ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules

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2010 SECTION VIII, DIVISION 2

4-85

1.0 1.5

CforM=≤ (4.4.24)

c) STEP 3 – Calculate the predicted buckling stress,

F .

2.439

ic y

FS for

=≥

(4.4.25)

0.4

0.7 0.552 2.439

he he

ic y

FS for

⎛⎞

=<<

⎜⎟

⎝⎠

(4.4.26)

0.552

ic he

FF for

=≤

(4.4.27)

d) STEP 4 – Calculate the value of design factor,

FS , per paragraph 4.4.2.

e) STEP 5 – Calculate the allowable external pressure,

P .

aha

⎛⎞

⎜⎟

⎝⎠

(4.4.28)

where,

(4.4.29)

f) STEP 6 – If the allowable external pressure,

P , is less than the design external pressure, increase the

shell thickness or reduce the unsupported length of the shell (i.e. by the addition of a stiffening rings) and

go to STEP 2. Repeat this process until the allowable external pressure is equal to or greater than the

design external pressure.

4.4.5.2 Stiffening Ring Size – The following equations shall be used to determine the size of a stiffening

ring.

a) Stiffening Ring Configuration – A combination of large and small stiffening rings may be used along the

length of a shell. If a single size stiffener is used, then it shall be sized as a small stiffener. Alternatively,

a combination of large and small stiffeners can be used to reduce the size of the intermittent small

stiffening rings. The large stiffening rings may be sized to function as end stiffeners or bulkheads with

small stiffeners spaced as required between end rings based on the shell thickness selected and loading

combinations considered in the design.

b) Small Stiffening Ring – The required moment of inertia of the effective stiffening ring (i.e. actual stiffening

ring plus the effective length of shell, see Figure 4.4.3) shall satisfy Equation (4.4.30). The parameter

F shall be evaluated using the equations in paragraph 4.4.5.1 with

LRt= .

()

1.5

he s c

FLRt

≥

−

(4.4.30)

where,

2010 SECTION VIII, DIVISION 2

4-86

1.5

0.5

;22,

10 10

n where n is an integer for n use n

and for n use n

=<=

(4.4.31)

The actual moment of inertia of the composite section comprised of the small stiffening ring and effective

length of the shell about the centroidal axis shall be calculated using Equation (4.4.32):

ssss

Lt Lt

IIAZ

ALt

⎛⎞

=+ +

⎜⎟

⎝⎠

(4.4.32)

where,

1.1

LDt= (4.4.33)

c) Large Stiffening Ring or Bulkhead – The required moment of inertia of the effective stiffening ring (i.e.

actual stiffening ring plus the effective length of shell) shall satisfy Equation (4.4.34). The parameter

hef

F is the average value of the hoop buckling stress,

F , over length

L evaluated using the

equations in paragraph 4.4.5.1 with

LRt= .

hef F c

FLRt

≥

(4.4.34)

The actual moment of inertia of the composite section comprised of the large stiffening ring and effective

length of the shell about the centroidal axis shall be calculated using Equation (4.4.35):

LLLL

Lt Lt

IIAZ

ALt

⎛⎞

=+ +

⎜⎟

⎝⎠

(4.4.35)

where,

1.1

ALt

LDt

ALt

⎛⎞

⎜⎟

⎝⎠

(4.4.36)

d) Local Stiffener Geometry Requirements for all Loading Conditions – The following equations shall be

met to assure the stability of a stiffening ring.

1) Flat bar stiffener, flange of a tee section and the outstanding leg of an angle stiffener (see Figure

4.4.3)

0.5

0.375

⎛⎞

≤

⎜⎟

⎝⎠

(4.4.37)

2) Web of a tee stiffener or leg of an angle stiffener attached to the shell (see Figure 4.4.3).

0.5

⎛⎞

≤

⎜⎟

⎝⎠

(4.4.38)

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2010 SECTION VIII, DIVISION 2

4-87

e) Stiffener Size To Increase Allowable Longitudinal Compressive Stress – ring stiffeners can be used to

increase the allowable longitudinal compressive stress for cylindrical or conical shells subject to uniform

axial compression and axial compression due to bending. The required size of the stiffener shall satisfy

the following equations. In addition, the spacing of the stiffeners must result in a value of

≤

where

is given by Equation (4.4.42).

0.6

0.334

0.063

ALt

⎛⎞

≥−

⎜⎟

⎝⎠

(4.4.39)

0.06

ALt≥ (4.4.40)

1.8

5.33

≥

(4.4.41)

(4.4.42)

f) Stiffener Size For Shear – The required size of the stiffener shall satisfy the following equation where

is evaluated using Equations (4.4.81) thru (4.4.84) with

is given by Equation (4.4.42).

0.8 3

0.184

vs s

CM tL≥ (4.4.43)

g) Arrangement Of Stiffening Rings

1) As shown in Figure 4.4.4, any joints between the ends or sections of such rings, at locations (A) and

(B), and any connection between adjacent portions of a stiffening ring lying inside or outside the

shell, at location (C), shall be made so that the required moment of inertia of the combined ring-

shell section is maintained. For a section with a strut at location (D), the required moment of inertia

shall be supplied by the strut alone.

2) As shown in Figure 4.4.4, stiffening rings placed on the inside of a vessel may be arranged as

shown at locations (E) and (F) provided that the required moment of inertia of the ring at location (E)

or of the combined ring-shell section at location (F) is maintained within the sections indicated.

Where the gap at locations (A) or (E) does not exceed eight times the thickness of the shell plate,

the combined moment of inertia of the shell and stiffener may be used.

3) Stiffening rings shall extend completely around the vessel except as provided below.

Any gap in

that portion of a stiffening ring supporting the shell, as shown in Figure 4.4.4 at locations (D) and

(E), shall not exceed the length of arc given in Figure 4.4.5 unless additional reinforcement is

provided as shown at location (C), or unless all of the following conditions are met:

i) only one unsupported shell arc is permitted per ring

ii) the length of unsupported shell arc does not exceed 90 deg

iii) the unsupported shell arcs in adjacent stiffening rings are staggered 180 deg

iv) the dimension

is taken as the larger of the distance between alternate stiffening rings or

the distance from the head-bend line to the second stiffening ring plus one-third of the head

depth

2010 SECTION VIII, DIVISION 2

4-88

4) When internal plane structures perpendicular to the longitudinal axis of the cylinder, such as bubble

trays or baffle plates, are used in a vessel, they may also be considered to act as stiffening rings

provided they are designed to function as such.

5) Any internal stays or supports used shall bear against the shell of the vessel through the medium of

a substantially continuous ring.

h) Attachment Of Stiffening Rings – Stiffening rings shall be attached to either the outside or the inside of

the vessel by continuous welding, or if the component is not in cyclic service (i.e. a fatigue analysis is not

required in accordance with paragraph 4.1.1.4) intermittent welding. Where gaps occur in the stiffening

ring, the attachment weld shall conform to the details in paragraph 4.2.

4.4.5.3 Combined Loadings – cylindrical shells subject to external pressure and other loadings shall

satisfy the requirements of paragraph 4.4.12.

4.4.6 Conical Shell

4.4.6.1 Required Thickness – The required thickness of a conical shell subjected to external pressure

loading shall be determined using the equations for a cylinder by making the following substitutions:

a) The value of

t is substituted for t in the equations in paragraph 4.4.5.

b) For offset cones, the cone angle,

, shall satisfy the requirements of paragraph 4.3.4.

c) The value of

()

[

]

0.5 cos

+ is substituted for

D in the equations in paragraph 4.4.5.

d) The value of

[

]

cos

is substituted for L in the equations in paragraph 4.4.5 where

L is

determined as shown below.

1) For Sketches (a) and (e) in Figure 4.4.7

ce c

LL= (4.4.44)

2) For Sketch (b) in Figure 4.4.7

[

]

sin

ce k c

Lr L

=+ (4.4.45)

3) For Sketch (c) in Figure 4.4.7

[

]

sin

ce f c

Lr L

=+ (4.4.46)

4) For Sketch (d) in Figure 4.4.7

()

[

]

sin

ce k f c

Lrr L

=+ + (4.4.47)

e) Note that the half-apex angle of a conical transition can be computed knowing the shell geometry with

the following equations. These equations were developed with the assumption that the conical transition

contains a cone section, knuckle, or flare. If the transition does not contain a knuckle or flare, the radii of

these components should be set to zero when computing the half-apex angle (see Figure 4.4.7).

1) If

()( )

Lk S f

rRr−≥ +

βφ

=+ (4.4.48)

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2010 SECTION VIII, DIVISION 2

4-89

()( )

arctan

Lk S f

rRr

−− +

⎡⎤

⎢⎥

⎣⎦

(4.4.49)

2) If

()( )

Lk S f

rRr−< +:

φβ

=− (4.4.50)

()

(

)

arctan

Sf Lk

rRr

⎡⎤

+− −

⎢⎥

⎣⎦

(4.4.51)

3) In both cases shown above, the angle

is given by the following equation.

()cos

arcsin

⎡⎤

⎢⎥

⎣⎦

(4.4.52)

4.4.6.2 Small Stiffening Rings – Intermediate circumferential stiffening rings within the conical transition

shall be sized using Equation (4.4.30) where

L is determined from paragraph 4.4.6.1.d, and

t is the cone

thickness at the ring location.

4.4.6.3 Combined Loadings – conical shells subject to external pressure and other loadings shall satisfy

the requirements of paragraph 4.4.12.

4.4.7 Spherical Shell and Hemispherical Head

4.4.7.1 Required Thickness – The required thickness of a spherical shell or hemispherical head

subjected to external pressure loading shall be determined using the following procedure.

a) STEP 1 – Assume an initial thickness,

t for the spherical shell.

b) STEP 2 – Calculate the predicted elastic buckling stress,

F .

0.075

he y

⎛⎞

⎜⎟

⎝⎠

(4.4.53)

c) STEP 3 – Calculate the predicted buckling stress,

F .

6.25

ic y

FS for

=≥

(4.4.54)

1.31

1.6 6.25

1.15

Ffor

=<<

⎛⎞

⎜⎟

⎝⎠

(4.4.55)

0.18 0.45 0.55 1.6

ic he y

FFS for

=+ <≤

(4.4.56)

2010 SECTION VIII, DIVISION 2

4-90

0.55

ic he

FF for

=≤

(4.4.57)

d) STEP 4 – Calculate the value of design margin,

S , per paragraph 4.4.2.

e) STEP 5 – Calculate the allowable external pressure,

aha

⎛⎞

⎜⎟

⎝⎠

(4.4.58)

where,

(4.4.59)

f) STEP 6 – If the allowable external pressure,

P , is less than the design external pressure, increase the

shell thickness and go to STEP 2. Repeat this process until the allowable external pressure is equal to

or greater than the design external pressure.

4.4.7.2 Combined Loadings – spherical shells and hemispherical heads subject to external pressure and

other loadings shall satisfy the requirements of paragraph 4.4.12.

4.4.8 Torispherical Head

4.4.8.1 Required Thickness – the required thickness of a torispherical head subjected to external

pressure loading shall be determined using the equations for a spherical shell in paragraph 4.4.7 by

substituting the outside crown radius for

4.4.8.2 Restrictions On Torispherical Head Geometry – the restriction of paragraph 4.3.6 shall apply.

4.4.8.3 Torispherical Heads With Different Dome and Knuckle Thicknesses – heads with this

configuration shall be designed in accordance with Part 5.

4.4.8.4 Combined Loadings – torispherical heads subject to external pressure and other loadings shall

satisfy the requirements of paragraph 4.4.12.

4.4.9 Ellipsoidal Head

4.4.9.1 Required Thickness – the required thickness of an elliptical head subjected to external pressure

loading shall be determined using the equations for a spherical shell in paragraph 4.4.7 by substituting

for

where

K is given by the following equation:

0.25346 0.13995 0.12238 0.015297

22 2

⎛⎞ ⎛⎞ ⎛⎞

=+ + −

⎜⎟ ⎜⎟ ⎜⎟

⎝⎠ ⎝⎠ ⎝⎠

oo o

DD D

hh h

(4.4.60)

4.4.9.2 Combined Loadings – ellipsoidal heads subject to external pressure and other loadings shall

satisfy the requirements of paragraph 4.4.12.

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2010 SECTION VIII, DIVISION 2

4-91

4.4.10 Local Thin Areas

Rules for the evaluation of Local Thin Areas are covered in paragraph 4.14.

4.4.11 Drilled Holes Not Penetrating Through the Vessel Wall

Design requirements for partially drilled holes that do not penetrate completely through the vessel wall are

covered in paragraph 4.3.9.

4.4.12 Combined Loadings and Allowable Compressive Stresses

4.4.12.1 The rules in paragraphs 4.4.2 through 4.4.11 are applicable for external pressure loading. The

rules in this paragraph provide allowable compressive stresses that shall be used for the design of shells

subjected to supplemental loads that result in combined loadings. The allowable stresses of this paragraph

shall also be used as the acceptance criteria for shells subjected to compressive stress evaluated using Part

4.4.12.2 Cylindrical Shells – The allowable compressive stresses for cylindrical shells shall be computed

using the following rules that are based on loading conditions. The loading conditions are underlined for

clarity in the following paragraphs. Common parameters used in each of the loading conditions are given in

paragraph 4.4.12.2.k.

a) External Pressure Acting Alone

– the allowable hoop compressive membrane stress of a cylinder subject

to external pressure acting alone,

F , is computed using the equations in paragraph 4.4.5.1.

b) Axial Compressive Stress Acting Alone

– the allowable axial compressive membrane stress of a cylinder

subject to an axial compressive load acting alone,

F , is computed using the following equations.

1) For

0.15

≤ (Local Buckling):

[

]

min ,

xa xa xa

FFF= (4.4.61)

135

Ffor

FS t

=≤

(4.4.62)

466

135 600

331

Ffor

=<<

⎛⎞

⎜⎟

⎝⎠

(4.4.63)

0.5

600 2000

Ffor

FS t

=≤≤

(4.4.64)

(4.4.65)

CEt

(4.4.66)

2010 SECTION VIII, DIVISION 2

4-92

409

min , 0.9 1247

389

Cfor

⎡⎤

⎢⎥

⎛⎞

⎢⎥

⎜⎟

⎢⎥

⎝⎠

⎣⎦

(4.4.67)

0.25 1247 2000

Cc for

=≤≤

(4.4.68)

2.64 1.5

cforM=≤ (4.4.69)

0.42

3.13

1.5 15

cforM

=<< (4.4.70)

1.0 15

cforM=≥ (4.4.71)

2) For

0.15 200

cuug

and K L r

>< (Column Buckling):

()

0.3

1 0.74 0.15 0.15 1.147

ca xa c c

FF for

λλ

=− − <<⎡⎤

⎣⎦

(4.4.72)

0.88

1.147

ca c

Ffor

=≥ (4.4.73)

c) Compressive Bending Stress

– the allowable axial compressive membrane stress of a cylindrical shell

subject to a bending moment acting across the full circular cross section

F , is computed using the

following equations.

135 2000=≤≤

ba xa

FF for

(4.4.74)

466

100 135

331

Ffor

=≤<

⎛⎞

⎜⎟

⎝⎠

(4.4.75)

1.081

100 0.11

F for and

FS t

=<≥ (4.4.76)

()

1.4 2.9

100 0.11

F for and

FS t

−

=<<

(4.4.77)

= (4.4.78)

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2010 SECTION VIII, DIVISION 2

4-93

d) Shear Stress

– the allowable shear stress of a cylindrical shell ,

F , is computed using the following

equations.

vve

= (4.4.79)

ve v v y

FCE

⎛⎞

⎜⎟

⎝⎠

(4.4.80)

4.454 1.5

CforM=≤ (4.4.81)

()

0.5

9.64

1 0.0239 1.5 26

vxx

CMforM

⎛⎞

=+ <<

⎜⎟

⎝⎠

(4.4.82)

0.5

1.492

26 4.347

v x

CforM

⎛⎞

=≤<

⎜⎟

⎝⎠

(4.4.83)

0.5

0.716 4.347

CforM

⎛⎞

=≥

⎜⎟

⎝⎠

(4.4.84)

0.8 500

for

=≤ (4.4.85)

1.389 0.218log 500

for

⎛⎞

=− >

⎜⎟

⎝⎠

(4.4.86)

1.0 0.48

for

=≤ (4.4.87)

0.43 0.1 0.48 1.7

ve y

for

⎛⎞

=+ <<

⎜⎟

⎝⎠

(4.4.88)

0.6 1.7

ve y

for

⎛⎞

=≥

⎜⎟

⎝⎠

(4.4.89)

e) Axial Compressive Stress And Hoop Compression

– the allowable compressive stress for the

combination of uniform axial compression and hoop compression,

F , is computed using the following

equations:

1) For

0.15

≤ ;

F is computed using the following equation with

F and

F evaluated using the

equations in paragraphs 4.4.12.2.a and 4.4.12.2.b.1, respectively.

2010 SECTION VIII, DIVISION 2

4-94

0.5

222

xha

xa xa ha ha

FCFFCF

−

⎡⎤

⎛⎞⎛ ⎞⎛ ⎞

=− +

⎢⎥

⎜⎟⎜ ⎟⎜ ⎟

⎝⎠⎝ ⎠⎝ ⎠

⎣⎦

(4.4.90)

()

1.0

xa ha

FFSFFS

⋅+⋅

=−

(4.4.91)

(4.4.92)

aq x xha

ff forfF=+ ≤ (4.4.93)

The parameters

and

are defined in paragraph 4.4.12.2.k.

2) For

0.15 1.2

<≤ :

F is computed from the following equation with

1ah xha

FF= evaluated using

the equations in paragraph 4.4.12.2.e.1 with

, and

F evaluated using the equations in

paragraph 4.4.12.2.b.2. As noted, the load on the end of a cylinder due to external pressure does

not contribute to column buckling and therefore

1ah

F is compared with

rather than

. The

stress due to the pressure load does, however, lower the effective yield stress and the quantity in

()

S−

accounts for this reduction

[

]

min ,

xha ah ah

FFF= (4.4.94)

ah ca

⎛⎞

=−

⎜⎟

⎝⎠

(4.4.95)

3) For

0.15

≤ , the allowable hoop compressive membrane stress,

hxa

F , is given by the following

equation:

hxa

(4.4.96)

f) Compressive Bending Stress And Hoop Compression

– the allowable compressive stress for the

combination of axial compression due to a bending moment and hoop compression,

bha

F , is computed

using the following equations.

1) An iterative solution procedure is utilized to solve these equations for

with

F and

F evaluated using the equations in paragraphs 4.4.12.2.a and 4.4.12.2.c, respectively.

34bha ba

FCCF= (4.4.97)

bha

hba

⎛⎞⎛ ⎞

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

(4.4.98)

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