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

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2010 SECTION VIII, DIVISION 2

4-95

()

22 2

34 4 3

0.6 1 0

CC C C++−= (4.4.99)

FFS

⋅

=−

(4.4.100)

2) The allowable hoop compressive membrane stress,

hba

F , is given by the following equation:

hba bha

⎛⎞

⎜⎟

⎝⎠

(4.4.101)

g) Shear Stress And Hoop Compression

– The allowable compressive stress for the combination of shear,

vha

F , and hoop compression is computed using the following equations.

1) The allowable shear stress is given by the following equation with

F and

F evaluated using the

equations in paragraphs 4.4.12.2.a and 4.4.12.2.d, respectively.

0.5

va va

vha va

ha ha

CF CF

⎡⎤

⎛⎞

⎢⎥

=+−

⎜⎟

⎢⎥

⎝⎠

⎣⎦

(4.4.102)

= (4.4.103)

2) The allowable hoop compressive membrane stress,

hva

F , is given by the following equation:

vha

hva

(4.4.104)

h) Axial Compressive Stress, Compressive Bending Stress, Shear Stress, And Hoop Compression

– The

allowable compressive stress for the combination of uniform axial compression, axial compression due

to a bending moment, and shear in the presence of hoop compression is computed using the following

interaction equations.

1) The shear coefficient is determined using the following equation with

F from paragraph

4.4.12.2.d.

1.0

⎛⎞

=−

⎜⎟

⎝⎠

(4.4.105)

2) For

0.15

≤ ; the acceptability of a member subject to compressive axial and bending stresses,

and

, respectively, is determined using the following interaction equation with

F and

bha

evaluated using the equations in paragraphs 4.4.12.2.e.1 and 4.4.12.f.1, respectively.

1.7

1.0

s xha s bha

KF KF

⎛⎞⎛⎞

+≤

⎜⎟⎜⎟

⎝⎠⎝⎠

(4.4.106)

2010 SECTION VIII, DIVISION 2

4-96

3) For

0.15 1.2

<≤ ; the acceptability of a member subject to compressive axial and bending

stresses,

and

, respectively, is determined using the following interaction equation with

and

bha

F evaluated using the equations in paragraphs 4.4.12.2.e.2 and 4.4.12.2.f.1, respectively.

1.0 0.2

ab a

s xha s bha s xha

ff f

for

KF KF KF

⎛⎞⎛ ⎞

+≤ ≥

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

(4.4.107)

1.0 0.2

ab a

s xha s bha s xha

ff f

for

KF KF KF

⎛⎞⎛⎞

+≤ <

⎜⎟⎜⎟

⎝⎠⎝⎠

(4.4.108)

Δ=

⎛⎞

⋅

−

⎜⎟

⎝⎠

(4.4.109)

⎛⎞

⎜⎟

⎝⎠

(4.4.110)

i) Axial Compressive Stress, Compressive Bending Stress, And Shear

– The allowable compressive stress

for the combination of uniform axial compression, axial compression due to a bending moment, and

shear in the absence of hoop compression is computed using the following interaction equations:

1) The shear coefficient is determined using the equation in paragraph 4.4.12.2.h.1 with

F from

paragraph 4.4.12.2.d.

2) For

0.15

≤ ; the acceptability of a member subject to compressive axial and bending stresses

and

, respectively, is determined using the following interaction equation with,

F and

F evaluated using the equations in paragraphs 4.4.12.2.b.1 and 4.4.12.2.c, respectively

1.7

1.0

sxa sba

KF KF

⎛⎞⎛⎞

+≤

⎜⎟⎜⎟

⎝⎠⎝⎠

(4.4.111)

3) For

0.15 1.2

<≤ ; the acceptability of a member subject to compressive axial and bending

stresses,

and

, respectively, is determined using the following interaction equation with,

F and

F evaluated using the equations in paragraphs 4.4.12.2.b.2 and 4.4.12.2.c respectively.

The coefficient Δ is evaluated using the equations in paragraph 4.4.12.2.h.3

1.0 0.2

ab a

sca sba sca

ff f

for

KF KF KF

⎛⎞⎛ ⎞

+≤ ≥

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

(4.4.112)

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2010 SECTION VIII, DIVISION 2

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1.0 0.2

ab a

sca sba sca

ff f

for

KF KF KF

⎛⎞⎛⎞

+≤ <

⎜⎟⎜⎟

⎝⎠⎝⎠

(4.4.113)

j) The maximum deviation,

e may exceed the value of

e given in paragraph 4.4.4.2 if the maximum axial

stress is less than

F for shells designed for axial compression only, or less than

F for shells

designed for combinations of axial compression and external pressure. The change in buckling stress,

F , is given by Equation (4.4.114). The reduced allowable buckling stress,

()

a reduced

F , is determined

using Equation (4.4.115) where

e is the new maximum deviation,

F is determined using Equation

(4.4.61), and

FS is the value of the stress reduction factor used to determine

F .

0.0005

0.944 0.286 log

⎛⎞

⎡⎤

⎛⎞

=−

⎜⎟

⎢⎥

⎜⎟

⎝⎠

⎣⎦

⎝⎠

(4.4.114)

()

axaxe

xa reduced

FFS F

⋅−

(4.4.115)

The quantity

()

0.286log 0.0005

ee⎡⎤

⎣⎦

in Equation (4.4.114) is an absolute number (i.e. the log of a

very small number is negative). For example, if

, then the change in the buckling stress computed

using Equation (4.4.114) is

(

)

0.086

xe y

FEtR= .

k) Section Properties, Stresses, Buckling Parameters – equations for section properties, nominal shell

stresses, and buckling parameters that are used in paragraphs 4.12.2.a through 4.12.2.i are provided

below.

()

−

(4.4.116)

()

−

(4.4.117)

= (4.4.118)

= (4.4.119)

= (4.4.120)

= (4.4.121)

2010 SECTION VIII, DIVISION 2

4-98

[

]

sin

= (4.4.122)

0.25

rDD=+ (4.4.123)

= (4.4.124)

0.5

uu xa

KL F FS

⎛⎞

⋅

⎜⎟

⎝⎠

(4.4.125)

4.4.12.3 Conical Shells – Unstiffened conical transitions or cone sections between stiffening rings of

conical shells with a half-apex angle,

, less than 60° shall be evaluated as an equivalent cylinder using the

equations in paragraph 4.4.12.2 with the substitutions shown below. Both the shell tolerances and stress

criteria in this paragraph shall be satisfied at all cross-sections along the length of the cone.

a) The value of

t is substituted for t to determine the allowable compressive stress.

b) The value of

cosD

is substituted for

to determine the allowable compressive stress where D is

the outside diameter of the cone at the point under consideration.

c) The value of

cos

, is substituted for L where

L is the distance along the cone axis between

stiffening rings.

4.4.12.4 Spherical Shells and Formed Heads – The allowable compressive stresses are based on the

ratio of the biaxial stress state.

a) Equal Biaxial Stresses

– The allowable compressive stress for a spherical shell subject to a uniform

external pressure,

F , is given by the equations in paragraph 4.4.7.

b) Unequal Biaxial Stresses, Both Stresses Are Compressive

– The allowable compressive stress for a

spherical shell subject to unequal biaxial stresses,

and

, where both

and

are compressive

stresses resulting from the applied loads is given by the equations shown below. In these equations,

F is determined using paragraph 4.4.7.

F is the allowable compressive stress in the direction of

and

F is the allowable compressive stress in the direction of

0.6

10.4

−

(4.4.126)

21aa

FkF= (4.4.127)

kwhere

(4.4.128)

c) Unequal Biaxial Stresses, One Stress Is Compressive And The Other Is Tensile

– The allowable

compressive stress for a spherical shell subject to unequal biaxial stresses,

and

, where

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2010 SECTION VIII, DIVISION 2

4-99

compressive and

is tensile resulting from the applied loads is given by the equations shown below.

In these equations,

F is the allowable compressive stress in the direction of

, and is the value of

F determined using paragraph 4.4.7 with

F computed using the following equations.

()

opy

CCEt

(4.4.129)

102.2

622

195

Cfor

(4.4.130)

0.125 622 1000=≤≤

Cfor

(4.4.131)

1.06

3.24

⎛⎞

⎜⎟

⎝⎠

(4.4.132)

4.4.13 Cylindrical-To-Conical Shell Transition Junctions Without A Knuckle

4.4.13.1 The design rules in paragraph 4.3.11 shall be satisfied. In these calculations, a negative value of

pressure shall be used in all applicable equations.

4.4.13.2 If a stiffening ring is provided at the cone-to-cylinder junction, the design shall be made in

accordance with Part 5.

4.4.14 Cylindrical-To-Conical Shell Transition Junctions With A Knuckle

4.4.14.1 The design rules in paragraph 4.3.12 shall be satisfied. In these calculations, a negative value of

pressure shall be used in all applicable equations.

4.4.14.2 If a stiffening ring is provided within the knuckle, the design shall be made in accordance with

Part 5.

4.4.15 Nomenclature

A cross-sectional area of cylinder.

A cross-sectional area of a small ring stiffener.

A cross-sectional area of a large ring stiffener that acts as a bulkhead.

one-half of the conical shell apex angle (degrees).

C coefficient whose value is established as follows:

0.85 for compression members in frames subject to joint translation (sideway).

()

0.6 0.4

M−

for rotationally restrained members in frames braced against joint

translation and not subject to transverse loading between their supports in the plane of

bending; in this equation,

M is the ratio of the smaller to large bending moment at the

ends of the portion of the member that is unbraced in the plane of bending under

2010 SECTION VIII, DIVISION 2

4-100

consideration (

M is positive when the member is bent in reverse curvature and

negative when the member is bent in single curvature).

0.85 for compression members in frames braced against joint translation and subject to

transverse loading between support points, the member ends are restrained against rotation

in the plane of bending.

1.0

for compression members in frames braced against joint translation and subject to

transverse loading between support points, the member ends are unrestrained against

rotation in the plane of bending.

1.0 for an unbraced skirt supported vessel.

c distance from the neutral axis to the point under consideration.

D diameter to the centroid of the composite ring section for an external ring; or the inside

diameter for and internal ring (see Figure 4.4.6)

D Inside diameter of cylinder (including the effects of corrosion.

D outside diameter of cylinder.

D outside diameter of an assumed equivalent cylinder for the design of cones or conical

sections.

D outside diameter of at the small end of the cone or conical section between lines of support.

D outside diameter of at the large end of the cone or conical section between lines of support.

E modulus of elasticity of material at the design temperature from Part 3 .

E tangent modulus of elasticity of material at the design temperature from Part 3 .

F applied net-section axial compression load.

axial compressive membrane stress resulting from applied axial load.

axial compressive membrane stress resulting from applied bending moment.

hoop compressive stress in the cylinder from external pressure.

axial compressive membrane stress resulting from the pressure load,

, on the end of the

cylinder.

shear stress from applied loads.

FS design factor.

F allowable compressive membrane stress of a cylinder subject to a net-section bending

moment in the absence of other loads.

F allowable compressive membrane stress of a cylinder due to an axial compressive load with

0.15

= .

bha

F allowable axial compressive membrane stress of a cylinder subject to bending in the

presence of hoop compression.

hba

F allowable hoop compressive membrane stress of a cylinder in the presence of longitudinal

compression due to net-section bending moment.

F elastic hoop compressive membrane failure stress of a cylinder or formed head subject to

external pressure only.

F allowable hoop compressive membrane stress of a cylinder or formed head subject to

external pressure only.

hef

F average value of the hoop buckling stress,

F , averaged over the length

L where

F is

determined from Equation (4.4.19).

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2010 SECTION VIII, DIVISION 2

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hva

F allowable hoop compressive membrane stress of a cylinder in the presence of shear stress.

hxa

F allowable hoop compressive membrane stress of a cylinder in the presence of axial

compression.

F predicted buckling stress, which is determined by letting

1.0FS

in the allowable stress

equations.

F allowable tensile stress from paragraph 3.8.

F allowable shear stress of a cylinder subject only to shear loads.

F elastic shear buckling stress of a cylinder subject only to shear loads.

vha

F allowable shear stress of a cylinder subject to shear stress in the presence of hoop

compression.

F allowable compressive membrane stress of a cylinder due to an axial compressive load with

0.15

≤ .

F elastic axial compressive failure membrane stress (local buckling) of a cylinder in the

absence of other loads.

F allowable axial compressive membrane stress of a cylinder in the presence of hoop

compression for

0.15

≤ .

Buckling parameter.

h height of the elliptical head measured to the outside surface.

h length of a flat bar stiffener, or leg of an angle stiffener, or flange of a tee stiffener, as

applicable.

h length of the angle leg or web of the stiffener, as applicable.

moment of inertia of the cylinder or cone cross section.

actual moment of inertia of the large stiffening ring.

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

effective length of the shell about the centroidal axis.

actual moment of inertia of the small stiffening ring.

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

effective length of the shell about the centroidal axis.

K elliptical head factor:

K coefficient based on end conditions of a member subject to axial compression:

2.10 for a member with one free end and the other end fixed. In this case, “member” is

the unbraced cylindrical shell or cylindrical shell section as defined in the Nomenclature.

1.00 for a member with both ends pinned,

0.80 for a member with one end pinned and the other end fixed,

0.65 for a member with both ends fixed.

, , ,...LL L design lengths of the unstiffened vessel sections between lines of support (see Figure 4.4.2).

A line of support is (1) a circumferential line on a head (excluding conical heads) at one-third

the depth of the head measured from the tangent line, (2) a small stiffening ring that meets

the requirements of paragraph 4.4.5.2.b, or (3) a tubesheet.

, , ,...

BB B

LL L design lengths of the cylinder between bulkheads or large rings designated to act as

bulkheads (see Figure 4.4.2).

2010 SECTION VIII, DIVISION 2

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L axial length of a cone or conical section for an unstiffened cone, or the length from the cone-

to-cylinder junction to the first stiffener in the cone for a stiffened cone (see Figures 4.4.6 and

4.4.7).

L effective length of the shell.

L one-half of the sum of the distances,

L , from the center line of a large ring to the next large

ring of head line of support on either side of the large ring.

L one-half of the sum of the distances from the centerline of a stiffening ring to the next line of

support on either side of the ring measured parallel to the axis of the cylinder. A line of

support is defined in the definition of

L .

L overall length of the vessel.

L laterally unbraced length of cylindrical member that is subject to column buckling, equal to

zero when evaluating the shell of a vessel under external pressure only.

slenderness factor for column buckling.

applied net-section bending moment.

shell parameter.

applied external pressure.

P allowable external pressure in the absence of other loads.

angle measured around the circumference from the direction of the applied shear force to the

point under consideration.

r inside radius of the flare.

r radius of gyration.

r inside radius of the knuckle.

inside or outside radius of cylindrical, conical, and spherical shells, as applicable

radius to the centerline of the shell.

radius to the centroid of the combined ring stiffener and effective length of the shell,

RZ=+ (see Figure 4.4.3)

inside radius of the cylinder at the large end of a cone to cylinder junction.

outside radius of a cylinder or spherical shell.

inside radius of the cylinder at the small end of a cone to cylinder junction.

S section modulus of the shell.

S minimum specified yield strength from Annex 3.D at specified design metal temperature.

principal compressive stress in the 1-direction.

principal compressive stress in the 2-direction.

V net-section shear force.

t shell thickness.

t cone thickness.

t shell thickness of large end cylinder at a conical transition.

t shell thickness of small end cylinder at a conical transition.

t thickness of a flat bar stiffener, or leg of an angle stiffener, or flange of a tee stiffener, as

applicable.

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2010 SECTION VIII, DIVISION 2

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t thickness of the angle leg or web of the stiffener, as applicable.

radial distance from the centerline of the shell to the combined section of the ring stiffener

and effective length of the shell.

radial distance from the centerline of the shell to the centroid of the large ring stiffener.

radial distance from the centerline of the shell to the centroid of the small ring stiffener.

2010 SECTION VIII, DIVISION 2

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4.4.16 Tables

Table 4.4.1 – Maximum Metal Temperature For Compressive Stress Rules

Materials

Temperature Limit

Carbon and Low Alloy Steels - Table 3.A.1 425 800

High Alloy Steels - Table 3.A.3 425 800

Quenched and Tempered Steels - Table 3.A.2 370 700

Aluminum and Aluminum Alloys - Table 3.A.4 150 300

Copper and Copper Alloys - Table 3.A.5 65 150

Nickel and Nickel Alloys - Table 3.A.6 480 900

Titanium and Titanium Alloys - Table 3.A.7 315 600

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