ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules
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2010 SECTION VIII, DIVISION 2
4-55
4.3.11.3 The length of the conical shell, measured parallel to the surface of the cone shall be equal to or
greater than the following value.
[] []
2.0 1.4
cos cos
LC SC
C
R
tRt
L
α
α
≥+ (4.3.46)
4.3.11.4 The procedure that shall be used to design the large end of a cylinder-to-cone junction without a
knuckle is described below.
a) STEP 1 – Compute the large end cylinder thickness,
L
t , using paragraph 4.3.3.
b) STEP 2 – Determine the cone half-apex angle,
α
, and compute the cone thickness,
C
t , at the large
end using paragraph 4.3.4.
c) STEP 3 – Proportion the cone geometry such that Equation (4.3.46) and the following equations are
satisfied. If all of these equations are not satisfied, then the cylinder-to-cone junction shall be designed
in accordance with Part 5. In the calculations, if
010
α
°< ≤ °, then use 10
α
=
° .
20 500
L
L
R
t
≤≤
(4.3.47)
12
C
L
t
t
≤≤
(4.3.48)
60
α
≤°
(4.3.49)
d) STEP 4 – Determine the net section axial force,
L
F , and bending moment ,
L
M
, applied to the conical
transition. The thrust load due to pressure shall not be included as part of the axial force,
L
F .
Determine an equivalent,
L
X
, using Equation (4.3.50).
2
2
LL
L
LL
FM
X
R
R
π
π
=±
(4.3.50)
e) STEP 5 – Compute the junction transition design parameters. For calculated values of
n other than
those presented in Table 4.3.3 and Table 4.3.4, linear interpolation of the equation coefficients,
i
C , is
permitted.
C
L
t
n
t
=
(4.3.51)
L
L
R
H
t
=
(4.3.52)
[
]
tanB
α
= (4.3.53)
2010 SECTION VIII, DIVISION 2
4-56
f) STEP 6 – Compute the stresses in the cylinder and cone at the junction using the equations in Table
4.3.1. The allowable stress criterion for a tensile stress is provided in Table 4.3.1. If either the hoop
membrane stress,
m
θ
σ
, or axial membrane stress,
s
m
σ
, at the junction is compressive, then the
condition of local buckling shall be considered. Local buckling is not a concern if the limits given in
Equations (4.3.54) and (4.3.55) are satisfied.
ha
F is evaluated using paragraph 4.4.5.1, but substituting
0.4
he
o
t
FE
D
⎛⎞
=
⎜⎟
⎝⎠
.
x
a
F is evaluated using paragraph 4.4.12.2.b with
0.15
λ
=
. If the stresses of the
acceptance criteria are satisfied, the design of the junction is complete.
mha
F
θ
σ
≤ (4.3.54)
s
mxa
F
σ
≤ (4.3.55)
g) STEP 7 – If the stress acceptance criterion in STEP 6 is satisfied, then the design is complete. If the
stress acceptance criterion in STEP 6 is not satisfied, the cylinder thickness or cone thickness near the
junction may be increased until the stress acceptance criterion is satisfied. The section of increased
thickness for the cylinder and cone shall extend a minimum distance from the junction as shown in
Figure 4.3.9. Proceed to STEP 3 to repeat the calculation with the new wall thickness.
4.3.11.5 The procedure that shall be used to design the small end of a cylinder-to-cone junction without a
flare is described below.
a) STEP 1 – Compute the small end cylinder thickness,
S
t , using paragraph 4.3.3.
b) STEP 2 – Determine the cone half-apex angle,
α
, and compute the cone thickness,
C
t , at the small
end using paragraph 4.3.4.
c) STEP 3 – Proportion the cone geometry such that Equation (4.3.46) and the following equations are
satisfied. If all of these equations are not satisfied, then the cylinder-to-cone junction shall be designed
in accordance with Part 5. In the calculations, if
010
α
°< ≤ °, then use 10
α
=
° .
20 500
S
S
R
t
≤≤
(4.3.56)
12
C
S
t
t
≤≤
(4.3.57)
60
α
≤° (4.3.58)
d) STEP 4 – Determine the net section axial force,
S
F , and bending moment ,
S
M
, applied to the conical
transition. The thrust load due to pressure shall not be included as part of the axial force,
S
F .
Determine an equivalent,
S
X
, line using Equation (4.3.59).
2
2
SS
S
SS
FM
X
R
R
ππ
=± (4.3.59)
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2010 SECTION VIII, DIVISION 2
4-57
e) STEP 5 – Compute the junction transition design parameters. For calculated values of
n other than
those presented in Table 4.3.5 and Table 4.3.6, linear interpolation of the equation coefficients,
i
C , is
permitted.
C
S
t
n
t
=
(4.3.60)
S
S
R
H
t
= (4.3.61)
[
]
tanB
α
=
(4.3.62)
f) STEP 6 – Compute the stresses in the cylinder and cone at the junction using the equations in Table
4.3.2. The allowable stress criterion for a tensile stress is provided in Table 4.3.2. If either the hoop
membrane stress,
m
θ
σ
, or axial membrane stress,
s
m
σ
, at the junction is compressive, then the
condition of local bucking shall be considered. Local buckling is not of concern if the limits given in
Equations (4.3.54) and (4.3.55) are satisfied, using the procedure provided in paragraph 4.3.11.4.f. If the
stresses of the acceptance criteria are satisfied, the design of the junction is complete.
g) STEP 7 – If the stress acceptance criterion in STEP 6 is satisfied, then the design is complete. If the
stress acceptance criterion in STEP 6 is not satisfied, the cylinder thickness or cone thickness near the
junction may be increased until the stress acceptance criterion is satisfied. The section of increased
thickness for the cylinder and cone shall extend a minimum distance from the junction as shown in
Figure 4.3.9. Proceed to STEP 3 to repeat the calculation with the new wall thickness.
4.3.12 Cylindrical-To-Conical Shell Transition Junctions with a Knuckle
4.3.12.1 General – The following rules are applicable for the design of conical transitions of circular cross-
section with a knuckle at the large end or flare at the small end under loadings of internal pressure and
applied net section axial force and bending moment. Acceptable conical transition details are shown in
Figure 4.3.10. Design rules for transition junctions without a knuckle at the large end or flare at the small end
are provided in paragraph 4.3.11.
4.3.12.2 The procedure that shall be used to design the large end of a cylinder-to-cone junction with a
knuckle is described below.
a) STEP 1 – Compute the large end cylinder thickness,
L
t , using paragraph 4.3.3.
b) STEP 2 – Determine the cone half-apex angle,
α
, and compute the cone thickness,
C
t , at the large
end using paragraph 4.3.4.
c) STEP 3 – Proportion the transition geometry by assuming a value for the knuckle radius,
k
r , and knuckle
thickness,
k
t , such that the following equations are satisfied. If all of these equations cannot be
satisfied, then the cylinder-to-cone junction shall be designed in accordance with Part 5.
kL
tt≥ (4.3.63)
3
kk
rt> (4.3.64)
2010 SECTION VIII, DIVISION 2
4-58
0.03
k
L
r
R
>
(4.3.65)
60
α
≤° (4.3.66)
d) STEP 4 – Determine the net section axial force,
L
F , and bending moment ,
L
M
, applied to the conical
transition at the location of the knuckle. The thrust load due to pressure shall not be included as part of
the axial force,
L
F .
e) STEP 5 – Compute the stresses in the cylinder, knuckle, and cone at the junction using the equations in
Table 4.3.7. The allowable stress criterion for a tensile stress is provided in Table 4.3.7. If either the
hoop membrane stress,
m
θ
σ
, or axial membrane stress,
s
m
σ
, at the junction is compressive, then the
condition of local buckling shall be considered. Local buckling is not a concern if the limits given in
Equations (4.3.67) and (4.3.68) are satisfied.
ha
F is evaluated using paragraph 4.4.5.1, but substituting
0.4
he
o
t
FE
D
⎛⎞
=
⎜⎟
⎝⎠
.
x
a
F is evaluated using paragraph 4.4.12.2.b with
0.15
λ
=
. If the stresses of the
acceptance criteria are satisfied, the design of the junction is complete.
mha
F
θ
σ
≤ (4.3.67)
s
mxa
F
σ
≤ (4.3.68)
f) STEP 6 – If the stress acceptance criterion in STEP 5 is satisfied, then the design is complete. If the
stress acceptance criterion in STEP 5 is not satisfied, the knuckle thickness, cylinder thickness, or cone
thickness near the junction may be increased until the stress acceptance criterion is satisfied. If the
cylinder or cone thickness is increased, the section of increased thickness shall extend a length given by
Equations (4.3.69) and (4.3.70), respectively. Proceed to STEP 3 to repeat the calculation with the
new
wall thicknesses.
rcy m L L
LKRt= (4.3.69)
rco m k C
LKLt= (4.3.70)
4.3.12.3 The procedure that shall be used to design the small end of a cylinder-to-cone junction with a
flare is described below.
a) STEP 1 – Compute the small end cylinder thickness,
S
t , using paragraph 4.3.3.
b) STEP 2 – Determine the cone half-apex angle,
α
, and compute the cone thickness,
C
t , at the small
end using paragraph 4.3.4.
c) STEP 3 – Proportion the transition geometry by assuming a value for the flare radius,
f
r , and flare
thickness,
f
t , such that the following equations are satisfied. If all of these equations cannot be
satisfied, then the cylinder-to-cone junction shall be designed in accordance with Part 5.
f
S
tt≥ (4.3.71)
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2010 SECTION VIII, DIVISION 2
4-59
3
f
f
rt> (4.3.72)
0.03
f
S
r
R
>
(4.3.73)
60
α
≤° (4.3.74)
d) STEP 4 – Determine the net section axial force,
S
F , and bending moment ,
S
M
, applied to the conical
transition at the location of the knuckle. The thrust load due to pressure shall not be included as part of
the axial force,
S
F .
e) STEP 5 – Compute the stresses in the cylinder, flare, and cone at the junction using the equations in
Table 4.3.8. The allowable stress criterion for a tensile stress is provided in Table 4.3.8. If either the
hoop membrane stress,
m
θ
σ
, or axial membrane stress,
s
m
σ
, at the junction is compressive, then the
condition of local buckling shall be considered. Local buckling is not of concern if the limits given in
Equation (4.3.67) and (4.3.68) are satisfied, using the procedure provided in paragraph 4.3.12.2.e. If the
stresses of the acceptance criteria are satisfied, the design of the junction is complete.
f) STEP 6 – If the stress acceptance criterion in STEP 5 is satisfied, then the design is complete. If the
stress acceptance criterion in STEP 5 is not satisfied, the knuckle thickness, cylinder thickness, or cone
thickness near the junction may be increased until the stress acceptance criterion is satisfied. If the
cylinder or cone thickness is increased, the section of increased thickness shall extend a length given by
Equations (4.3.75) and (4.3.76), respectively. Proceed to STEP 3 to repeat the calculation with the
new
wall thicknesses.
rcy m S S
LKRt= (4.3.75)
rco m f C
LKLt= (4.3.76)
4.3.13 Nomenclature
R
A cross-sectional area of the stiffening ring at the junction.
α
one-half of the apex angle of a conical shell.
1
α
cone angle in an offset transition.
2
α
cone angle in an offset transition.
B
curve-fit geometric constant
co
β
geometric factor for the cone.
cy
β
geometric factor for the cylinder.
f
β
angle used in the conical transition calculation when a flare is present.
1
f
β
angle used in the conical transition calculation when a flare is present.
2
f
β
angle used in the conical transition calculation when a flare is present.
k
β
angle used in the conical transition calculation when a knuckle is present.
1k
β
angle used in the conical transition calculation when a knuckle is present.
2k
β
angle used in the conical transition calculation when a knuckle is present.
th
β
angle used in the torispherical head calculation.
2010 SECTION VIII, DIVISION 2
4-60
1
C angle constant used in the torispherical head calculation.
2
C angle constant used in the torispherical head calculation.
3
C strength parameter used in the torispherical head calculation.
d diameter of a drilled hole that does not completely penetrate a shell.
D inside diameter of a shell or head.
o
D outside diameter of a shell or head.
T
E modulus of elasticity at maximum design temperature.
RT
E modulus of elasticity at room temperature.
E weld joint factor (see paragraph 4.2.4), the ligament efficiency (see paragraph 4.10.2), or the
casting quality factor (see Part 3), as applicable, for the weld seam being evaluated (i.e.
longitudinal or circumferential).
F net-section axial force acting at the point of consideration, a positive force produces an axial
tensile stress in the cylinder.
L
F net-section axial force acting on the large end cylindrical shell, a positive force produces an
axial tensile stress in the cylinder.
S
F net-section axial force acting on the small end cylindrical shell, a positive force produces an
axial tensile stress in the cylinder.
ha
F allowable compressive hoop membrane stress as given in paragraph 4.4.
x
a
F allowable compressive axial membrane stress as given in paragraph 4.4.
G constant used in the torispherical head calculation.
H curve-fit geometric constant.
h height of the ellipsoidal head measured to the inside surface.
R
I
moment of inertia of the stiffening ring at the junction.
k
j
number of locations around the knuckle that shall be evaluated, used in the conical transition
stress calculation when a non-compact knuckle is present.
f
j
number of locations around the flare that shall be evaluated, used in the conical transition
stress calculation when a non-compact flare is present.
k
angle constant used in the torispherical and elliptical head calculation.
m
K length factor used in the conical transition calculation when a flare or knuckle is present.
pc
K cylinder-to-cone junction plasticity correction factor.
λ
compressive stress factor.
L
inside crown radius of a torispherical head.
c
L projected length of a conical shell.
f
L length used in the conical transition stress calculation when a flare is present.
1
f
L length used in the conical transition stress calculation when a flare is present.
1
j
f
L length used in the conical transition stress calculation when a flare is present.
k
L length used in the conical transition stress calculation when a knuckle is present.
1k
L length used in the conical transition stress calculation when a knuckle is present.
1
j
k
L length used in the conical transition stress calculation when a knuckle is present.
M
net-section bending moment acting at the point of consideration.
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2010 SECTION VIII, DIVISION 2
4-61
cs
M
total resultant meridional moment acting on the cone.
csP
M cylinder-to-cone junction resultant meridional moment acting on the cone, due to internal
pressure.
csX
M cylinder-to-cone junction resultant meridional moment acting on the cone, due to an
equivalent line load.
s
M
total resultant meridional moment acting on the cylinder.
s
P
M
cylinder-to-cone junction resultant meridional moment acting on the cylinder, due to internal
pressure.
s
X
M
cylinder-to-cone junction resultant meridional moment acting on the cylinder, due to an
equivalent line load.
s
N
M
normalized curve-fit resultant meridional moment acting on the cylinder.
L
M
net-section bending moment acting at the large end cylindrical shell.
S
M
net-section bending moment acting at the small end cylindrical shell.
t
M
net-section torsional moment acting on a shell section.
cs
N resultant meridional membrane force acting on the cone, due to pressure plus an equivalent
line load.
c
N
θ
resultant circumferential membrane force acting on the cone, due to pressure plus an
equivalent line load.
s
N resultant meridional membrane force acting on the cylinder, due to pressure plus an
equivalent line load.
N
θ
resultant circumferential membrane force acting on the cylinder, due to pressure plus an
equivalent line load.
n ratio of the thickness of the cone to the thickness of the cylinder.
P internal design pressure.
a
P maximum allowable internal pressure of a torispherical head
ac
P allowable internal pressure of a torispherical head based on the rupture of the crown.
ak
P allowable internal pressure of a torispherical head based on a buckling failure of the knuckle.
ck
P value of internal pressure expected to result in a buckling failure of the knuckle in a
torispherical head.
e
P equivalent design pressure used in the conical transition stress calculation when a knuckle or
flare is present.
j
e
P
equivalent design pressure at locations around the knuckle or flare, used in the conical
transition stress calculation when a knuckle or flare is present.
eth
P value of internal pressure expected to produce elastic buckling of the knuckle in a
torispherical head.
y
P value of the internal pressure expected to result in a maximum stress equal to the material
yield strength in a torispherical head.
φ
angle to locate a circumferential section in a spherical shell.
f
φ
angle used in the conical transition calculation when a flare is present.
j
f
φ
angle used in the conical transition calculation when a non-compact flare is present.
e
f
φ
angle used in the conical transition calculation when a non-compact flare is present.
2010 SECTION VIII, DIVISION 2
4-62
s
f
φ
angle used in the conical transition calculation when a non-compact flare is present.
k
φ
angle used in the conical transition calculation when a knuckle is present.
j
k
φ
angle used in the conical transition calculation when a non-compact knuckle is present.
e
k
φ
angle used in the conical transition calculation when a non-compact knuckle is present.
s
k
φ
angle used in the conical transition calculation when a non-compact knuckle is present.
th
φ
angle used in the torispherical head calculation.
Q total resultant shear force acting on the cylinder.
c
Q total resultant shear force acting on the cone.
N
Q normalized curve-fit resultant shear force acting on the cylinder.
P
Q cylinder-to-cone junction resultant shear force acting on the cylinder, due to internal pressure.
X
Q cylinder-to-cone junction resultant shear force acting on the cylinder, due to an equivalent line
load.
C
R
equivalent radius of the cone.
f
R
radius to the center of curvature for the flare.
k
R
radius to the center of curvature for the knuckle.
L
R
inside radius of the large end of a conical transition.
m
R
mean radius of the cylinder.
S
R
inside radius of the small end of a conical transition.
th
R
radius used in the torispherical head calculation.
r inside knuckle radius used in torispherical head calculation.
hr
r minimum hole radius.
k
r inside knuckle radius of the large end of a toriconical transition.
f
r inside flare radius of the small end of a toriconical transition.
R
s distance measured along the cylinder from the centroid of the stiffening ring centroid to the
intersection of the cylinder and cone.
S allowable stress value from Annex 3.A evaluated at the design temperature.
a
S allowable stress amplitude.
P
S
S allowable primary plus secondary stress evaluated using paragraph 5.5.6.1.d at the design
temperature.
y
S
yield strength from Annex 3.D evaluated at the design temperature.
s
σ
axial (longitudinal) stress in a shell.
s
m
σ
axial (longitudinal) membrane stress in a shell.
s
b
σ
axial (longitudinal) bending stress in a shell.
θ
σ
hoop (circumferential) stress in a shell.
m
θ
σ
hoop (circumferential) membrane stress in a shell.
b
θ
σ
hoop (circumferential) bending stress in a shell.
1
σ
principal stress in the 1-direction.
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2010 SECTION VIII, DIVISION 2
4-63
2
σ
principal stress in the 2-direction.
t minimum required thickness of a shell.
C
t thickness of the cone in a conical transition.
L
t thickness of the large end cylinder in a conical transition.
S
t thickness of the small end cylinder in a conical transition.
j
t thickness of the cylinder, knuckle, or flue, as applicable, at the junction of a toriconical
transition,
j
tt≥ and
j
c
tt≥ .
rw
t remaining wall thickness at the location of a partially drilled hole.
1rw
t limit for the remaining wall thickness at the location of a partially drilled hole.
τ
torsional shear stress in a shell.
pd
τ
average shear stress in a shell at the location of a partially drilled hole. location where stress
is computed for shells subject to supplemental loads. A value of zero defines the location of
maximum positive longitudinal stress from net-section bending moment.
ν
Poisson’s ratio.
L
X
equivalent line load acting on the large end cylinder, due to an axial force and bending
moment.
S
X
equivalent line load acting on the small end cylinder, due to an axial force and bending
moment.
2010 SECTION VIII, DIVISION 2
4-64
4.3.14 Tables
Table 4.3.1 – Large End Junction
Cylinder Cone
Stress Resultant Calculation
2
s
PLsN
M
Pt M= , see Table 4.3.3
s
XLLsN
M
XtM= , see Table 4.3.4
s
sP sX
M
MM=+
P
LN
QPtQ= , see Table 4.3.3
XLN
QXQ= , see Table 4.3.4
P
X
QQ Q=+
0.25
2
22
3(1 )
cy
LL
Rt
ν
β
⎡⎤
−
=
⎢⎥
⎣⎦
2
L
s
L
PR
NX
=+
()
2
LcyLscy
NPR RM Q
θ
ββ
=+ − +
1.0
pc
K =
Stress Calculation
s
sm
L
N
t
σ
=
2
6
s
sb
Lpc
M
tK
σ
=
m
L
N
t
θ
θ
σ
=
2
6
s
b
Lpc
M
tK
θ
ν
σ
=
Acceptance Criteria
1.5
sm
S
σ
≤
s
msbPS
S
σ
σ
±≤
1.5
m
S
θ
σ
≤
mbPS
S
θθ
σ
σ
±≤
Stress Resultant Calculation
csP sP
M
M
=
csX sX
M
M
=
cs csP csX
MM M
=
+
[
]
[
]
cos sin
α
α
=+
cs
QQ N (1)
[]
cos
α
=
L
C
R
R
0.25
2
22
3(1 )
co
CC
Rt
ν
β
⎡
⎤
−
=
⎢
⎥
⎣
⎦
[
]
[
]
cos sin
α
α
=−
cs s
NN Q (2)
[]
()
2
cos
L
ccoCcscoc
PR
NRMQ
θ
ββ
α
=+ −−
1.0
cpc
K
=
Stress Calculation
cs
sm
C
N
t
σ
=
2
6
cs
sb
Ccpc
M
tK
σ
=
c
m
C
N
t
θ
θ
σ
=
2
6
cs
b
Ccpc
M
tK
θ
ν
σ
=
Acceptance Criteria
1.5
sm
S
σ
≤
s
msbPS
S
σ
σ
±
≤
1.5
m
S
θ
σ
≤
mbPS
S
θθ
σ
σ
±
≤
Notes:
1. The
Q and
s
N values used to determine the resultant shear force in the cone,
c
Q , are the
same as those defined for the cylinder.
2. The
Q
and
s
N values used to determine the resultant meridional membrane force in the
cone,
cs
N , are the same as those defined for the cylinder.
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