Hybrid Post Processing Procedure for Displacement Based Plane Elements

(1)

http://dx.doi.org

ABSTRAC

In the analysi internal forces loss of precisi present accura variational pri best stress fie with drilling D two displacem

Keywords: Fi

1. Introduc

For tradition from the princ functions are displacements be derived by of formulation curate displac procedures of most situation tant than the high-rise buil are depended stresses. For t the accuracy placement res overcoming t formulations Taylor [2].

On the othe veloped from Washisu, such Pian and Sum ment-based el internal force first. Without

g/10.4236/jamp.

Hy

Di

2_{Department o}

CT

is of high-rise s of the shear w ion induced by acy enough for inciple is used eld, three diffe DOF. Numeric ment-based plan

inite Element;

ction

nal displacem

ciple of minim usually assum s, and then the y stress-strain n, these kinds cement results f these formul ns, the stress r displacement ldings. The re d on the inte the sake of di of stresses is sults. Many e this shortage, developed by er hand, the hy the modified h as the hybrid mihara [4]. It i

lements. To f field or stres the differentia

.2013.16004

brid Po

isplacem

Xiaoming C

1_{China St}

of Engineering M E

building, trad walls by stress y using differen

r design. In thi d for improving

erent forms ar cal results show

ne elements ca

Displacement

ment-based ele mum potential med as a polyn

internal forces relationship. B s of elements s, otherwise th

ations are also results are muc ts such as th einforcements ernal forces i ifferential with s usually lowe

efforts have b such as the y MacNeal [ ybrid/mixed e d variational p method propo s different fro formulate thes s field should al process whic

ost-Proc

ment-Ba

Chen1_{, Song C}

tate Construction Mechanics, Scho Email: chenxiao

Receiv

ditional displac s integration. L ntial method to is paper, the hy g the stress pr e assumed for w that by usin an be improved

t-Based Plane

ements derive energy, the tri nomial of nod s or stresses ca By this proce

can present a he constructio o easier. But i ch more impo he designing o

of shear wal integrated from

h displacemen er than the di

been made fo assumed strai 1], Piltner an elements are d principle of Hu osed by Pian [3 om the displac

e elements, th d be assumed

ch is needed fo

cessing

ased Pl

Cen2_{, Jianyu}

n Technical Cen ool of Aerospac oming00@tsingh ved August 2013

cement-based Limited by the

o derive strain ybrid post-proc recision of two r the displacem ng the proposed

d.

Element; Hyb

ed ial dal an ss c- on in or- of lls m nt, is- for in nd e- u- 3], e- he at for

the dis they ca sent ob tion pro tion of dable, f usually elemen By c displac brid po method placem internal Equatio function mined p be solv calculat ing the paper, t ments.

2. Hyb

For the tional c

Proced

lane Ele

un Sun1_{, Yun}

nter, Beijing, Ch ce, Tsinghua Un

hua.org.cn 3

plane element e singular prob ns, the displace cessing proced o quadrilateral ment-based pl d method, the

rid Post-Proce

splacement-bas an exhibit bett bvious disadva ocedures are m matrix inversi furthermore, th y are not as a nts.

combining the ement-based e ost procedure d, the nodal d ment-based plat l force field w on is assumed nal, by the p parameters of ved out, and fin

ted easily. It i e internal force

this method is

brid Post-P

e hybrid plate can be written

dure for

ements

ngui Li1

hina

niversity, Beijing

ts are often us blem produced ement-based el dure based on

plane elemen ane elements accuracy of st

essing Procedu

sed elements ter stress resul antages. For e much more com

ion and conden he accuracy o accurate as th

merits of both elements, a ne

was proposed displacements

te elements at which should s to substitute i principle of st the assumed i nally the new s seemed very e accuracy of s extended to i

Processing P

elements, the as follows [5,6

r

g, China

sed to get the d by wall holes lements canno the Hellinger-nts. In order to

AQ4 and tress solutions

ure

to derive str lts. But they a example, the c mplex, and the nse are usually of displacemen he displaceme

h hybrid elem ew method na d by Cen [5] are derived fr

first, after thi satisfy the equ into the hybrid ationary, the internal force f

internal force y effective for

plate element improve the pl

Procedure

discrete energ 6]:

in-plane s and the ot always Reissner find the AQ4

of these

rains, so also pre- construc- calcula- y unavoi- nt results ent-based

ments and med hy- . In this rom dis- is, a new uilibrium d energy undeter- field can es can be improv- t. In this lane ele-

(2)

1

{ } [ ] { } 2

1 _{{ } [} _{] { }} 2

{ } { } { } { }

( )

e

e e e

e T

R _A b

T s A

T T

z

A A A

n xn x yn y s

dA

dA dA f wdA

T w M  M  ds



  



  

  





M D M

T D T

M κ T γ

(1)

where Wexp is the work of external forces:

exp _Ae z _s( n xn x yn y)

W 

_

f wdA

_

T w M  M  ds (2) Assume the new internal force field as follows:

, ,

{ } [ ]{ }

M M

x x x xy y

T M y xy x y y

T M M





   

   

_{  } _



   

   

M P α

T P α (3)

where





11 12 11 12

21 22 21 22

11 12 11 12

1 2 11 12

{ } [ ]

1 0 0 0 0 0 0 0 0

[ ] 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0

[ ]

0 0 0 0 0

0 0

{ }

T x y xy

M

T

T M

M M M

j j j j

  

 

   



 

 

  

 

 

 

  _



 



 _

 M

P

α 

(4)

and j11, j12, j21, j22 are components of the inver-

sion matrix of Jacobian .

After substituting Equation (3) into Equation (1), the new form of energy functional can be written as:

1

exp

[ ] [ ] [ ] 1

{ } { }

2 _{[ ] [} _{] [ ]}

{ } ([ ] [ ]

[ ] [ ]) { } e

e

T

M b M

A

e T

R M _T M

T s T

A

T T

M _A M b

T e

T s

dA dA

dA W



 

 

  

_ 

 



  



P D P

α α

P D P

α P B

P B q

(5)

By the principle of stationary:

0

{ }

e R M

 _

 α (6) the parameters of the assumed internal force field can be written as:

1

{ } [ ] [ ]{ }e

M MM Mq



 

α K K q (7)

where

1

[ ] ([ ] [ ] [ ]

[ ] [ ] [ ])

[ ] ([ ] [ ] [ ] [ ])

e

T

MM _A M b M

T

T s T

T T

Mq _A M b T s

dA



  

 



K P D P

P D P

K P B P B

(8)

Substitute Equation (7) into Equation (3), the new internal force can be obtained.

Compared with plate elements, it is much easier to ex- tend this method to plane elements. The according hybrid discrete energy functional of plane elements can be written as [6]:

1

{ } { } { } [ ] { } V 2

e

e T T

R _V D d



 

  _  _

 



σ u σ D σ (9)

Similar to Equation (4), a new form of stress field of displacement-based plane element can be assumed as:

{ } [ ]{σ  P_ α_} (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:

1

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

e e

T A

T q _A b

tdA

  

 



  



K P D P

K P B (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].

3. Introduction of AQ4θ and AQ4θλ

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 distor- tions. 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:

 

       

1 2 3 4

T T

e _ T T T_

  

q q q q q (12)

where

 





T

i  ui vi i

q



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 ₍₁₄₎

where

 

_u0 _{is a polynomial about}

i

(3)

the additional displacement field induced only by the rigid rotation at nodes denoted as _i.

In order to determine the displacement field

 

_u0 _{, the}

shape functions in reference [10] is used, they are as follows: 4 0 0 1 4 0 0 1 i i i i i i

u N u

v N v

   



(15) where:





0

2 1 2

1 2 1, 2,3, 4

i i i i i i i

N g L L g P

i            (16) and

3 1 4 2 2 3 3 1 1 2 4 2 2 4 1 3

1 3 2 4

1

3( )( ) ( )( ) ( )( ) ( )

2 1

P

L L L L g g L L g g L L g g gg

gg g g



         

 

(17)

Assume the rotational displacement field as polyno- mials of quadrilateral area coordinates:

1 2 3 1 3 4 2

4 3 1 4 2

5 1 3 6 2 4

1 2 3 1 3 4 2

4 3 1 4 2

5 1 3 6 2 4

( ) ( )

( )( )

( ) ( )

( )( )

u L L L L

L L L L

L L L L

v L L L L

L L L L

L L L L

                                  (18)

with the conforming Equations as follows, the rotational displacement can be obtained.

) 41 , 34 , 23 , 12 ( 0 ) ( 0 ) ( 0 ) ( 4 1 4 1       





  ij s d u u u u u u ij l i i i i i i i i       

 (19)

where                                                                         _                1 2 1 2 4 4 4 2 1 4 4 2 3 3 41 41 4 1 4 2 3 3 3 2 4 3 3 1 2 2 34 34 3 4 3 2 2 2 2 2 3 2 4 2 1 1 23 23 2 3 2 2 1 1 1 2 2 1 3 1 4 4 12 12 1 1 1 1 1 1 1 1                 L g g L g g L L c b v u L g g L g g L L c b v u L g g L g g L L c b v u L g g L g g L L c b v u (20)

and b_iy_i_₁y_i_₂, c_ix_i_₂x_i_₁

Then the stiffness matrix of element AQ4 can be solved out easily, and the strain matrix is:

 

 

 

e q q

B

ε  (21) where

] [

]

[B_q  B₁ B₂ B₃ B₄ (22) and                                     x N y N x N y N y N y N x N x N i v i u i i i v i i u i i     0 0 0 0 0 0 ]

[B (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:

1 2 3 1 3 4 2

4 1 3 1 2 4 2 3 1 4 2

1 2 3 1 3 4 2

' '

4 1 3 1 2 4 2 3 1 4 2

( ) ( )

( )( )

( ) ( )

( )( )

u L L L L

L L L L L L L L

v L L L L

L L L L L L L L

                                  (24)

where ' '

1, , ,1 2 2

    are the internal parameters.

In order to solve the undetermined parameters, the conforming Equations are taken as:

{ } 0

ij

l  ds



u (25) Substitute Equation (24) into (25), the shape functions of { }u can be formulated out, they are:

1 2 3 4 1 3 2 4

1

1 4 2 3

1 2

3 1

1 4 1 4 2 3

4 2

1 4 2 3

1 3 2 4

1 4 2 3

2 3 2 3 1 1 2 4 2

3 1 4 2

(6 ) 6(1 ) ( ) 6 ( )( )₍ ₎ 6(1 ) 1

1₍ ₎₍ _{) 2(} ₎₍ ₎

3

( )( )

g g g g g g g g N

g g g g g g

L L

g g g g g g _L _L g g g g

g g g g

L L L L g g g g

N g g L L g g L L

L L L L

                              (26)

For element AQ4, N1 is taken as the final inter-

(4)

 

N

 

λ u_} _

{ (27) where

 

 

'

1 1

1

[ ]

0 0

T

N N  



  

 

  

 

λ

N (28) Through the condense calculation, stiffness matrix of AQ4 can be written as:

 

   

 

q

T q qq e

 



k





1 (29)

where

   

 

    

 

   

 

tdA

tdA tdA

q T q

T q T q qq

B D B k



  

 

 (30)

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   Bq 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.

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]:

 

2 2

1 3 1

2 2

1 3

1 1 3 3 5

1 1

1

a a

b b

a b a b

  

 

  

   

   

_ __{ }

 

 _{  }

 

σ  (32)

where

4 4 4

1 2 3

1 1 1

4 4 4

1 2 3

1 1 1

4 4 4

1 1 1

4 4 4

i i i i i i i

i i i

i i i i i i i

i i i

a x a x a x

b y b y b y

   

  



(33)

4.2. Stress Field II

In order to raise the complete order of stress field, the second form is assumed as a polynomial of analytical trial function methodpresented by Fu and Long [11]:

 

1

2 2

9

0 0 1 0 2 0 2 0 6

0 1 0 2 0 2 0 6 0

1 0 0 0 0 2 2 3 3

y x xy

x y xy

x y x y



  

 

 

_ __{ }

 

     

  

σ  (34)

4.3. Stress Field III

Try to make the three stress components to be indepen- dent, the third form of stress field is assumed as follows:

 

1

12

1

x y xy



  

 

 

_ __{ }

 

 

  

σ  (35)

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 Post- processing, 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

(5)

y

4 2

2

2.5 1.5

4

E = 1500, μ= 0.25

7 3

5 6

[image:5.595.61.286.85.394.2]

1000

Figure 1. Patch test.

48

P=1

E=1500, μ=1/3 44

44

A

x

y _B

[image:5.595.59.283.422.543.2]

Figure 2. Cook’s skew beam.

Table 1. Stress at point A and B of Cook’s Beam of AQ4θ.

[image:5.595.59.285.571.692.2]

Method σAmax σBmin 2 × 2 4 × 4 8 × 8 2 × 2 4 × 4 8 × 8 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. 0.2362 −0.2023

Table 2. Stress at point A and B of Cook’s Beam of AQ4θλ.

Method σAmax σBmin 2 × 2 4 × 4 8 × 8 2 × 2 4 × 4 8 × 8 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 −0.2023

the Hellinger-Reissner variational principle, hybrid post- process procedure can take advantage of the merits of

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.

REFERENCES

[1] R. H. MacNeal, “Derivation of Element Stiffness Matric- es by Assumed Strain Distributions,” Nuclear Engineer- ing 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-En- hanced Finite Elements for Plane Elasticity Problems,” International Journal for Numerical Methods in Engi- neering, 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

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