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An exact zooming method
Finite Elements in Analysis and Design 1 (1985) 61-69 North-Holland
61
AN EXACT ZOOMING METHOD Itio HIRAI Kumamoto University, Kumamoto, 860 Japan
Yoshihiro UCHIYAMA and Yoji MIZUTA Yatsushiro National College of Technology, Kumamoto, 866 Japan
Walter D. PlLKEY Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22901, U.S.A. Received June 1984 Abstract. This paper presents an efficient and exact formulation for finding stress concentration factors using a zooming analysis. A favorable characteristic of this method is its ability to perform a zooming analysis with computations taking place only in the zooming area under consideration with no need to treat the region outside the zooming area. At the final zooming step, the flexibility of a limited zooming area is determined for all nodes, including the new nodes created in the zooming processes as well as those of the original system before zooming.
Introduction Finite element methods used to study the stress distribution in a solid often require fine meshes to obtain accurate results, especially where rapid stress changes may be expected. Normally, due to computer storage limitations, a stress concentration factor is found by first using a coarse mesh and then the local region where stress concentration may occur is isolated from the original system for division into finer meshes [1-8]. For such a procedure, the boundary values of the freer mesh are the displacements obtained using the first coarse mesh. This results in an approximation because the effects on the complete system of the new nodes created while refining meshes are not taken into consideration. Thus, these' traditional' methods are not "exact' techniques for finding the stress concentration factor of the solid. Also, it will not be possible to proceed to further zooming steps to obtain more accurate results as long as the analyses are approximate. Finally, it should be mentioned that there is no measure available to judge the accuracy of the resulting stress concentration factor. To find exact results, it is possible to employ static condensation and the reanalysis techniques [9] to form a zooming analysis as described in [10]. Although this method is exact and involves the solution of a system of equations of small order, this is not always satisfactory since all the previous zooming processes are needed to proceed to each new zooming step. The method presented in this paper, which is similar to the above method, can perform an exact zooming analysis utilizing the results of only the previous level of zooming for the calculations. Rather than the re-analysis approach of [10], an expanded stiffness matrix is employed with the proposed method. Since the analysis is exact, in theory the zooming process can be continued until satisfactory results are obtained. Simple numerical examples are presented to illustrate the proposed method. 0168-874X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
L Hirai et aL / An exact zoe.rning method
62
Theory The equilibrium equations of the original system (system 0) before zooming (see Fig. 1) are generally given in terms of the stiffness matrix K (°) as K '°~{ X `°) } = ( F `°, }, (1) where { X (°) } and { F m~ } are node displacement and force vectors of order N (°~, respectively. Eq. (i) can be expressed in the partitioned form:
K(O) K(°) !1 12 K(O) 21
i,.(o) a~, 22
x;0,
FgO,}
with
(2) / FJ°'}
xB(O) ,
{ F `°, } = [ r;o,
(3)
and { X~°) } = the node displacement vector of order N~°) in region R~ ) outside the zooming area, { X~°) } = the node displacement vector of order N2¢°) in region R~ ) to be zoomed, { F~°) } = the force vector in region R~ ), { Fs¢°)} = the force vector in region R~ ). The subscripts A and B represent outside and inside the zooming region, respectively. Eq. (2) can be solved [11] to form the flexibility relations
x~O) where
/~o, flo,
=
F~o)j"
[/:o, f(o,] = ,/""f"'°'-' -- -"'°"(°'-'v'°'.. . Lf ,o,
L
-z'°'-'Y'°'
Zm)_- - r" =(2o2 ) _ tr(o)v(o)-ltr(o) =L21 J X l l ~xi2 V(O)ffi v(o)-ztr(o) • Lll -L12
,
y(o) = v(ow(o)-, "~'21 "LII
,
(4)
_ v(O)z(O)-
(5)
z(O)- ]
(6a)
,
(6b)
(6c)
V(°)= -f~°)f2m)-l, y(o)= _f~o)- lf2(o)"
(6d) (6e)
System 1
System 0
Io
o
o
IR~O'
Io
,.,
o
i o
o
o
t
o o
o
o
T:o,
node
,:. o old node •
Fig. 1. Original system.
new node
Fig. 2. First z o o m i n g model.
63
L Hirai et al. / A n exact zooming method
If zooming in region R~ ) is to be performed, then the stiffness matrix will be expanded to account for the newly chosen nodes (shown by solid circles in Fig. 2). Assume thai the degrees of freedom of region R~ ) are increased by N °). Represent the stiffness matrix K (!) of the total system (system 1) by i,,-(o) "'11
K(, ) =
0 ]
I,,- (o) "'i2
~.(o) tr(o)K~]) "'2t "'2: -
0
K~] ) , K /7(I'
K]] )
(7)
which can be expressed in the compact form
l'r(o) KO)= I " l l
ro)l "'12 1
I i-o)
~-o.~I'
L"~!
(8)
--22 j
where
[ I,'O)= I,'(o) "']2
0
""12
]
[I,-(o)] v(')--"'21 •
"'21
--
0
t,'O)-"
"'Z2
[K~)+K~i l) __,~c.1.~] |K _K-'-"_, K~7'] .
(9)
- - L/
From (5), the solution to the displacement equations formed by using the expanded stiffness matrix of (7) is given by ix,,)}
(10)
=
where { F~ ~ } is a force vector for region R~);
[f(l:) f(ll)
[I~) f~)
=
~.,o,-, + VO,ZO,-,yO)
_ Vo)ZO,-, ] '
--II
-zc"-'v
yO' = •I,"0)1"(o'-' _ [ Y(°)] "21 " ' i i 0
=
"~
[ - t.(, ,°, ) 0- '
z"'-' f~o,
l'
VO) ffi i-(o)-,I,-,,,_ "ffill "*12 -- [V,O) 0],
(11)
" (12)
(13)
Z(i)_ ~{1) ~-(I)&,-(o)-1"*12 jz(I) a L 2 2 -- a~21 "*II --
+ o
[r1,,, !:.o,," --]] j
In establishing (10), the condensed matrix multiplications Z o )- t yO ) _ Z o )- t y(O), V(')ZO)-' = [Z(')-"-"~y(°']T,
(15)
V(1)Z(1)- IyO) ffi v(O)zO)- i y(O) havebee..__~nemployed, where the superscript T represents the transpose of a matrix, and Z ° ) - t and Z °)-! represent N2(°)× N2(°~ and (N2(°) + N °)) × N2(°) matrices, respectively. The displacement vector { Xa(t~} of the zooming area is found from
{ x;,} =.t,':,{
} _-,--,,,
F;"} .
(16)
64
i. Hirai et al. / An exact zooming method
where A,I , = -
z"
,- ' Y'°',
' = z"
(17)
'- ' .
In general, { FB°) } = 0 so that (16) becomes
{x;,}
F:'}.
(]8)
W h e n a new region R ~ ~ within R~ ) is to be zoomed (Fig. 3), system 0 is separated into the region R ~ ~ outside the zooming area and the region R ~ ) to be zoomed. In performing this new zooming, treat system 2 as if it were system I and follow the zooming analysis, equation (4) to equation (18). This zooming analysis can be implemented, provided that the flexibilityof the system before zooming (the old system I which is now designated as system 0) is known. The following equation will be obtained:
(19) where - { Xa¢2)} and { X~2' } are the displacement vectors in regions R~' and R~ ), respectively. -- { F~ 2) } a n d { F J 2) } are thc force vectors in regions R~ ~ and R~ ), respectively. After several zooming processes, the degrees of freedom of the system increase. But, as can be seen from (12) to (14), the size of the matrices in these equations is limited to the degrees of freedom of the zooming area. The zooming region is normally small, and the stiffness matrix in each zooming step is represented by a banded matrix. It is important to note that the proposed zooming analysis proceeds for a zooming region on the basis of the flexibility of the previous zooming step. For instance, { X~!~} in (16) can be calculated from f2~°)-1 and f2~°~ without using f~0~ of the flexibility of the original system prior to zooming.
Numerical, examples The rectangular plate of Fig. 4 with a center hole and in-plane loading is used as a numerical model. Since the plate is symmetric about the x- and y-axis, a quarter of the plate will be
t
P
X
System 2
D
a..-
......
BI "-
........
le
12)
e
Fig. 3. Second zooming model.
Y Fig. 4. Rectangular plate to be zoomed.
!. Hirai et al. / An exact zooming method P
l,J T I ] I I"] ID IP
IC
Fig. 5. Model for numerical calculation. Table 1 Dimensions used for numerical calculation Uniform load p Length of plate L Width of plate B Thickness of plate h Diameter of hole D Young's modulus E Poisson's ratio v D/B
9.306 N / r a m 2 40 mm 20 mm 1 mm 4 mm 9.806 X l06 N / m m 2 0.3 0.2
No. of
Elements =t02
No. of
Elements =t03
No. of Nodes
No. of 'Nodes
= 67
=68
Fig. 6. Model 1 (original system).
Fig. 7. Model 2 (original system).
65
66
!. Hirai et aL / An exact zooming method
No. of Elements = t 0 5 No. of Nodes = 68
F E
B
E
E m
H
E B
No. of Nodes
= 71
C 6 mm
G
D[
J t . 3 5 mm
G
I
H
1
No. of Elements =102 No. of Nodes =66
V/N/V
C 10"41_Mmm
c
~J
0"2335
P mm
d
Fig. 8. (a) Mesh l(a).(b)Mesh l(b). (c) Mesh l(c). (d) Mesh l(d). (e) Mesh l(e).
,I M
!. Hirai et aL / An exact zooming method
67
~ .-[-
////
No. of Elements = 84
////// I///\ /VV// / /tt/~\
No. of Nodes =55
/f~,--.d
c V
c5
\
0-05 mm e
Fig. 8. (continued).
utilized in the numerical model (Fig. 5). The physical characteristics chosen for this numerical example are shown in Table 1. The stress concentration factor a in this model is "- O'max. y / O 0
where oread. ,.-- the maximum stress in the y-direction, oo = the average stress in the y-direction at the smallest cross section at y = 0. ................
"1 | I ! ! I
l
.,,, ,1,.,1, ,ib, ~
!
I
"~
".
E[-.~
I
' ~,."x
J
C F ~
....
C
D
J
D
"-,,
B~"--,,~
\\
-.,-.,, "-,',',, _
"-,,
\
'~:,
\\ \ !
c l ~ ..... ~F
I
c~, K
Fig. 9. Zooming areas for Model 2.
L Hirai et ai. / An exact zooming method
68
2"55 aT=2.5t2
/
2"50
--..o
....
2"
//
.
2"45
I i/
(~
/ / ! /
2"40
/ /
0 ....
I;
//
o....
mode,11 (
model 2 J
proposed method
----~---- model2 oldmethod
2"35 0
100
200
3 0
400
Number of Elements of EachModel
Fig. 10. Stress concentration factors vs. number of elements of each model.
The stress concentration factor a T theoretically obtained [12] is a T --' 2 . 5 1 2 .
Numerical calculations were performed using several kinds of zooming areas, since the resulting stress concentration factors are dependent on the zooming areas and meshes. As prototypes, two numerical models, Model 1 (Fig. 6) and Model 2 (Fig. 7), are used. Calculatioas for Model 1 use the zooming parts and meshes shown in Fig. 8(a)-(e). The area to be zoomed is shown in each former mesh. For Model 2, the zooming areas only are illustrated in Fig. 9. The resulting stres~ concentration factors are shown in Fig. 10. Each model at each level of zooming was also solved using the direct stiffness method to check the numerical accuracy. The results obtained by the two methods are the same as indicated in Table 2. In this calculation, sixteen digits were used. Table 2 Check of accuracy of stress concentration factors by direct stiffness method Model
Zooming steps
Proposed method
Direct stiffness method
Model 1
0 1 2 3 4 5
2.411 2.492 2.547 2.560 2.537 2.542
2.411 2.492 2.547 2.560 2.537 2.542
Model 2
0 1 2 3 4
2.293 2.465 2.507 2.524 2.526
2.293 2.465 2.507 2.524 2.526
!. Hirai el al. / An exact zooming method
69
For comparison, the stress concentration factors were also computed using a traditional ('old') zooming method, with these results also plotted in Fig. 10. As is to be expected, despite the use of an exact analysis, it is evident from Fig. 10 that the stress concentration factors vary according to the zooming areas, copfigurations, and meshes.
Concluding remarks The proposed method is a zooming technique that for successive zooming involves characteristics only of the area to be zoomed. The flexibility matrix of the zooming region is found for the load vector of the final zooming system, which involves both the region outside the zooming area and the new nodes created in the previous zooming steps. Therefore, this method is applicable to cases wherein loads within and outside of the zoomed area are modified as the zooming proceeds. The zooming area becomes smaller at every zooming step. As far as stress concentration is concerned, the overall stress distribution of a system is not of interest. Thus, the proposed method is suitable for an efficient, in-depth study of local stress distributions. The proposed zooming method is theoretically exact in the sense that no new approximations are introduced by the zooming procedures obtained. However, as is to be expected, the stress concentration factors do depend on the chosen combination of zooming areas, configurations, and meshes. For particular problems, this may indicate that the zooming areas employed are not large enough or finer meshes may be needed. Finally, it should be mentioned that, although the analysis developed in this paper uses an expanded stiffness matrix, the same analytical process could have been followed using the static condensation and reanalysis techniques of the sort found in [10].
References [l] HOLAND, !. and K. BELL, "'Finite element method in stress analysis", TAPIR, Technical University of Norway, Trondheim, p. 10L 1969. [2] DESAI,C.S. and J.F. ABEL, Introduction to the Finite Element Method Van Nostrand Reinhold, New York, p. 169, 1972. [3] Coog, R.D., Concepts and Applications of Finite Element Analysis, Wiley, New York, Section 2, Chapter 18, 1974. [41 A~u¥osm, K., M. NAKAYAMAand K. MITSUNAGA,"'Stress concentration factor of two-dimensional problems by finite ~iement method", Bulletin of the Kyushu Institute of Technology, 26, pp. 35-39, 1973. [5] FUHmNO, H., "Parametrische Kerbspannungsuntersuchungen an der Lochschribe mit der Methode der finiten Elemente", Stahlbau 9, pp. 272-279, 1975. [6] MARUYAMA,K., "Stress analysis of a bolt-nut joint by the finite element method and the copper-electroplating method", Bulletin of Japan Society of Mechanical Engineers 16 (94) pp. 671-678, 1973. [7] SUGAWARA,K., et al., "A study on core discing of rock", Mining and Metallurgical Institute of Japan 94, pp. 19-25, 1978. [8] BATHE,K.J. and E.L. WILSON, Numerical I~,ethods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N J, pp. 259-263, 1976. [9] WANG, B.P., A.B. PALAZZOLOand W.D. PtLgEY, "Reanalysis, model synthesis, and dynamic design", State-ofthe-Art Surveys in Finite Element Technology, edited by A. NOOR and W. PILKEY,ASME, New York, Chapter 8, 1983. [10] HIItAI, !., B.P. WANG and W.D. PILKEY,"An efficient zooming method for finite element analysis", lnternat. J. Numer. Meths. Eng. 20, pp. 1671-1683, 1984. [1 l] WESTLAKE,J.R., A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, "Aqley,New York, p. 26, 1968. [12] SAViN,G.N., Stress Concentration Around Holes, Pergamon Press, Oxford. p. 109, 1961.
61
AN EXACT ZOOMING METHOD Itio HIRAI Kumamoto University, Kumamoto, 860 Japan
Yoshihiro UCHIYAMA and Yoji MIZUTA Yatsushiro National College of Technology, Kumamoto, 866 Japan
Walter D. PlLKEY Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22901, U.S.A. Received June 1984 Abstract. This paper presents an efficient and exact formulation for finding stress concentration factors using a zooming analysis. A favorable characteristic of this method is its ability to perform a zooming analysis with computations taking place only in the zooming area under consideration with no need to treat the region outside the zooming area. At the final zooming step, the flexibility of a limited zooming area is determined for all nodes, including the new nodes created in the zooming processes as well as those of the original system before zooming.
Introduction Finite element methods used to study the stress distribution in a solid often require fine meshes to obtain accurate results, especially where rapid stress changes may be expected. Normally, due to computer storage limitations, a stress concentration factor is found by first using a coarse mesh and then the local region where stress concentration may occur is isolated from the original system for division into finer meshes [1-8]. For such a procedure, the boundary values of the freer mesh are the displacements obtained using the first coarse mesh. This results in an approximation because the effects on the complete system of the new nodes created while refining meshes are not taken into consideration. Thus, these' traditional' methods are not "exact' techniques for finding the stress concentration factor of the solid. Also, it will not be possible to proceed to further zooming steps to obtain more accurate results as long as the analyses are approximate. Finally, it should be mentioned that there is no measure available to judge the accuracy of the resulting stress concentration factor. To find exact results, it is possible to employ static condensation and the reanalysis techniques [9] to form a zooming analysis as described in [10]. Although this method is exact and involves the solution of a system of equations of small order, this is not always satisfactory since all the previous zooming processes are needed to proceed to each new zooming step. The method presented in this paper, which is similar to the above method, can perform an exact zooming analysis utilizing the results of only the previous level of zooming for the calculations. Rather than the re-analysis approach of [10], an expanded stiffness matrix is employed with the proposed method. Since the analysis is exact, in theory the zooming process can be continued until satisfactory results are obtained. Simple numerical examples are presented to illustrate the proposed method. 0168-874X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
L Hirai et aL / An exact zoe.rning method
62
Theory The equilibrium equations of the original system (system 0) before zooming (see Fig. 1) are generally given in terms of the stiffness matrix K (°) as K '°~{ X `°) } = ( F `°, }, (1) where { X (°) } and { F m~ } are node displacement and force vectors of order N (°~, respectively. Eq. (i) can be expressed in the partitioned form:
K(O) K(°) !1 12 K(O) 21
i,.(o) a~, 22
x;0,
FgO,}
with
(2) / FJ°'}
xB(O) ,
{ F `°, } = [ r;o,
(3)
and { X~°) } = the node displacement vector of order N~°) in region R~ ) outside the zooming area, { X~°) } = the node displacement vector of order N2¢°) in region R~ ) to be zoomed, { F~°) } = the force vector in region R~ ), { Fs¢°)} = the force vector in region R~ ). The subscripts A and B represent outside and inside the zooming region, respectively. Eq. (2) can be solved [11] to form the flexibility relations
x~O) where
/~o, flo,
=
F~o)j"
[/:o, f(o,] = ,/""f"'°'-' -- -"'°"(°'-'v'°'.. . Lf ,o,
L
-z'°'-'Y'°'
Zm)_- - r" =(2o2 ) _ tr(o)v(o)-ltr(o) =L21 J X l l ~xi2 V(O)ffi v(o)-ztr(o) • Lll -L12
,
y(o) = v(ow(o)-, "~'21 "LII
,
(4)
_ v(O)z(O)-
(5)
z(O)- ]
(6a)
,
(6b)
(6c)
V(°)= -f~°)f2m)-l, y(o)= _f~o)- lf2(o)"
(6d) (6e)
System 1
System 0
Io
o
o
IR~O'
Io
,.,
o
i o
o
o
t
o o
o
o
T:o,
node
,:. o old node •
Fig. 1. Original system.
new node
Fig. 2. First z o o m i n g model.
63
L Hirai et al. / A n exact zooming method
If zooming in region R~ ) is to be performed, then the stiffness matrix will be expanded to account for the newly chosen nodes (shown by solid circles in Fig. 2). Assume thai the degrees of freedom of region R~ ) are increased by N °). Represent the stiffness matrix K (!) of the total system (system 1) by i,,-(o) "'11
K(, ) =
0 ]
I,,- (o) "'i2
~.(o) tr(o)K~]) "'2t "'2: -
0
K~] ) , K /7(I'
K]] )
(7)
which can be expressed in the compact form
l'r(o) KO)= I " l l
ro)l "'12 1
I i-o)
~-o.~I'
L"~!
(8)
--22 j
where
[ I,'O)= I,'(o) "']2
0
""12
]
[I,-(o)] v(')--"'21 •
"'21
--
0
t,'O)-"
"'Z2
[K~)+K~i l) __,~c.1.~] |K _K-'-"_, K~7'] .
(9)
- - L/
From (5), the solution to the displacement equations formed by using the expanded stiffness matrix of (7) is given by ix,,)}
(10)
=
where { F~ ~ } is a force vector for region R~);
[f(l:) f(ll)
[I~) f~)
=
~.,o,-, + VO,ZO,-,yO)
_ Vo)ZO,-, ] '
--II
-zc"-'v
yO' = •I,"0)1"(o'-' _ [ Y(°)] "21 " ' i i 0
=
"~
[ - t.(, ,°, ) 0- '
z"'-' f~o,
l'
VO) ffi i-(o)-,I,-,,,_ "ffill "*12 -- [V,O) 0],
(11)
" (12)
(13)
Z(i)_ ~{1) ~-(I)&,-(o)-1"*12 jz(I) a L 2 2 -- a~21 "*II --
+ o
[r1,,, !:.o,," --]] j
In establishing (10), the condensed matrix multiplications Z o )- t yO ) _ Z o )- t y(O), V(')ZO)-' = [Z(')-"-"~y(°']T,
(15)
V(1)Z(1)- IyO) ffi v(O)zO)- i y(O) havebee..__~nemployed, where the superscript T represents the transpose of a matrix, and Z ° ) - t and Z °)-! represent N2(°)× N2(°~ and (N2(°) + N °)) × N2(°) matrices, respectively. The displacement vector { Xa(t~} of the zooming area is found from
{ x;,} =.t,':,{
} _-,--,,,
F;"} .
(16)
64
i. Hirai et al. / An exact zooming method
where A,I , = -
z"
,- ' Y'°',
' = z"
(17)
'- ' .
In general, { FB°) } = 0 so that (16) becomes
{x;,}
F:'}.
(]8)
W h e n a new region R ~ ~ within R~ ) is to be zoomed (Fig. 3), system 0 is separated into the region R ~ ~ outside the zooming area and the region R ~ ) to be zoomed. In performing this new zooming, treat system 2 as if it were system I and follow the zooming analysis, equation (4) to equation (18). This zooming analysis can be implemented, provided that the flexibilityof the system before zooming (the old system I which is now designated as system 0) is known. The following equation will be obtained:
(19) where - { Xa¢2)} and { X~2' } are the displacement vectors in regions R~' and R~ ), respectively. -- { F~ 2) } a n d { F J 2) } are thc force vectors in regions R~ ~ and R~ ), respectively. After several zooming processes, the degrees of freedom of the system increase. But, as can be seen from (12) to (14), the size of the matrices in these equations is limited to the degrees of freedom of the zooming area. The zooming region is normally small, and the stiffness matrix in each zooming step is represented by a banded matrix. It is important to note that the proposed zooming analysis proceeds for a zooming region on the basis of the flexibility of the previous zooming step. For instance, { X~!~} in (16) can be calculated from f2~°)-1 and f2~°~ without using f~0~ of the flexibility of the original system prior to zooming.
Numerical, examples The rectangular plate of Fig. 4 with a center hole and in-plane loading is used as a numerical model. Since the plate is symmetric about the x- and y-axis, a quarter of the plate will be
t
P
X
System 2
D
a..-
......
BI "-
........
le
12)
e
Fig. 3. Second zooming model.
Y Fig. 4. Rectangular plate to be zoomed.
!. Hirai et al. / An exact zooming method P
l,J T I ] I I"] ID IP
IC
Fig. 5. Model for numerical calculation. Table 1 Dimensions used for numerical calculation Uniform load p Length of plate L Width of plate B Thickness of plate h Diameter of hole D Young's modulus E Poisson's ratio v D/B
9.306 N / r a m 2 40 mm 20 mm 1 mm 4 mm 9.806 X l06 N / m m 2 0.3 0.2
No. of
Elements =t02
No. of
Elements =t03
No. of Nodes
No. of 'Nodes
= 67
=68
Fig. 6. Model 1 (original system).
Fig. 7. Model 2 (original system).
65
66
!. Hirai et aL / An exact zooming method
No. of Elements = t 0 5 No. of Nodes = 68
F E
B
E
E m
H
E B
No. of Nodes
= 71
C 6 mm
G
D[
J t . 3 5 mm
G
I
H
1
No. of Elements =102 No. of Nodes =66
V/N/V
C 10"41_Mmm
c
~J
0"2335
P mm
d
Fig. 8. (a) Mesh l(a).(b)Mesh l(b). (c) Mesh l(c). (d) Mesh l(d). (e) Mesh l(e).
,I M
!. Hirai et aL / An exact zooming method
67
~ .-[-
////
No. of Elements = 84
////// I///\ /VV// / /tt/~\
No. of Nodes =55
/f~,--.d
c V
c5
\
0-05 mm e
Fig. 8. (continued).
utilized in the numerical model (Fig. 5). The physical characteristics chosen for this numerical example are shown in Table 1. The stress concentration factor a in this model is "- O'max. y / O 0
where oread. ,.-- the maximum stress in the y-direction, oo = the average stress in the y-direction at the smallest cross section at y = 0. ................
"1 | I ! ! I
l
.,,, ,1,.,1, ,ib, ~
!
I
"~
".
E[-.~
I
' ~,."x
J
C F ~
....
C
D
J
D
"-,,
B~"--,,~
\\
-.,-.,, "-,',',, _
"-,,
\
'~:,
\\ \ !
c l ~ ..... ~F
I
c~, K
Fig. 9. Zooming areas for Model 2.
L Hirai et ai. / An exact zooming method
68
2"55 aT=2.5t2
/
2"50
--..o
....
2"
//
.
2"45
I i/
(~
/ / ! /
2"40
/ /
0 ....
I;
//
o....
mode,11 (
model 2 J
proposed method
----~---- model2 oldmethod
2"35 0
100
200
3 0
400
Number of Elements of EachModel
Fig. 10. Stress concentration factors vs. number of elements of each model.
The stress concentration factor a T theoretically obtained [12] is a T --' 2 . 5 1 2 .
Numerical calculations were performed using several kinds of zooming areas, since the resulting stress concentration factors are dependent on the zooming areas and meshes. As prototypes, two numerical models, Model 1 (Fig. 6) and Model 2 (Fig. 7), are used. Calculatioas for Model 1 use the zooming parts and meshes shown in Fig. 8(a)-(e). The area to be zoomed is shown in each former mesh. For Model 2, the zooming areas only are illustrated in Fig. 9. The resulting stres~ concentration factors are shown in Fig. 10. Each model at each level of zooming was also solved using the direct stiffness method to check the numerical accuracy. The results obtained by the two methods are the same as indicated in Table 2. In this calculation, sixteen digits were used. Table 2 Check of accuracy of stress concentration factors by direct stiffness method Model
Zooming steps
Proposed method
Direct stiffness method
Model 1
0 1 2 3 4 5
2.411 2.492 2.547 2.560 2.537 2.542
2.411 2.492 2.547 2.560 2.537 2.542
Model 2
0 1 2 3 4
2.293 2.465 2.507 2.524 2.526
2.293 2.465 2.507 2.524 2.526
!. Hirai el al. / An exact zooming method
69
For comparison, the stress concentration factors were also computed using a traditional ('old') zooming method, with these results also plotted in Fig. 10. As is to be expected, despite the use of an exact analysis, it is evident from Fig. 10 that the stress concentration factors vary according to the zooming areas, copfigurations, and meshes.
Concluding remarks The proposed method is a zooming technique that for successive zooming involves characteristics only of the area to be zoomed. The flexibility matrix of the zooming region is found for the load vector of the final zooming system, which involves both the region outside the zooming area and the new nodes created in the previous zooming steps. Therefore, this method is applicable to cases wherein loads within and outside of the zoomed area are modified as the zooming proceeds. The zooming area becomes smaller at every zooming step. As far as stress concentration is concerned, the overall stress distribution of a system is not of interest. Thus, the proposed method is suitable for an efficient, in-depth study of local stress distributions. The proposed zooming method is theoretically exact in the sense that no new approximations are introduced by the zooming procedures obtained. However, as is to be expected, the stress concentration factors do depend on the chosen combination of zooming areas, configurations, and meshes. For particular problems, this may indicate that the zooming areas employed are not large enough or finer meshes may be needed. Finally, it should be mentioned that, although the analysis developed in this paper uses an expanded stiffness matrix, the same analytical process could have been followed using the static condensation and reanalysis techniques of the sort found in [10].
References [l] HOLAND, !. and K. BELL, "'Finite element method in stress analysis", TAPIR, Technical University of Norway, Trondheim, p. 10L 1969. [2] DESAI,C.S. and J.F. ABEL, Introduction to the Finite Element Method Van Nostrand Reinhold, New York, p. 169, 1972. [3] Coog, R.D., Concepts and Applications of Finite Element Analysis, Wiley, New York, Section 2, Chapter 18, 1974. [41 A~u¥osm, K., M. NAKAYAMAand K. MITSUNAGA,"'Stress concentration factor of two-dimensional problems by finite ~iement method", Bulletin of the Kyushu Institute of Technology, 26, pp. 35-39, 1973. [5] FUHmNO, H., "Parametrische Kerbspannungsuntersuchungen an der Lochschribe mit der Methode der finiten Elemente", Stahlbau 9, pp. 272-279, 1975. [6] MARUYAMA,K., "Stress analysis of a bolt-nut joint by the finite element method and the copper-electroplating method", Bulletin of Japan Society of Mechanical Engineers 16 (94) pp. 671-678, 1973. [7] SUGAWARA,K., et al., "A study on core discing of rock", Mining and Metallurgical Institute of Japan 94, pp. 19-25, 1978. [8] BATHE,K.J. and E.L. WILSON, Numerical I~,ethods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N J, pp. 259-263, 1976. [9] WANG, B.P., A.B. PALAZZOLOand W.D. PtLgEY, "Reanalysis, model synthesis, and dynamic design", State-ofthe-Art Surveys in Finite Element Technology, edited by A. NOOR and W. PILKEY,ASME, New York, Chapter 8, 1983. [10] HIItAI, !., B.P. WANG and W.D. PILKEY,"An efficient zooming method for finite element analysis", lnternat. J. Numer. Meths. Eng. 20, pp. 1671-1683, 1984. [1 l] WESTLAKE,J.R., A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, "Aqley,New York, p. 26, 1968. [12] SAViN,G.N., Stress Concentration Around Holes, Pergamon Press, Oxford. p. 109, 1961.