Journal of Appplied Mathematiics and Physics,, 2013, 1, 15-199
Published Onlinne November 20013 (http://www
w.scirp.org/journnal/jamp)
http://dx.doi.orgg/10.4236/jamp..2013.16004
Hybrid Po
ost-Proccessing Proced
dure forr
Diisplacem
ment-Baased Pllane Eleements
1
Xiaoming Chen
C
, Song C
Cen2, Jianyu
un Sun1, Yun
ngui Li1
1
2
China Sttate Constructionn Technical Cennter, Beijing, Ch
hina
Department oof Engineering Mechanics,
M
Schoool of Aerospacce, Tsinghua Un
niversity, Beijingg, China
Email:
E
[email protected]
Receivved August 20133
ABSTRAC
CT
In the analysiis of high-rise building, trad
ditional displaccement-based plane elementts are often ussed to get the in-plane
internal forcess of the shear w
walls by stresss integration. L
Limited by thee singular prob
blem producedd by wall holess and the
loss of precisiion induced byy using differen
ntial method too derive strainns, the displaceement-based ellements cannoot always
present accuraacy enough forr design. In thiis paper, the hyybrid post-proccessing proced
dure based on the Hellinger-Reissner
variational priinciple is usedd for improving
g the stress prrecision of twoo quadrilateral plane elemennts. In order to find the
best stress fieeld, three diffeerent forms are assumed forr the displacem
ment-based plane elements AQ4 and AQ4
with drilling DOF.
D
Numericcal results show
w that by usinng the proposedd method, the accuracy of sttress solutions of these
two displacem
ment-based planne elements caan be improvedd.
Keywords: Fiinite Element; Displacementt-Based Plane Element; Hybrid Post-Proceessing Proceduure
1. Introducction
For traditionnal displacem
ment-based eleements deriveed
from the princciple of minim
mum potential energy, the triial
functions are usually assum
med as a polyn
nomial of noddal
displacementss, and then the internal forcess or stresses caan
be derived byy stress-strain relationship. By
B this process
of formulationn, these kindss of elements can present accurate displaccement resultss, otherwise th
he constructioon
procedures off these formulations are also
o easier. But iin
most situationns, the stress rresults are mucch more impoortant than the displacementts such as th
he designing oof
high-rise builldings. The reeinforcements of shear wallls
are dependedd on the inteernal forces integrated
i
from
m
stresses. For the
t sake of diifferential with
h displacemennt,
the accuracy of stresses iss usually loweer than the diisplacement ressults. Many eefforts have been
b
made fo
for
overcoming this
t
shortage, such as the assumed straiin
formulations developed byy MacNeal [1], Piltner annd
Taylor [2].
On the otheer hand, the hyybrid/mixed elements
e
are developed from the modifiedd variational principle
p
of HuuWashisu, suchh as the hybrid method propo
osed by Pian [33],
Pian and Sum
mihara [4]. It is different fro
om the displacement-based ellements. To fformulate these elements, thhe
internal force field or stress field should
d be assumed at
first. Without the differentiaal process whicch is needed fo
for
Open Access
the dissplacement-bassed elements to derive strrains, so
they caan exhibit bettter stress resullts. But they aalso present obbvious disadvaantages. For example,
e
the cconstruction proocedures are much
m
more com
mplex, and the calculation of matrix inversiion and condennse are usuallyy unavoidable, furthermore,
f
th
he accuracy of
o displacemennt results
usuallyy are not as accurate
a
as thhe displacemeent-based
elemennts.
By combining
c
the merits of bothh hybrid elem
ments and
displacement-based elements,
e
a neew method named hybrid poost procedure was proposedd by Cen [5]. In this
are derived frrom dismethodd, the nodal displacements
d
placem
ment-based platte elements at first, after thiis, a new
internall force field which
w
should satisfy
s
the equuilibrium
Equatioon is assumed to substitute into
i
the hybridd energy
functionnal, by the principle
p
of stationary, the undetermined parameters
p
of the assumed internal
i
force ffield can
be solvved out, and fin
nally the new internal forcees can be
calculatted easily. It is seemed veryy effective for improving thee internal forcee accuracy of plate elementt. In this
paper, this
t method iss extended to improve
i
the pllane elements.
2. Hyb
brid Post-P
Processing Procedure
P
For thee hybrid plate elements, the discrete energgy functional can
c be written as follows [5,66]:
JAMP
X. M. CHEN ET AL.
16
 eR  

where
1
T
1
e {M} [ Db ] {M} dA
2 A
[K MM ]    e ([PM ]T [Db ]1[PM ]
A
1
T
1
e {T} [ D s ] {T}dA
2 A
 [PT ]T [Ds ]1[PT ])dA
(1)
 Ae {M}T {κ} dA  Ae {T}T {γ} dA  Ae f z wdA
 s (Tn w  M xn x  M yn y )ds
where Wexp is the work of external forces:
Wexp  Ae f z wdA  s (Tn w  M xn x  M yn y )ds
(2)
1


 eR   e {σ}T D{u}  {σ}T [D]1{σ} dV
V
2


{M}  [PM ]{α M }
(3)
{σ}  [P ]{α  }
j11
0
j12
0
j11  j12 0
0
0
0
0
j21
j11
j22
j12
j21  j22 
j11  j12 
0
j21
0
j22
0
j21  j22
(10)
Substitute Equation (10) into Equation (9), the parameters  i in [P ] can be obtained by using the principle of stationary. The according matrix named [K  ]
and [K  q ] for plane elements are as follows:
{M}  [M x M y M xy ]T
0
[PT ]  
0
(9)
Similar to Equation (4), a new form of stress field of
displacement-based plane element can be assumed as:
where
1    0 0 0 0 0 0 0 0 
[PM ]  0 0 0 0 1    0 0 0 0 
0 0 0 0 0 0 0 0 1    
T
Substitute Equation (7) into Equation (3), the new internal force can be obtained.
Compared with plate elements, it is much easier to extend this method to plane elements. The according hybrid
discrete energy functional of plane elements can be written as [6]:
Assume the new internal force field as follows:
Tx   M x , x  M xy , y 
{T}     
  [PT ]{α M }
Ty   M xy , x  M y , y 
(8)
[K Mq ]  Ae ([PM ] [Bb ]  [PT ] [B s ])dA
T
[K  ]   Ae [P ]T [D]1[P ]tdA
(4)
[K  q ]  Ae [P ]T [Bb ]tdA
(11)
With different matrix of [P ] , the stress field will be
different too. This new method will be used to try to improve the stress accuracy of plane element named
AQ4 and AQ4 [7].
{α M}  1 2  11 12 
T
and j11 , j12 , j21 , j22 are components of the inversion matrix of Jacobian .
After substituting Equation (3) into Equation (1), the
new form of energy functional can be written as:
  e [PM ]T [Db ]1[PM ]dA 
1
 {α }
 eR   {α M }T  A
  e [PT ]T [Ds ]1[PT ]dA  M
2
 A

{α M }T Ae ([PM ]T [Bb ]
(5)
e
By the principle of stationary:

0
{α M }
Open Access
 q1

T
q 2 
T
q3
T
q 4 
T


T
(12)
where
q i   ui
(6)
the parameters of the assumed internal force field can be
written as:
{α M }  [K MM ]1[K Mq ]{q}e
AQ4 and AQ4 are two plane elements with drilling DOF formulated using quadrilateral area coordinate
methods presented by Long et al. [8,9]. They have the
merits of high accuracy and robust against mesh distortions. After having been programmed into the software
for the analysis of high-rise buildings, the results show
that better accuracy is still needed for the stress of shear
walls with holes.
The definition of DOF of these two elements is:
q
[PT ]T [B s ])dA  {q}e  Wexp
e
R
3. Introduction of AQ4θ and AQ4θλ
(7)
vi  i 
T
 i  1, 2,3, 4 
(13)
and i is the additional rigid rotation at element node.
The displacements of element are as follows:
u  u0   u 
where
u 
0
is a polynomial about
(14)
ui and vi , u  is
JAMP
X. M. CHEN ET AL.
the additional displacement field induced only by the
rigid rotation at nodes denoted as  i .
In order to determine the displacement field u 0  , the
shape functions in reference [10] is used, they are as follows:
17
Then the stiffness matrix of element AQ4 can be
solved out easily, and the strain matrix is:
i 1
4
(15)
v N v
0
i 1
0
i i
where:
1
N i 0   gi  2  Li  Li 1  ii gi  2 P
2
 i  1, 2,3, 4 
(16)
and
P
1
(17)
3(L3 L1)(L4 L2)(g2 g3)(L3 L1)(g1 g2)(L4 L2) (g2g4 gg
1 3)
2
1gg
1 3 g2g4
Assume the rotational displacement field as polynomials of quadrilateral area coordinates:
u  1   2 ( L3  L1 )   3 ( L4  L2 )
v  1   2 ( L3  L1 )  3 ( L4  L2 )
(18)
with the conforming Equations as follows, the rotational
displacement can be obtained.
 (u  u )  0
i 1
(19)
   (u  u )  0
i
i 1

lij
i
i
i
(u  u )ds  0
(ij  12,23,34,41)
v  1   2 ( L3  L1 )   3 ( L4  L2 )
 1

u 34   b2 
1
L3 3  2 L1 4 


 L1 L3 
2
g3 g4
v 34   c 2 
 g3 g 4

 1

u 41   b3 
1
L4 4  2 L2 1 


 L2 L4 
2

v
c
g
g
g
g
  41   3 
4 1
 4 1

and bi  yi 1  yi  2 , ci  xi  2  xi 1
(24)
where 1 , 1' , 2 , 2' are the internal parameters.
In order to solve the undetermined parameters, the
conforming Equations are taken as:
l {u }ds  0
(25)
Substitute Equation (24) into (25), the shape functions
of {u } can be formulated out, they are:
(6 g1 g 2 g3 g 4  g1 g3  g 2 g 4 )
6(1  g1 g 4  g 2 g3 )
g1  g 2
( L3  L1 )
6
( g  g 4 )( g1 g 4  g 2 g3 )
 1
( L4  L2 )
6(1  g1 g 4  g 2 g3 )

 1

u 12   b4 
1
L1 1  2 L3 2 


 L1 L3 
2
g1 g 2
v 12   c 4 
 g1 g 2

 1

u 23   b1 
1
L2 2  2 L4 3 


 L2 L4 
2
g2 g3
v 23   c1 
 g2 g3

(23)
After substituting Equation (21) into the stress-strain
relationship, the stress of AQ4 can be obtained.
AQ4 is an improved element based on AQ4 by
adding a displacement field which is induced by internal
parameters.
The additional displacement fields mentioned above
are as follows:
N 1 
where
Open Access

 N i0
N ui
0


x

 x
0
N i
N vi


[Bi ]   0

y
y

 0
0
 N i N i N ui  N vi 
 y
x
y
x 
ij
i
4
and
  4 L1 L3  1' L2 L4  2' ( L3  L1 )( L4  L2 )
 5 L1 L3   6 L2 L4
i
(22)
 4 L1 L3  1 L2 L4  2 ( L3  L1 )( L4  L2 )
  4 ( L3  L1 )( L4  L2 )
4
[ B q ]  [ B1 B 2 B3 B 4 ]
u  1   2 ( L3  L1 )   3 ( L4  L2 )
 4 ( L3  L1 )( L4  L2 )
 5 L1L3   6 L2 L4
(21)
where
4
u 0   N i0ui
ε  Bq qe
(20)

(26)
g1 g 4  g 2 g3
L1L3  L2 L4
1  g1 g 4  g 2 g3
1
N  2  ( g3  g 2 )( L3  L1 )  2( g1  g 2 )( L4  L2 )
3
( L3  L1 )( L4  L2 )
For element AQ4 , N 1 is taken as the final internal shape function, then:
JAMP
X. M. CHEN ET AL.
18
{u  }  N  λ
(27)
In order to raise the complete order of stress field, the
second form is assumed as a polynomial of analytical
trial function method presented by Fu and Long [11]:
where
λ  [1 1' ]T
 N 1
 0
N   
(28)
0 
N 1 
Through the condense calculation, stiffness matrix of
AQ4 can be written as:
k e  k qq   k q T k  1 k q 
(29)
k    B  DB tdA
T
q
q
k     B  DB tdA
T
(30)
k    B  DB tdA
T
q

0
6xy 1 
 0 0 1 0 2y 0 2x

σ   0 1 0 2x 0 2y 0 6xy 0     (34)
1 0 0 0 0 2x 2y 3x2 3y2 
 9 
4.3. Stress Field III
Try to make the three stress components to be independent, the third form of stress field is assumed as follows:
where
qq
4.2. Stress Field II
1 x y xy
 1 

    (35)
1 x y xy
σ  
 

 12 
1 x y xy
q
and  B   is the strain matrix of additional displacement
field. The according stress fields are as follows:
ε    B λ λ
(31)
Combined Equation (21) with (31), the stress field of
AQ4 can be obtained.
4. Hybrid Post-Processing of AQ4θ and
AQ4θλ
In the formulating of stress in elements AQ4 and
AQ4 , the strain matrix B q  and  B   are the
differential results of displacements to coordinates. The
differential calculation lowered the accurate order of
stress. To avoid the differential, Hybrid Post-processing
procedures can be used to improve the stress accuracy.
Based on this theory, three forms of stress fields are presented for these two displacement-based elements.
5. Numerical Examples
5.1. Strict Patch Test
The constant strain/stress patch test using irregular mesh
is shown in Figure 1. Let Young’s modulus E = 1000,
Poisson’s ratio  = 0.25, and thickness of the patch t = 1.
After modified by three different forms of Hybrid Postprocessing, these two elements can produce exact solutions without any problem.
5.2. Cook’s Skew Beam
This example was proposed by Cook et al. [12]. As
shown in Figure 2, a skew cantilever beam subjected to
distributed shear load along its free edge. The results of
max at point A and min at point B are listed in Tables 1
and 2.
6. Conclusion
4.1. Stress Field I
The first form of stress field is assumed as Equation (32).
It is the stress field of a hybrid element developed by
Pian and Wu [6]:
1
a12

b12
σ   1

1 a1b1

a32   1 
 
b32    
a3b3   5 
(32)
Using the traditional method to calculate stress of displacement-based elements, the accuracy will descend for
the reason of differential when strains are derived from
the displacement fields. For improving the stress accuracy, hybrid/mix elements are very effective. However, the
formulations of the hybrid elements are more complicated than those displacement-based elements. Based on
where
a1 
1 4
 i xi
4 i 1
a2 
1 4
 ii xi
4 i 1
b1 
1 4
 i yi
4 i 1
b2 
1 4
 ii yi
4 i 1
Open Access
1 4
i xi (33)
4 i 1
1 4
b3  i yi
4 i 1
a3 
JAMP
X. M. CHEN ET AL.
y
2
3
2
7
E = 1500, μ = 0.25
1000
4
2
6
5
4
2.5
1.5
these two kinds of elements to establish the relationship
between the displacement and the stress or internal force
fields. In this paper, based on this theory, three forms of
stress fields are used to improve the stress of plane elements with drilling DOF. Through the numerical results,
for element AQ4θ , only the second form of stress field
is effective, but for AQ4θ , except for the first form of
stress, the other two forms can present better results than
the source elements. It is proved that the method of hybrid post-process procedure is workable.
Figure 1. Patch test.
y
44
REFERENCES
P=1
B
44
A
E=1500, μ=1/3
x
48
[1]
R. H. MacNeal, “Derivation of Element Stiffness Matrices by Assumed Strain Distributions,” Nuclear Engineering and Design, Vol. 70, No. 1, 1982, pp. 3-12.
ttp://dx.doi.org/10.1016/0029-5493(82)90262-X
[2]
R. Piltner and R. L. Taylor, “A Systematic Constructions
of B-Bar Functions for Linear and Nonlinear Mixed-Enhanced Finite Elements for Plane Elasticity Problems,”
International Journal for Numerical Methods in Engineering, Vol. 44, No. 5, 1997, pp. 615-639.
http://dx.doi.org/10.1002/(SICI)1097-0207(19990220)44:
5<615::AID-NME518>3.0.CO;2-U
[3]
T. H. H. Pian, “Derivation of Element Stiffness Matrices
by Assumed Stress Distributions,” AIAA Journal, Vol. 2,
No. 7, 1964, pp. 1333-1336.
http://dx.doi.org/10.2514/3.2546
[4]
T. H. H. Pian and K. Sumihara, “Rational Approach for
Assumed Stress Finite Elements,” International Journal
for Numerical Methods in Engineering, Vol. 20, No. 9,
1984, pp. 1685-1695.
http://dx.doi.org/10.1002/nme.1620200911
[5]
S. Cen, Y. Q. Long, et al., “Application of the Quadrilateral Area Co-Ordinate Method: A New Element for
Mindlin-Reissner Plate,” International Journal for Numerical Methods in Engineering, Vol. 66, No. 1, 2006, pp.
1-45. http://dx.doi.org/10.1002/nme.1533
[6]
T. H. H. Pian and C. C. Wu, “Hybrid and Incompatible
Finite Element Methods,” Chapman & Hall/CRC, Boca
Raton, 2006.
[7]
X. M. Chen, Y. Q. Long and Y. Xu, “Construction of
Quadrilateral Membrane Elements with Drilling DOF Using Area Coordinate Method,” Engineering Mechanics,
Vol. 20, No. 6, 2003, pp. 6-11.
[8]
Y. Q. Long, J. X. Li, Z. F. Long and S. Cen, “Area Coordinates Used in Quadrilateral Elements,” Communications in Numerical Methods in Engineering, Vol. 15, No.
8, 1999, pp. 533-545.
http://dx.doi.org/10.1002/(SICI)1099-0887(199908)15:8<
533::AID-CNM265>3.0.CO;2-D
[9]
Z. F. Long, J. X. Li, S. Cen and Y. Q. Long, “Some Basic
Formulae for Area Coordinates Used in Quadrilateral
Elements,” Communications in Numerical Methods in
Engineering, Vol. 15, No. 12, 1999, pp. 841-852.
http://dx.doi.org/10.1002/(SICI)1099-0887(199912)15:12
<841::AID-CNM290>3.0.CO;2-A
Figure 2. Cook’s skew beam.
Table 1. Stress at point A and B of Cook’s Beam of AQ4θ.
Method
σAmax
σBmin
2×2
4×4
Field I
0.1791
0.2261
0.2338 −0.1700 −0.1929 −0.2002
Field II
0.1951
0.2298
0.2348 −0.1942 −0.1933 −0.2010
Field III
0.1914
0.2240
0.2319 −0.1769 −0.1938 −0.2009
Source Val.
0.1917
0.2241
0.2377 −0.1877 −0.1939 −0.2060
Ref. Val.
8×8
2×2
0.2362
4×4
8×8
−0.2023
Table 2. Stress at point A and B of Cook’s Beam of AQ4θλ.
Method
σBmin
σAmax
2×2
4×4
Field I
0.1913
0.2271
0.2342 −0.1748 −0.1919 −0.2009
Field II
0.2145
0.2358
0.2364 −0.2084 −0.2032 −0.2027
Field III
0.2147
0.2358
0.2364 −0.2092 −0.2033 −0.2027
Source Val.
0.2498
0.2338
0.2358 −0.1729 −0.1896 −0.2018
Ref. Val.
0.2362
8×8
2×2
4×4
8×8
−0.2023
the Hellinger-Reissner variational principle, hybrid postprocess procedure can take advantage of the merits of
Open Access
19
JAMP
20
X. M. CHEN ET AL.
[10] Z. F.Long, X. M. Chen and Y. Q. Long, “Second-Order
Quadrilateral Plane Element Using Area Coordinates,”
Engineering Mechanics, Vol. 18, No. 4, 2001, pp. 95-101.
[11] X. G. Fu and Y. Q. Long, “Generalized Conforming Quadrilateral Plane Elements Based on Analytical Trial Func-
Open Access
tions,” Engineering Mechanics, Vol. 19, No. 4, 2002, pp.
12-16.
[12] R. D. Cook, D. S. Malkus and M. E. Plesha, “Concepts
and Applications of Finite Element Analysis,” 3rd Edition,
John Wiley & Sons, Inc., New York, 1989.
JAMP