elastic analysis, the most accurate results are obtained

by using the LINTRI element, however, it requires the longest computational time. The best element for this type of analysis was

found to be the QUAD9NOD element considering its accuracy

and computational time. The second choice was the QUAD8NOD

element. It is worth noting that in spite of its bad performance in

elastic analysis, the LINTRI element was found to be more consistent to use with the Vortex Lattice Method in the aero-elastic analysis, but it requires long computation time. In laminated composite

plate analysis, the best recommended elements are either the LINTRI or QUAD9NOD. However, the LINTRI element requires dense

mesh, and subsequently longer computational time. The present

results can serve many researchers and engineers interested in

elastic and aero-elastic analyses, especially those who find difficulties in finding the detailed formulation of the finite elements, and

those who are confused in selecting specific elements for specific

application.

648

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Conflict of Interest

The authors have declared no conflict of interest.

Compliance with Ethics requirements

This article does not contain any studies with human or animal

subjects.

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Contents lists available at ScienceDirect

Journal of Advanced Research

journal homepage: www.elsevier.com/locate/jare

Original Article

A comparison between different finite elements for elastic and

aero-elastic analyses

Mohamed Mahran a,⇑, Adel ELsabbagh b, Hani Negm a

a

b

Aerospace Engineering Department, Cairo University, Giza 12613, Egypt

Asu Sound and Vibration Lab, Design and Production Engineering Department, Ain Shams University, Abbaseya, Cairo 11517, Egypt

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history:

Received 8 April 2017

Revised 23 June 2017

Accepted 28 June 2017

Available online 1 July 2017

Keywords:

Finite element method

Triangular element

Quadrilateral element

Free vibration analysis

Stress analysis

Aero-elastic analysis

a b s t r a c t

In the present paper, a comparison between five different shell finite elements, including the Linear

Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation

modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented. The shape functions and the element equations related to each element were presented through a detailed mathematical formulation. Additionally, the Jacobian matrix for the second order derivatives was simplified and

used to derive each element’s strain-displacement matrix in bending. The elements were compared using

carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements

needed to reach convergence, the resulting accuracy, and the needed computation time. The best suitable

element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with

deformation-based shape functions, whereas the most suitable element for stress analysis was the 8Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node

Quadrilateral Element. Although the linear triangular element was the last choice for modal and stress

analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer

computation time. Additionally, the nine-node quadrilateral element was found to be the best choice

for laminated composite plates analysis.

Ó 2017 Production and hosting by Elsevier B.V. on behalf of Cairo University. This is an open access article

under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer review under responsibility of Cairo University.

⇑ Corresponding author.

E-mail address: abdu_aerospace@eng.cu.edu.eg (M. Mahran).

http://dx.doi.org/10.1016/j.jare.2017.06.009

2090-1232/Ó 2017 Production and hosting by Elsevier B.V. on behalf of Cairo University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

636

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Nomenclature

Symbol

A

As

Asd

Avlm

B

d

D

J

K

M

N

w

W

coefficient matrix for in-plane action

steady aerodynamic coefficient matrix in structural

coordinates

unsteady aerodynamic coefficient matrix in structural

coordinates

steady aerodynamic coefficient matrix

strain-displacement matrix

displacement in global coordinates

stress-strain matrix (isotropic material properties matrix)

Jacobian Matrix for first order derivatives

stiffness matrix

mass matrix

shape function matrix

structural bending displacement field

structural bending nodal displacements

Introduction

Numerical methods are usually the first choice for many

researchers and engineers to analyze complicated systems because

of their accessibility, flexibility and ability to solve complex systems. The Finite Element Method (FEM) as one of the powerful

numerical methods for structural analysis comes at the top of the

list of all numerical methods. As introduced in many Refs. [1–6],

the method is mainly based on dividing the whole structure into

a finite number of elements connected at nodes. The properties of

the whole structure such as mass and stiffness, which are continuous in nature, are discretized over the elements and approximate

solutions are obtained for the governing equations. The elements

equations are assembled together to reach a global system of algebraic equations, which can be solved for the unknown solution variables of the structure. The accuracy of the FEM solution depends on

many factors, such as the interpolation polynomials and subsequently the element shape functions, the number of degrees of freedoms selected for each element, the mesh size, and the type of

element used. The model accuracy is a result of the deep understanding of the effect of each factor on the final results.

The selection of the element interpolation functions is a key factor in the accuracy of the FEM solution. For this reason, intensive

researches have been made to develop new finite elements having

different shapes and interpolation functions. There are numerous

types of elements for different structural problems. In this paper,

the main focus is on two-dimensional shell elements. Finite shell

elements such as triangular elements [7–9], quadrilateral elements

[10,11], higher order elements [12–17], and improved elements

[18] are all tested and approved to achieve an acceptable level of

accuracy. Although a vast number of elements are available in literature, researchers cannot easily figure out which element is the

most suitable to select for their particular problem. The selection

problem is even more difficult for engineers who are mainly interested in the application rather than the theoretical background.

Additionally, the detailed mathematical formulation of some thin

shell bending elements, especially the higher order ones, cannot

be easily found in the literature.

Considering aero-elasticity in which the structural model is

coupled to an aerodynamic model adds more complications to

the problem, and makes the choice of the suitable element more

challenging. Aero-elasticity is crucial for structures such as aircraft,

wind turbines, and several other applications in which divergence

and flutter phenomena may occur leading to catastrophic failures

x, y, z

d

X, Y, Z

V

E

V inf

q1

r

br

JJ

k

t

1

x

e

n, g

q

element local coordinates

displacement vector in local coordinates

structural global coordinates

volume

elasticity modulus of the wing material

flow speed

dynamic pressure

stress

reference length (half the wing root chord)

Jacobian matrix for second order derivatives

reduced frequency

plate wing thickness

wing damping ratio

flutter frequency

strain vector

reference element coordinates

air density

of the structure. Therefore, designers of these structures are constrained by the design limits and definitely need accurate FEM

without being computationally expensive.

Therefore, the aim of the present work is to present a detailed

mathematical formulation for different thin shell finite elements

along with a complete comparison between them for specific problems in structures and aero-elasticity. The results of the selected

elements are compared based on (1) solution accuracy of each element, (2) number of elements needed to achieve convergence, and

(3) computational time. The comparison is for free vibration analysis, stress analysis, aero-elastic analysis, and laminated composite

analysis. Five different elements are selected for the present comparison with different nature. These finite elements are

(1) Three-node linear triangular element [1] denoted as LINTRI.

(2) Four-node linear quadrilateral element [1] denoted as

LINQUAD.

(3) Four-node linear quadrilateral element based on deformation modes (MKQ12 [18]).

(4) Eight-node quadrilateral element denoted as QUAD8NOD.

(5) Nine-node quadrilateral element denoted as QUAD9NOD.

These elements are selected with different nature ranging from

linear to higher order, triangular to quadrilateral, and improved to

regular elements to provide wide range of variety to the present

comparison. All these elements are tested using bench mark problems from the literature [19,20] for elastic and aero-elastic analyses with analytical results and/or experimental measurements.

The element shape functions are derived using MATHEMATICA

[21] software and then implemented into MATLAB [22] codes to

solve the selected problems.

The finite elements’ formulation

The present finite element model is based on either the classical

plate theory for metallic materials or laminated plate theory for composite materials. Both are based on the Kirchhoff assumptions which

neglect the transverse shear and transverse normal effects [2].

To formulate a finite shell element there is a standard procedure

that is usually followed.

(1) Start from the weak (integral) form of the governing

equation.

637

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

(2) Assume suitable interpolation polynomials for both the inplane Pp and bending Pb displacement fields.

(3) Calculate the coefficients of these polynomials by applying

the nodal movement conditions.

(4) Determine the shape functions for both the in-plane Np and

bending Nb actions.

(5) Derive the strain-displacement matrix B from the shape

functions’ derivatives.

(6) Integrate to obtain the element stiffness matrix K knowing

the material elasticity matrix and the strain displacement

relationships.

(7) Calculate the element mass matrix Me from the element

shape functions and the material density q, and finally,

(8) The structural matrices K and M can be obtained by assembling the element matrices obtained in steps 6 and 7.

All these steps were followed for each element considered in

the current study to derive the element shape functions, straindisplacement relationships, stiffness, and mass matrices for both

the in-plane and bending actions. The element shape functions,

presented in this section, are derived using MATHEMATICA software. All the strain-displacement matrices, stiffness, and mass

matrices are numerically integrated using MATLAB software.

General formulation

In the present section, general formulation of the element shape

functions and strain displacement matrices is developed. Based on

this formulation all the shell elements’ shape functions and subsequently the elements’ equations are derived

In-plane action

u ¼ Pp a

ð1Þ

where u is the in-plane displacement field at any point through the

element and a is a vector of constants to be determined from the

nodal in-plane displacements U.

U ¼ Aa

ð2Þ

8

9

u1 >

>

>

>

>

>

>

>

>

>

v

>

1 >

>

>

<

=

..

U¼

. >

>

>

>

>

>

>

> unnod >

>

>

>

>

>

:

;

ð3Þ

v nnod

nnod represents the total number of nodes in an element. The size

of U equals the total element in-plane degrees of freedoms.

Finally, the in-plane shape functions can be obtained,

À1

N p ¼ Pp A

ð4Þ

The in-plane strain displacement matrix can be obtained from the

shape functions’ derivatives by using the Jacobian matrix definition

9

8

9

8

9

ex >

Np;x

>

= >

< u;x >

= >

<

=

ey ¼

u;y

Np;y

¼

Bp ¼

>

>

>

:c >

; >

:

; >

:

;

u;y þ v ;x

Np;y þ Np;x

xy

2

¼

J1

J3

J2

J4

!

J4

1 6

4 ÀJ 3

J det

J4 À J3

ÀJ 2

J1

J1 À J2

3

&

'

7 Np;n

U

5

Np;g

ð5Þ

where J 1 ; J2 ; J3 ; J 4 are the elements of the Jacobian matrix and J det is

the Jacobian determinant

¼

x;n

y;n

x; g

y ;g

#

; J det ¼ J 1 J 4 À J 3 J 2

ð6Þ

Bending action

An interpolation function is chosen either from Pascal’s Triangle

or based on the displacements modes,

w ¼ Pb a

ð7Þ

where w is the bending displacement at any point on the element,

from which we can obtain the rotation around the x-axis (hx) and

the y-axis (hy) using the Jacobian matrix.

hx ¼

dw

dw

; and hy ¼ À

dy

dx

ð8Þ

Note that the Jacobian matrix elements are rearranged in JÃ so

that the displacement rotations are defined as,

&

hx

hy

'

¼

ÀJ 3

J1

J det ÀJ 4

J2

1

!&

w;n

'

w;g

¼ JÀ1

Ã

&

w;n

'

ð9Þ

w;g

Then, the three bending displacements can be calculated from

the equation,

8 9

8

9

>

! > Pb >

!

=

=

1 0 <

1 0

Pb;n a ¼

hx ¼

À1

À1 c a

>

>

0 JÃ >

0 JÃ

: >

:

;

;

Pb;g

hy

ð10Þ

a is the coefficients vector to be determined from the out of plane

nodal displacements W. The bending shape functions will have

the form

Nb ¼ Pb CÀ1

An interpolation function is chosen from Pascal’s Triangle [1,2],

8

>

<

J¼

"

1

0

!

ð11Þ

0 JÃ

where the C matrix is calculated from the c matrix after applying

the nodal boundary conditions, and

8

9

w1 >

>

>

>

>

>

>

>

>

hx1 >

>

>

>

>

>

>

8 9

>

>

>

h

>

>

y1 >

w

>

>

>

>

< =

<

=

..

hx ¼ Nb

¼ Nb W

.

>

>

>

: >

;

>

>

>

>

hy

>

>

>

> wnnod >

>

>

>

>

>

>

>

>

h

> xnnod >

>

>

>

:

;

hynnod

ð12Þ

Then the strain-displacement matrix can be derived and simplified from the Jacobian definition for second order derivatives. They

were derived and simplified by the authors to have the form

8

8

9

9

>

>

< wxx >

< wnn >

=

=

wyy

Bb ¼

¼ JJÀ1 wgg

>

>

>

>

:

:

;

;

2wng

2wxy

ð13Þ

where JJ is the Jacobian Matrix for the second order derivatives,

which can be calculated using the elements of the regular Jacobian

matrix to have the form,

8

9 2 2

J1

>

< wnn >

=

6

wgg

¼ 4 J 23

>

>

:

;

2wng

2J 1 J 3

J 22

J 24

2J 2 J 4

8

9

9

38

>

>

< wxx >

< wxx >

=

=

7

¼ JJ: wyy

5 wyy

J3 J4

>

>

>

>

:

:

;

;

2wxy

2wxy

J2 J3 þ J1 J4

J1 J2

ð14Þ

The inverse of the Jacobian Matrix for the second order derivatives is

638

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

2

À1

JJ

1 6

¼ 2 4

J det

J 24

J 22

J 23

J 21

ÀJ 1 J 3

À2J 3 J 4

À2J 1 J 2

J2 J3 þ J1 J4

ÀJ 2 J 4

3

7

5

Pp ¼ f 1; n; g; gn g

and subsequently the in-plane shape functions have the form

Based on this simple and detailed mathematical implementation, the considered elements’ equations can be derived. All the

shape functions for those elements are presented in the following

sections.

Notice that Pb and Pp are represented as row vectors all over the

present paper.

The LINTRI thin-shell element has three nodes. The element has

six degrees of freedom per node with a total of 18 degrees of freedom. Fig. 1a shows a schematic of the element with the element

global, local, and reference coordinates. The element interpolation

and shape functions are derived in the following.

For in-plane action

The interpolation polynomial for in-plane action has the form

ð16Þ

and subsequently the in-plane shape functions have the form

Np ¼ f 1 À g g À n n g

ð17Þ

For bending action

The interpolation polynomial for bending action based on the

element area coordinates has the form

(

Pb ¼

n; g; 1 À g À n; gn; ÀgðÀ1 þ g þ nÞ; ÀnðÀ1 þ g þ nÞ; gn2 ;

)

Àg2 ðÀ1 þ g þ nÞ; nðÀ1 þ g þ nÞ2

ð18Þ

and subsequently the bending shape functions are

N b1 ¼ ðÀ1 þ g þ nÞð2g2 þ gðÀ1 þ 2nÞ þ ðÀ1 þ nÞð1 þ 2nÞÞ

N b2 ¼ ðÀ1 þ g þ nÞðJ 4 ðÀ1 þ gÞg þ J 2 nðÀ1 þ g þ nÞÞ

N b3 ¼ ÀðÀ1 þ g þ nÞðJ 3 ðÀ1 þ gÞg þ J 1 nðÀ1 þ g þ nÞÞ

N b4 ¼ Àgð2g2 þ 2ðÀ1 þ nÞn þ gðÀ3 þ 2nÞÞ

N b5 ¼ gðÀJ 2 ðÀ1 þ nÞn þ J 4 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

1

Np ¼ f ð1 À gÞð1 À nÞ ð1 À gÞð1 þ nÞ ð1 þ gÞð1 þ nÞ ð1 þ gÞð1 À nÞ g

4

ð21Þ

For bending action

The interpolation basis functions for bending action selected

from Pascal’s Triangle has the form

The linear triangular element (LINTRI)

Pp ¼ f 1; n; g g

ð20Þ

ð15Þ

ð19Þ

Pb ¼ f1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; gn3 ; g3 ng

ð22Þ

1

Nb1 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðÀ2 þ g þ g2 þ n þ n2 Þ

8

1

Nb2 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðJ 4 ðÀ1 þ g2 Þ þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb3 ¼ ðÀ1 þ gÞðÀ1 þ nÞðJ 3 ðÀ1 þ g2 Þ þ J 1 ðÀ1 þ n2 ÞÞ

8

1

Nb4 ¼ ðÀ1 þ gÞð1 þ nÞðg þ g2 þ ðÀ2 þ nÞð1 þ nÞÞ

8

1

Nb5 ¼ ðÀ1 þ gÞð1 þ nÞðJ 2 þ J 4 ðÀ1 þ g2 Þ À J 2 n2 Þ

8

1

Nb6 ¼ ðÀ1 þ gÞð1 þ nÞðJ 3 À J 3 g2 þ J 1 ðÀ1 þ n2 ÞÞ

8

ð23Þ

and subsequently the bending shape functions are

1

Nb7 ¼ À ð1 þ gÞð1 þ nÞððÀ1 þ gÞg þ ðÀ2 þ nÞð1 þ nÞÞ

8

1

Nb8 ¼ ð1 þ gÞð1 þ nÞðJ 4 ðÀ1 þ g2 Þ þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb9 ¼ À ð1 þ gÞð1 þ nÞðJ 3 ðÀ1 þ g2 Þ þ J 1 ðÀ1 þ n2 ÞÞ

8

1

Nb10 ¼ ð1 þ gÞðÀ1 þ nÞðÀ2 þ ðÀ1 þ gÞg þ n þ n2 Þ

8

1

Nb11 ¼ ð1 þ gÞðÀ1 þ nÞðJ 4 À J 4 g2 þ J 2 ðÀ1 þ n2 ÞÞ

8

1

Nb12 ¼ ð1 þ gÞðÀ1 þ nÞðJ 1 þ J 3 ðÀ1 þ g2 Þ À J 1 n2 Þ

8

ð24Þ

N b6 ¼ ÀgðÀJ 1 ðÀ1 þ nÞn þ J 3 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

N b7 ¼ ÀnðÀ3n þ 2ðg2 þ gðÀ1 þ nÞ þ n2 ÞÞ

N b8 ¼ nðJ 4 gn þ J 2 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

N b9 ¼ ÀnðJ 3 gn þ J 1 ðg2 þ gðÀ1 þ nÞ þ ðÀ1 þ nÞnÞÞ

The linear quadrilateral element (LINQUAD)

The LINQUAD element consists of four nodes. It has six degrees

of freedom per node with a total of 24 degrees of freedom. Fig. 1b

shows a schematic of the element with the global, local, and reference coordinates. The element interpolation and shape functions

are derived to be as follows.

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

The linear quadrilateral element based on deformation modes

(MKQ12)

The MKQ12 element has four nodes. It has six degrees of freedom per node with a total of 24 degrees of freedom. It has the global, local, and reference coordinates shown in Fig. 1b. This element

was introduced by Karkon and Rezaiee-Pajand [18]. It has the same

in-plane shape functions of the LINQUAD element but with

improved bending shape functions based on the deformation

modes.

8

9

2

2

2

2

>

< 1;n; g; gn;0:5ðÀ1 þ n Þ; 0:5ðÀ1 þ g Þ; 0:5nðÀ1 þ n Þ; 0:5gðÀ1 þ g Þ; >

=

2

2

Pb ¼ 0:25ðÀ1 þ g2 Þnð3 À n Þ; 0:25gð3 À g2 ÞðÀ1 þ n Þ;

>

>

:

;

0:25gðÀ1 þ g2 Þnð3 À n2 Þ;0:25gð3 À g2 ÞnðÀ1 þ n2 Þ

ð25Þ

The shape functions are then

639

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Fig. 1. Finite elements local and reference coordinates.

1

N b1 ¼ ðÀ1 þ gÞðÀ1 þ nÞð2 À g À g2 À n þ gn þ g2 n À n2 þ gn2 þ g2 n2 Þ

8

1

N b2 ¼ ðÀ1 þ gÞ2 ðÀ1 þ nÞ2 ð2J1 þ 2J 3 þ J 1 g þ 2J 3 g þ 2J1 n þ J3 n þ J 1 gn þ J3 gnÞ

16

1

N b3 ¼ ðÀ1 þ gÞ2 ðÀ1 þ nÞ2 ð2J2 þ 2J 4 þ J 2 g þ 2J 4 g þ 2J2 n þ J4 n þ J 2 gn þ J4 gnÞ

16

1

N b4 ¼ ðÀ1 þ gÞð1 þ nÞðÀ2 þ g þ g2 À n þ gn þ g2 n þ n2 À gn2 À g2 n2 Þ

8

1

N b5 ¼ ðÀ1 þ gÞ2 ð1 þ nÞ2 ðÀ2J1 þ 2J 3 À J 1 g þ 2J 3 g þ 2J1 n À J3 n þ J 1 gn À J3 gnÞ

16

1

N b6 ¼ ðÀ1 þ gÞ2 ð1 þ nÞ2 ðÀ2J2 þ 2J 4 À J 2 g þ 2J 4 g þ 2J2 n À J4 n þ J 2 gn À J4 gnÞ

16

1

N b7 ¼ ð1 þ gÞð1 þ nÞð2 þ g À g2 þ n þ gn À g2 n À n2 À gn2 þ g2 n2 Þ

8

1

N b8 ¼ ð1 þ gÞ2 ð1 þ nÞ2 ðÀ2J1 À 2J3 þ J1 g þ 2J 3 g þ 2J 1 n þ J3 n À J1 gn À J 3 gnÞ

16

1

N b9 ¼ ð1 þ gÞ2 ð1 þ nÞ2 ðÀ2J2 À 2J4 þ J2 g þ 2J 4 g þ 2J 2 n þ J4 n À J2 gn À J 4 gnÞ

16

1

N b10 ¼ ð1 þ gÞðÀ1 þ nÞðÀ2 À g þ g2 þ n þ gn À g2 n þ n2 þ gn2 À g2 n2 Þ

8

1

N b11 ¼ ð1 þ gÞ2 ðÀ1 þ nÞ2 ð2J 1 À 2J 3 À J1 g þ 2J3 g þ 2J1 n À J 3 n À J1 gn þ J3 gnÞ

16

1

N b12 ¼ ð1 þ gÞ2 ðÀ1 þ nÞ2 ð2J 2 À 2J 4 À J2 g þ 2J4 g þ 2J2 n À J 4 n À J2 gn þ J4 gnÞ

16

ð26Þ

The eight-node quadrilateral element (QUAD8NOD)

The QUAD8NOD element has eight nodes. It has six degrees of

freedom per node with a total of 48 degrees of freedom. Fig. 1c

shows a schematic of the element with the global, local, and reference coordinates. The element interpolation and shape functions

were derived in the following.

Pp ¼

1; n; g; gn; n2 ; g2 ; n2 g; g2 n

É

ð27Þ

and subsequently the in-plane shape functions have the form

1

1

Np1 ¼ ð1 À gÞð1 À nÞðÀ1 À g À nÞ; Np2 ¼ ð1 À gÞð1 þ nÞðÀ1 À g þ nÞ

4

4

1

1

Np3 ¼ ð1 þ gÞð1 þ nÞðÀ1 þ g þ nÞ; Np4 ¼ ð1 þ gÞð1 À nÞðÀ1 þ g À nÞ

4

4

1

1

Np5 ¼ ð1 À gÞð1 À nÞð1 þ nÞ; Np6 ¼ ð1 À gÞð1 þ gÞð1 þ nÞ

2

2

1

1

Np7 ¼ ð1 þ gÞð1 À nÞð1 þ nÞ; Np8 ¼ ð1 À gÞð1 þ gÞð1 À nÞ

2

2

ð28Þ

For bending action

The interpolation polynomial for bending action was selected

carefully from the well-known Pascal Triangle. Initially, the following basis functions were selected;

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g2 n2

)

ð29Þ

g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; g2 n4 ; g3 n3 ; g4 n2

Using the above basis functions yields a singular C matrix. The

rank of the matrix turns out to be 22 instead of 24, which indicates

that two terms result in repeated equations. Different terms have

been replaced with higher order terms to detect the reason for

the singularity. The analysis revealed that the bilinear term gn,

the biquadratic term g2 n2 , and the bicubic term g3 n3 all yield similar equations. Therefore, the biquadratic and bicubic terms were

replaced with gn5 and g5 n. Finally, the bending interpolation function for the QUAD8NOD element is;

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

È

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n;

g4 ; n5 ; gn4 ; g2 n3 ; g3 n2 ; g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n

)

ð30Þ

640

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

This choice eliminates all the singularities, and subsequently

the bending shape functions are

bal, local, and reference coordinates shown in Fig. 1d. The element

interpolation and shape functions were derived in the following.

1

N b1 ¼ À ðÀ1 þ gÞðÀ1 þ nÞðg3 þ 3g4 À gð1 þ nÞ2 À g2 ð5 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 ðÀ2 þ 3nÞÞ

8

!

1

J 4 ðÀ1 þ g2 ÞðÀ1 þ nÞðÀ3g3 þ g2 ð1 À 2nÞ þ 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

N b2 ¼

24 þJ 2 ðÀ1 þ gÞðÀ1 þ n2 ÞðÀ1 þ g2 þ g3 þ gðÀ1 þ ð3 À 2nÞnÞ þ nð3 þ n À 3n2 ÞÞ

N b3 ¼

1

J 3 ð1 þ gÞð1 þ 3g3 þ n À 3gð1 þ nÞ À n2 ð1 þ nÞ þ g2 ðÀ1 þ 2nÞÞ

ðÀ1 þ gÞðÀ1 þ nÞ

24

ÀJ 1 ð1 þ nÞðÀ1 þ g2 þ g3 þ gðÀ1 þ ð3 À 2nÞnÞ þ nð3 þ n À 3n2 ÞÞ

!

1

ðÀ1 þ gÞð1 þ nÞðg3 þ 3g4 þ g2 ðÀ5 þ nÞ À gðÀ1 þ nÞ2 þ ðÀ1 þ nÞ2 ð1 þ nÞð2 þ 3nÞÞ

8

!

1

J 4 ð1 þ gÞð3g3 þ 3gðÀ1 þ nÞ þ ðÀ1 þ nÞ2 ð1 þ nÞ À g2 ð1 þ 2nÞÞ

ðÀ1 þ gÞð1 þ nÞ

¼

24

ÀJ 2 ðÀ1 þ nÞðg2 þ g3 À gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

N b4 ¼

N b5

N b6

1

J 3 ð1 þ gÞð3g3 þ 3gðÀ1 þ nÞ þ ðÀ1 þ nÞ2 ð1 þ nÞ À g2 ð1 þ 2nÞÞ

¼ À ðÀ1 þ gÞð1 þ nÞ

24

ÀJ 1 ðÀ1 þ nÞðg2 þ g3 À gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

1

N b7 ¼ À ð1 þ gÞð1 þ nÞðÀg3 þ 3g4 þ g2 ðÀ5 þ nÞ þ gðÀ1 þ nÞ2 þ ðÀ1 þ nÞ2 ð1 þ nÞð2 þ 3nÞÞ

8

!

1

J 4 ðÀ1 þ gÞðÀ1 þ gðÀ3 þ g þ 3g2 Þ þ n þ gð3 þ 2gÞn þ n2 À n3 Þ

ð1 þ gÞð1 þ nÞ

N b8 ¼

24

þJ 2 ðÀ1 þ nÞðg2 À g3 þ gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

N b9 ¼ À

1

J 3 ðÀ1 þ gÞðÀ1 þ gðÀ3 þ g þ 3g2 Þ þ n þ gð3 þ 2gÞn þ n2 À n3 Þ

ð1 þ gÞð1 þ nÞ

24

þJ 1 ðÀ1 þ nÞðg2 À g3 þ gð1 þ nÞð1 þ 2nÞ þ ðÀ1 þ nÞð1 þ nÞð1 þ 3nÞÞ

!

ð31Þ

!

1

ð1 þ gÞðÀ1 þ nÞððÀ1 þ gÞ2 ð1 þ gÞð2 þ 3gÞ À ðÀ1 þ gÞ2 n þ ðÀ5 þ gÞn2 þ n3 þ 3n4 Þ

8

!

J 2 ð1 þ gÞðÀ1 þ n2 ÞððÀ1 þ gÞ2 ð1 þ gÞ þ 3ðÀ1 þ gÞn À ð1 þ 2gÞn2 þ 3n3 Þ

1

¼

24 ÀJ 4 ðÀ1 þ g2 ÞðÀ1 þ nÞð3g3 þ g2 ð1 À 2nÞ À 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

!

2

1 ÀJ 1 ð1 þ gÞðÀ1 þ n2 ÞððÀ1 þ gÞ ð1 þ gÞ þ 3ðÀ1 þ gÞn À ð1 þ 2gÞn2 þ 3n3 Þ

¼

24 þJ 3 ðÀ1 þ g2 ÞðÀ1 þ nÞð3g3 þ g2 ð1 À 2nÞ À 3gð1 þ nÞ þ ðÀ1 þ nÞð1 þ nÞ2 Þ

N b10 ¼

N b11

N b12

1

N b13 ¼ À ðÀ1 þ gÞðÀ1 þ n2 ÞðÀ2 þ g þ g2 þ 2n2 Þ

4

1

ðÀ1 þ gÞðÀ1 þ n2 ÞðJ 4 ð1 þ gÞð1 þ gðÀ3 þ 2gÞ À n2 Þ À 6J 2 nðÀ1 þ n2 ÞÞ

N b14 ¼

12

1

N b15 ¼ À ðÀ1 þ gÞðÀ1 þ n2 ÞðJ 3 ð1 þ gÞð1 þ gðÀ3 þ 2gÞ À n2 Þ À 6J 1 nðÀ1 þ n2 ÞÞ

12

1

Nb16 ¼ ðÀ1 þ g2 Þð1 þ nÞð2g2 þ ðÀ2 þ nÞð1 þ nÞÞ

4

1

Nb17 ¼ ðÀ1 þ g2 Þð1 þ nÞð6J 4 gðÀ1 þ g2 Þ þ J 2 ðÀ1 þ nÞðg2 À ð1 þ nÞð1 þ 2nÞÞÞ

12

1

Nb18 ¼ À ðÀ1 þ g2 Þð1 þ nÞð6J 3 gðÀ1 þ g2 Þ þ J 1 ðÀ1 þ nÞðg2 À ð1 þ nÞð1 þ 2nÞÞÞ

12

1

Nb19 ¼ ð1 þ gÞðÀ1 þ n2 ÞðÀ2 À g þ g2 þ 2n2 Þ

4

1

Nb20 ¼ À ð1 þ gÞðÀ1 þ n2 ÞðJ4 ðÀ1 þ gÞð1 þ gð3 þ 2gÞ À n2 Þ À 6J 2 nðÀ1 þ n2 ÞÞ

12

1

N b21 ¼ ð1 þ gÞðÀ1 þ n2 ÞðJ3 ðÀ1 þ gÞð1 þ gð3 þ 2gÞ À n2 Þ À 6J1 nðÀ1 þ n2 ÞÞ

12

1

Nb22 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞðÀ2 þ 2g2 þ n þ n2 Þ

4

1

Nb23 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞð6J4 gðÀ1 þ g2 Þ þ J 2 ð1 þ nÞðÀ1 þ g2 þ ð3 À 2nÞnÞÞ

12

1

N b24 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞð6J3 gðÀ1 þ g2 Þ þ J1ð1 þ nÞðÀ1 þ g2 þ ð3 À 2nÞnÞÞ

12

ð32Þ

For in-plane action

The interpolation polynomial for in-plane action selected from

Pascal’s Triangle has the form

Pp ¼

È

1; n; g; gn; n2 ; g2 ; n2 g; g2 n; n2 g2

É

ð33Þ

and subsequently the in-plane shape functions have the form

1

1

ðÀg þ g2 ÞðÀn þ n2 Þ; Np2 ¼ ðÀg þ g2 Þðn þ n2 Þ

4

4

1

1

2

2

¼ ðg þ g Þðn þ n Þ; N p4 ¼ ðg þ g2 ÞðÀn þ n2 Þ

4

4

1

1

2

2

¼ ðÀg þ g Þð1 À n Þ; Np6 ¼ ð1 À g2 Þðn þ n2 Þ

2

2

1

1

2

2

¼ ðg þ g Þð1 À n Þ; Np8 ¼ ð1 À g2 ÞðÀn þ n2 Þ

2

2

¼ ð1 À g2 Þð1 À n2 Þ

Np1 ¼

Np3

Np5

Np7

Np9

ð34Þ

For bending action

The nine-node quadrilateral element (QUAD9NOD)

The QUAD9NOD element has nine nodes. It has six degrees of

freedom per node with a total of 54 degrees of freedom. It has the glo-

The interpolation polynomial for bending action was selected

carefully from the well-known Pascal Triangle. Initially, the following basis functions were selected;

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

(

Pb ¼

1; n; g; n2 ; gn; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; g2 n2 ; n5

4

2 3

3 2

5

2 4

4 2

4 2

)

2 4

; gn ; g n ; g n ; g n; g ; gn ; g n ; g n ; g n; g n ; g n ;

4

5

5

ð35Þ

Using the above basis functions yielded singular C matrix. The

rank of the matrix turned out to be 23 instead of 24, which indicated that two terms result in repeated equations. Different terms

have been replaced with higher order terms to detect the origin of

the singularity. Finally, the bending interpolation function for the

QUAD9NOD element is;

(

Pb ¼

1; n; g; gn; n2 ; g2 ; n3 ; gn2 ; g2 n; g3 ; n4 ; gn3 ; g3 n; g4 ; n5 ; gn4 ;

641

1

Nb23 ¼ À ðÀ1 þ g2 ÞðÀ1 þ nÞnðÀ2J 4 gðÀ1 þ g2 Þ þ J 2 ðg þ ðÀ1 þ gÞn þ n3 ÞÞ

4

1

Nb24 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞnðÀ2J3 gðÀ1 þ g2 Þ þ J 1 ðg þ ðÀ1 þ gÞn þ n3 ÞÞ

4

Nb25 ¼ ÀðÀ1 þ g2 ÞðÀ1 þ n2 ÞðÀ1 þ g2 À 3gn þ n2 Þ

Nb26 ¼ ÀðÀ1 þ gÞð1 þ gÞðÀ1 þ nÞð1 þ nÞðJ 4 gðÀ1 þ g2 Þ þ J2 nðÀ1 þ n2 ÞÞ

Nb27 ¼ ðÀ1 þ gÞð1 þ gÞðÀ1 þ nÞð1 þ nÞðJ3 gðÀ1 þ g2 Þ þ J 1 nðÀ1 þ n2 ÞÞ

ð38Þ

)

The test problems

g4 n; g5 ; gn5 ; g2 n4 ; g4 n2 ; g5 n; g2 n5 ; g5 n2 ; 12 ðg2 n2 þ g3 n3 Þ

ð36Þ

and subsequently the bending shape functions are

It has been mentioned earlier that the aim of the present work

was to compare between different shell finite elements with different behavior for elastic and aero-elastic analyses. This will enable

1

ðÀ1 þ gÞgðÀ1 þ nÞnð4 þ g2 þ 3g3 þ gð2 þ 6nÞ þ nð2 þ n þ 3n2 ÞÞ

8

1

Nb2 ¼ ðÀ1 þ gÞgðÀ1 þ nÞnðJ 4 ð1 þ g3 þ n þ gnÞ þ J 2 ð1 þ g þ gn þ n3 ÞÞ

8

1

Nb3 ¼ À ðÀ1 þ gÞgðÀ1 þ nÞnðJ 3 ð1 þ g3 þ n þ gnÞ þ J 1 ð1 þ g þ gn þ n3 ÞÞ

8

1

Nb4 ¼ ðÀ1 þ gÞgnð1 þ nÞðÀ8 þ gð2 þ gÞðÀ5 þ 3gÞ þ 10n þ 6gn þ n2 À 3n3 Þ

8

1

Nb5 ¼ ðÀ1 þ gÞgnð1 þ nÞðJ 4 ðÀ1 þ g3 þ gðÀ2 þ nÞ þ nÞ þ J 2 ð1 þ g À ð2 þ gÞn þ n3 ÞÞ

8

1

Nb6 ¼ À ðÀ1 þ gÞgnð1 þ nÞðJ 3 ðÀ1 þ g3 þ gðÀ2 þ nÞ þ nÞ þ J 1 ð1 þ g À ð2 þ gÞn þ n3 ÞÞ

8

1

Nb7 ¼ À gð1 þ gÞnð1 þ nÞðÀg2 þ 3g3 þ gð2 À 6nÞ þ ðÀ1 þ nÞð4 þ nð2 þ 3nÞÞÞ

8

1

Nb8 ¼ gð1 þ gÞnð1 þ nÞðJ 4 ðÀ1 þ g3 þ n À gnÞ þ J 2 ðÀ1 þ g À gn þ n3 ÞÞ

8

1

Nb9 ¼ À gð1 þ gÞnð1 þ nÞðJ 3 ðÀ1 þ g3 þ n À gnÞ þ J 1 ðÀ1 þ g À gn þ n3 ÞÞ

8

1

Nb10 ¼ À gð1 þ gÞðÀ1 þ nÞnð8 þ ðÀ2 þ gÞgð5 þ 3gÞ þ 10n À 6gn À n2 À 3n3 Þ

8

1

Nb11 ¼ gð1 þ gÞðÀ1 þ nÞnðJ 2 ðÀ1 þ g þ ðÀ2 þ gÞn þ n3 Þ þ J 4 ð1 þ g3 þ n À gð2 þ nÞÞÞ

8

1

Nb12 ¼ À gð1 þ gÞðÀ1 þ nÞnðJ 1 ðÀ1 þ g þ ðÀ2 þ gÞn þ n3 Þ þ J 3 ð1 þ g3 þ n À gð2 þ nÞÞÞ

8

1

Nb13 ¼ À ðÀ1 þ gÞgðgðÀ4 þ g þ 3g2 Þ þ 6ð1 þ gÞn À 2n2 ÞðÀ1 þ n2 Þ

4

1

Nb14 ¼ À ðÀ1 þ gÞgðÀ1 þ n2 ÞðJ 4 ðg3 þ gðÀ1 þ nÞ þ nÞ À 2J 2 nðÀ1 þ n2 ÞÞ

4

1

Nb15 ¼ ðÀ1 þ gÞgðÀ1 þ n2 ÞðJ 3 ðg3 þ gðÀ1 þ nÞ þ nÞ À 2J 1 nðÀ1 þ n2 ÞÞ

4

1

Nb16 ¼ ðÀ1 þ g2 Þnð1 þ nÞð2g2 À 6gðÀ1 þ nÞ þ nð1 þ nÞðÀ4 þ 3nÞÞ

4

Nb1 ¼

1

Nb17 ¼ ðÀ1 þ g2 Þnð1 þ nÞð2J 4 gðÀ1 þ g2 Þ þ J 2 ðgðÀ1 þ nÞ þ n À n3 ÞÞ

4

1

Nb18 ¼ À ðÀ1 þ g2 Þnð1 þ nÞð2J 3 gðÀ1 þ g2 Þ þ J1 ðgðÀ1 þ nÞ þ n À n3 ÞÞ

4

1

Nb19 ¼ gð1 þ gÞðÀ1 þ n2 Þðgð1 þ gÞðÀ4 þ 3gÞ À 6ðÀ1 þ gÞn þ 2n2 Þ

4

1

Nb20 ¼ À gð1 þ gÞðÀ1 þ n2 ÞðJ 4 ðÀ1 þ gÞðg þ g2 À nÞ À 2J 2 nðÀ1 þ n2 ÞÞ

4

1

Nb21 ¼ gð1 þ gÞðÀ1 þ n2 ÞðJ 3 ðÀ1 þ gÞðg þ g2 À nÞ À 2J 1 nðÀ1 þ n2 ÞÞ

4

1

Nb22 ¼ ðÀ1 þ g2 ÞðÀ1 þ nÞnð2g2 À 6gð1 þ nÞ À nðÀ4 þ n þ 3n2 ÞÞ

4

ð37Þ

any researcher to select the shell element which best suits his/

her specific application. Different test benchmark problems are

considered for elastic and aero-elastic analyses. These problems

are described in detail in this section in addition to their mathematical models. These models are implemented in the next section

into MATLAB codes, which are carefully constructed and validated.

For each problem, a suitable number of elements was selected

based on convergence analysis. The number of elements was

increased till the response converged to a certain value. Then the

results were compared with published experimental or analytical

solutions.

642

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Study of the natural frequencies of free vibration of an elastic

square plate

Problem formulation

This problem was presented by Safizadeh et al. [20] in which an

analytical solution was provided. It deals with a square plate

(1 m Â 1 m) with thickness t equal 0.003 m and the material properties, elastic modulus, mass density, and Poisson’s ratio

E ¼ 71 Gpa; q ¼ 2700 kg=m3 ; and

The mathematical model

m ¼ 0:3

The plate is fixed along all sides.

The mathematical model

The plate natural frequencies are calculated by solving the

eigenvalue problem

ðKb À x Mb ÞW ¼ 0

ð39Þ

2

where the subscript b refers to the bending action and the mass and

stiffness matrices are calculated from the strain displacement

matrix and the shape functions

Z

Keb ¼

Meb

¼

2

Z

V

V

t

BTb Ds Bb dV ¼

NTb Nb dV

0

0

3

t

12

ZZ

First the stiffness matrix for both the in-plane and bending

actions are derived. The stiffness matrix of the bending action Keb

is given in Eq. (40) and the in-plane stiffness matrix Kep has the

form

Z

Kep ¼

BTb Ds Bb J det dndg

NTb INb J det dnd

g

3

V

ZZ

BTp Ds Bp dV ¼ t

BTp Ds Bp J det dndg

8

9

>

>

< rx =

ry ¼ Ds ðBp Á dp À zBb Á db Þ

>

sxy ;

ð40Þ

t3

12

07

5

Wing aero-elastic analysis

0

0

t3

12

Problem formulation

The superscript e means that these matrices are calculated over

each element and then assembled in the global coordinates.

Ds is the isotropic material stiffness matrix

Ds ¼

1

m

E 6

4m 1

1 À m2

0 0

3

0

0 7

5

ð43Þ

where dp is the local in-plane displacements vector and db is the

local bending displacements vector.

6

I ¼ 40

2

ð42Þ

The elastic problem was solved to find the displacements, then

the stresses are calculated using the equation

>

:

ZZ

¼q

formed to select the suitable number of elements for both the aerodynamic and finite element analyses.

The material properties are

Young’s Modulus = 98E9 Pa, Poison’s ratio = 0.28, plate

thickness = 0.001 m

The flow properties are

Speed = 30 m/s, density = 1.225 kg/m3, AOA = 3°

AOA is the flow angle of attack.

ð41Þ

1Àm

2

Stress and deformation analysis of metallic plate wing

The problem formulation

The elastic stress and displacement analysis performance of the

considered elements was tested by analyzing a plate-like straightrectangular wing under aerodynamic load. The wing geometry was

shown in Fig. 2. The aerodynamic analysis was performed by using

the Doublet lattice method [23]. A convergence analysis was per-

The finite element selection in aero-elastic analysis is always a

problem. The point is to select the suitable element for accurate

aero-elastic calculations and load transformation between the

aerodynamic model and the structural model and vice versa. For

this purpose, a comparison was made between different finite elements for aero-elastic analysis.

The results are compared with published experimental results.

For all the finite elements, the shape functions are used in the

aero-elastic coupling, rather than the conventional spline interpolation, to make the model more accurate and consistent [24].

A straight-rectangular plate wing model made of laminated

composite materials was analyzed with different laminate configurations. The wing has the same plane form shown in Fig. 2. The

suitable number of elements was selected for both the aerodynamic and finite element analyses throughout convergence analyses. The lamina material properties were EL = 98 Gpa, ET = 7.9 Gpa,

GLT = 5.6 Gpa, tL = 0.28, q = 1520 kg/m3, t = 0.134eÀ3m

Mathematical model

Fig. 2. Plate wing plane form geometry.

The need to decrease the aircraft structural weight for economic

purposes leads to an increase in the aircraft flexibility, and subsequently the tendency for aero-elastic instability. The wing aeroelastic instability was categorized into divergence and flutter analyses [25–29]. In the former, analysts were interested in determining the minimum speed at which wing static torsional instability

takes place. In the latter, analysts were interested in determining

the minimum speed at which wing dynamic instability flutter

takes place. For both analyses, the doublet lattice method was used

for the steady and unsteady aerodynamic analyses [23] for all elements. An exception was the linear triangular element where the

vortex lattice method [30] was used in the steady aerodynamic

analysis because it produces more accurate results, although it

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

needs more elements and subsequently longer computation time.

For this reason, it was not considered for the rest of elements, as

the doublet lattice method was enough and smaller computational

time.

The problem was solved by developing two models; one for the

structural analysis and the other for the aerodynamic analysis.

Then, the aerodynamic coefficient or stiffness matrix was transformed into the structural nodes by means of either spline interpolation or by the same shape functions of the finite element. The use

of the shape functions of the finite element in the connection

between the finite element model and the aerodynamic model

was found to be more accurate and consistent than the spline

method [24]. The mathematical models for both the divergence

and flutter analyses are presented in this section.

The divergence analysis

Divergence can be regarded as a static torsional instability that

occurs for aircraft wings at a certain flight speed. The divergence

speed can be calculated by solving the eigenvalue problem

ðKb À q1 As ÞW ¼ 0

ð44Þ

where As is the aerodynamic stiffness transformed from the aerodynamic control points into the finite element nodes by using the element shape functions [31], and qdiv represents the dynamic

pressure at which divergence takes place.

qdiv ¼

1

qV 2div

2

ð45Þ

where V div is the velocity at which the divergence occurs.

The flutter analysis

Flutter can be regarded as a dynamic instability that occurs to

aircraft wings at a certain flight speed. The flutter speed was determined by solving the eigenvalue problem

KÀ1

Mb þ

b

qb2r

2k

A

2 sd

!

À

1 þ i1

x2

!

I W¼0

ð46Þ

where br is a reference length (chosen to be half the wing root

chord), Asd is the unsteady aerodynamic stiffness matrix transformed from the aerodynamic control points to the finite elements

nodes by using the element shape functions [31], while k is known

as the reduced frequency which is defined as

k¼

br x

Vf

ð47Þ

Eq. (46) comprises two unknowns; the speed V and the frequency x, and both can be obtained by iteration where the flutter

occurs at zero damping coefficient 1.

It is worth noting that Eq. (46) is nested and solved using the kmethod.

The aerodynamic coefficient matrix As for steady aerodynamic

analysis was calculated using either the Vortex Lattice Method

(VLM) [30] or the Doublet Lattice Method (DLM) [23]. Then, they

were transformed to the structural coordinates as following

ð48Þ

where Asteady is the steady aerodynamic coefficient matrix at the

aerodynamic control points. GN1 and GN2d are transformation

matrices calculated from the element bending shape functions.

ZZ

GN1 ¼ TT

NTb dxdyK

T and K are geometric transformation matrices that connect

between the global structural coordinates and the element local

coordinates.

GN2d ¼ TT

n

X

NT K

ð50Þ

i¼1

n represents the number of aerodynamic control points in each

element.

In flutter analysis, the unsteady aerodynamic coefficient matrix

Asd was calculated using the DLM.

Asd ¼ GN1 AÀ1

unsteady BcGN2f

ð51Þ

Aunsteady is the unsteady aerodynamic coefficient matrix at the aerodynamic control points. GN2f is calculated as GN2d, by considering

the lateral displacement w. Bc is a boundary conditions matrix calculated as

Bc ¼

iÃk

; À1

br

!

ð52Þ

Laminated plate elastic analysis

Problem formulation

To present a complete picture regarding the differences

between the considered finite elements, a comparison between

them for the analysis of a composite laminated plate was established. A square plate was considered with 25 cm side length and

1 cm total thickness [32]. The lamina material properties were

EL = 52.5 MPa, ET = 2.1 MPa, GLT = 1.05 MPa, tL = 0.25. Reddy [32]

presents an analytical solution for the maximum displacement of

the square plate subject to a distributed pressure of 1 N/cm2.

Two laminate configurations were considered as well as two different boundary conditions. The analytical results are represented in

the following section.

The mathematical model for this problem is exactly the same as

that of the elastic deformation problem, but with imposing the

effect of composite material on the stress-strain relation. More

details can be found in Reddy [32].

Results and discussion

The selected finite elements are tested by the four problems

described in the previous section, and the results are demonstrated

in this section. All the analyses have been performed on a personal

computer with an i7 processor CPU @ 3.6 GHz, intel core and 16 GB

RAM. The results for each analysis are shown in the next

subsection.

The dynamic elastic analysis

Aerodynamic analysis and aero-elastic coupling

As ¼ GN1 AÀ1

steady GN2d

643

ð49Þ

The problem of the square plate presented in the previous section was implemented on a computer code using MATLAB software.

Table 1 shows the predicted natural frequencies. The first column

contains the analytical solution [20] for the first five natural frequencies, while the finite element results are listed in the rest of

the columns. Results have shown that the natural frequencies predicted by the different types of finite elements are in general less

than those predicted by the analytical model. The frequency values

obtained by using the Linear Triangular Element are farthest from

the analytical ones and the number of elements (N_elem) needed

to reach convergence is a maximum. On the other hand, the frequencies obtained by using the Linear Quadrilateral Element based

on deformation modes are closest to the analytical solution. The

644

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 1

The natural frequencies of clamped square plate [Hz].

Mode

Analytical (20)

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

1

2

3

4

5

Nelem

28.47

58.06

58.06

85.66

104.09

–

26.2

53.35

53.35

78.35

95.54

800

26.54

54.06

54.06

79.09

96.91

225

26.81

54.78

54.78

81.49

98.29

100

26.68

54.39

54.39

80.3

97.42

100

26.61

54.27

54.25

80.01

97.24

100

Fig. 3. The average error and execution time of each finite element in the frequency analysis.

Table 2

The product of the average error percentage and processing time for various finite

elements.

Mode

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

avg Â Time

125.5

23.8

7.7

25.8

101

Table 3

Max. displacement and stress over the plate wing under aerodynamic load.

Element

LINTRI

LINQUAD

MKQ12

QUAD8NO

QUAD9NOD

dmax [mm]

rmax [MPa]

Nelem

Time [s]

19.1

28.9

192

20.4

20.3

32.9

120

5.2

20.3

33.2

120

8.1

20.4

34.4

60

6

19.9

34.1

60

7.9

MKQ12 performed even better than the higher order elements,

which were expected to produce accurate results in bending analysis. The number of elements needed to reach convergence in the

linear quadrilateral element based on deformation modes is also

the same as those of higher order elements.

The average error percentage and processing time of each element are demonstrated in Fig. 3. The product of the average error

percent and processing time can be used as a measure of excellence of the finite element, where the best element has the minimum value for the product. This product is given in Table 2,

which shows that the linear quadrilateral element based on deformation modes is the best element to use in this type of problems.

Plate wing stress and displacement analysis

The elastic performance of a plate wing under steady aerodynamic load was studied using the five finite elements under consideration. The values of the maximum Von Mises stresses and

maximum displacements are tabulated in Table 3 and plotted

in Fig. 4 together with the execution time. Fig. 5 shows the distribution of the Von Mises stress in the wing for each element

type. The stresses are calculated in the case of triangular elements over the mid-side points and then averaged over the element. In case of the quadrilateral elements, the stresses are

determined at the element integration points, and then averaged

over the element.

Since there was neither analytical nor experimental data available for this model, the error percentage cannot be computed for

this particular problem. However, it is clear from Fig. 5 that the

stress distribution resulting from using higher order elements

(QUAD8NOD and QUAD9NOD) is the smoothest and most realistic.

This can be attributed to the higher order interpolation functions

for displacements, which render the stress distribution (derived

from the displacement derivatives) continuous. Hence, if all the

results are considered together, the best performing element can

be considered to be the QUAD8NOD element, which results in

accurate displacement and stress distributions, together with a

reasonable computational time. Following this element comes

the QUAD9NOD in the second place. On the other hand, the LINTRI

element comes as the worst element for wing stress analysis from

the point of view of computation time and stress distribution as

seen in Fig. 5.

645

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Tme

32.9

LINTRI

LINQUAD

7.9

19.9

20.4

6

8.1

20.3

5.2

20.3

19.1

20.4

28.9

34.1

34.4

σ

33.2

d

MKQ12

QUAD8NOD

QUAD9NOD

Fig. 4. Max. displacement and stress of the plate wing in addition to the executing time.

Fig. 5. The Von Mises stresses for each element model.

646

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 4

Divergence speed of the plate wing related to different laminate configurations.

Laminate configuration

[0 0 90 90 0 0]

[45 À45 0 0 À45 45]

[45 45 0 0 45 45]

[À45 À45 0 0 À45 À45]

[30 30 0 0 30 30]

[À30 À30 0 0 À30 À30]

Nelem

FE mesh

DLM mesh

Divergence/Flutter Speed [m/s]

Exp [19]

LINTRI [31]

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

25F

>32

28F

12.5D

27F

11.7D

–

–

–

25.4/26.4

47.5F

27.8F

12.7/29.1

27.4F

12.8/48.1

48

3Â8

8 Â 12

25.4/24.47

43.8F

26.1F

11.4/26.9

26.1F

11.58/33.7

96

6 Â 16

6 Â 12

25.5/25.9

46.8F

27.6F

11.5/29

27.16F

11.67/35.09

96

6 Â 16

6 Â 12

25.4F

45.6F

29.2F

12.86/23.5

28.1F

13/31.4

12

2Â6

7 Â 12

52.6/25.3

46.6F

29F

12.88/32.4

27.8F

13/30.6

12

2Â6

6 Â 12

Table 5

The error % in each analysis and the computation time.

Laminate configuration

[0 0 90 90 0 0]

[45 À45 0 0 À45 45]

[45 45 0 0 45 45]

[À45 À45 0 0 À45 À45]

[30 30 0 0 30 30]

[À30 À30 0 0 À30 À30]

avg %

Time [s]

avg Â Time

Error %

LINTRI

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

5.6

–

0.7

1.6

1.5

5.6

2.3

120

276

2.1

–

6.8

8.8

3.3

2.1

5.2

20.3

106

3.7

–

1.4

7.7

0.6

3.7

3.4

22.8

77.5

1.6

–

4.4

2.9

4.2

1.6

3.3

17.6

58.1

1.3

–

3.6

3.1

3.2

1.3

2.8

18.6

52.1

choice for wing aero-elastic analysis, followed by the QUAD8NOD

element.

Composite plate wing aero-elastic analysis

The static and dynamic aero-elastic analysis of a composite

plate wing was carried out using the five elements. The results

are listed in Table 4 for the smaller of the divergence and Flutter

speeds. The wing aero-elastic analysis was performed for different

laminate configurations. The subscript D refers to the Divergence

speed while subscript F refers to the Flutter speed. The error percent, the average error, and the computation time are listed in

Table 5 and plotted in Fig. 6. Considering the minimum value of

the product of the average error and execution time to be the sign

of excellence, we find that the QUAD9NOD element was the best

Laminated plate elastic analysis

The square laminated plate was analyzed for [0, 90]o and

[À45, 45]° laminate configurations. Two boundary conditions were

considered; in the first all the plate sides were simply supported,

and in the second all the plate sides were clamped [32]. The analytical results are listed in Table 6, and the average error and computation time for each element are depicted in Fig. 7. The analyses are

obtained for the maximum normalized bending displacement,

Fig. 6. The average error and computation time for each element.

647

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Table 6

Laminated elastic plate maximum normalized displacement.

BC’s

Laminate config

Analytical

LINTRI

LINQUAD

MKQ12

QUAD8NOD

QUAD9NOD

Simply supported

[0, 90]

[À45, 45]

[0, 90]

[À45, 45]

Nelem

1.6955

0.6773

0.3814

0.3891

–

1.719

0.6902

0.3952

0.3901

200

1.61

0.8554

0.3806

0.294

144

1.606

0.8721

0.371

0.2865

144

1.6958

0.7198

0.4096

0.4229

36

1.6996

0.6925

0.3968

0.4078

25

Fixed

Fig. 7. The average error and computational time of laminated plate elastic analysis.

¼

w

100wmax ET t3tot

L4 P

ð53Þ

where wmax is the plate maximum thickness, t tot is the total

laminate thickness, L is the side length, and P is the applied pressure

load.

The results have shown that the linear triangular and the

9-node quadrilateral elements are the best elements for laminated

composite analysis from the accuracy point of view. However, the

linear triangular element (LINTRI) needs higher number of elements, and subsequently, longer computational time. On the other

hand, the worst elements for laminated composite analysis were

the LINQUAD and the MKQ12 elements as they have the maximum

relative average error.

Conclusions

In the present paper, five different thin shell finite elements

were considered. The five elements were the Linear Triangular Element (LINTRI), the Linear Quadrilateral Element (LINQUAD), the

Linear Quadrilateral Element Based on Deformation Modes

(MKQ12), the 8-Node Quadrilateral Element (QUAD8NOD), and

the 9-Node Quadrilateral Element (QUAD9NOD). A simple and

detailed mathematical model to derive the interpolation functions

and the stiffness matrix of each element was presented. The basis

functions were selected from the well-known Pascal Triangle to

minimize the order of the interpolation functions. However, singularities existed and specific terms had to be removed and replaced

with other terms to eliminate the source of singularities. The five

elements were tested using several elastic and aero-elastic analyses through three carefully selected bench mark problems with

analytical or experimental results available in the literature. In

order to have a fair comparison, a convergence analysis was conducted for each element and the minimum number of elements

needed for convergence was used in the comparison.

From the present investigation, it was found that the MKQ12

element was the best choice for elastic free vibration analysis of

a plate, since it yields the most accurate results with the minimum

execution time. The second choice is the QUAD8NOD element, and

the worst results are produced by the LINTRI element. In case of

elastic thin shells, and if the stress analysis was sought, the most

accurate elements are naturally the higher order elements. From

the point of view of time and accuracy, the best element was found

to be the QUAD8NOD element, and the QUAD9NOD element comes

out second. The worst element for this kind of analysis was the

LINTRI element, which requires longer computational times and

produces discontinuous stress distributions.

For aero-elastic analysis, the most accurate results are obtained

by using the LINTRI element, however, it requires the longest computational time. The best element for this type of analysis was

found to be the QUAD9NOD element considering its accuracy

and computational time. The second choice was the QUAD8NOD

element. It is worth noting that in spite of its bad performance in

elastic analysis, the LINTRI element was found to be more consistent to use with the Vortex Lattice Method in the aero-elastic analysis, but it requires long computation time. In laminated composite

plate analysis, the best recommended elements are either the LINTRI or QUAD9NOD. However, the LINTRI element requires dense

mesh, and subsequently longer computational time. The present

results can serve many researchers and engineers interested in

elastic and aero-elastic analyses, especially those who find difficulties in finding the detailed formulation of the finite elements, and

those who are confused in selecting specific elements for specific

application.

648

M. Mahran et al. / Journal of Advanced Research 8 (2017) 635–648

Conflict of Interest

The authors have declared no conflict of interest.

Compliance with Ethics requirements

This article does not contain any studies with human or animal

subjects.

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