ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules
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2010 SECTION VIII, DIVISION 2
5-109
Table 5.E.3 – Values of
E
∗
For Perforated Tubesheets With A Square Pattern
Coefficients
hp
0.1 0.25 0.5 2.0
0
A
-4.085600E-05 -4.828500E-04 8.925870E-05 1.589308E-03
1
A
5.231431E+02 8.372334E+00 1.783577E+01 1.814855E+01
2
A
-6.376136E+00 2.947067E+00 2.896931E+00 2.693793E+00
3
A
7.857623E+03 -9.963856E+00 -1.945644E+01 -2.374421E+01
4
A
2.284635E+03 3.710254E+00 1.377083E+01 9.002308E+00
5
A
-7.366960E+03 7.250874E+00 8.840783E+00 1.627106E+01
6
A
4.844156E+03 0.0 -8.457560E+00 0.0
7
A
6.092597E+03 0.0 0.0 0.0
Note:
1.
23
02 4 6
234
1357
1
AA A A
EE
AA A A
μμ μ
μμ μ μ
∗∗ ∗
∗
∗∗ ∗ ∗
⎡⎤
++ +
=
⎢⎥
++ + +
⎣⎦
2. These coefficients are valid for 01.0
μ
∗
≤≤, data for 2.0
μ
∗
= is provided for information only.
3. If
0.1hp< , then use 0.1hp= .
4. If
2.0hp> , then use 2.0hp= .
2010 SECTION VIII, DIVISION 2
5-110
Table 5.E.4 – Values of
*
ν
For Perforated Tubesheets With A Square Pattern
Coefficients
hp
0.1 0.15 0.25 0.5 1.0 2.0
0
B
-7.288894E-02 2.089015E+00 1.397869E+03 4.915542E-01 3.475156E-01 3.337833E-01
1
B
-1.127111E+01 1.199162E+02 8.678686E+04 2.202591E+01 -1.094083E+00 2.923681E+00
2
B
1.325618E+00 -1.535964E+01 5.003924E+03 4.392312E+00 -5.737278E-01 1.132010E+00
3
B
3.949776E+01 -7.543170E+02 -1.857192E+05 -5.969066E+01 4.985966E+00 -1.209380E+01
4
B
-8.530838E+00 6.129051E+01 -2.174219E+04 -1.407781E+01 1.875825E+00 -4.921427E+00
5
B
-2.359313E+01 1.921504E+03 2.363003E+05 6.022030E+01 6.108286E-01 1.656896E+01
6
B
2.130422E+01 -1.346693E+02 5.674145E+04 1.627461E+01 0.0 6.722375E+00
7
B
5.916208E+00 -1.921418E+03 0.0 0.0 0.0 2.496675E+00
8
B
-1.056089E+01 1.808025E+02 0.0 0.0 0.0 0.0
9
B
0.0 9.465777E+02 0.0 0.0 0.0 0.0
Note:
1.
234
02 4 6 8
2345
13 5 7 9
1
BB B B B
BB B B B
μμ μ μ
ν
μ
μμμμ
∗∗ ∗ ∗
∗
∗∗ ∗ ∗ ∗
++ + +
=
++ + + +
2. These coefficients are valid for 01.0
μ
∗
≤≤, data for 2.0
μ
∗
= is provided for information only.
3. If
0.1hp< , then use 0.1hp= .
4. If
2.0hp>
, then use
2.0hp=
.
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2010 SECTION VIII, DIVISION 2
5-111
Table 5.E.5 – Effective Elastic Modulus, Poisson’s Ratio, And Shear Modulus For A Perforated Plate
With A Triangular Hole Pattern
Plane Stress
**2
*
*3 *4 *5
0.000710796 0.138017 4.22004
7.55885 7.08667 2.88588
EE
μμ
μμμ
⎡⎤
−++−
=
⎢⎥
+−
⎣⎦
* * *2 *3 *4
0.999804 3.94540 8.47377 8.02434 2.79724
ν
μμ μ μ
=− + − +
()
*
*
*
21
E
G
ν
=
+
Generalized Plane Strain
()
1
2
2
*
**
1
z
E
EE
ν
ν
−
⎡⎤
−
⎢⎥
=+
⎢⎥
⎣⎦
()
2*
3
**
2
**
1
11
z
E
EEEE
νν
ν
νν
⎡⎤
−
⎛⎞
⎢⎥
=+++
⎜⎟
⎢⎥
⎝⎠
⎣⎦
*
*
GG=
Plane Stress and Generalized Plane Strain
()
2
*
*
1
1
23
z
EE
πμ
⎡⎤
−
⎢⎥
=−
⎢⎥
⎣⎦
23
*
23
0.000209547 3.00960 * 32.7845 * 16.57921 *
1 34.7831 * 33.7539 * 17.1860 *
z
GG
μμ μ
μμ μ
⎡⎤
++ −
=
⎢⎥
+− +
⎣⎦
Notes: These coefficients are only valid for
0.1 * 1.0
μ
≤
≤
.
2010 SECTION VIII, DIVISION 2
5-112
Table 5.E.6 – Effective Elastic Modulus, Poisson’s Ratio, And Shear Modulus For A Perforated Plate
With A Square Hole Pattern – Pitch Direction
Plane Stress
**2 *3
*
*4 *5 *6 *7
0.00381121 2.98282 17.5077 79.3742
193.933 258.876 177.201 48.4060
p
EE
μμ μ
μμμμ
⎡⎤
+− + −
=
⎢⎥
+−+
⎣⎦
**2*3
*
**2 *3*4
0.000479230 2.22119 6.92863 7.35650
1 22.8486 37.2914 46.4082 26.9865
p
μ
μμ
ν
μ
μμμ
++ −
=
+− + −
()
*
*
*
21
d
p
d
E
G
ν
=
+
Generalized Plane Strain
()
1
2
2
*
**
1
p
pz
E
EE
ν
ν
−
⎡⎤
−
⎢⎥
=+
⎢⎥
⎣⎦
()
2*
3
**
2
**
1
11
p
p
pz
E
EEEE
νν
ν
νν
⎡⎤
−
⎛⎞
⎢⎥
=+++
⎜⎟
⎢⎥
⎝⎠
⎣⎦
*
*
p
p
GG=
Plane Stress and Generalized Plane Strain
()
2
*
*
1
1
4
z
EE
πμ
⎡⎤
−
⎢⎥
=−
⎢⎥
⎣⎦
23
*
23
0.0017 4.42892 * 23.3350 * 12.8075 *
1 25.7230 * 22.3355 * 10.5712 *
z
GG
μμ μ
μμ μ
⎡⎤
++ −
=
⎢⎥
+− +
⎣⎦
Notes:
1. These coefficients are only valid for
0.1 * 1.0
μ
≤
≤
.
2. The constants
*
d
E and
*
d
ν
can be determined using the equations in Table 5.E.7.
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2010 SECTION VIII, DIVISION 2
5-113
Table 5.E.7 – Effective Elastic Modulus, Poisson’s Ratio, And Shear Modulus For A Perforated Plate
With A Square Hole Pattern – Diagonal Direction
Plane Stress
**2*3
*
**2*3
0.000399053 0.165345 2.02690 1.15319
1 0.493831 0.354744 0.177698
d
EE
μ
μμ
μμ μ
⎡⎤
−++−
=
⎢⎥
−+ +
⎣⎦
**2
*
**2 *3
0.997998 2.27993 1.96210
1 0.250614 1.330 2.85019
d
μμ
ν
μ
μμ
−+
=
−−+
()
*
*
*
21
p
d
p
E
G
ν
=
+
Generalized Plane Strain
()
1
2
2
*
**
1
d
dz
E
EE
ν
ν
−
⎡⎤
−
⎢⎥
=+
⎢⎥
⎣⎦
()
2*
3
**
2
**
1
11
d
d
dz
E
EEEE
νν
ν
νν
⎡⎤
−
⎛⎞
⎢⎥
=+++
⎜⎟
⎢⎥
⎝⎠
⎣⎦
*
*
d
d
GG=
Plane Stress and Generalized Plane Strain
()
2
*
*
1
1
4
z
EE
πμ
⎡⎤
−
⎢⎥
=−
⎢⎥
⎣⎦
23
*
23
0.0017 4.42892 * 23.3350 * 12.8075 *
1 25.7230 * 22.3355 * 10.5712 *
z
GG
μμ μ
μμ μ
⎡⎤
++ −
=
⎢⎥
+− +
⎣⎦
Notes:
1. These coefficients are only valid for
0.1 * 1.0
μ
≤
≤ .
2. The constants
*
p
E
and
*
p
ν
can be determined using the equations in Table 5.E.6.
2010 SECTION VIII, DIVISION 2
5-114
Table 5.E.8 – Orthotropic Effective Elasticity Matrix For A Perforated Plate With An Equilateral
Triangular Hole Pattern
**
11 12 13
**
21 22 23
**
31 32 33
**
44
**
55
**
66
000
000
000
000 00
0000 0
00000
rr
zz
zz
rz rz
rr
EEE
EEE
EEE
E
E
E
θθ
θθ
θ
θ
σ
ε
σ
ε
σ
ε
τ
γ
τ
γ
τ
γ
⎡⎤ ⎡⎤
⎡⎤
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
=⋅
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎣⎦
⎣⎦ ⎣⎦
**
2
11 22
**
1
(1 )
z
EE
EE
HE
ν
ν
⎡⎤
⎛⎞
== −
⎢⎥
⎜⎟
⋅+
⎝⎠
⎣⎦
**
*2
12 21
**
(1 )
z
EE
EE
HE
ν
ν
ν
⎡⎤
⎛⎞
== −
⎢⎥
⎜⎟
⋅+
⎝⎠
⎣⎦
*
13 31 32 23
E
EEEE
H
ν
====
**
33
(1 )
z
E
E
H
ν
−
=
*
44 55 z
E
EG==
*
66
*
2(1 )
E
E
ν
=
+
where:
*
*2
*
12
z
E
H
E
ν
ν
⎛⎞
=− −
⎜⎟
⎝⎠
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2010 SECTION VIII, DIVISION 2
5-115
Table 5.E.9 – Orthotropic Effective Elasticity Matrix For A Perforated Plate With A Square Hole
Pattern
**
11 12 13
**
21 22 23
**
31 32 33
**
44
**
55
**
66
000
000
000
000 00
0000 0
00000
rr
zz
zz
rz rz
rr
EEE
EEE
EEE
E
E
E
θθ
θθ
θ
θ
σ
ε
σ
ε
σ
ε
τ
γ
τ
γ
τ
γ
⎡⎤ ⎡⎤
⎡⎤
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
=⋅
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎣⎦
⎣⎦ ⎣⎦
** *
2*
11 22
** *
1
1
(1 ) 2 2(1 )
pp p
p
pz p
EE E
EE G
HE
ν
νν
⎡⎤
⎡
⎤
⎛⎞
== − + −
⎢⎥
⎜⎟
⎢
⎥
⎜⎟
⋅+ −
⎢
⎥
⎢⎥
⎝⎠
⎣
⎦
⎣⎦
** *
*2 *
12 21
** *
1
(1 ) 2 2(1 )
pp p
pp
pz p
EE E
EE G
HE
νν
νν
⎡⎤
⎡
⎤
⎛⎞
== + − −
⎢⎥
⎜⎟
⎢
⎥
⎜⎟
⋅+ −
⎢
⎥
⎢⎥
⎝⎠
⎣
⎦
⎣⎦
*
13 31 32 23
p
E
EEEE
H
ν
====
**
33
(1 )
zp
E
E
H
ν
−
=
*
44 55 z
E
EG==
*
*
66
*
2(1 )
d
p
d
E
EG
ν
==
+
where:
*
*2
*
12
p
p
z
E
H
E
ν
ν
⎛⎞
=− −
⎜⎟
⎜⎟
⎝⎠
Note: The effective constants in diagonal directions
***
,,
ddd
EG
ν
can be equivalently used instead of pitch
direction constants
***
,,
ppp
EG
ν
in the elasticity matrix shown above.
2010 SECTION VIII, DIVISION 2
5-116
Table 5.E.10 – Equations For Determining Stress Components Based On The Results From An
Equivalent Plate Analysis For An Equilateral Rectangular Hole Pattern
()
()
()
()
()
()
*
*
11 12 13
*
*
22 22 23
*
*
33 33 33
*
*
44
*
55*
*
66
*
000
000
000
000 00
0000 0
00000
r
a
r
a
z
a
z
z
z
a
rz
rz
a
r
r
a
EEE
EEE
EEE
E
E
E
θ
θ
θ
θ
θ
θ
σ
ε
σ
ε
σ
ε
γ
τ
γ
τ
γ
τ
⎡⎤
⎢⎥
⎡
⎤
⎡⎤
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
=⋅
⎢
⎥
⎢⎥
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
⎢
⎥
⎢⎥
⎢⎥
⎢⎥
⎢
⎥
⎣⎦
⎣
⎦
⎢⎥
⎢⎥
⎣⎦
** *
222*
11 22
** *
14sincos
(1 ) 2(1 )
pp p
p
pz p
EE E
EE G
HE
νθθ
νν
⎡⎤
⎡
⎤
⎛⎞
== − + −
⎢⎥
⎜⎟
⎢
⎥
⎜⎟
⋅+ +
⎢
⎥
⎢⎥
⎝⎠
⎣
⎦
⎣⎦
** *
*2 2 2 *
12 21
** *
4sin cos
(1 ) 2(1 )
pp p
pp
pz p
EE E
EE G
HE
νν θ θ
νν
⎡⎤
⎡
⎤
⎛⎞
== + + −
⎢⎥
⎜⎟
⎢
⎥
⎜⎟
⋅+ +
⎢
⎥
⎢⎥
⎝⎠
⎣
⎦
⎣⎦
*
13 31 32 23
p
E
EEEE
H
ν
====
()
()
*
*22
14 41 24 42
*
2sincoscossin
21
p
P
p
E
EE E E G
θ
θθ θ
ν
⎡⎤
⎢⎥
= =−=−= − −
+
⎢⎥
⎣⎦
**
33
(1 )
zp
E
E
H
ν
−
=
*
44 55 z
E
EG==
*
*22*
66
*
4sin cos
2(1 )
p
pp
p
E
EG G
θθ
ν
⎡⎤
=− −
⎢⎥
+
⎢⎥
⎣⎦
Note:
1. The constants
** * *
,,, ,,
pP z z
EEGH
ν
ν
are evaluated from Table 5.E.5 through 5.E.9.
2. The angle
θ
defines the orientation of the radial direction in the equivalent solid plate with respect to
the pitch direction of the square hole pattern.
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2010 SECTION VIII, DIVISION 2
5-117
Table 5.E.11 – Stress Factor
x
K Coefficients – Triangular Hole Pattern
*
μ
1
C
2
C
3
C
4
C
5
C
0.05 -2.48810E-03 0.00000E+00 2.29959E-04 0.00000E+00 -8.60430E-07
0.1 -4.15591E-03 0.00000E+00 3.99110E-04 0.00000E+00 -1.66750E-06
0.15 9.71786E-03 0.00000E+00 2.27507E-04 0.00000E+00 -2.02240E-06
0.2 2.44173E-02 -2.05029E-03 -1.50600E-04 2.92198E-06 2.80179E-08
0.25 4.97018E-03 -4.07320E-04 -4.68940E-04 1.34485E-05 -7.98040E-08
0.3 -4.38107E-02 1.63067E-03 -6.65130E-04 1.92456E-05 -1.14750E-07
0.333 -9.22413E-02 3.29146E-03 -6.93000E-04 2.32530E-05 -2.12050E-07
0.4 -2.09998E-01 6.59092E-03 -5.32060E-04 1.63467E-05 -1.04550E-07
0.5 -4.01999E-01 -3.94910E-04 2.45399E-04 -1.44260E-05 1.19709E-06
0.6 -5.75989E-01 -1.66130E-04 5.74246E-04 -4.90600E-06 3.73271E-07
0.7 -7.14696E-01 -8.35510E-05 8.53270E-04 -7.23000E-06 5.22499E-07
0.8 -8.19200E-01 -8.61900E-04 1.26476E-03 -3.60460E-05 2.33690E-06
0.9 -9.07202E-01 -3.10980E-04 1.16915E-03 -6.47650E-06 2.87057E-07
1 -1.00000E+00 -6.01550E-04 1.40134E-03 -2.02820E-05 1.00488E-06
*
μ
6
C
7
C
8
C
9
C
10
C
0.05 0.00000E+00 3.79570E-10 0.00000E+00 -5.00380E-14 0.00000E+00
0.1 0.00000E+00 7.52024E-10 0.00000E+00 -9.80750E-14 0.00000E+00
0.15 0.00000E+00 9.96594E-10 0.00000E+00 -1.38990E-13 0.00000E+00
0.2 -3.48070E-10 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.25 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.3 -6.26280E-11 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.333 4.79506E-10 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.4 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.5 -3.91660E-08 8.96544E-10 -1.42550E-11 1.38091E-13 -7.22820E-16
0.6 -1.21420E-08 2.55297E-10 -3.89990E-12 3.65760E-14 -1.81670E-16
0.7 -2.46600E-08 6.39236E-10 -1.03660E-11 1.01715E-13 -5.48690E-16
0.8 -9.59670E-08 2.34704E-09 -3.58780E-11 3.34855E-13 -1.74180E-15
0.9 -1.57740E-08 4.06586E-10 -6.50940E-12 6.42221E-14 -3.52630E-16
1 -3.55490E-08 6.67355E-10 -7.22950E-12 4.25909E-14 -1.05160E-16
*
μ
11
C
12
C
13
C
14
C
15
C
0.05 2.03746E-18 0.00000E+00 -1.02414E-03 0.00000E+00 6.61817E-07
0.1 4.37065E-18 0.00000E+00 -6.92440E-04 0.00000E+00 6.50223E-07
0.15 7.88278E-18 0.00000E+00 5.49459E-07 0.00000E+00 2.94340E-07
0.2 0.00000E+00 -7.40767E-02 2.51869E-03 -4.42910E-05 3.93678E-07
0.25 0.00000E+00 -5.46478E-02 1.59424E-03 -2.61460E-05 2.33088E-07
0.3 0.00000E+00 -3.98100E-02 1.04897E-03 -1.75510E-05 1.75088E-07
0.333 0.00000E+00 -3.57930E-02 8.10224E-04 -1.19240E-05 1.11006E-07
0.4 0.00000E+00 -3.13608E-02 7.61981E-04 -1.15130E-05 1.06015E-07
0.5 1.57382E-18 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.6 3.65531E-19 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.7 1.24975E-18 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.8 3.86817E-18 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.9 8.21600E-19 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
2010 SECTION VIII, DIVISION 2
5-118
Table 5.E.11 – Stress Factor
x
K Coefficients – Triangular Hole Pattern
*
μ
16
C
17
C
18
C
19
C
0.05 0.00000E+00 -2.21770E-10 3.53954E-14 -1.97610E-18
0.1 0.00000E+00 -2.34390E-10 3.92678E-14 -2.04780E-18
0.15 0.00000E+00 -8.74970E-11 1.41889E-14 -6.43590E-20
0.2 -1.36300E-09 0.00000E+00 0.00000E+00 0.00000E+00
0.25 -8.31760E-10 0.00000E+00 0.00000E+00 0.00000E+00
0.3 -7.17880E-10 0.00000E+00 0.00000E+00 0.00000E+00
0.333 -4.62140E-10 0.00000E+00 0.00000E+00 0.00000E+00
0.4 -4.07500E-10 0.00000E+00 0.00000E+00 0.00000E+00
0.5 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.6 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.8 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
0.9 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00
Notes:
1.
2345678 9 10
1234567891011
23456810
12 13 14 15 16 17 18 19
1
x
CCCCCCCCCC C
K
CC C C C C C C
θ
θθθθθθθ θ θ
θθ θ θ θ θ θ θ
+++++++++ +
=
++ + + + + + +
2. See Figure 5.E.4 for definition of
x
K and
θ
(
θ
is in Degrees,
090
θ
≤
≤
)
3.
Valid range is 0.1 1.0
μ
∗
≤≤, data for 0.05
μ
∗
= is provided for information only.
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