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

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2010 SECTION VIII, DIVISION 2

5-59

5.A.4.1.2 Stress Linearization Procedure

The methods to derive the membrane, bending, and peak components of a stress distribution are shown

below, and in Figure 5.A.4. The component stresses used for the calculations shall be based on a local

coordinate system defined by the orientation of the SCL, see Figure 5.A.2.

a) STEP 1 – Calculate the membrane stress tensor. The membrane stress tensor is the tensor comprised

of the average of each stress component along the stress classification line, or:

ij m ij

σσ

∫

(5.A.1)

b) STEP 2 – Calculate the bending stress tensor.

1) Bending stresses are calculated only for the local hoop and meridional (normal) component

stresses, and not for the local component stress parallel to the SCL or in-plane shear stress.

2) The linear portion of shear stress needs to be considered only for shear stress distributions that

result in torsion of the SCL (out-of-plane shear stress in the normal-hoop plane, see Figure 5.A.2).

3) The bending stress tensor is comprised of the linear varying portion of each stress component along

the stress classification line, or:

ij b ij

σσ

⎛⎞

=−

⎜⎟

⎝⎠

∫

(5.A.2)

c) STEP 3 – Calculate the peak stress tensor. The peak stress tensor is the tensor whose components are

equal to:

() ()

()

,,,

ij F ij ij m ij b

σσσσ

=−+

(5.A.3)

() ()

()

,,,ij F ij ij m ij b

xt xt

σσσσ

=−− (5.A.4)

d) Step 4 – Calculate the three principal stresses at the ends of the SCL based on components of

membrane and membrane plus bending stresses.

e) Step 5 – Calculate the equivalent stresses using Equation (5.1) at the ends of the SCL based on

components of membrane and membrane plus bending stresses.

5.A.4.2 Shell Elements

5.A.4.2.1 Overview

Stress results derived from a finite element analysis utilizing two-dimensional or three-dimensional shells are

obtained directly from the analysis results. Using the component stresses, the equivalent stress shall be

computed per Equation (5.1).

5.A.4.2.2 Stress Linearization Procedure

The methods to derive the membrane, bending, and peak components of a stress distribution are shown

below.

a) The membrane stress tensor is the tensor comprised of the average of each stress component along the

stress classification line, or:

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ij in ij out

ij m

= (5.A.5)

b) The bending stress tensor is the tensor comprised of the linear varying portion of each stress component

along the stress classification line, or:

ij in ij out

ij b

−

= (5.A.6)

c) The peak stress tensor is the tensor whose components are equal to:

()()

,,,

ij F ij m ij b f

σσσ

=+ − (5.A.7)

5.A.5 Structural Stress Method Based on Nodal Forces

5.A.5.1 Overview

Stress results derived from a finite element analysis utilizing continuum or shell elements may be processed

using the Structural Stress Method based on nodal forces. The mesh-insensitive structural stress method

provides a robust procedure for capturing the membrane and bending stresses and can be directly utilized in

fatigue design of welded joints. With this method, the structural stress normal to a hypothetical cracked plane

at a weld is evaluated. For typical pressure vessel component welds, the choice of possible crack

orientations is straightforward (e.g. toe of fillet weld). Two alternative calculation procedures for the structural

stress method are presented for continuum elements; a procedure based on nodal forces and a procedure

based on stress integration. A typical finite element continuum model and stress evaluation line for this type

of analysis is shown in Figure 5.A.5.

5.A.5.2 Continuum Elements

a) Stress results derived from a finite element analysis utilizing two-dimensional or three-dimensional

continuum elements may be processed using the structural stress method and nodal forces as described

below. The membrane and bending stresses can be computed from element nodal internal forces using

the equations provided in Table 5.A.1. The process is illustrated in Figure 5.A.6. This method is

recommended when internal force results can be obtained as part of the finite element output because

the results are insensitive to the mesh density.

b) When using three-dimensional continuum elements, forces and moments must be summed with respect

to the mid-thickness of a member from the forces at nodes in the solid model at a through-thickness

cross section of interest. For a second order element, three summation lines of nodes are processed

along the element faces through the wall thickness. The process is illustrated in Figure 5.A.7.

c) For a symmetric structural stress range, the two weld toes have equal opportunity to develop fatigue

cracks. Therefore, the structural stress calculation involves establishing the equilibrium equivalent

membrane and bending stress components with respect to one-half of the plate thickness. The

equivalent structural stress calculation procedure for a symmetric stress state is illustrated in Figure

5.A.8.

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2010 SECTION VIII, DIVISION 2

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5.A.5.3 Shell Elements

a) Stress results derived from a finite element analysis utilizing shell elements may be processed using the

structural stress method and nodal forces. The membrane and bending stresses can be computed from

element nodal internal forces using the equations provided in Table 5.A.2. A typical shell model is

illustrated in Figure 5.A.9.

b) When using three-dimensional shell elements, forces and moments with respect to the mid-thickness of

a member must be obtained at a cross section of interest. The process is illustrated in Figure 5.A.10.

5.A.6 Structural Stress Method Based on Stress Integration

As an alternative to the nodal force method above, stress results derived from a finite element analysis

utilizing two-dimensional or three-dimensional continuum elements may be processed using the Structural

Stress Method Based on Stress Integration. This method utilizes the Stress Integration Method of paragraph

5.A.3, but restricts the set of elements that contribute to the line of nodes being processed. The elements

applicable to the SCL for the region being evaluated shall be included in the post-processing, as is illustrated

in Figure 5.A.11.

5.A.7 Nomenclature

structural stress

Δ structural stress range.

line force at element location position i .

NF nodal force at node

, normal to the section.

nodal force at node

, normal to the section, for element location position i .

NM in-plane nodal moment at node

, normal to the section, for a shell element.

F nodal force resultant for element location position i .

K fatigue strength reduction factor used to compute the cyclic stress amplitude or range.

m line moment at element location position i .

number of nodes in the through-wall thickness direction.

nodal moment resultant for element location position

membrane stress.

bending stress.

stress tensor at the point under evaluation.

,ij m

membrane stress tensor at the point under evaluation.

,ij b

bending stress tensor at the point under evaluation.

,ij F

peak stress component.

,ij in

stress tensor on the inside surface of the shell.

,ij out

stress tensor on the outside surface of the shell.

membrane stress for element location position

bending stress for element location position

r radial coordinate of node

for an axisymmetric element.

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s local coordinate, parallel to the stress classification line, that defines the location of nodal force

relative to the mid-thickness of the section.

P primary equivalent stress.

secondary equivalent stress.

local

axis, oriented parallel to the stress classification line.

Y local Y axis, oriented normal to the stress classification line.

global

axis.

Y global Y axis.

t minimum wall thickness in the region under consideration, or the thickness of the vessel, as

applicable.

w width of the element to determine structural stresses from Finite Element Analysis.

through-wall thickness coordinate.

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5.A.8 Tables

Table 5.A.1 – Structural Stress Definitions For Continuum Finite Elements

Element Type Membrane Stress Bending Stress

Two-Dimensional

Axisymmetric

Second Order (8-

Node) Continuum

Elements

∑

NF s

⋅

∑

Two-Dimensional

Second Order

Plane Stress or

Plane Strain (8-

Node) Continuum

Elements

∑

NF s

⋅

∑

Three-

Dimensional

Second Order

(20-Node)

Continuum

Elements

Note:

represents the line force

corresponding to the element location

positions (

1, 2, 3i = ) along the element

width (

); position

corresponds to

the mid-side of the element (see Figure

5.A.7):

132

3(6 2 )

3(2 2 3 )

3(2 6 )

FFF

−

−+−

+−

In the above,

F ,

F , and

F are the

nodal force resultants (producing normal

membrane stress to Section A-A) through

the thickness and along the width (

w ) of

the group of elements

iij

NF=

∑

summed over the nodes from

(number of nodes in the through-thickness

direction) at Section A-A (see Figure

5.A.7)

⋅

Note:

m represents the line moment

corresponding to the element location

positions (

1, 2, 3i

) along the element

width (

); position

2i =

corresponds to

the mid-side of the element (see Figure

5.A.7):

132

3(6 2 )

3(2 2 3 )

3(2 6 )

MMM

+−

−+−

+−

In the above,

, and

are the

nodal moment resultants (producing normal

bending stress to Section A-A) calculated

based on nodal forces with respect to the

mid-thickness

()

s along the width ( w ) of

the group of elements

iijj

NF s=⋅

∑

summed over the nodes from

(number of nodes in the through-thickness

direction) at Section A-A (see Figure 5.A.7)

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Table 5.A.2 – Structural Stress Definitions For Shell Or Plate Finite Elements

Element Type Membrane Stress Bending Stress

Three-Dimensional

Second Order (8-

Node) Shell

Elements

Note:

represents the force

corresponding to the element

location positions (

1, 2, 3i

) along

the element width (

w ); position

2i = corresponds to the mid-side of

the element (see Figure 5.A.10):

132

3(6 2 )

3(2 2 3 )

3(2 6 )

NF NF NF

+−

−+−

+−

In the above,

NF ,

NF , and

are the internal nodal forces (in the

direction normal to Section A-A) from

the shell model along a weld (see

Figure 5.A.10)

⋅

Note:

m represents the moment

corresponding to the element location

positions (

1, 2, 3i

) along the element

width (

w ); position 2i = corresponds

to the mid-side of the element (see

Figure 5.A.10):

132

3(6 2 )

3(2 2 3 )

3(2 6 )

NM NM NM

+−

−+−

+−

In the above,

NM ,

NM , and

are the internal nodal moments

(producing normal bending stresses to

Section A-A) from the shell model along

a weld (see Figure 5.A.10)

Three-Dimensional

First Order (4-

Node) Shell

Elements

Note:

represents the force

corresponding to the element corner

node location positions (

1, 2i

)

along the element width (

2(2 )

NF NF

−

⋅

Note:

m represents the moment

corresponding to the element corner

node location positions (

1, 2i = ) along

the element width (

w ):

2(2 )

NM NM

−

Axisymmetric

Linear and

Parabolic Shell

Finite Element

⋅

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2010 SECTION VIII, DIVISION 2

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5.A.9 Figures

SCL

SCP

Figure 5.A.1

Stress Classification Line (SCL) and Stress Classification Plane (SCP)

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Normal

Tangential

Hoop

SCL

Mid-plane

Normal

Tangential

Hoop

SCL

(a) SCL Orientation, Three-Dimensional Model

(b) SCL Orientation, Two-Dimensional Model

Figure 5.A.2

Stress Classification Lines (SCLs)

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Figure 5.A.3

Stress Classification Line Orientation and Validity Guidelines

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Stress Distribution Transformed to y

direction

Memb

Figure 5.A.4

Computation of Membrane and Bending Equivalent Stresses by the Stress Integration Method Using

the Results from a Finite Element Model with Continuum Elements

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