ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules
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2010 SECTION VIII, DIVISION 2
3-99
ANNEX 3.C
ISO MATERIAL GROUP NUMBERS
(CURRENTLY NOT USED)
2010 SECTION VIII, DIVISION 2
3-100
ANNEX 3.D
STRENGTH PARAMETERS
(NORMATIVE)
3.D.1 Yield Strength
Values for the yield strength as a function of temperature are provided in Table Y-1 in the ASME B&PV Code,
Section II, Part D.
3.D.2 Ultimate Tensile Strength
Values for the ultimate tensile strength as a function of temperature are provided in Table U in the ASME
B&PV Code, Section II, Part D.
3.D.3 Stress Strain Curve
The following model for the stress-strain curve shall be used in design calculations where required by this
Division when the strain hardening characteristics of the stress-strain curve are to be considered. The yield
strength and ultimate tensile strength in paragraphs 3.D.1 and 3.D.2 may be used in this model to determine
a stress-strain curve at a specified temperature.
12
t
t
y
E
σ
ε
γγ
=++
(3.D.1)
where
[]
()
1
1
1.0 tanh
2
H
ε
γ
=− (3.D.2)
[]
()
2
2
1.0 tanh
2
H
ε
γ
=+
(3.D.3)
1
1
1
1
m
t
A
σ
ε
⎛⎞
=
⎜⎟
⎝⎠
(3.D.4)
()
()
1
1
1
ln 1
ys ys
m
ys
A
σε
ε
+
=
⎡⎤
+
⎣⎦
(3.D.5)
[
]
()
1
ln
ln 1
ln
ln 1
pys
p
ys
R
m
ε
ε
ε
ε
+−
=
⎡⎤
⎡⎤
+
⎣⎦
⎢⎥
⎡⎤
+
⎢⎥
⎣⎦
⎣⎦
(3.D.6)
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2010 SECTION VIII, DIVISION 2
3-101
2
1
2
2
m
t
A
σ
ε
⎛⎞
=
⎜⎟
⎝⎠
(3.D.7)
[
]
2
2
2
2
exp
uts
m
m
A
m
σ
= (3.D.8)
{
}
()
()
2
tys utsys
uts ys
K
H
K
σσ σσ
σσ
⎡⎤
−+ −
⎣⎦
=
−
(3.D.9)
ys
uts
R
σ
σ
= (3.D.10)
0.002
ys
ε
=
(3.D.11)
1.5 2.5 3.5
1.5 0.5KR RR=−− (3.D.12)
The parameters
2
m , and
p
ε
are provided in Table 3.D.1.The development of the stress strain curve should
be limited to a value of true ultimate tensile stress at true ultimate tensile strain. The stress strain curve
beyond this point should be perfectly plastic. The value of true ultimate tensile stress at true ultimate tensile
strain is calculated as follows:
[
]
,2
exp
uts t uts
m
σσ
= (3.D.13)
3.D.4 Cyclic Stress Strain Curve
The cyclic stress-strain curve of a material (i.e. strain amplitude versus stress amplitude) may be represented
by the Equation (3.D.14). The material constants for this model are provided in Table 3.D.2.
1
css
n
aa
ta
ycss
EK
σσ
ε
⎡⎤
=+
⎢⎥
⎣⎦
(3.D.14)
The hysteresis loop stress-strain curve of a material (i.e. strain range versus stress range) obtained by scaling
the cyclic stress-strain curve by a factor of two is represented by the Equation (3.D.15). The material
constants provided in Table 3.D.2 are also used in this equation.
1
2
2
css
n
rr
tr
ycss
EK
σσ
ε
⎡⎤
=+
⎢⎥
⎣⎦
(3.D.15)
3.D.5 Tangent Modulus
3.D.5.1 Tangent Modulus Based on the Stress-Strain Curve Model
The tangent modulus based on the stress-strain curve model in paragraph 3.D.3 is given by the following
equation.
2010 SECTION VIII, DIVISION 2
3-102
1
1
1234
1
tt
t
tt y
EDDDD
E
σε
εσ
−
−
⎛⎞
⎛⎞
∂∂
== =++++
⎜⎟
⎜⎟
⎜⎟
∂∂
⎝⎠
⎝⎠
(3.D.16)
where
1
1
1
1
1
1
11
2
m
t
m
D
mA
σ
⎛⎞
−
⎜⎟
⎝⎠
⎛⎞
⎜⎟
⎝⎠
= (3.D.17)
()
[]
{}
[]
1 1
1
11
1
2
2
1
1
1
11 2 1
1 tanh tanh
2
mm
tt
uts ys
m
DHH
m
K
A
σσ
σσ
⎛⎞ ⎛ ⎞
−
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
⎛⎞
⎜⎟
⎝⎠
⎛⎞
⎛⎞
⎧⎫
⎜⎟
⎪⎪
⎜⎟
=− ⋅ − +
⎨⎬
⎜⎟
⎜⎟
−
⎪⎪
⎜⎟
⎩⎭
⎝⎠
⎝⎠
(3.D.18)
2
2
1
1
3
1
22
2
m
t
m
D
mA
σ
⎛⎞
−
⎜⎟
⎝⎠
⎛⎞
⎜⎟
⎝⎠
= (3.D.19)
()
[]
{}
[]
2 2
2
11
1
2
4
1
2
2
11 2 1
1tanh tanh
2
mm
tt
uts ys
m
DHH
m
K
A
σσ
σσ
⎛⎞ ⎛ ⎞
−
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
⎛⎞
⎜⎟
⎝⎠
⎛⎞
⎛⎞
⎧⎫
⎜⎟
⎪⎪
⎜⎟
=⋅ −+
⎨⎬
⎜⎟
⎜⎟
−
⎪⎪
⎜⎟
⎩⎭
⎝⎠
⎝⎠
(3.D.20)
The parameter K is given by Equation (3.D.12)
3.D.5.2 Tangent Modulus Based on External Pressure Charts
An acceptable alternative for calculating the Tangent Modulus is to use the External Pressure charts in
Section II, Part D, Subpart 3, including the notes to Subpart 3. The appropriate chart for the material under
consideration is assigned in the column designated External Pressure Chart Number given in Tables 1A or
1B. The tangent modulus,
t
E , is equal to 2
B
A , where
A
is the strain given on the abscissa and
B
is the
stress value on the ordinate of the chart.
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2010 SECTION VIII, DIVISION 2
3-103
3.D.6 Nomenclature
A Section II, Part D, Subpart 3 external pressure chart A-value.
1
A curve fitting constant for the elastic region of the stress-strain curve.
2
A curve fitting constant for the plastic region of the stress-strain curve.
B
Section II, Part D, Subpart 3 external pressure chart B-value.
1
D coefficient used in the tangent modulus.
2
D coefficient used in the tangent modulus.
3
D coefficient used in the tangent modulus.
4
D coefficient used in the tangent modulus.
p
ε
stress-strain curve fitting parameter.
t
ε
total true strain
ta
ε
total true strain amplitude.
tr
ε
total true strain range.
ys
ε
0.2% engineering offset strain.
1
ε
true plastic strain in the micro-strain region of the stress-strain curve.
2
ε
true plastic strain in the macro-strain region of the stress-strain curve.
t
E tangent modulus of elasticity evaluated at the temperature of interest.
y
E modulus of elasticity evaluated at the temperature of interest, see Annex 3.E.
1
γ
true strain in the micro-strain region of the stress-strain curve.
2
γ
true strain in the macro-strain region of the stress-strain curve.
H stress-strain curve fitting parameter.
K material parameter for stress-strain curve model
css
K material parameter for the cyclic stress-strain curve model.
1
m curve fitting exponent for the stress-strain curve equal to the true strain at the proportional
limit and the strain hardening coefficient in the large strain region.
2
m curve fitting exponent for the stress-strain curve equal to the true strain at the true ultimate
stress.
css
n material parameter for the cyclic stress-strain curve model.
a
σ
total stress amplitude.
r
σ
total stress range.
t
σ
true stress at which the true strain will be evaluated, may be a membrane, membrane plus
bending, or membrane, membrane plus bending plus peak stress depending on the
application.
ys
σ
engineering yield stress evaluated at the temperature of interest, see paragraph 3.D.1.
uts
σ
engineering ultimate tensile stress evaluated at the temperature of interest, see paragraph
3.D.2.
,uts t
σ
true ultimate tensile stress evaluated at the true ultimate tensile strain
R
engineering yield to engineering tensile ratio.
2010 SECTION VIII, DIVISION 2
3-104
3.D.7 Tables
Table 3.D.1 – Stress-Strain Curve Parameters
Material Temperature Limit
2
m
p
ε
Ferritic Steel
480°C (900°F)
(
)
0.60 1.00
R
−
2.0E-5
Stainless Steel and
Nickel Base Alloys
480°C (900°F)
(
)
0.75 1.00
R
−
2.0E-5
Duplex Stainless Steel
480°C (900°F)
(
)
0.70 0.95
R
−
2.0E-5
Precipitation
Hardenable Nickel
Base
540°C (1000°F)
(
)
1.90 0.93
R
−
2.0E-5
Aluminum
120°C (250°F)
(
)
0.52 0.98
R
−
5.0E-6
Copper
65°C (150°F)
(
)
0.50 1.00
R
−
5.0E-6
Titanium and
Zirconium
260°C (500°F)
(
)
0.50 0.98
R
−
2.0E-5
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2010 SECTION VIII, DIVISION 2
3-105
Table 3.D.2 – Cyclic Stress-Strain Curve Data
Material Description
Temperature (°F)
css
n
css
K (ksi)
Carbon Steel (0.75 in. – base metal)
70 0.128 109.8
390 0.134 105.6
570 0.093 107.5
750 0.109 96.6
Carbon Steel (0.75 in. – weld metal)
70 0.110 100.8
390 0.118 99.6
570 0.066 100.8
750 0.067 79.6
Carbon Steel (2 in. – base metal)
70 0.126 100.5
390 0.113 92.2
570 0.082 107.5
750 0.101 93.3
Carbon Steel (4 in. – base metal)
70 0.137 111.0
390 0.156 115.7
570 0.100 108.5
750 0.112 96.9
1Cr–1/2Mo (0.75 in. – base metal)
70 0.116 95.7
390 0.126 95.1
570 0.094 90.4
750 0.087 90.8
1Cr–1/2Mo (0.75 in. – weld metal)
70 0.088 96.9
390 0.114 102.7
570 0.085 99.1
750 0.076 86.9
1Cr–1/2Mo (0.75 in. – base metal)
70 0.105 92.5
390 0.133 99.2
570 0.086 88.0
750 0.079 83.7
1Cr–1Mo–1/4V
70 0.128 156.9
750 0.128 132.3
930 0.143 118.2
1020 0.133 100.5
1110 0.153 80.6
2-1/4Cr–1/2Mo
70 0.100 115.5
570 0.109 107.5
750 0.096 105.9
930 0.105 94.6
1110 0.082 62.1
2010 SECTION VIII, DIVISION 2
3-106
Table 3.D.2 – Cyclic Stress-Strain Curve Data
Material Description
Temperature (°F)
css
n
css
K (ksi)
9Cr–1Mo
70 0.177 141.4
930 0.132 100.5
1020 0.142 88.3
1110 0.121 64.3
1200 0.125 49.7
Type 304
70 0.171 178.0
750 0.095 85.6
930 0.085 79.8
1110 0.090 65.3
1290 0.094 44.4
Type 304 (Annealed) 70 0.334 330.0
800H
70 0.070 91.5
930 0.085 110.5
1110 0.088 105.7
1290 0.092 80.2
1470 0.080 45.7
Aluminum (Al–4.5Zn–0.6Mn) 70 0.058 65.7
Aluminum (Al–4.5Zn–1.5Mg) 70 0.047 74.1
Aluminum (1100-T6) 70 0.144 22.3
Aluminum (2014-T6) 70 0.132 139.7
Aluminum (5086) 70 0.139 96.0
Aluminum (6009-T4) 70 0.124 83.7
Aluminum (6009-T6) 70 0.128 91.8
Copper 70 0.263 99.1
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2010 SECTION VIII, DIVISION 2
3-107
Table 3.D.2M – Cyclic Stress-Strain Curve Data
Material Description Temperature (C)
css
n
css
K (MPa)
Carbon Steel (20 mm – base metal)
20 0.128 757
200 0.134 728
300 0.093 741
400 0.109 666
Carbon Steel (20 mm – weld metal)
20 0.110 695
200 0.118 687
300 0.066 695
400 0.067 549
Carbon Steel (50 mm – base metal)
20 0.126 693
200 0.113 636
300 0.082 741
400 0.101 643
Carbon Steel (100 mm – base metal)
20 0.137 765
200 0.156 798
300 0.100 748
400 0.112 668
1Cr–1/2Mo (20 mm – base metal)
20 0.116 660
200 0.126 656
300 0.094 623
400 0.087 626
1Cr–1/2Mo (20 mm – weld metal)
20 0.088 668
200 0.114 708
300 0.085 683
400 0.076 599
1Cr–1/2Mo (50 mm – base metal)
20 0.105 638
200 0.133 684
300 0.086 607
400 0.079 577
1Cr–1Mo–1/4V
20 0.128 1082
400 0.128 912
500 0.143 815
550 0.133 693
600 0.153 556
2-1/4Cr–1/2Mo
20 0.100 796
300 0.109 741
400 0.096 730
500 0.105 652
600 0.082 428
9Cr–1Mo
20 0.117 975
500 0.132 693
550 0.142 609
600 0.121 443
2010 SECTION VIII, DIVISION 2
3-108
Table 3.D.2M – Cyclic Stress-Strain Curve Data
Material Description Temperature (C)
css
n
css
K (MPa)
650 0.125 343
Type 304
20 0.171 1227
400 0.095 590
500 0.085 550
600 0.090 450
700 0.094 306
Type 304 (Annealed) 20 0.334 2275
800H
20 0.070 631
500 0.085 762
600 0.088 729
700 0.092 553
800 0.080 315
Aluminum (Al–4.5Zn–0.6Mn) 20 0.058 453
Aluminum (Al–4.5Zn–1.5Mg) 20 0.047 511
Aluminum (1100-T6) 20 0.144 154
Aluminum (2014-T6) 20 0.132 963
Aluminum (5086) 20 0.139 662
Aluminum (6009-T4) 20 0.124 577
Aluminum (6009-T6) 20 0.128 633
Copper 20 0.263 683
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