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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
40, 1399–1411 (1997)
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL
HYBRID STRESS ELEMENTS
ZONGSHU TIAN∗ , JINSONG LIU† AND LIN YE†
Department of Physics, Graduate School, Chinese Academy of Sciences, P.O. Box 2706, Beijing 100080, China
THEODORE H. H. PIAN‡
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
SUMMARY
The stresses around rectangular holes of rounded corners and rounded notches are analysed by using the
combination of three kinds of three-dimensional special hybrid stress nite elements. One kind of threedimensional hybrid stress nite element with a traction-free inclined surface has been developed here based
on an extended Hellinger–Reissner principle by a rational approach. Other two kinds of 12-node threedimensional hybrid stress elements with a traction-free cylindrical surface and a traction-free perpendicular
surface, respectively, were derived based on the Hellinger–Reissner principle. Examples have indicated that
the combination of these special solid elements is far superior in predicting the stress concentration factors,
the distributions of circumferential stresses and normal stresses for solids with holes and notches. ? 1997 by
John Wiley & Sons, Ltd.
KEY WORDS: special hybrid stress element; stress concentration; rectangular hole; V-shaped notch
INTRODUCTION
Stress concentrations in the vicinity of dierent holes and notches have been the subject of numerous analytical and experimental investigations. Closed-form theoretical solutions have been found
only for a few geometries and loadings. For most geometries, an approximate numerical solution
or an experimental investigation is necessary.
The stress concentration eects of V-shaped, U-shaped and hyperbolic notches in a long thin tensile plate were studied by using experimental methods,1−5 Neuber’s theory and Shawen’s method,6;7
Peterson’s design curves8 and singular integral method.9 Scanning these theoretical calculations,
experimental determinations and numerical methods, most of them are limited in two-dimensional
problems, simple geometries and simple boundary traction. Naturally, the nite element methods
were considered to be used for the problem. But the accuracy of the ordinary assumed displacement
elements and the ordinary assumed stress elements is not desirable.
For this purpose a series of three-dimensional hybrid elements with a traction-free cylindrical surface and with a traction-free perpendicular surface, respectively, were developed based on
the Hellinger–Reissner principle.10−14 Numerical results show that the special elements not only
∗ Professor
†
‡
Graduate student
Professor
CCC 0029–5981/97/081399–13$17.50
? 1997 by John Wiley & Sons, Ltd.
Received 27 March 1995
Revised 3 August 1996
1400
Z. TIAN ET AL.
Figure 1. Geometry of special 12-node element with a traction-free inclined surface
provide much more accurate stress concentration factors than those obtained by using conventional
assumed displacement elements and ordinary hybrid stress elements, but also provide accurate
distributions of circumferential stresses and normal stresses z along the rim of hole.
One objective of the present study is to develop one kind of 12-node special three-dimensional
element based on an extended Hellinger–Reissner principle which contains a traction-free inclined
surface. (Figure 1; The two planes 1234 and 5678 are parallel to each other and are perpendicular
to the Z-axis.) Another objective is to study the stress concentration of solids with holes and
notches consisting of inclined faces, straight faces and cylindrical faces (such as rectangular and
octagonal holes of rounded corners, U-shaped and V-shaped notches of rounded corners and semicircular notches, etc.) under complex boundary traction by using the combination of these three
kinds of special hybrid stress elements.
ELEMENT STIFFNESS MATRICES
In formulating the element stiness matrix by the Hellinger–Reissner principle, the energy functional for an individual element is given by equation (1), when the terms corresponding to applied
loads are neglected:
Z
1 T
(1)
− 2 S + T (Du) dV = Stationary
R (u; ) =
vn
where vn is the volume of the individual element, the stresses, S the elastic compliance matrix,
u the displacements and Du the strains.
By introducing the common displacements ũ along the interelement boundary, the extended
Hellinger–Reissner principle mR can be obtained:15
Z
Z
1 T
mR (u; ; ũ) =
TT (u − ũ) dS
(2)
− 2 S + T (Du) dV −
vn
@vn
where ũ is the common displacements along the interelement boundary, T the surface traction
(T = ), the matrix of the directional cosine of the surface normal and @vn boundary surface
of the individual element.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
? 1997 by John Wiley & Sons, Ltd.
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
1401
If the element displacements u are separated into two components: the compatible
displacements uq and the added incompatible displacements u which can be eliminated in the
element level,
u = u q + u
u = u − ũ
(3)
on @vn
The functional mR becomes
Z
Z
1 T
mR (uq ; ; u ) =
− 2 S + T (Duq ) + T (Du ) dV −
vn
TT u dS
(4)
@vn
In the rational approach, the desirable stress eld ∗∗ is obtained by following the two steps:
(a) The stresses are separated into the constant terms c and the high-order terms h ,
= c + h
(5)
The constraint conditions for stresses are chosen as16
I
cT T u dS = 0 −→ u∗
I
@vn
@vn
(6)
hT T u∗
dS = 0 −→ ∗
It means that the virtual work along the element boundary due to the surface traction and the
incompatible displacements is equal to zero. By the use of above rst constraint condition the
displacement u∗ is obtained. Then substituting the determined u∗ into the second constraint
condition the stress assumption ∗ can be obtained.
(b) The traction-free boundary condition is then used to determine the nal expression of ∗∗ .
By using the transformation based on divergence theorem, the variational functional is of the
form
Z h
i
mR (uq ; u∗ ; ∗∗ ) =
− 12 ∗∗ T S∗∗ + ∗∗ T (Duq ) − (DT c )T u∗ + h ∗∗T (Du∗ ) dV
(7)
vn
If the displacements uq are also interpolated in terms of nodal displacements q and the additional
displacements u∗ are interpolated in terms of internal parameters , the stresses ∗∗ are expressed
in terms of stress parameters ∗ :
uq = N q
u∗ = M ∗∗
(8)
∗ ∗
=P =
Pc∗∗
+
Ph∗∗
The following expression can be obtained for the element stiness matrix:
T
k = G H−1 G
? 1997 by John Wiley & Sons, Ltd.
(9)
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
1402
Z. TIAN ET AL.
where
Z
G=
vn
G = G − G (GT H−1 G )−1 GT H−1 G
Z
Z
∗T
∗T
P (DN) dV; G =
Ph (DM) dV; H =
P∗T SP∗ dV
vn
(10)
(11)
vn
On the other hand, if the introduction of the incompatible displacement u∗ is considered only
as a means to obtain the desirable stress eld, then the element stiness matrix can be obtained
by using the functional
Z h
i
∗∗
R (uq ; ) =
(12)
− 12 ∗∗ T S∗∗ + ∗∗ T (Duq ) dV
vn
The element stiness matrix for this compatible element is then given by
k = GT H−1 G
(13)
where the matrices G and H are determined by equation (11). In the following numerical examples the stiness matrices k of these kinds of element are calculated by using the simple equation (13).
ELEMENT STRESS ASSUMPTIONS
For 12-node special element with a traction-free inclined surface, the initial stresses with 60
-parameters are chosen as
 0

P
0
0
0
0
0  



 1 

 0 P0



0
0
0
0




2 













0
 0



0
0
0
0
P
.
..


(14)
h = Ph h = 

 0



0
0 
0
0 P0






 .. 




 . 
0

 0


0 
0
0
0 P







0
54
0
0
0
0
0 P
(15)
P0 = ; ; ; 2 ; 2 ; ; ; ; 2







c = 






1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)


55 










 56 












57 







 58 










59 








60
(16)
? 1997 by John Wiley & Sons, Ltd.
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
The following initial incompatible displacements u are chosen:

0
0
0 0
0
2 2 2 2 0 0


u =  0 0
0
0
0 2 2 2 2 0 0

0 0
0
0
0 0
0
0 2 2 2
By using the constraint conditions (6) the additional
stresses ∗∗ are obtained:





N 0
0 








0
0
N
u∗ = 








0
0 N 










0 




0 




2 








1 




2 





.. 
.



.. 


. 





12
1403
(17)
displacements u∗ and then the assumed

1 




2 


..
. 

.. 

. 



12
(18)
where
N = [N1 ; N2 ; N3 ; N4 ]
N 1 = 2 +
N2 = 2
(19)
2(a2 b4 − a4 b2 ) + 2(a3 b4 − a4 b3 )
3(a2 b3 − a3 b2 )
(20)
N3 = 2 − (=3)
N4 = 2 − (=3)
and

a1
 a2

 a3
a4
 
b1
1 1
 −1 1
b2 
=
b3   −1 −1
b4
1 −1

1 1
x1
 x2
1 −1 

1 1   x3
x4
1 −1
x = ( − 2 ) 1 + ( + 2 ) 2 + 2 − 4 3 +
8
5

y1
y2 

y3 
y4
(21)
+ 2 4 + (1 + ) 5 + 6
+4 5 7 + 5 8 − 5 9 + 5 2 10 − (1 − 4 ) 28 + 4 5 29
y = −
×
2
−
5
5
1 +
2
−
5
5
2 − 6 3 − (21 − 8a2 ) − 5a2 2 + 21 1
4 + 6 + 2 + 6 5 7 + 8 + 9 + 2 10 − 3 28 + (1 + 5 6 ) 29
5a2 5
5
? 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
1404
Z. TIAN ET AL.
z = 8 6 + 11 + 12 + 13 + 2 14 + 15 + 16 + 2 17 + 9 24 − 9 25
− 52 9 2 27 + 30
(22)
h
a3 b3 − b23 d
10
1 +
2 −
3 − (21 − 8a2 ) −
xy = −
3 5
3
i 1
1
10
−5a2 2 + 21 4 − 6 +
+ 11 7 − d8 + d9 + d2 10
5a2 d
d
3
a3 b3 − b23 d
10
10
2
+ − 1 18 + (1 + ) 19 −
−
+ 11 29
28 +
3 5
3
3
2
−
d
d
2
−
d
d
2h13
59 2
4 + h13 + h13 −
6 + 20 + 2 21
yz = 12 1 − 12 2 −
5
2d
+ 22 + 2 23 +
1
24 − 25 + 27 + 31
d
d
d
2
2
zx = + 9 1 − 9 2 − dh13 4 − 9 81 + 5b3 d2 + 81 6 + (15 − d + 15 ) 20
5
2h
+ d21 + d22 + d2 23 − (14 − + 14 ) 24 + [( + 14 ) + 14 ] 25 + 2 − 1
×26 + 27 + d31
where
1 = b3 d − a3 ;
2 =
4a2
;
1
3 = b23 d2 − a23
4 =
a23
;
3
5 = d2 ;
8 =
1 h2
;
a2 b23 5
9 =
h
;
b3 d
10 = a3 b3 d2
13 =
11 =
a3 b3 d2 − a23 d
;
3
12 =
9
;
d
14 =
4a2
;
b3 d
15 =
a3
;
b3
6 =
b23
3
(23)
1
a2 b3 d2
d = ctg The stresses ∗∗ are chosen to satisfy the traction-free condition along the inclined surface.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
? 1997 by John Wiley & Sons, Ltd.
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
1405
NUMERICAL RESULTS
Thin rectangular plate with V-shaped rounded notches under tension
A thin plate with symmetric V-shaped rounded notches under tension is shown in Figure 2. The
radius of the arc is R and the thickness of plate is equal to 0·1R. The angles of the V-shaped
notch equal to 30; 60; 90 and 120◦ , respectively.
The problem is analysed using only one layer of elements. The mesh patterns are shown in
Figure 3. The computed stress concentration factors (SCF) obtained by the following three
arrangements are shown in Table I.
1. Combining the present special elements and the ordinary isoparametric elements derived by
the conventional assumed displacement approach.
2. Using the ordinary assumed displacement isoparametric elements everywhere.
3. Using experimental formulae17 obtained by photoelastic method.
Figure 2. Thin plate with V-shaped rounded notches
Figure 3. Mesh patterns for 1/4 of the plate with V-shaped notches
? 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
1406
Z. TIAN ET AL.
Table I. Computed stress concentration factors SCF. Thin plate with symmetric V-shaped rounded
notches under tension T (L=R = 24; h=R = W=R = 8)
Angle of the V-shaped notches
30◦
60◦
90◦
120◦
Degree of freedom (DOF)
Present special elements degenerated into 2D,
and ordinary assumed displacement elements
Ordinary assumed displacement elements (ODE)
Photoelastic method
Reference solutions9
40
20
19
24
2·83
3·01
2·86
2·81
2·67
3·26
2·86
2·72
2·57
3·25
2·83
2·65
2·67
2·76
2·61
2·43
Figure 4. Thin plate with a horizontal hole
It is seen that in comparison with the ordinary assumed displacement element and the photo
elastic method, the present special elements provide the stress concentration factors most close to
the reference solutions even by using the coarse meshes.
Thin rectangular plate with a horizontal hole under tension
A thin rectangular plate of 12R × 4R × 0·1R with a centre horizontal hole is acted by uniform
tension along two opposite edges. The problem is analysed here also using only one layer of
elements in two dierent meshes with 5 and 19 elements, respectively, for one-quarter of the plate
(Figure 4). The computed stress concentration factors are given by Table II.
It can be seen that the stress concentration factors obtained by the special elements are also in
good agreement with the reference solution.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
? 1997 by John Wiley & Sons, Ltd.
1407
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
Table II. Computed stress concentration factors SCF. Thin plate with a horizontal hole under
tension T
Mesh I
Mesh II
Type of elements
SCF
Error %
SCF
Error %
Present special elements and ODE
Ordinary assumed displacement elements (ODE)
Reference solution17
1·75
1·11
4·17
−33·93
1·67
1·46
−0·59
−13·10
1·68
Figure 5. Thin plate with a rectangular hole under bending
Thin plate with a rectangular hole under bending
A thin plate of 19R × 8R × 0 · 1R with a rectangular hole of rounded corners is subjected to
pure bending (Figure 5). The problem is analysed here using two dierent meshes with 10 and 21
elements, respectively, for one-quarter of the plate. The mesh 2 is shown in Figure 5. For a thin
plate with a rectangular hole in pure bending, the largest circumferential stress is acting either
at point A (near the intersection point of the circular arc and the straight side) or at point E (outer
edge of the plate). Which one of A and E is the largest will be decided by the sizes of the
plate and the hole, for the problem the largest stress is acting at point E. The stress concentration
factors are calculated by the following equations:
SCFA =
A
;
0
SCFE =
E
;
0
0 =
3bM
2t B3 − b3
(24)
Here A is the stress acting at point A. E is the stress acting at point E. t is the thickness of the
plate.
? 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
1408
Z. TIAN ET AL.
Table III. Computed stress concentration factors SCF. Thin plate with a rectangular hole under bending M
SCF at point A (SCFA )
Mesh 1
Type of elements
Special elements and ODE
ODE
Photoelastic method
Reference solution
A
17·15
9·30
1
SCFA
◦
89·1
90·0◦
2·00
1·08
Mesh 2
Error %
13·64
−38·64
A
14·98
13·25
1
◦
90·0
84·4◦
∼85·0◦
80·7◦
SCFA
Error %
1·75
1·55
1·90
1·76
− 0·57
−11·93
7·95
SCF at point E (SCFE )
Mesh 1
Type of elements
Special elements and ODE
ODE
Photoelastic method
Reference solution
E
17·06
16·29
2
SCFE
◦
28·8
14·0◦
1·99
1·90
Mesh 2
Error %
0·51
−4·04
E
16·51
16·53
2
◦
14·0
14·0◦
∼50◦
39·7◦
SCFE
Error %
1·93
1·93
2·00
1·98
−2·53
−2·53
1·01
There are no analytical solutions for the problem. The results obtained by the ne mesh with
336 elements for one-quarter of the plate are considered as the references. The per cent errors as
compared to the references are tabulated in Table III.
As shown, the special elements provide more accurate factors SCFA than those obtained by
the use of the ordinary assumed displacement elements and the photoelastic method. The factors
SCFE obtained by all of the three methods are very close to the reference solution. However, the
location 2 of the largest stress E given by the present method in coarse meshes 1 and 2 are not
in good agreement with the reference solution.
Square block with a square hole of rounded corners under tension
A square block of 110a × 110a × 40a with a square hole of rounded corners is acted by uniform
tension over two opposite faces. The top view of the block is shown in Figure 6. The radius R
of the arc is equal to 0·616a. The Poisson ratio for the material is taken as 0·25. The problem
is analysed by using ve dierent meshes with 16, 32, 64, 128 and 256 elements for one-eighth
of the block. The mesh 3 is shown in Figure 7.
The resulting solutions for at point P of the middle plane and at point Q of the surface are
given in Table IV. It shows that the SCF are equal to 6·1 in the middle plane and equal to 6·4
on the face for the three-dimensional problem.
Thick plate with a vertical hole or with a square hole under tension
A thick plate of 22R × 8R × 8R with a square hole or with a vertical hole is acted by uniform
tension over two opposite faces. The top views are shown in Figure 8. The Poisson ratio is also
taken as 0·25. Each of the cases is analysed by using three dierent meshes. The mesh 3 with
240 elements for one-eighth of the thick plate is shown in Figure 9.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
? 1997 by John Wiley & Sons, Ltd.
1409
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
Figure 6. Top view of a square block with a square hole
Figure 7. Mesh 3 for 1/8 of the square block with a square hole
Table IV. Computed stress concentration factors SCF (3-D problem). Thick square block with a
square hole under tension T (t = 40a, = 0·25)
Meshes
Degrees of freedom (DOF)
Stress concentration factors of middle plane
Stress concentration factors of face
1
2
3
4
5
155
5·980
6·020
250
6·052
6·192
440
6·070
6·291
820
6·068
6·357
1580
6·064
6·386
The results of the largest circumferential stresses and the normal stresses z at point A (in
middle plane) and point D (on the face) along the rim of the hole are given in Table V. The
reference solutions are also obtained by the ne meshes of each case.
? 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
1410
Z. TIAN ET AL.
Figure 8. Top view of a rectangular thick plate with a vertical hole or with a square hole
Figure 9. Mesh 3 for 1/8 of the thick plate with a square hole
Table V. Computed =0 and z =0 at the rim of the hole (3-D problem). Thick plate with a vertical hole
or with a square hole under tension T (t = 8R; = 0·25)
Vertical hole
=0
Square hole
=0
z =0
z =0
Mesh
DOF
Middle
plane A
Face
D
Middle
plane A
DOF
Middle
plane A
Face
D
Middle
plane A
1
2
3
Reference
solutions
268
472
880
2·506
2·527
2·545
2·405
2·326
2·263
0·39
0·41
0·41
332
584
1088
1·946
1·952
1·962
1·894
1·870
1·847
0·23
0·25
0·25
2·55
2·26
0·41
1·97
1·84
0·25
It is shown that for this thick plate with a vertical hole or with a square hole of rounded corners
the SCF are smaller than those of corresponding two-dimensional problems. But the normal stresses
z at the middle plane are quite large. It also can be seen that a fair SCF can be obtained by the
present method with coarse mesh for three-dimensional problems.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
? 1997 by John Wiley & Sons, Ltd.
STUDIES OF STRESS CONCENTRATION BY USING SPECIAL HYBRID STRESS ELEMENTS
1411
CONCLUSIONS
Three kinds of 12-node special solid hybrid stress elements are derived for stress concentration
analysis in a solid with many holes and rounded notches consisting of straight faces, inclined faces
and cylindrical faces under complex boundary traction. Numerical results have demonstrated that
the special solid elements are not only superior to the ordinary assumed displacement elements and
ordinary assumed stress elements, but also are more convenient than the photoelastic experiments
in predicting stress concentration factors and the distributions of stresses. It is particularly suitable
for three-dimensional problems of stress concentration which are dicult to be analysed by the
use of theoretical calculations and experimental methods.
ACKNOWLEDGEMENT
The work was sponsored by the National Natural Science Foundation of China.
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? 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1399–1411 (1997)
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