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

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Figure 4.11.3 – Half Pipe Jackets

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4.12 Design Rules for NonCircular Vessels

4.12.1 Scope

4.12.1.1 The procedures in paragraph 4.12 cover the design requirements for single wall vessels having a

rectangular or obround cross section. The design rules cover the walls and parts of the vessels subject to

pressure stresses including stiffening, reinforcing and staying members. All other types of loadings shall be

evaluated in accordance with the design-by-analysis rules of Part 5.

4.12.1.2 The design rules in this paragraph cover noncircular vessels of the types shown in Table 4.12.1.

Vessel configurations other than Types 1 to 12, illustrated in Figures 4.12.1 through 4.1.2.13, may be used.

However, in this case, the design-by-analysis rules of Part 5 shall be used.

4.12.2 General Design Requirements

4.12.2.1 In the noncircular vessel configurations covered in this paragraph, the walls of the vessel can

have different thicknesses. Therefore, the design of a noncircular vessel requires an iterative approach

where the vessel configuration and wall thickness are initially set and the stresses at locations on the cross

section are computed and compared to allowable values. If the allowable values are exceeded, the

configuration and/or wall thickness are changed, and the stresses are revaluated. This process is continued

until a final configuration including wall thickness is obtained where all allowable stress requirements are

satisfied.

4.12.2.2 In the design rules of this paragraph, both membrane and bending stresses shall be computed at

locations on the cross section. The membrane stress is added algebraically to the bending stress at both the

outermost surface of the shell plate or reinforcement (when used) and the innermost surface of the shell plate

to obtain two values of total stress. The total stresses at the section shall be compared to the allowable

stress.

4.12.2.3 The total stresses (membrane plus bending) at the cross section of a vessel with and without

reinforcement shall be calculated as follows.

a) For a vessel without reinforcement, the total stresses shall be determined at the inside and outside

surfaces of the cross section of the shell plate.

b) For a vessel with reinforcement, when the reinforcing member has the same allowable stress as the

vessel, the total stress shall be determined at the inside and outside surfaces of the composite cross

section. The appropriate value of c (the location from the neutral axis) for the composite section

properties shall be used in the bending equations. The total stresses at the inside and outside surfaces

shall be compared to the allowable stress.

c) For a vessel with reinforcement, when the reinforcing member does not have the same allowable stress

as the vessel, the total stresses shall be determined at the inside and outside surfaces of each

component of the composite cross section. The appropriate value of c (the location from the neutral

axis) for the composite section properties shall be used in the bending equations considering location of

desired stress with respect to the composite section neutral axis. The total stresses at the inside and

outside surfaces shall be compared to the allowable stress.

1) For locations of stress below the neutral axis, the bending equation used to compute the stress

shall be that considered acting on the inside surface.

2) For locations of stress above the neutral axis, the bending equation used to compute the stress

shall be that considered acting on the outside surface.

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4.12.2.4 Particular attention shall be given to the effects of local internal and external loads and expansion

differentials at design temperature, including reactions at supporting lugs, piping, and other types of

attachments (see paragraph 4.12.1.1).

4.12.2.5 Except as otherwise specified in paragraph 4.12.8, vessel parts of noncircular cross section

subject to external pressure shall be designed in accordance with Part 5.

4.12.2.6 The end closures for noncircular vessels covered in this paragraph shall be designed in

accordance with the provisions of Part 5 except in cases where the ends are flat plates. For this case, the

design rules of paragraph 4.6 shall be used except that 0.20 shall be used for the value of the C factor in all

of the calculations.

4.12.2.7 The design equations in this paragraph are based on vessels in which the ratio of the long side to

short side length is greater than four. These equations are conservatively applicable to vessels of aspect

ratio less than four. Vessel side plates with aspect ratios less than four are strengthened by the interaction of

the end closures and may be designed in accordance with the provisions of Part 5. Short unreinforced or

unstayed vessels of rectangular cross section having an aspect ratio smaller than two may be designed in

accordance with paragraph 4.12.5.

4.12.2.8 Bolted full side or end plates and flanges may be provided for vessels of rectangular cross

section. Many acceptable configurations are possible. Therefore, rules for specific designs are not provided,

and these parts shall be designed in accordance with Part 5. The analysis of the components shall consider

thermal loads, gasket reactions, bolting forces, and resulting moments, as well as pressure and other

mechanical loading.

4.12.2.9 Openings may be provided in vessels of noncircular cross section as follows:

a) Openings in noncircular vessels do not require reinforcement other than that inherent in the construction,

provided they meet the conditions given in paragraph 4.5.2.

b) Compensation for openings in noncircular vessels must account for the bending strength as well as the

membrane strength of the side with the opening. In addition, openings may significantly affect the

stresses in adjacent sides. Because many acceptable configurations are possible, rules for specific

designs are not provided and the design shall be in accordance with Part 5.

4.12.3 Requirements for Vessels With Reinforcement

4.12.3.1 Design rules are provided for Types 4, 5, and 6 (see Table 4.12.1) where the welded on

reinforcement members are in a plane perpendicular to the long axis of the vessel; however, the spacing

between reinforcing members need not be uniform. All reinforcement members attached to two opposite

plates shall have the same moment of inertia. The design for any other type of reinforced rectangular cross

section vessel shall be in accordance with Part 5 .

4.12.3.2 For a Type 4 vessel, when the side plate thicknesses are equal, the plates may be formed to a

radius at the corners. The analysis is, however, carried out in the same manner as if the corners were not

rounded. For corners that are cold formed, the provisions of Part 6 shall apply. For the special case where

0.0L = , the analysis methodology for a Type 11 vessel shall be used.

4.12.3.3 A Type 5 vessel has rounded corners and non-continuous reinforcement. If continuous

reinforcement is provided that follows the contour of the vessel, the design requirements for a Type 4 vessel

shall be used.

4.12.3.4 For a Type 6 vessel, the corner region consists of a flat, chamfered segment joined to the

adjacent sides by curved segments with constant radii. The chamfered segments shall be perpendicular to

diagonal lines drawn through the points where the sides would intersect if they were extended.

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4.12.3.5 Reinforcing members shall be placed on the outside of the vessel and shall be attached to the

plates of the vessel by welding on each side of the reinforcing member. For continuous reinforcement,

welding may be either continuous or intermittent. The total length of intermittent welding on each side of the

reinforcing member shall be not less than one-half the length being reinforced on the shell. Welds on

opposite sides of the reinforcing member may be either staggered or in-line and the distance between

intermittent welds shall be no more than eight times the plate thickness of the plate being reinforced. For

assuring the composite section properties, for non-continuous reinforcement, the welds must be capable of

developing the necessary shear (see Manual of Steel Construction, AISC, American Institute of Steel

Construction).

4.12.3.6 The maximum distance between reinforcing members is computed as follows.

a) The maximum distance between any reinforcing member centerlines is given by Equation (4.12.1). In

the equations for calculating

stresses for reinforced noncircular vessels, the value of

shall be taken

as the sum of one-half the distances to the next reinforcing member on each side.

[

]

min ,

pp= (4.12.1)

where

tforHp

=≥ (4.12.2)

tSJ

for H p

=< (4.12.3)

or rectangular vessels

= (4.12.4)

or obround vessels

(4.12.5)

2.1

(4.12.6)

()

()()()()

2345

1max

1max 1max 1max 1max

24.222 99.478 194.59 169.99 55.822

0.26667J

ββββ

=− + − + − +

(4.12.7)

1max 1

min max , , 4.0

ββ

⎡⎤

⎢⎥

⎣⎦

(4.12.8)

tforhp

=≥ (4.12.9)

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tSJ

for h p

=< (4.12.10)

or rectangular vessels

(4.12.11)

or obround vessels

= (4.12.12)

()

()()()()

2345

2max

2 max 2 max 2 max 2 max

24.222 99.478 194.59 169.99 55.822

0.26667

ββββ

=− + − + − + (4.12.13)

2max 2

min max , , 4.0

ββ

⎡⎤

⎢⎥

⎣⎦

(4.12.14)

2.1

(4.12.15)

b) The allowable effective widths of the shell plate,

w and

w , shall not be greater than the value given by

Equation (4.12.16) or Equation (4.12.17) nor greater than the actual value of

if this value is less than

that computed in paragraph 4.12.3.6.a. One-half of

w shall be considered to be effective on each side

of the reinforcing member centerline, but the effective widths shall not overlap. The effective width shall

not be greater than the actual width available.

[

]

1max1

min ,wwp= (4.12.16)

[

]

2max2

min ,wwp=

(4.12.17)

where

max

⎛⎞

⎜⎟

⎝⎠

(4.12.18)

c) At locations, other than in the corner regions where the shell plate is in tension, the effective moments of

inertia

and

of the composite section (reinforcement and shell plate acting together) shall be

computed based on the values of

w and

w computed in paragraph 4.12.3.6.b. The equations given in

paragraph 4.12.3.6.b do not include the effects of high-localized stresses. In the corner regions of some

Type 4 configurations, the localized stresses may significantly exceed the calculated stress. Only a very

small width of the shell plate may be effective in acting with the composite section in the corner region.

The localized stresses in this region shall be evaluated using the principles of Part 5.

4.12.4 Requirements for Vessels With Stays

4.12.4.1 Three types of stayed construction are considered, Types 7, 8, and 11. In these types of

construction the staying members may be plates welded to the side plates for the entire length of the vessel.

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In this case, the stay plates shall not be constructed so as to create pressure-containing partitions.

Alternatively, the stays may be bars of circular cross section fastened to the side plates on a uniform pitch

designed in accordance with paragraph 4.9.

4.12.4.2 The Type 12 noncircular vessel is comprised of a cylindrical shell with a single stay plate that

divides the cylinder into two compartments. The design rules ensure that the various vessel members will not

be overstressed when there is full pressure in both vessel compartments or when there is full pressure in one

compartment and zero pressure in the other compartment. Stresses may be computed only at the shell-plate

junction since this is the location of maximum stress.

4.12.5 Requirements for Rectangular Vessels With Small Aspect Ratios

4.12.5.1 Type 1 and Type 2 noncircular vessels with aspect ratios of

LH or

Lh between 1.0 and

2.0, and with flat heads welded to the sides may be designed using the procedure in paragraph 4.12.7 except

that the following plate parameters shall be utilized in the calculations.

JJx

⎛⎞

⎜⎟

⎝⎠

(4.12.19)

JJx

⎛⎞

⎜⎟

⎝⎠

(4.12.20)

JJx

⎛⎞

⎜⎟

⎝⎠

(4.12.21)

JJx

⎛⎞

⎜⎟

⎝⎠

(4.12.22)

where

()

23 4

-0.65979 1.0052 0.86072 - 0.82362 0.17483Jx x x x x=++ + (4.12.23)

()

23 4

-0.37508 0.66706 0.99709 - 0.84305 0.17483Jx x x x x=+ + + (4.12.24)

Note in the above nomenclature,

JJx

⎛⎞

⎜⎟

⎝⎠

is defined as computing

J using the function

(

)

evaluated at

4.12.5.2 For vessels with aspect ratios of

LH or

Lh less than 1.0, the axis of the vessel shall be

rotated so that the largest dimension becomes the length

L , and the new ratios

LH or

Lh are greater

than or equal to 1.0. All stresses shall be recalculated using the new orientation.

4.12.6 Weld Joint Factors and Ligament Efficiency

4.12.6.1 The stress calculations for the noncircular vessel shall include a weld joint factor for weld

locations and ligament efficiency for those locations containing holes. In the stress calculations two factors

E and

E are used to account for the weld joint factor and ligament efficiency that is to be applied to the

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membrane and bending stresses, respectively. The weld joint factor shall be determined from paragraph 4.2

and the ligament efficiency shall be determined from paragraph 4.12.6.3. The correct combination of weld

joint factor and ligament efficiencies to be used in the design is shown below.

a) If there is not a weld or a hole pattern at the stress calculation location, then:

1.0

E = (4.12.25)

1.0

E = (4.12.26)

b) If there is a weld, and there is not a hole pattern at the stress calculation location, then:

EE= (4.12.27)

EE= (4.12.28)

c) If there is not a weld, and there is a hole pattern at the stress calculation location, then:

Ee= (4.12.29)

Ee= (4.12.30)

d) If there is a weld and a hole pattern at the stress calculation location, then:

1) If

e and

e are less than the joint efficiency,

, which would be used if there were no ligaments

in the plate, then use Equations (4.12.29) and(4.12.30).

2) If

e and

e are greater than the weld joint factor, E , which would be used if there were no

ligaments in the plate, then use Equations (4.12.27) and(4.12.28).

4.12.6.2 Cases may arise where application of a weld joint factor,

E , at non-welded locations results in

unnecessarily increased plate thicknesses. If the butt weld occurs at one of the locations for which equations

are provided in this paragraph, then no relief can be provided. However, if the weld occurs at some

intermediate location, it is permissible to calculate the bending stress at the weld location and compare it to

the allowable stress considering the weld joint factor in the calculation. An alternate location for computing

stresses is provided for some of the noncircular geometry types, and is identified as “Maximum Membrane

And Bending Stresses – Defined Locations” in the stress calculation tables. The value of

or Y to be used

in the equations is the distance from the midpoint of the side to the location of the weld joint.

4.12.6.3 The ligament efficiency factors

e and

e , for membrane and bending stresses, respectively,

shall only be applied to the calculated stresses for the plates containing the ligaments.

a) For the case of uniform diameter holes, the ligament efficiency factors

e and

e shall be the same and

computed in accordance with paragraph 4.10.

b) For the case of multi-diameter holes, the neutral axis of the ligament may no longer be at mid-thickness

of the plate; the bending stress is higher at one of the plate surfaces than at the other surface.

Therefore, for multi-diameter holes, the ligament efficiency factor shall be computed using the following

equations.

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1) The ligament efficiency of plate with multi-diameter holes subject to membrane stress is computed

as follows.

()

−

(4.12.31)

where

()

00 11 22

.....

Enn

DdTdTdT dT

=++++

(4.12.32)

2) The ligament efficiency and location from the neutral axis of a plate with multi-diameter holes (see

Figure 4.12.14) subject to bending stress is computed as follows.

()

−

(4.12.33)

where

=− (4.12.34)

()

33 3 3

00 11 22

00 1 2 11 2

.....

..... .....

.....

Enn

nnn

IbTbTbT bT

bT T T T X bT T T X

bT T X bT X

=+++++

⎛⎞

+++ +− + ++ +− +

⎜⎟

⎝⎠

⎛⎞

++− + −

⎜⎟

⎝⎠

(4.12.35)

[]

00 1 2

11 2 0 0 11 2 2

.....

..... .....

.....

nnn

bT T T T

bT T T bT bT bT b T

bT T bT

−

⎡⎤

⎛⎞

+++ + +

⎜⎟

⎢⎥

⎝⎠

⎢⎥

⎛⎞

=++++⋅++++

⎢⎥

⎜⎟

⎝⎠

⎢⎥

⎛⎞

++ +

⎢⎥

⎜⎟

⎝⎠

⎣⎦

(4.12.36)

where

00h

bpd=− (4.12.37)

11h

bpd=− (4.12.38)

22h

bpd=− (4.12.39)

nhn

bpd=− (4.12.40)

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()

max ,

cXtX

⎡⎤

=−

⎣⎦

(4.12.41)

T is measured from the inside surface, then

cX= (4.12.42)

ctX=− (4.12.43)

T is measured from the outside surface, then

ctX=− (4.12.44)

cX=

(4.12.45)

c) Rows of holes may be located in regions of relatively low bending moments to keep the required plate

thickness to a minimum. Therefore, it is permissible to calculate the stresses at the centerline of each

row of holes closest to the locations where the highest bending moments occurs (i.e. at the midpoint of

the sides and at the corners). If the diameter of all the holes is not the same, the stresses must be

calculated for each set of

e and

e values.

d) The applied gross area stresses may be calculated using the same procedure as for calculating the

stresses at a weld joint (see paragraph 4.12.3.2). The value of

or Y to be used in the equations is the

distance from the midpoint of the side to the plane containing the centerlines of the holes.

4.12.7 Design Procedure

4.12.7.1 A procedure that can be used to design a noncircular vessel subject to internal pressure is shown

below.

a) STEP 1 – Determine the design pressure and temperature.

b) STEP 2 – Determine the configuration of the noncircular vessel by choosing a Type from Table 4.12.1.

c) STEP 3 – Determine the initial configuration (i.e. width, height, length, etc.) and wall thicknesses of the

pressure containing plates.

1) If the vessel has stiffeners, then determine the spacing (see paragraph 4.12.3) and size of the

stiffeners.

2) If the vessel has stays, then determine the stay type and configuration (see paragraph 4.12.4), and

check the stay plate welds using paragraph 4.2.

3) If the vessel aspect ratio is less than two, then determine the plate parameters in paragraph 4.12.5.

d) STEP 4 – Determine the location of the neutral axis from the inside and outside surfaces.

1) If the section under evaluation has stiffeners, then

c and

c are determined from the cross section

of the combined plate and stiffener section using strength of materials concepts.

2) If the section under evaluation has multi-diameter holes, then

c and

c are determined from

paragraph 4.12.6.3.

3) If the section under evaluation does not have a stiffener, does not have holes, or has uniform

diameter holes, then

cct== where t is the thickness of the plate.

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e) STEP 5 – Determine the weld joint factor and ligaments efficiencies, as applicable (see paragraph

4.12.6), and determine the factors

E or

E .

f) STEP 6 – Complete the stress calculation for the selected noncircular vessel Type (see Table 4.12.1),

and check the acceptance criteria. If the criteria are satisfied, then the design is complete. If the criteria

are not satisfied, then modify the plate thickness and/or stiffener size and go to STEP 3 and repeat the

calculation. Continue this process until a design is achieved that satisfies the acceptance criteria.

4.12.7.2 If the vessel is subject to external pressure, the additional requirements of paragraph 4.12.8 shall

be satisfied.

4.12.8 Noncircular Vessels Subject to External Pressure

4.12.8.1 Rectangular vessel Types 1 and 2 subject to external pressure shall meet the following

requirements.

a) The stresses shall be calculated in accordance with Tables 4.12.2 and 4.12.3 except that the design

external pressure shall be substituted for

P . These computed stresses shall meet the acceptance

criteria defined in these tables.

b) The four side plates and the two end plates shall be checked for stability in accordance with Equation

(4.12.46). The required calculations for

S ,

crA

crB

and

crB

are shown in Table

4.12.15. In the equations, the subscript

A is used to identify stress or load acting in a direction parallel

to the long dimension of the panel being considered and the subscript

is used to identify stress or

load acting in a direction parallel to the short dimension of the panel being considered. In the

calculations, the plate thickness

t shall be adjusted if the plate is perforated. This can be accomplished

by multiplying

t by

e in the equations for

S and

S . It is not necessary to make this adjustment in

the equations for

crA

S and

crB

S .

1.0

mA mB

crA crB

+≤

(4.12.46)

where

crA crA crA

SS whenS=≤

(4.12.47)

** *

crA crA crA

SS whenS=>

(4.12.48)

crB crB crB

SS whenS=≤

(4.12.49)

** *

crB crB crB

SS whenS=>

(4.12.50)

c) In addition to checking each of the four side plates and the two end plates for stability, the cross section

shall be checked for column stability using the following equations. Equation (4.12.52) applies to

vessels where the long plate thicknesses are equal. If the thicknesses are not equal, replace

2t with

()

222

tt+ in the equation.

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