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 8- Node 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.
Trang 1Original Article
A comparison between different finite elements for elastic and
aero-elastic analyses
Mohamed Mahrana,⇑, Adel ELsabbaghb, Hani Negma
a Aerospace Engineering Department, Cairo University, Giza 12613, Egypt
b
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 func-tions and the element equafunc-tions related to each element were presented through a detailed mathemat-ical 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 8-Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-8-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/)
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.
Peer review under responsibility of Cairo University.
⇑ Corresponding author.
E-mail address: abdu_aerospace@eng.cu.edu.eg (M Mahran).
Contents lists available atScienceDirect
Journal of Advanced Research
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j a r e
Trang 2Numerical 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
sys-tems 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
continu-ous 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
alge-braic equations, which can be solved for the unknown solution
vari-ables of the structure The accuracy of the FEM solution depends on
many factors, such as the interpolation polynomials and
subse-quently the element shape functions, the number of degrees of
free-doms selected for each element, the mesh size, and the type of
element used The model accuracy is a result of the deep
under-standing of the effect of each factor on the final results
The selection of the element interpolation functions is a key
fac-tor 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
lit-erature, 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
inter-ested 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
of the structure Therefore, designers of these structures are con-strained 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 prob-lems in structures and aero-elasticity The results of the selected elements are compared based on (1) solution accuracy of each ele-ment, (2) number of elements needed to achieve convergence, and (3) computational time The comparison is for free vibration anal-ysis, stress analanal-ysis, aero-elastic analanal-ysis, and laminated composite analysis Five different elements are selected for the present com-parison 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 deforma-tion 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 prob-lems from the literature[19,20]for elastic and aero-elastic analy-ses 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 com-posite 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
Nomenclature
Symbol
A coefficient matrix for in-plane action
As steady aerodynamic coefficient matrix in structural
coordinates
Asd unsteady aerodynamic coefficient matrix in structural
coordinates
Avlm steady aerodynamic coefficient matrix
B strain-displacement matrix
d displacement in global coordinates
D stress-strain matrix (isotropic material properties
ma-trix)
J Jacobian Matrix for first order derivatives
K stiffness matrix
M mass matrix
N shape function matrix
w structural bending displacement field
W structural bending nodal displacements
x, y, z element local coordinates
d displacement vector in local coordinates
X, Y, Z structural global coordinates
V volume
E elasticity modulus of the wing material
Vinf flow speed
q1 dynamic pressure
r stress
br reference length (half the wing root chord)
JJ Jacobian matrix for second order derivatives
k reduced frequency
t plate wing thickness
1 wing damping ratio
x flutter frequency
e strain vector
n,g reference element coordinates
q air density
Trang 3(2) Assume suitable interpolation polynomials for both the
in-plane Ppand bending Pbdisplacement fields
(3) Calculate the coefficients of these polynomials by applying
the nodal movement conditions
(4) Determine the shape functions for both the in-plane Npand
bending Nbactions
(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 densityq, and finally,
(8) The structural matrices K and M can be obtained by
assem-bling 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,
strain-displacement 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
soft-ware 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
subse-quently the elements’ equations are derived
In-plane action
An interpolation function is chosen from Pascal’s Triangle[1,2],
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
U¼
u1
unnod
8
>
>
>
>
>
>
9
>
>
>
>
>
>
ð3Þ
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,
The in-plane strain displacement matrix can be obtained from the
shape functions’ derivatives by using the Jacobian matrix definition
Bp¼
ex
ey
cxy
8
>
>
9
>
>¼
u;x
u;y
u;yþv;x
8
>
>
9
>
>¼
Np;x
Np;y
Np;yþ Np;x
8
>
>
9
>
>
Jdet
J4 J2
J3 J1
J4 J3 J1 J2
2
64
3
75 Np ;n
Np ; g
U
ð5Þ
where J1; J2; J3; J4are the elements of the Jacobian matrix and Jdetis
the Jacobian determinant
J¼ J1 J2
J3 J4
¼ x;n y;n
x;g y;g
; Jdet¼ J1J4 J3J2 ð6Þ
Bending action
An interpolation function is chosen either from Pascal’s Triangle
or based on the displacements modes,
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
hx¼dw
dy; and hy¼ dw
Note that the Jacobian matrix elements are rearranged in Jso that the displacement rotations are defined as,
hx
hy
¼ 1
Jdet
J3 J1
J4 J2
;n
w;g
¼ J1
w;n
w;g
ð9Þ
Then, the three bending displacements can be calculated from the equation,
w
hx
hy
8
>
>
9
>
>¼
0 J1
Pb ;n
Pb ; g
8
>
>
9
>
>a¼
0 J1
a is the coefficients vector to be determined from the out of plane nodal displacements W The bending shape functions will have the form
Nb¼ PbC1 1 0
0 J
ð11Þ
where the C matrix is calculated from the c matrix after applying the nodal boundary conditions, and
w
hx
hy
8
>
>
9
>
>¼ Nb
w1
hx1
hy1
wnnod
hxnnod
hynnod
8
>
>
>
>
<
>
>
>
>
:
9
>
>
>
>
=
>
>
>
>
;
Then the strain-displacement matrix can be derived and simpli-fied from the Jacobian definition for second order derivatives They were derived and simplified by the authors to have the form
Bb¼
wxx
wyy
2wxy
8
>
>
9
>
>¼ JJ1
wnn
wgg
2wn g
8
>
>
9
>
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,
wnn
wgg
2wn g
8
>
>
9
>
>¼
J21 J22 J1J2
J2 J2 J3J4 2J1J3 2J2J4 J2J3þ J1J4
2 64
3
75 wwyyxx
2wxy
8
>
>
9
>
>¼ JJ:
wxx
wyy
2wxy
8
>
>
9
>
> ð14Þ
The inverse of the Jacobian Matrix for the second order deriva-tives is
Trang 4JJ1¼ 1
J2det
J24 J22 J2J4
J2 J2 J1J3
2J3J4 2J1J2 J2J3þ J1J4
2
64
3
Based on this simple and detailed mathematical
implementa-tion, the considered elements’ equations can be derived All the
shape functions for those elements are presented in the following
sections
Notice that Pband Ppare represented as row vectors all over the
present paper
The linear triangular element (LINTRI)
The LINTRI thin-shell element has three nodes The element has
six degrees of freedom per node with a total of 18 degrees of
free-dom.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
and subsequently the in-plane shape functions have the form
For bending action
The interpolation polynomial for bending action based on the
element area coordinates has the form
Pb¼ n;g; 1 g n;gn; gð1 þgþ nÞ; nð1 þgþ nÞ;gn2;
g2ð1 þgþ nÞ; nð1 þgþ nÞ2
ð18Þ
and subsequently the bending shape functions are
Nb1¼ ð1 þgþ nÞð2g2þgð1 þ 2nÞ þ ð1 þ nÞð1 þ 2nÞÞ
Nb2¼ ð1 þgþ nÞðJ4ð1 þgÞgþ J2nð1 þgþ nÞÞ
Nb3¼ ð1 þgþ nÞðJ3ð1 þgÞgþ J1nð1 þgþ nÞÞ
Nb4¼ gð2g2þ 2ð1 þ nÞn þgð3 þ 2nÞÞ
Nb5¼gðJ2ð1 þ nÞn þ J4ðg2þgð1 þ nÞ þ ð1 þ nÞnÞÞ
Nb6¼ gðJ1ð1 þ nÞn þ J3ðg2þgð1 þ nÞ þ ð1 þ nÞnÞÞ
Nb7¼ nð3n þ 2ðg2þgð1 þ nÞ þ n2ÞÞ
Nb8¼ nðJ4gnþ J2ðg2þgð1 þ nÞ þ ð1 þ nÞnÞÞ
Nb9¼ nðJ3gnþ J1ðg2þgð1 þ nÞ þ ð1 þ nÞnÞÞ
ð19Þ
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
refer-ence 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
and subsequently the in-plane shape functions have the form
Np¼1
4fð1 gÞð1 nÞ ð1 gÞð1 þ nÞ ð1 þgÞð1 þ nÞ ð1 þgÞð1 nÞg
ð21Þ
For bending action The interpolation basis functions for bending action selected from Pascal’s Triangle has the form
Pb¼ f1; n;g; n2;gn;g2; n3;gn2;g2n;g3;gn3;g3ng ð22Þ
Nb1¼ 1
8ð1 þgÞð1 þ nÞð2 þgþg2þ n þ n2Þ
Nb2¼ 1
8ð1 þgÞð1 þ nÞðJ4ð1 þg2Þ þ J2ð1 þ n2ÞÞ
Nb3¼1
8ð1 þgÞð1 þ nÞðJ3ð1 þg2Þ þ J1ð1 þ n2ÞÞ
Nb4¼1
8ð1 þgÞð1 þ nÞðgþg2þ ð2 þ nÞð1 þ nÞÞ
Nb5¼1
8ð1 þgÞð1 þ nÞðJ2þ J4ð1 þg2Þ J2n2Þ
Nb6¼1
8ð1 þgÞð1 þ nÞðJ3 J3g2þ J1ð1 þ n2ÞÞ
ð23Þ
and subsequently the bending shape functions are
Nb7¼ 1
8ð1 þgÞð1 þ nÞðð1 þgÞgþ ð2 þ nÞð1 þ nÞÞ
Nb8¼1
8ð1 þgÞð1 þ nÞðJ4ð1 þg2Þ þ J2ð1 þ n2ÞÞ
Nb9¼ 1
8ð1 þgÞð1 þ nÞðJ3ð1 þg2Þ þ J1ð1 þ n2ÞÞ
Nb10¼1
8ð1 þgÞð1 þ nÞð2 þ ð1 þgÞgþ n þ n2Þ
Nb11¼1
8ð1 þgÞð1 þ nÞðJ4 J4g2þ J2ð1 þ n2ÞÞ
Nb12¼1
8ð1 þgÞð1 þ nÞðJ1þ J3ð1 þg2Þ J1n2Þ
ð24Þ
The linear quadrilateral element based on deformation modes (MKQ12)
The MKQ12 element has four nodes It has six degrees of free-dom per node with a total of 24 degrees of freefree-dom It has the glo-bal, local, and reference coordinates shown inFig 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
Pb¼
1;n;g;gn;0:5ð1 þn2Þ;0:5ð1 þg2Þ;0:5nð1 þ n2Þ;0:5gð1 þg2Þ; 0:25ð1 þg2Þnð3 n2Þ;0:25gð3 g2Þð1 þ n2Þ;
0:25gð1 þg2Þnð3 n2
Þ;0:25gð3 g2Þnð1 þ n2
Þ
8
>
>
9
>
> ð25Þ The shape functions are then
Trang 5Nb1¼18ð1 þgÞð1 þ nÞð2 gg2 n þgnþg2n n2þgn2þg2
n2Þ
Nb2¼1
16ð1 þgÞ2ð1 þ nÞ2ð2J1þ 2J3þ J1gþ 2J3gþ 2J1nþ J3nþ J1gnþ J3gnÞ
Nb3¼1
16ð1 þgÞ2
ð1 þ nÞ2
ð2J2þ 2J4þ J2gþ 2J4gþ 2J2nþ J4nþ J2gnþ J4gnÞ
Nb4¼18ð1 þgÞð1 þ nÞð2 þgþg2 n þgnþg2nþ n2
gn2g2n2Þ
Nb5¼1
16ð1 þgÞ2
ð1 þ nÞ2
ð2J1þ 2J3 J1gþ 2J3gþ 2J1n J3nþ J1gn J3gnÞ
Nb6¼161ð1 þgÞ2
ð1 þ nÞ2
ð2J2þ 2J4 J2gþ 2J4gþ 2J2n J4nþ J2gn J4gnÞ
Nb7¼1
8ð1 þgÞð1 þ nÞð2 þgg2þ n þgng2n n2gn2þg2n2Þ
Nb8¼1
16ð1 þgÞ2
ð1 þ nÞ2
ð2J1 2J3þ J1gþ 2J3gþ 2J1nþ J3n J1gn J3gnÞ
Nb9¼161ð1 þgÞ2
ð1 þ nÞ2
ð2J2 2J4þ J2gþ 2J4gþ 2J2nþ J4n J2gn J4gnÞ
Nb10¼1
8ð1 þgÞð1 þ nÞð2 gþg2þ n þgng2nþ n2þgn2g2
n2Þ
Nb11¼161ð1 þgÞ2
ð1 þ nÞ2
ð2J1 2J3 J1gþ 2J3gþ 2J1n J3n J1gnþ J3gnÞ
Nb12¼1
16ð1 þgÞ2ð1 þ nÞ2ð2J2 2J4 J2gþ 2J4gþ 2J2n J4n J2gnþ J4gnÞ
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
refer-ence coordinates The element interpolation and shape functions
were derived in the following
For in-plane action
The interpolation polynomial for in-plane action selected from
Pascal’s Triangle has the form
Pp¼ 1; n; g; gn; n2; g2; n2
g; g2n
ð27Þ
and subsequently the in-plane shape functions have the form
Np1¼1
4ð1 gÞð1 nÞð1 g nÞ;Np2¼1
4ð1 gÞð1 þ nÞð1 gþ nÞ
Np3¼1
4ð1 þgÞð1 þ nÞð1 þgþ nÞ;Np4¼1
4ð1 þgÞð1 nÞð1 þg nÞ
Np5¼1
2ð1 gÞð1 nÞð1 þ nÞ;Np6¼1
2ð1 gÞð1 þgÞð1 þ nÞ
Np7¼1
2ð1 þgÞð1 nÞð1 þ nÞ;Np8¼1
2ð1 gÞð1 þgÞð1 nÞ
ð28Þ
For bending action The interpolation polynomial for bending action was selected carefully from the well-known Pascal Triangle Initially, the follow-ing basis functions were selected;
Pb¼ 1; n;g; n2;gn;g2; n3;gn2;g2n;g3; n4;gn3;g3n;g2n2
g4; n5;gn4;g2n3;g3n2;g4n;g5;g2n4;g3n3;g4n2
ð29Þ
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 termg2n2, and the bicubic termg3n3all yield sim-ilar equations Therefore, the biquadratic and bicubic terms were replaced withgn5andg5n Finally, the bending interpolation func-tion for the QUAD8NOD element is;
Pb¼ 1; n;g; n2
;gn;g2; n3
;gn2;g2n;g3; n4
;gn3;g3n;
g4; n5
;gn4;g2
n3;g3
n2;g4n;g5;gn5;g2
n4;g4
n2;g5
n
ð30Þ
Fig 1 Finite elements local and reference coordinates.
Trang 6This choice eliminates all the singularities, and subsequently
the bending shape functions are
Nb16¼1
4ð1 þg2Þð1 þ nÞð2g2þ ð2 þ nÞð1 þ nÞÞ
Nb17¼1
12ð1 þg2Þð1 þ nÞð6J4gð1 þg2Þ þ J2ð1 þ nÞðg2 ð1 þ nÞð1 þ 2nÞÞÞ
Nb18¼ 1
12ð1 þg2Þð1 þ nÞð6J3gð1 þg2Þ þ J1ð1 þ nÞðg2 ð1 þ nÞð1 þ 2nÞÞÞ
Nb19¼1
4ð1 þgÞð1 þ n2Þð2 gþg2þ 2n2Þ
Nb20¼ 1
12ð1 þgÞð1 þ n2ÞðJ4ð1 þgÞð1 þgð3 þ 2gÞ n2Þ 6J2nð1 þ n2ÞÞ
Nb21¼1
12ð1 þgÞð1 þ n2ÞðJ3ð1 þgÞð1 þgð3 þ 2gÞ n2Þ 6J1nð1 þ n2ÞÞ
Nb22¼ 1
4ð1 þg2Þð1 þ nÞð2 þ 2g2þ n þ n2Þ
Nb23¼ 1
12ð1 þg2Þð1 þ nÞð6J4gð1 þg2Þ þ J2ð1 þ nÞð1 þg2þ ð3 2nÞnÞÞ
Nb24¼1
12ð1 þg2Þð1 þ nÞð6J3gð1 þg2Þ þ J1ð1 þ nÞð1 þg2þ ð3 2nÞnÞÞ
ð32Þ
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-bal, local, and reference coordinates shown inFig 1d The element interpolation and shape functions were derived in the following
For in-plane action The interpolation polynomial for in-plane action selected from Pascal’s Triangle has the form
Pp¼ 1; n; g; gn; n2; g2; n2g; g2n; n2g2
ð33Þ
and subsequently the in-plane shape functions have the form
Np1¼1
4ðgþg2Þðn þ n2
Þ; Np2¼1
4ðgþg2Þðn þ n2Þ
Np3¼1
4ðgþg2Þðn þ n2Þ; Np4¼1
4ðgþg2Þðn þ n2Þ
Np5¼1
2ðgþg2Þð1 n2Þ; Np6¼1
2ð1 g2Þðn þ n2Þ
Np7¼1
2ðgþg2Þð1 n2
Þ; Np8¼1
2ð1 g2Þðn þ n2Þ
Np9¼ ð1 g2Þð1 n2Þ
ð34Þ
For bending action
The interpolation polynomial for bending action was selected carefully from the well-known Pascal Triangle Initially, the follow-ing basis functions were selected;
Nb1¼ 1
8ð1 þgÞð1 þ nÞðg3þ 3g4gð1 þ nÞ2g2ð5 þ nÞ þ ð1 þ nÞð1 þ nÞ2ð2 þ 3nÞÞ
Nb2¼ 1
24
J4ð1 þg2Þð1 þ nÞð3g3þg2ð1 2nÞ þ 3gð1 þ nÞ þ ð1 þ nÞð1 þ nÞ2Þ
þJ2ð1 þgÞð1 þ n2Þð1 þg2þg3þgð1 þ ð3 2nÞnÞ þ nð3 þ n 3n2ÞÞ
!
Nb3¼ 1
24ð1 þgÞð1 þ nÞ J3ð1 þgÞð1 þ 3g3þ n 3gð1 þ nÞ n2ð1 þ nÞ þg2ð1 þ 2nÞÞ
J1ð1 þ nÞð1 þg2þg3þgð1 þ ð3 2nÞnÞ þ nð3 þ n 3n2ÞÞ
!
Nb4¼1
8ð1 þgÞð1 þ nÞðg3þ 3g4þg2ð5 þ nÞ gð1 þ nÞ2þ ð1 þ nÞ2ð1 þ nÞð2 þ 3nÞÞ
Nb5¼ 1
24ð1 þgÞð1 þ nÞ J4ð1 þgÞð3g3þ 3gð1 þ nÞ þ ð1 þ nÞ2ð1 þ nÞ g2ð1 þ 2nÞÞ
J2ð1 þ nÞðg2þg3gð1 þ nÞð1 þ 2nÞ þ ð1 þ nÞð1 þ nÞð1 þ 3nÞÞ
!
Nb6¼ 1
24ð1 þgÞð1 þ nÞ J3ð1 þgÞð3g3þ 3gð1 þ nÞ þ ð1 þ nÞ2ð1 þ nÞ g2ð1 þ 2nÞÞ
J1ð1 þ nÞðg2þg3gð1 þ nÞð1 þ 2nÞ þ ð1 þ nÞð1 þ nÞð1 þ 3nÞÞ
!
Nb7¼ 1
8ð1 þgÞð1 þ nÞðg3þ 3g4þg2ð5 þ nÞ þgð1 þ nÞ2þ ð1 þ nÞ2ð1 þ nÞð2 þ 3nÞÞ
Nb8¼ 1
24ð1 þgÞð1 þ nÞ J4ð1 þgÞð1 þgð3 þgþ 3g2Þ þ n þgð3 þ 2gÞn þ n2 n3Þ
þJ2ð1 þ nÞðg2g3þgð1 þ nÞð1 þ 2nÞ þ ð1 þ nÞð1 þ nÞð1 þ 3nÞÞ
!
Nb9¼ 1
24ð1 þgÞð1 þ nÞ J3ð1 þgÞð1 þgð3 þgþ 3g2Þ þ n þgð3 þ 2gÞn þ n2 n3Þ
þJ1ð1 þ nÞðg2g3þgð1 þ nÞð1 þ 2nÞ þ ð1 þ nÞð1 þ nÞð1 þ 3nÞÞ
!
Nb10¼1
8ð1 þgÞð1 þ nÞðð1 þgÞ2ð1 þgÞð2 þ 3gÞ ð1 þgÞ2
nþ ð5 þgÞn2þ n3þ 3n4Þ
Nb11¼ 1
24
J2ð1 þgÞð1 þ n2Þðð1 þgÞ2ð1 þgÞ þ 3ð1 þgÞn ð1 þ 2gÞn2þ 3n3Þ
J4ð1 þg2Þð1 þ nÞð3g3þg2ð1 2nÞ 3gð1 þ nÞ þ ð1 þ nÞð1 þ nÞ2Þ
!
Nb12¼ 1
24
J1ð1 þgÞð1 þ n2Þðð1 þgÞ2ð1 þgÞ þ 3ð1 þgÞn ð1 þ 2gÞn2þ 3n3Þ
þJ3ð1 þg2Þð1 þ nÞð3g3þg2ð1 2nÞ 3gð1 þ nÞ þ ð1 þ nÞð1 þ nÞ2Þ
!
Nb13¼ 1
4ð1 þgÞð1 þ n2Þð2 þgþg2þ 2n2Þ
Nb14¼ 1
12ð1 þgÞð1 þ n2ÞðJ4ð1 þgÞð1 þgð3 þ 2gÞ n2Þ 6J2nð1 þ n2ÞÞ
Nb15¼ 1
12ð1 þgÞð1 þ n2ÞðJ3ð1 þgÞð1 þgð3 þ 2gÞ n2Þ 6J1nð1 þ n2ÞÞ
ð31Þ
Trang 7Pb¼ 1; n;g; n2;gn;g2; n3;gn2;g2n;g3; n4;gn3;g3n;g4;g2n2; n5
;gn4;g2n3;g3n2;g4n;g5;gn5;g2n4;g4n2;g5n;g4n2;g2n4;
ð35Þ
Using the above basis functions yielded singular C matrix The
rank of the matrix turned out to be 23 instead of 24, which
indi-cated 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;
Pb¼ 1; n;g;gn; n2
;g2; n3
;gn2;g2n;g3; n4
;gn3;g3n;g4; n5
;gn4;
g4n;g5;gn5;g2n4;g4n2;g5n;g2n5;g5n2;1ðg2n2þg3n3Þ
ð36Þ
and subsequently the bending shape functions are
Nb17¼1
4ð1 þg2Þnð1 þ nÞð2J4gð1 þg2Þ þ J2ðgð1 þ nÞ þ n n3ÞÞ
Nb18¼ 1
4ð1 þg2Þnð1 þ nÞð2J3gð1 þg2Þ þ J1ðgð1 þ nÞ þ n n3ÞÞ
Nb19¼1
4gð1 þgÞð1 þ n2Þðgð1 þgÞð4 þ 3gÞ 6ð1 þgÞn þ 2n2Þ
Nb20¼ 1
4gð1 þgÞð1 þ n2ÞðJ4ð1 þgÞðgþg2 nÞ 2J2nð1 þ n2ÞÞ
Nb21¼1
4gð1 þgÞð1 þ n2ÞðJ3ð1 þgÞðgþg2 nÞ 2J1nð1 þ n2ÞÞ
Nb22¼1
4ð1 þg2Þð1 þ nÞnð2g2 6gð1 þ nÞ nð4 þ n þ 3n2ÞÞ
Nb23¼ 1
4ð1 þg2Þð1 þ nÞnð2J4gð1 þg2Þ þ J2ðgþ ð1 þgÞn þ n3ÞÞ
Nb24¼1
4ð1 þg2Þð1 þ nÞnð2J3gð1 þg2Þ þ J1ðgþ ð1 þgÞn þ n3ÞÞ
Nb25¼ ð1 þg2Þð1 þ n2Þð1 þg2 3gnþ n2Þ
Nb26¼ ð1 þgÞð1 þgÞð1 þ nÞð1 þ nÞðJ4gð1 þg2Þ þ J2nð1 þ n2ÞÞ
Nb27¼ ð1 þgÞð1 þgÞð1 þ nÞð1 þ nÞðJ3gð1 þg2Þ þ J1nð1 þ n2ÞÞ
ð38Þ
The test problems
It has been mentioned earlier that the aim of the present work was to compare between different shell finite elements with differ-ent behavior for elastic and aero-elastic analyses This will enable
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 mathe-matical 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
Nb1¼1
8ð1 þgÞgð1 þ nÞnð4 þg2þ 3g3þgð2 þ 6nÞ þ nð2 þ n þ 3n2ÞÞ
Nb2¼1
8ð1 þgÞgð1 þ nÞnðJ4ð1 þg3þ n þgnÞ þ J2ð1 þgþgnþ n3ÞÞ
Nb3¼ 1
8ð1 þgÞgð1 þ nÞnðJ3ð1 þg3þ n þgnÞ þ J1ð1 þgþgnþ n3ÞÞ
Nb4¼1
8ð1 þgÞgnð1 þ nÞð8 þgð2 þgÞð5 þ 3gÞ þ 10n þ 6gnþ n2 3n3Þ
Nb5¼1
8ð1 þgÞgnð1 þ nÞðJ4ð1 þg3þgð2 þ nÞ þ nÞ þ J2ð1 þg ð2 þgÞn þ n3ÞÞ
Nb6¼ 1
8ð1 þgÞgnð1 þ nÞðJ3ð1 þg3þgð2 þ nÞ þ nÞ þ J1ð1 þg ð2 þgÞn þ n3ÞÞ
Nb7¼ 1
8gð1 þgÞnð1 þ nÞðg2þ 3g3þgð2 6nÞ þ ð1 þ nÞð4 þ nð2 þ 3nÞÞÞ
Nb8¼1
8gð1 þgÞnð1 þ nÞðJ4ð1 þg3þ n gnÞ þ J2ð1 þggnþ n3ÞÞ
Nb9¼ 1
8gð1 þgÞnð1 þ nÞðJ3ð1 þg3þ n gnÞ þ J1ð1 þggnþ n3ÞÞ
Nb10¼ 1
8gð1 þgÞð1 þ nÞnð8 þ ð2 þgÞgð5 þ 3gÞ þ 10n 6gn n2 3n3Þ
Nb11¼1
8gð1 þgÞð1 þ nÞnðJ2ð1 þgþ ð2 þgÞn þ n3Þ þ J4ð1 þg3þ n gð2 þ nÞÞÞ
Nb12¼ 1
8gð1 þgÞð1 þ nÞnðJ1ð1 þgþ ð2 þgÞn þ n3Þ þ J3ð1 þg3þ n gð2 þ nÞÞÞ
Nb13¼ 1
4ð1 þgÞgðgð4 þgþ 3g2Þ þ 6ð1 þgÞn 2n2Þð1 þ n2Þ
Nb14¼ 1
4ð1 þgÞgð1 þ n2ÞðJ4ðg3þgð1 þ nÞ þ nÞ 2J2nð1 þ n2ÞÞ
Nb15¼1
4ð1 þgÞgð1 þ n2ÞðJ3ðg3þgð1 þ nÞ þ nÞ 2J1nð1 þ n2ÞÞ
Nb16¼1
4ð1 þg2Þnð1 þ nÞð2g2 6gð1 þ nÞ þ nð1 þ nÞð4 þ 3nÞÞ
ð37Þ
Trang 8Study 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 1 m) with thickness t equal 0.003 m and the material
prop-erties, elastic modulus, mass density, and Poisson’s ratio
E¼ 71 Gpa;q¼ 2700 kg=m3; andm¼ 0:3
The plate is fixed along all sides
The mathematical model
The plate natural frequencies are calculated by solving the
eigenvalue problem
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
Keb¼
Z
V
BTbDsBbdV ¼t3
12
ZZ
BTbDsBbJdetdndg
Me¼
Z
V
NTNbdV¼q
ZZ
NTINbJdetdndg
I¼
0 t 3
0 0 t 3
12
2
64
3
75
ð40Þ
The superscript e means that these matrices are calculated over
each element and then assembled in the global coordinates
Dsis the isotropic material stiffness matrix
Ds¼ E
1m2
0 0 1m
2
2
64
3
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
straight-rectangular wing under aerodynamic load The wing geometry was
shown inFig 2 The aerodynamic analysis was performed by using
the Doublet lattice method[23] A convergence analysis was
per-formed to select the suitable number of elements for both the aero-dynamic 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
The mathematical model First the stiffness matrix for both the in-plane and bending actions are derived The stiffness matrix of the bending action Ke
b
is given in Eq.(40) and the in-plane stiffness matrix Ke
p has the form
Ke
p¼ Z
V
BT
pDsBpdV¼ t
ZZ
BT
The elastic problem was solved to find the displacements, then the stresses are calculated using the equation
rx
ry
sxy
8
>
>
9
>
where dpis the local in-plane displacements vector and db is the local bending displacements vector
Wing aero-elastic analysis
Problem formulation 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 ele-ments 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 interpo-lation, 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 config-urations The wing has the same plane form shown inFig 2 The suitable number of elements was selected for both the aerody-namic and finite element analyses throughout convergence analy-ses The lamina material properties were EL= 98 Gpa, ET= 7.9 Gpa,
GLT= 5.6 Gpa,tL= 0.28,q= 1520 kg/m3, t = 0.134e3m Mathematical model
The need to decrease the aircraft structural weight for economic purposes leads to an increase in the aircraft flexibility, and subse-quently the tendency for elastic instability The wing aero-elastic instability was categorized into divergence and flutter anal-yses[25–29] In the former, analysts were interested in determin-ing the minimum speed at which wdetermin-ing 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 ele-ments 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
Trang 9needs 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
trans-formed into the structural nodes by means of either spline
interpo-lation 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
where Asis the aerodynamic stiffness transformed from the
aerody-namic control points into the finite element nodes by using the
ele-ment shape functions [31], and qdiv represents the dynamic
pressure at which divergence takes place
qdiv¼1
where Vdiv 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
deter-mined by solving the eigenvalue problem
K1b Mbþqb
2
r
2k2Asd
!
1þ i1
x2 I
!
where br is a reference length (chosen to be half the wing root
chord), Asd is the unsteady aerodynamic stiffness matrix
trans-formed 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
k¼brx
Vf
ð47Þ
Eq (46) comprises two unknowns; the speed V and the
fre-quencyx, and both can be obtained by iteration where the flutter
occurs at zero damping coefficient1
It is worth noting that Eq.(46)is nested and solved using the
k-method
Aerodynamic analysis and aero-elastic coupling
The aerodynamic coefficient matrix Asfor 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
where Asteadyis the steady aerodynamic coefficient matrix at the
aerodynamic control points GN1 and GN2d are transformation
matrices calculated from the element bending shape functions
GN1¼ TT
ZZ
T and K are geometric transformation matrices that connect between the global structural coordinates and the element local coordinates
GN2d¼ TTXn
i¼1
NT
n represents the number of aerodynamic control points in each element
In flutter analysis, the unsteady aerodynamic coefficient matrix
Asdwas calculated using the DLM
Aunsteadyis the unsteady aerodynamic coefficient matrix at the aero-dynamic control points GN2fis calculated as GN2d,by considering the lateral displacementw Bc is a boundary conditions matrix cal-culated as
Bc¼ i k
br ; 1
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 estab-lished 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 differ-ent boundary conditions The analytical results are represdiffer-ented 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 The problem of the square plate presented in the previous sec-tion was implemented on a computer code using MATLAB software
Table 1shows the predicted natural frequencies The first column contains the analytical solution[20]for the first five natural fre-quencies, while the finite element results are listed in the rest of the columns Results have shown that the natural frequencies pre-dicted 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 fre-quencies obtained by using the Linear Quadrilateral Element based
on deformation modes are closest to the analytical solution The
Trang 10MKQ12 performed even better than the higher order elements,
which were expected to produce accurate results in bending
anal-ysis 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
ele-ment are demonstrated inFig 3 The product of the average error
percent and processing time can be used as a measure of
excel-lence of the finite element, where the best element has the
mini-mum value for the product This product is given in Table 2,
which shows that the linear quadrilateral element based on
defor-mation 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 aerody-namic load was studied using the five finite elements under con-sideration The values of the maximum Von Mises stresses and maximum displacements are tabulated in Table 3 and plotted
inFig 4together with the execution time.Fig 5shows the dis-tribution of the Von Mises stress in the wing for each element type The stresses are calculated in the case of triangular ments over the mid-side points and then averaged over the ele-ment 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 avail-able for this model, the error percentage cannot be computed for this particular problem However, it is clear fromFig 5that 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 inFig 5
Table 1
The natural frequencies of clamped square plate [Hz].
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.
Table 3
Max displacement and stress over the plate wing under aerodynamic load.