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

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2010 SECTION VIII, DIVISION 2

5-79

5.B.6 Nomenclature

range

SΔ von Mises equivalent stress range between time points

and

stress tensor at the point under evaluation.

stress tensor at the point under evaluation at time point

t .

stress component range between time points

t and

t .

stress range associated with the normal stress component in the 1-direction between time

points

t and

t .

Δ stress range associated with the normal stress component in the 2-direction between time

points

and

Δ stress range associated with the normal stress component in the 3-direction between time

points

t and

t .

stress range associated with the shear stress component in the 1-direction between time

points

t and

t .

Δ stress range associated with the shear stress component in the 2-direction between time

points

and

Δ stress range associated with the shear stress component in the 3-direction between time

points

t and

t .

time point under consideration with the highest peak or lowest valley.

t time point under consideration that forms a range with time point

t .

N specified number of repetitions of the event associated with time point

t .

specified number of repetitions of the event associated with time point

absolute value of the range (load or stress) under consideration using the Rainflow Cycle

Counting Method.

Y absolute value of the adjacent range (load or stress) to previous

using the Rainflow Cycle

Counting Method.

2010 SECTION VIII, DIVISION 2

5-80

ANNEX 5.C

ALTERNATIVE PLASTICITY ADJUSTMENT FACTORS AND

EFFECTIVE ALTERNATING STRESS FOR ELASTIC FATIGUE

ANALYSIS

(NORMATIVE)

5.C.1 Scope

5.C.1.1 This Annex contains procedures for the determination of plasticity correction factors and effective

alternating equivalent stress for elastic fatigue analysis. These procedures include a modified Poisson’s ratio

adjustment for local thermal and thermal bending stresses, a notch plasticity adjustment factor that is applied

to thermal bending stresses, and a non-local plastic strain redistribution adjustment that is applied to all

stresses except local thermal and thermal bending stresses. These procedures are an alternative to effective

alternating stress calculations in Step 4 of paragraph 5.5.3.2.

5.C.2 Definitions

5.C.2.1 Thermal Bending Stress - Thermal bending stress is caused by the linear portion of the through-wall

temperature gradient. Such stresses shall be classified as secondary stresses.

5.C.2.2 Local Thermal Stress - Local thermal stress is associated with almost complete suppression of the

differential expansion and thus produces no significant distortion. Such stresses shall be considered only

from the fatigue standpoint and are therefore classified as peak stresses. Examples of local thermal stresses

are the stress in a small hot spot in a vessel wall, the non-linear portion of a through-wall temperature

gradient in a cylindrical shell, and the thermal stress in a cladding material that has a coefficient of expansion

different from that of the base metal. Local thermal stresses are characterized by having two principal

stresses that are approximately equal.

5.C.3 Effective Alternating Stress for Elastic Fatigue Analysis

5.C.3.1 The effective total equivalent stress amplitude is used to evaluate the fatigue damage for results

obtained from a linear elastic stress analysis. The controlling stress for the fatigue evaluation is the effective

total equivalent stress amplitude, defined as one-half of the effective total equivalent stress range

()

PPQF+++

calculated for each cycle in the loading histogram.

5.C.3.2 The following procedure shall be used to determine plasticity correction factors for elastic fatigue

analysis and the effective alternating equivalent stress.

a) STEP 1 – At the point of interest, determine the following stress tensors and associated equivalent

stresses at the start and end points (time points

and

, respectively) for the

cycle counted in

Step 2 of paragraph 5.5.3.2.

1) Calculate the component stress range between time points

t and

t and compute an equivalent

stress range due to primary plus secondary plus peak stress as given below.

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2010 SECTION VIII, DIVISION 2

5-81

,,,

ij k ij k ij k

σσ

Δ= − (5.C.1)

()()

()

0.5

11, 22, 11, 33,

222

22, 33, 12, 13, 23,

kk kk

kk kkk

σσ σσ

σσ σσσ

⎡⎤

Δ−Δ +Δ−Δ +

⎢⎥

Δ=

⎢⎥

Δ−Δ +Δ+Δ+Δ

⎣⎦

(5.C.2)

2) Using the linearized stress results due to primary plus secondary stresses, compute the component

stress range using Equation (5.C.1). Compute the equivalent stress range using Equation (5.C.2)

and designate this quantity as

,nk

3) Determine the stress tensor due to local thermal and thermal bending stresses at the start and end

points for the

k cycle. It may be difficult to calculate the local thermal stress from stress

distributions obtained from numerical methods. If this is the case, the procedure below can be used

to calculate the local thermal and thermal bending stresses due to a non-linear temperature

distribution. This method is based on calculating a thermal stress difference range associated with

the linearized temperature distribution along the SCL for the time steps of interest. Consistent with

that method, consider the distribution of the temperature from numerical method as a function of the

local through thickness direction. The temperature distribution for each time step can be separated

into three parts.

i) A constant temperature equal to the average of the temperature distribution

avg

TTdx

∫

(5.C.3)

ii) The linearly varying portion of the temperature distribution

TTxdx

⎛⎞

=−

⎜⎟

⎝⎠

∫

(5.C.4)

iii) The non-linear portion of the temperature distribution

()

pavgb

TTT Tt=− + (5.C.5)

By assuming full suppression of the differential expansion of the cross-section, the associated local

thermal stress parallel to the surface for each time step may be calculated as given below where

is given by Equation (5.C.5).

()

1, 2

avg b

ij k

ETT Tt

for i j

⎡⎤

−−+

⎣⎦

===

−

(5.C.6)

ij k

or i j and i j

=≠== (5.C.7)

Using Equations (5.C.6) and (5.C.7), determine the local thermal component stress ranges using

Equation (5.C.1) and designate this quantity as

,ij k

Δ . The thermal bending component stress

range,

,ij k

Δ is determined by linearizing the through-wall stress distribution due to thermal effects

only.

2010 SECTION VIII, DIVISION 2

5-82

4) Compute the equivalent stress ranges due to primary plus secondary plus peak stress minus the

local thermal stress using Equations (5.C.8) and 5.C.9).

(

)

(

)

,,, ,,

mmLTnnLT

ij k ij k ij k ij k ij k

σσσ σσ

Δ= − − −

(5.C.8)

()

()()

()

0.5

11, 22, 11, 33,

222

22, 33, 12, 13, 23,

kk kk

Pk LTk

kk kkk

σσ σσ

σσ σσσ

⎡⎤

Δ−Δ +Δ−Δ +

⎢⎥

Δ−Δ =

⎢⎥

Δ−Δ +Δ+Δ+Δ

⎣⎦

(5.C.9)

5) Compute the equivalent stress ranges due to local thermal plus thermal bending stress using

Equations (5.C.10) and (5.C.11).

()()

,,, ,,

mTB m LT nTB nLT

ij k ij k ij k ij k ij k

σσσσσ

Δ= + − +

(5.C.10)

()

()()

()

0.5

11, 22, 11, 33,

222

22, 33, 12, 13, 23,

kk kk

LT k TB k

kk kkk

σσ σσ

σσ σσσ

⎡⎤

Δ−Δ +Δ−Δ +

⎢⎥

Δ+Δ =

⎢⎥

Δ−Δ +Δ+Δ+Δ

⎣⎦

(5.C.11)

6) If required, see Equation (5.C.32), compute the stress tensor due to non-thermal effects (all

loadings except local thermal and thermal bending),

ij k

, at the start and end points for the

cycle.

b) STEP 2 - Determine the Poisson’s ratio adjustment,

,vk

K to adjust local thermal and thermal bending

stresses for the

k cycle based on the equivalent stress ranges in STEP 1 using the following equations,

(

S is defined in paragraph 5.5.6.1):

1.0

vk Pk PS

KforSS=Δ≤ (5.C.12)

()

()()

,, ,

0.6 1.0

Pk PS

LT k TB k

LTk TBk Pk PS

S S and

Kfor

SS SS

Δ>

⎡⎤

Δ−

⎢⎥

Δ+Δ

Δ+Δ >Δ−

⎢⎥

⎣⎦

(5.C.13)

()()

,, ,

1.6

Pk PS

LTk TBk Pk PS

S S and

Kfor

SS SS

Δ>

Δ+Δ ≤Δ−

(5.C.14)

c) STEP 3 – Determine the non-local plastic strain redistribution adjustment,

,nl k

K to adjust all stresses

except local thermal and thermal bending for the

k cycle. In these equations, the parameters m and

are defined in Table 5.13.

1.0

nl k n k PS

KforSS=Δ≤

(5.C.15)

()

, ,

1.0 1

nl k PS n k PS

KforSSmS

nm S

−

⎛⎞

=+ − <Δ<

⎜⎟

−

⎝⎠

(5.C.16)

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2010 SECTION VIII, DIVISION 2

5-83

nl k n k PS

KforSmS

=Δ≥ (5.C.17)

d) STEP 4 – Determine the notch plasticity adjustment factor,

,np k

based on the equivalent stress ranges

in STEP 1 to adjust thermal bending stresses to account for additional local strain concentration due to a

geometric stress riser for the

k cycle. In these equations, the parameters n are defined in Table 5.13.

For numerical results used directly:

(

)

,,,

1.0

np k P k LT k PS

KforSSS=Δ−Δ≤

(5.C.18)

[

]

(

)

,12 ,,

min ,

np k P k LT k PS

KKK forSSS=Δ−Δ> (5.C.19)

(

)

()

1.0 1.0

Pk LTk PS

Pk LTk

SS S

−

⎛⎞

⎜⎟

⎝⎠

⎡⎤

Δ−Δ −

⎛⎞

Δ−Δ

=−⋅ +

⎢⎥

⎜⎟

Δ−Δ

⎢⎥

⎝⎠

⎣⎦

(5.C.20)

nl k

= (5.C.21)

For numerical results that are adjusted with a stress concentration factor (SCF):

(

)

1.0

np k n k PS

KforSSCFS=Δ⋅≤

(5.C.22)

[

]

(

)

,12 ,

min ,

np k n k PS

KKK forSSCFS=Δ⋅> (5.C.23)

()

1.0 1.0

nk PS

SSCFS

KSCF

SSCF

−

⎛⎞

⎜⎟

⎝⎠

⎡⎤

Δ⋅ −

=−⋅ +

⎢⎥

Δ⋅

⎢⎥

⎣⎦

(5.C.24)

nl k

(5.C.25)

Note that the

SCF and

,np k

values may be dependent upon the component stress direction.

e) STEP 5 – Apply the plasticity adjustment factors to the component stresses at the start and end points

for the

k cycle.

1) Compute the component stresses including plastic Poisson’s ratio and notch plasticity adjustments

as given below for time points

t and

t .

For numerical results used directly:

()

LT LT

ij ij k v k

adj

σσ

=⋅ (5.C.26)

()

(

)

,, , , ,

TB TB TB

ij ij k v k np k ij k NUM np k

adj

KK SCF K

σσ σ

=⋅⋅ +⋅ −⋅ (5.C.27)

2010 SECTION VIII, DIVISION 2

5-84

For numerical results that are adjusted with a stress concentration factor (SCF):

()

LT LT

ij ij k v k LT

adj

KSCF

σσ

=⋅⋅ (5.C.28)

()

(

)

,, , , ,

TB TB TB

ij ij k v k np k ij k NUM np k

adj

KK SCF SCF K

σσ σ

=⋅⋅ ⋅ +⋅ −⋅ (5.C.29)

2) Compute the component stresses including non-local plastic strain redistribution adjustment as

given below for time points

t and

t .

For numerical results used directly:

()

,, ,

NT TB

ij ij k ij k NUM np k

adj

SCF K

σσσ

⎡⎤

=− −⋅

⎣⎦

(5.C.30)

()

pk LT k

NUM

SCF

Δ−Δ

(5.C.31)

For numerical results that are adjusted with a stress concentration factor (SCF):

()

NT NT

ij ij k nl k

adj

KSCF

σσ

=⋅⋅ (5.C.32)

f) STEP 6 – Compute the adjusted component stress ranges between time points

t and

t as given

below.

()

()()()

LT NT TB

ij ij ij

adj adj adj

ij k

adj

LT NT TB

ij ij ij

adj adj adj

σσσ

⎧⎫

⎡⎤

++ −

⎪⎪

⎢⎥

⎪⎣ ⎦⎪

Δ=

⎨⎬

⎡⎤

⎪⎪

⎢⎥

⎪⎪

⎣⎦

⎩⎭

(5.C.33)

g) STEP 7 – Compute the effective equivalent stress range using the adjusted component stress ranges

from STEP 6 and Equation (5.C.2). Designate the adjusted effective equivalent stress range as

()

,Pk

adj

SΔ .

h) STEP 8 – Compute the effective alternating equivalent stress for the

k cycle as given below.

()

0.5

alt k P k

adj

SS=Δ (5.C.34)

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2010 SECTION VIII, DIVISION 2

5-85

5.C.4 Nomenclature

thermal expansion coefficient of the material at the point under consideration, evaluated at the

mean temperature of the

k cycle.

,nk

SΔ primary plus secondary equivalent stress range for the

k cycle.

SΔ

range of primary plus secondary plus peak equivalent stress for the

k cycle.

,LT k

SΔ primary plus secondary plus peak equivalent stress range due to local thermal effects for the

k cycle.

,TB k

SΔ

primary plus secondary plus peak equivalent stress range due to thermal bending effects for

the

k cycle.

,NT k

SΔ primary plus secondary plus peak equivalent stress range due to non-thermal effects for the

cycle.

()

,Pk

adj

SΔ adjusted range of primary plus secondary plus peak equivalent stress, including non-local

strain redistribution, notch plasticity, and plastic Poisson’s ratio adjustments for the

k cycle.

,ij k

stress component range between time points

t and

t for the

k cycle.

ij k

stress component range due to local thermal stress between time point

t and

t for the

cycle.

ij k

stress component range due to thermal bending stress between time points

t and

t for the

cycle.

adjusted stress tensor, including non-local strain redistribution, notch plasticity, and plastic

Poisson’s ratio adjustments at the location and time point under evaluation for the

cycle.

Young’s Modulus of the material evaluated at the mean temperature of the cycle.

K parameter used to compute

,np k

K .

K parameter used to compute

,np k

K ..

,nl k

K non-local strain redistribution adjustment factor for the

cycle.

,np k

notch plasticity adjustment factor for the

cycle.

plastic Poisson’s ratio adjustment factor for the

cycle.

m material constant used for the non-local strain redistribution adjustment factor, per Table 5.6.

n material constant used for the non-local strain redistribution adjustment factor, per Table 5.6.

Poisson’s ratio.

,alt k

S alternating equivalent stress for the

cycle.

S allowable limit on the primary plus secondary stress range.

SCF

stress concentration factor.

SCF stress concentration factor applicable for local thermal stress.

NUM

SCF stress concentration factor determined from the numerical model.

2010 SECTION VIII, DIVISION 2

5-86

ij k

stress tensor due to local thermal stress at the location and time point under evaluation for the

cycle.

ij k

stress tensor due to non-thermal stress at the location and time point under evaluation for the

cycle.

ij k

stress tensor due to thermal bending stress due to the linearly varying portion of the

temperature distribution at the location and time point under evaluation for the

cycle.

ij k

stress tensor at the point under evaluation at time point

for the

cycle.

ij k

stress tensor at the point under evaluation at time point

t for the

cycle.

mLT

ij k

stress tensor due to local thermal stress at the location under evaluation at time point

for

the

cycle.

nLT

ij k

stress tensor due to local thermal stress at the location under evaluation at time point

t for

the

cycle.

mTB

ij k

stress tensor due to thermal bending stress at the location under evaluation at time point

for

the

cycle.

nTB

ij k

stress tensor due to thermal bending stress at the location under evaluation at time point

t for

the

cycle.

()

adj

adjusted stress tensor due to local thermal stress at the location and time point under

evaluation for the

cycle.

()

adj

adjusted stress tensor due to non-thermal stress at the location and time point under

evaluation for the

cycle.

()

adj

adjusted stress tensor due to thermal bending stress at the location and time point under

evaluation for the

cycle.

t wall thickness.

time point under consideration with the highest peak or lowest valley.

t time point under consideration that forms a range with time point

temperature distribution.

avg

T average temperature component of temperature distribution T .

T equivalent linear temperature component of temperature distribution T .

T peak temperature component of temperature distribution T .

position through the wall thickness.

z local coordinate for the temperature distribution.

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2010 SECTION VIII, DIVISION 2

5-87

ANNEX 5.D

STRESS INDICES

(NORMATIVE)

5.D.1 General

5.D.1.1 In lieu of a detailed stress analysis, stress indices may be used to determine peak stresses around

a nozzle opening.

5.D.1.2 The term stress index, is defined as the numerical ratio of the stress components

, and

under consideration to the computed membrane hoop stress in the unreinforced vessel material; however, the

material which increases the thickness of a vessel wall locally at the nozzle shall not be included in the

calculation of these stress components. These stress directions are defined in Figure 5.D.1. When the

thickness of the vessel wall is increased over that required to the extent provided hereinafter, the values of

and

r in Figure 5.D.2 shall be referred to the thickened section.

5.D.1.3 The stress indices in these tables provide only the maximum stresses at certain general locations

due to internal pressure. In the evaluation of stresses in or adjacent to vessel openings and connections, it is

often necessary to consider the effect of stresses due to external loadings or thermal stresses. In such

cases, the combined stress at a given point may be determined by superposition. In the case of combined

stresses due to internal pressure and nozzle loading, the maximum stresses for a given location should be

considered as acting at the same point and added algebraically unless positive evidence is available to the

contrary.

5.D.2 Stress Indices for Radial Nozzles

5.D.2.1 The stress indices for radial nozzles in spherical shells and spherical portions of formed heads in

Table 5.D.1 and for nozzles in cylindrical shells in Table 5.D.2 may be used if all of the items listed below are

true.

a) The opening is for a circular nozzle whose axis is normal to the vessel wall. If the axis of the nozzle

makes an angle

with the normal to the vessel wall, an estimate of the

index on the inside may be

obtained from the equations shown below provided

0.15

ni i

≤

. In these equations,

K is the

stress index for a radial connection and

K is the

stress index for a non-radial connection.

()

12sin ( )K K for hillside nozzles in spheres or cylinders

=+ (5.D.1)

()

(

)

1tan ( )K K for lateral nozzles in cylinders

=+ (5.D.2)

b) The arc distance measured between the centerlines of adjacent nozzles along the inside surface of the

shell is not less than three times the sum of their inside radii for openings in a head or along the

longitudinal axis of a shell and is not less than two times the sum of their radii for openings along the

circumference of a cylindrical shell.

2010 SECTION VIII, DIVISION 2

5-88

c) For nozzles in cylindrical shells, the following dimensional limitations shown below are satisfied. In

addition, the total nozzle reinforcement area on the transverse plane of the nozzle including any outside

of the reinforcement limits shall not exceed 200% of that required for the longitudinal plane unless a

tapered transition section is incorporated into the reinforcement and the shell.

10 100

≤≤

(5.D.3)

0.50

≤

(5.D.4)

1.50

Dt r

≤

(5.D.5)

d) For nozzles in spherical shells, the following dimensional limitations shown below are satisfied. In

addition, at least 40% of the reinforcement is located on the outside surface of the nozzle-shell juncture.

10 100

≤≤

(5.D.6)

0.50

≤

(5.D.7)

0.80

≤

(5.D.8)

e) For nozzles in cylindrical and spherical shells, the following local geometry details shall be satisfied.

1) The inside corner radius

r (see Figure 5.D.2) is one-eighth to one-half of the shell thickness t .

2) The outer corner radius

r (see Figure 5.D.2) is large enough to provide a smooth transition

between the nozzles and the shell. In addition, for nozzle diameters greater than 1.5 shell

thicknesses in cylindrical shells and 2:1 ellipsoidal heads, or for nozzle diameters greater than 3

shell thicknesses in spherical shells, the value of

r shall satisfy the following:

max 2 ,

rrt

⎡⎤

≥

⎢⎥

⎣⎦

(5.D.9)

3) The value of

r shall satisfy the following:

max ,

rrt

⎡⎤

≥

⎢⎥

⎣⎦

(5.D.10)

5.D.2.2 Stress indices for radial nozzles in spherical shells and spherical portions of formed heads and

nozzle in cylindrical shells may be developed by stress analysis techniques consistent with the elastic design

analysis methods of Part 5, or obtained for other sources.

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