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COSMOSM Advanced Modules i

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Fatigue Module Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Cumulative Damage Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

Elastic-Plastic Formulation Definitions . . . . . . . . . . . . . . . . . . . . . . . . 2-4

Analysis Procedure for Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5Normal Procedure (at a Location) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

All-Nodes Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

Simplified Elastic-Plastic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

3 Description of Elements . . . . . . . . . . . . . . . . . . . . . . . . 3-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

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Contents

ii COSMOSM Advanced Modules

4 Brief Description of Commands . . . . . . . . . . . . . . . . . . .4-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-1

Commands Likely to be Used for a Given Analysis . . . . . . . . . . . . . . . . .4-1

Analysis Menu (Analysis > FATIGUE) . . . . . . . . . . . . . . . . . . . . . . . .4-2

Analysis Menu (Analysis > FATIGUE > FATIGUE LIST) . . . . . . . . .4-2

Analysis Menu (Analysis > FATIGUE > FATIGUE DELETION) . . .4-3

Results Menu (Results > PLOT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-3

5 Detailed Description of Examples . . . . . . . . . . . . . . . . . .5-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-1

How FSTAR Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-1

Cylinder Under Axial Cyclic Loading Example . . . . . . . . . . . . . . . . . . . .5-4

History Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-4Starting The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-4

Specifying the Fatigue Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-5

Defining the Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-5

Defining a Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-6

Specifying the Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-7

Specifying the Fatigue Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-8

Running the Fatigue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-10Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-11

Fatigue Caused by Pressure Loading Example . . . . . . . . . . . . . . . . . . . .5-12

Structural Modeling and Stress Analysis . . . . . . . . . . . . . . . . . . . . . . .5-13

Fatigue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-14

Fatigue Analysis (All-Nodes Option) . . . . . . . . . . . . . . . . . . . . . . . . . . .5-17

Fatigue Caused by Thermal Loading Example . . . . . . . . . . . . . . . . . . . .5-17

Creating the Model Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-18

Specifying and Running the Thermal Analysis . . . . . . . . . . . . . . . . . .5-19

Specifying and Running the Stress Analysis . . . . . . . . . . . . . . . . . . . .5-20

Specifying and Running the Fatigue Analysis . . . . . . . . . . . . . . . . . . .5-20

Running the Analysis Based on the Elastic-Plastic Formulation . . . . .5-21In

dex

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COSMOSM Advanced Modules iii

Contents FSTAR / Fatigue Analysis

6 A Brief Theoretical Background for Simplified Elastic-Plastic

Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

Section Orientation in Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

Stress Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Cartesian Formulation (Approximation) . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Axisymmetric Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

Simplified Elastic-Plastic Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

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Contents

iv COSMOSM Advanced Modules

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COSMOSM Advanced Modules 1-1

1 Introduction

Introduction

The suitability of a mechanical or structural component for specific serviceinvolving cyclic application of loads and thermal conditions is determined by the

use of the fatigue module. The basic postulate adopted for fatigue calculations with

spectrum loading is the cumulative damage theory based on the Miners rule.

Fatigue Module CapabilitiesThe Fatigue module, FSTAR, provides engineers with the capability to perform

fatigue analysis of their structural designs using the data base created with the stress

module of COSMOSM. Analysis can be performed easily and quickly to determine

the predicted life of a design and identify areas that are fatigue critical.

FSTAR calculates fatigue usage factor (fraction of life used up by a combination of

fatigue events) for any point of a structural model. The model can consist of 1D, 2Dplane and axisymmetric, 3D solid and shell elements. Stress conditions at any point

can be due to mechanical loadings and/or thermal loadings. All events and the

corresponding number of cycles are defined easily by the user. At each location

being evaluated, the stress can be associated with load conditions previously

calculated with COSMOSM (load cases/time-steps), or it can be directly input by

the user. The user has also the flexibility to modify the stored stresses in the dataIndex

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Chapter1 Introduction

1-2 COSMOSM Advanced Modules

base. The solution is based on the Miners rule approach and the ASME Boiler and

Pressure Vessel Code. There is also a capability to evaluate the fatigue usage factor

using the ASME Code for a simplified elastic-plastic formulation. The powerful

graphics capability of the COSMOSM software can be utilized to display colorplots of fatigue life and to identify the fatigue critical regions.

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2 Analysis

Introduction

This chapter covers background information, definitions and the procedure forconducting a fatigue analysis.

Cumulative Damage Theory

Service operation at any given cyclic stress amplitude produces fatigue damage, theseriousness of which will be related to the total number of cycles that would be

required to produce failure of an undamaged component at that stress amplitude. It

is also assumed that the damage incurred is permanent.

The method adopted here is a cumulative damage method based on the Miners

rule. To have a better understanding of the general theory, first consider the

following example:

Assume that operation at several different stress amplitudes S1, S2,..., St in

sequence for a number of cycles n1, n2,..., nt will result in an accumulation of total

damage equal to the sum of the damage increments accrued at each individual

stress level. Then if operation at a stress amplitude (level) S1 produces complete

damage (or failure) in N1 cycles, operation at stress amplitude S1 (event 1) for aIn

dex

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Chapter 2 Analysis


number of cycles n1 smaller than N1 will produce a smaller fraction of damage, say

D1. Factor D1 is termed damage fraction (usage factor). Operation over a spectrum

of different stress levels results in a usage factor Di for each of the different stress

levels Si in the spectrum (in the following development each one of the different

load level operations, which may consist of a number of cycles, is called an event).

When these factors sum to unity, failure is predicted; that is,

(2-1)

The linear damage rule states that the damage fraction (usage factor), D i, at stress

level Si is equal to the cycle ratio ni/Ni. Thus, the damage fraction D due to one

cycle of loading is 1/N. In other words, the application of one cycle of loading

consumes 1/N of the fatigue life. The failure criterion for variable amplitude

loading can now be stated as

(2-2)

In the above postulate it is assumed that there is no interaction between different

events, i.e. each event (consisting of a number of cycles of the same load level)

occurs in complete isolation from the other events. However, in practical

applications, every load cycle of the spectrum may contain multiple load levels.

The fatigue analysis of FSTAR also includes this latter effect as will be described in

more details in the section on Analysis Procedure for Cyclic Loading.

Definitions

General Definitions

Loading History

A series of load steps which may occur for a number of times.

Event

A portion of loading history which has a frequency of occurrence.

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Part 2 FSTAR / Fatigue Analysis

Fatigue Loading

A specified point in the loading history such as where the stress level is at the

extreme.

Location

A point on the structure for which the fatigue calculation is to be performed.

Stress Condition

Corresponds to the state of stress at a location for a particular (fatigue) loading.

Loading Combination (Set)

Is the combination of two loadings. There are as many as N(N-1)/2 loading

combinations, where N is the number of loadings.

Component Stress Range

Is computed for a loading combination, when the six components of stress field at

one (stress) condition are subtracted from the six components at the other condition.

Alternating Stress (Intensity)

If C1, C2 and C3 are the three principal stresses obtained from the component stress

range and S12 = C1 - C2, S23 = C2 - C3 and S31 = C3 - C1 are the three stress

differences, then the alternating stress intensity (Salt) is one-half of the largest

absolute magnitude of any stress difference.

Alternating Stress Intensity List

A list of all possible combinations of alternating stress intensities (at a location) in

decreasing order.

Partial Usage Factor

For loading combination i, the partial usage factor is equal to the cycle ratio n i/Ni,

where ni is the lower number of cycles remaining from either events E or F. EventsE and F contain the two loadings which constitute loading combination i, and N i is

the allowable number of cycles interpolated from the S-N curve for the alternating

stress intensity Salti (resulted from loading combination i).

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Chapter 2 Analysis


Cumulative Usage Factor

Sum of all partial usage factors.

Processing Phase

Initial COSMOSM finite element structural results (from STAR, HSTAR, ASTAR,

or NSTAR modules) applicable as input to the fatigue calculations.

Load Case (Time Step)

Stresses stored in the database during the processing phase. These stresses are

considered in fatigue calculations once a load case (time step) is associated with a

(fatigue) loading.

Scale Factor

Is a stress multiplier. It applies to stresses which are read for a fatigue loading from

the database stress file. The database stress file contains the finite element structural

solution obtained during the Processing Phase.

Stress Concentration Factor (in X, Y and Z directions)

Applies to the normal stress components at a specified location. This factor is to be

used more as a measure of the mesh refinement in the model used for finite element

analysis, in the areas where fatigue calculation is being performed. If the model is

fine enough, it takes a value of one in all directions. It may be increased to higher

values depending on the coarseness of the finite element mesh. This factor may also

be used to account for the existence of any local stress concentration effects other

than geometry effects.

Elastic-Plastic Formulation Definitions

Section

A straight path through the wall (in the thickness direction) of an axisymmetric

structure.

Linearized Stress

The equivalent linear stress distribution along a section which has the same net

bending moment as the actual distribution.

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Membrane Stress

The constant portion of stress such that pure moment acts on a section plane after the

membrane is subtracted from the equivalent linearized stress.

Bending Stress:

The variable portion of stress equal to the equivalent linear stress minus the

membrane stress.

Design Stress

The allowable design stress intensity value (Sm) at a particular temperature as

specified by the Sm-T curve. (These curves may be found in tables I-2.1 and I-2.2 of

Reference 5).

Sm-T Curve

Design stress versus temperature.

Analysis Procedure for Cyclic Loading

Normal Procedure (at a Location)

1. When a design fatigue curve (S-N curve) is not defined, the program evaluates

the alternating stress intensities according to the ASME Boiler and Pressure

Vessel Code [Reference 1]. Alternating stress intensities are evaluated betweenall possible pairs of loading combinations. That is, for a defined location, all six

stress components of loading A will be subtracted from the corresponding

components of loading B to yield component stress range from which the

alternating stress intensities are evaluated according to the General Definitions

section. These alternating stress intensities are listed for all possible

combinations in decreasing order together with their corresponding loading

pairs. No usage factor is evaluated in this case.

2. When a design fatigue curve (S-N curve) is defined, the program checks the

alternating stress intensity list from the top (highest value) to the bottom (lowest

value). It evaluates the partial usage factor Di for the ith alternating stress

intensity in that list by evaluating the cycle ratio ni/Ni (as its equivalence). The

ith alternating stress intensity is formed by the combination of loadings AE andIn

dex

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Chapter 2 Analysis


BF, where E and F are the corresponding events. Here, ni corresponds to the

lower number of cycles remaining from either events E and F, and Ni

corresponds to the allowable number of cycles interpolated from the design

fatigue curve (SN curve).

After evaluating the partial usage factor Di, the program updates the alternating

stress intensity list by reducing the number of cycles of both events E and F by

ni. Consequently, one of the two events E or F will be eliminated (or both if E

and F have the same number of cycles) and the other event will have ni cycles

less in the later calculations. Elimination of an event results in elimination of the

corresponding loadings. Once a loading is eliminated the corresponding stress

intensities (formed by combination of that loading with other loadings) will alsobe eliminated from the list. After updating the list the program checks the next

alternating stress intensity in the list and evaluates the corresponding partial

usage factor, adds that to the cumulative one and updates the list. This procedure

will be repeated for the next alternating stress intensity in the list and continues

until all stress intensities are considered.

3. An S-N curve is defined by log-log interpolation between the points (on the

curve) and linear interpolation among all the (S-N) curves with different stressratios if more than one curve is defined by the user. Also available are two

predefined (optional) curves for Carbon or Austenitic Steels which the user may

consider instead. For any alternating stress within the stress range S1 and S2 (the

first and last points) of an S-N curve the program uses log-log interpolation to

find the corresponding cycle. For any stress larger (smaller) than S1 (S2), the

program assigns N1 (N2) as the corresponding number of cycles, where N1 and

N2

correspond to the first and last points of the curve. Therefore, it is important

that the user defines the S-N curve for a wide range of cycles. For the same

reason, if an endurance limit is to be implemented into the curve, it is

appropriate to have a relatively large number of cycles assigned to the last point

(N2) on the curve.

All-Nodes Option

This option is similar to the normal procedure except that the fatigue calculation is

performed for a structure at all nodes. The program has the following limitations:

1. Since the calculation is done on the nodal basis, no location has to be defined

(all defined locations will be ignored).

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2. All defined concentration factors will be ignored.

3. All stresses are read either from stress files created during the processing phase

or considered to be zero if they are so defined (using command FT_LOAD(Analysis > FATIGUE > Fatigue Load). No modification on the nodal stresses

are allowed (i.e., any assignment made by the command FT_STREAD (Analysis

> FATIGUE > Apply Stress) is ignored).

4. No elastic-plastic formulation will be incorporated into the calculation at any

node.

Simplified Elastic-Plastic Formulation

If a simplified elastic-plastic formulation is desired, the following steps must be

taken:

1. For an axisymmetric structure

such as a pressure vessel and a

fatigue critical location (on theinner or outer surface), define a

thickness-through section with

one end at the desired location

(Figure 2-1). The section must

present a rational plane of

bending which, in most cases,

will be perpendicular to both

surfaces and the mid-plane. Formost of the structures with

parallel surfaces, such as pipes,

shells and external nozzles, this

criteria can be easily met.

However, in the irregular areas

such as in the nozzle to shell juncture, rational planes of bending may be

approximated so as to be perpendicular to the mid-plane and have the same

angle between the section and the surface on both sides (see Chapter 6).

2. Define an Sm-T curve. Typical values of Sm are given in Tables I-1.1 and I-1.2

of Reference 5.

Figure 2-1. Two Types of Sections Definedby Locations at Their Two Ends

1 2

3

4Y

X

CL

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Chapter 2 Analysis


3. If an Sm-T curve is defined, the calculation should proceed by evaluating the

equivalent linearized stresses along the defined section (Figure 2-2). The

linearized stresses are the sum of the bending stresses and the membrane

stresses as outlined in Chapter 1.

Figure 2-2. Equivalent Linearized Stress Along a Sections

4. Each alternating stress intensity (evaluated according to the Normal Procedure

section) is increased by a factor Ke. This factor is determined by considering

an equivalent alternating stress intensity based on linearized stresses (see

Chapter 6).

Membrane

Bending at X

ActualStress

Linear Stress(Membrane + Bend ing)

AlongSection

S

X

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3 Description of Elements

Introduction

The following table shows the elements that are used in the FSTAR program. Fordetailed descriptions of each element, you are referred to Chapter 4 of the

COSMOSM User Guide manual.

Table 3-1. Elements for FSTAR

Element TypeElement

Name

2D Spar/Truss TRUSS2D

2D Elastic Beam BEAM2D

3D Elastic Beam BEAM3D

3D Spar/Truss TRUSS3D

Elastic Straight Pipe PIPE

Boundary Element BOUND

General Mass Element MASS

Elastic Curved Pipe ELBOW

2D 4- to 8-Node Plane Stress, Strain, Body of Revolution PLANE2D3D 3- to 6-Node Plane Stress, Strain, Body of Revolution TRIANG

Triangular Thick Shell SHELL3T

Quadrilateral Thick Shell SHELL4T

3D 8- to 20-Node Continuum Brick SOLID

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Table 3-1. Elements for FSTAR (Concluded)

Element TypeElement

Name

3D 4-Node Tetrahedron Solid TETRA4

3D 10-Node Tetrahedron Solid TETRA10

2-Node Gap/with Friction GAP

Triangular Composite Shell SHELL3L

Quadrilateral Composite Shell SH3LL4L

Triangular Thin Shell SHELL3

Quadrilateral Thin Shell SHELL4

Spring Element SPRING

3D 4-Node Tetrahedron Solid with Rotation TETRA4R

Axisymmetric Shell SHELLAX

General Stiffness GENSTIF

8 or 9-Node Isoparametric Shell Element SHELL9

8 or 9-Node Isoparametric Composite Shell SHELL9L

8-Node Composite Solid SOLIDL

2-Node Rigid Bar RBAR3D 8- to 20-Node Isoparametric Piezoelectric Solid SOLIDPZ

6-Node Shell Element SHELL6

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4 Brief Description

of Commands

Introduction

This chapter outlines the commands most commonly used.

Commands Likely to be Used for a Given Analysis

The following section gives a brief description of commands that may be necessary

to run a given type of analysis once a proper finite element mesh is generated. This

is intended as a general guideline only because the problem at hand may not need

all the commands that are mentioned below or it may need some other commands

which are not mentioned.

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Chapter 4 Brief Description of Commands


Analysis Menu (Analysis > FATIGUE)

Analysis Menu (Analysis > FATIGUE > FATIGUE LIST)

Command Intended Use

FT_EVENT

( ...> Event Cycle) Specifies the number of cycles

FT_LOAD

( ...> Fatigue Load)Defines fatigue loading

FT_STREAD Defines stress conditions

( ...> Apply Stress)

FT_CURDEF Defines fatigue properties

( ...> Property Curve)

FT_LOC Defines a fatigue location( ...> Fatigue Location)

FT_SEC Defines a section for elastic-plastic

formulation( ...> Fatigue Section)

A_FATIGUE Specifies the face and layer number of multi-

layered shell or solids( ...> Analysis Options)

R_FATIGUE Performs the fatigue analysis

( ...> Run Fatigue Analysis)


FT_EVENTLIS List fatigue events and their specifications

( ...> Events)

FT_STLIST Lists stress conditions( ...> Stress Conditions)

FT_CURLIST Lists all defined fatigue properties

( ...> Property Curves)

FT_LOCLIST Lists specifications for a pattern of fatigue

locations( ...> Locations)

FT_SECLIST Lists specifications for a pattern of fatigue

sections( ...> Sections)

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Analysis Menu (Analysis > FATIGUE > FATIGUE DELETION)

Results Menu (Results > PLOT)


FT_EVENTDELDeletes a pattern of fatigue events

( ...> Events)

FT_LOADDEL Deletes a pattern of fatigue loadings

( ...> Loads)

FT_STDEL Deletes stresses associated with a pattern of

fatigue locations( ...> Stresses)

FT_CURDEL Deletes fatigue property specifications

( ...> Property Curves)

FT_LOCDEL Deletes a pattern of fatigue locations

( ...> Locations)

FT_SECDEL Deletes a pattern of fatigue sections

( ...> Sections)


ACTFTG Loads the cumulative fatigue usage factors

into the plot buffers( ...> Fatigue)

FTGPLOT Plots the previously loaded fatigue cumulative

factor( ...> Fatigue)

FTGLIST Lists the cumulative fatigue usage factors

( ...> LIST > Events)

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5 Detailed Description

of Examples

Introduction

These example are typical fatigue problems solved by the FSTAR module. Adetailed description of the steps required to set up and solve the problems are given.

How FSTAR Works

Typical input sequence prior to running the FATIGUE module is:

1. Define the required events with the FT_EVENT(Analysis > FATIGUE > EventCycle) command.

2. Define fatigue loadings using the FT_LOAD (Analysis > FATIGUE > Fatigue

Load) command and associate them with the stresses stored by the

COSMOSM structural solution. [Stresses are available for fatigue analysis from

the nonlinear module only for those time steps which are specified by

the NL_PLOT (Analysis > NONLINEAR > Plot Options) command and for

Advanced Dynamics, for those specified by the PD_PLOT (Analysis >

POST_DYNAMIC > PD_OUTPUT > Set Plot Options) command].

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3. You may use the FT_STREAD (Analysis > FATIGUE > Apply Stress)

command to input stresses directly. (This command may also be used to modify

stresses stored as the result of the initial finite element structural solution).

4. Define fatigue locations with the FT_LOC (Analysis > FATIGUE > Fatigue

Location) command. (Not required for all-nodes calculation option).

5. Define fatigue design curves (the S-N curves) with the FT_CURDEF (Analysis

> FATIGUE > Property Curve) command or use the A_FATIGUE (Analysis >

FATIGUE > Analysis Options) command to specify one of the two pre-defined

(optional) S-N curves. Modulus of elasticity must be defined in psi (using the

MPROP (Propsets > Material Property) command), if an optional design curveis considered. The predefined (optional) curves are defined in psi versus cycle.

(If no S-N curve is defined, the fatigue calculation will not produce usage

factors).

6. For shell elements specify the top or bottom face and the layer number with the

A_FATIGUE(Analysis > FATIGUE > Analysis Options) command.

7. Activate a location for fatigue calculation with the ACTSET, LOC, ...(Control >ACTIVATE > Set Entity, Loc) command (for the all-nodes option activate zero

location).

8. Run the FSTAR module by executing the R_FATIGUE(Analysis > FATIGUE >Run Fatigue Analysis) command.

9. Repeat steps (7) and (8) for other locations if the calculations are made on the

basis of one location at a time. If you intend to save the results of the previouscalculations in the output file, activate the append flag of the PRINT_OPS(Analysis > OUTPUT OPTIONS > Set Print Options) command.

You may review all your inputs at any time during the editing session or afterward

by using commands: FT_EVENTLIS(Analysis > FATIGUE > FATIGUE LIST >

Events), FT_LOCLIST(Analysis > FATIGUE > FATIGUE LIST > Locations),

FT_CURLIST (Analysis > FATIGUE > FATIGUE LIST > Property Curves),

FT_STLIST(Analysis > FATIGUE > FATIGUE LIST > Stress Conditions)orreview the result of fatigue calculations after each run with the command: FTGLIST

(Results > LIST > Fatigue Usage Factor)

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You can also check both the inputs and outputs by reviewing the file with the

extension .FTG. You may delete/ modify your inputs at anytime during the

editing session by using the commands: FT_EVENTDEL(Analysis > FATIGUE >

FATIGUE LIST > Events),FT_LOADDEL (Analysis >FATIGUE > FATIGUEDELETION > Loads), FT_LOCDEL (Analysis > FATIGUE > FATIGUE

DELETION > Locations), FT_STDEL (Analysis > FATIGUE > FATIGUE

DELETION > Stresses), FT_CURDEL (Analysis > FATIGUE > FATIGUE

DELETION > Property Curves).

You may have color plots of the results (for all-nodes option) by issuing ACTFTGand FTGPLOT (Results > PLOT > Fatigue) commands.

For the simplified elastic-plastic calculation (for axisymmetric models) add the

following steps:

10. Define a section through the wall thickness with the FT_SEC (Analysis >

FATIGUE > Fatigue Section) command (first define locations at the two ends

of the section by using the FT_LOC (Analysis > FATIGUE > Fatigue Location)

command)

11. Define the Sm-T curve and material parameters M and N with the FT_CURDEF

(Analysis > FATIGUE > Property Curve) command.

12. Activate one of the two end locations and run fatigue (steps g and h).

You may use the FT_SECLIST(Analysis > FATIGUE > FATIGUE LIST >Sections),FT_CURLIST(Analysis > FATIGUE > FATIGUE LIST > Property

Curves),FT_SECDEL(Analysis > FATIGUE > FATIGUE DELETION >Sections), and FT_CURDEL (Analysis > FATIGUE > FATIGUE DELETION >

Property Curves), commands for the listing and deleting of the inputs in steps j

and k.

The following three examples are designed to clarify the concept of cumulative

damage theory and to show how it is implemented in the COSMOSM Fatigue

module.

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p p p


Cylinder Under Axial Cyclic Loading Example

(Without the use of the processing phase modules.)

A cylindrical specimen is under axial cyclic loading with three different

amplitudes. One type of stress cycle (event 1) produces 800 cycles of a stress

difference variation from zero to +50,000 psi and a second type of stress cycle

(event 2) produces 2300 cycles of a stress difference variation from zero to -30,000

psi and a third type produces 1200 cycles of a stress difference variation from zero

to +20,000 psi (event 3). The cumulative effect shall be evaluated as stipulated in

steps 1 to 9 below.

History Loading

The load history is defined according to Figure 5-1. The number of cycles for each

event is specified by n1, n2 and n3.

Figure 5-1. Loading History

Starting The Problem

Move to the working directory and launch GEOSTAR. If you wish, you may use a

title for your problem using the command TITLE(Control > MISCELLANEOUS >Write Title).

t

P

(psi)

50,000

-30,000

20,000

Event 1 Event 2 Event 3

n = 800 n = 2300 n = 1200

x

1 2 3

x

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Specifying the Fatigue Events

Geo Panel: Analysis > FATIGUE > Event Cycle (FT_EVENT)

Reference number for the event > 1

Number of required cycles > 800







You may modify/correct any event specification by repeating command FT_EVENT

(Analysis > FATIGUE > Event Cycle) for that event. To delete an event use

commandFT_EVENTDEL (AnalysIs > FATIGUE > FATIGUE DELETION >Events).

Defining the Loading

Define loadings which

correspond to the extremes of

excursion within each event.

Considering the loading set in

Figure 5-2 we shall define the

following loadings:

Geo Panel: Analysis >

FATIGUE > FatigueLoad (FT_LOAD)

Reference number> 1

Associated event >1

Associated load case > 0

No stresses for

this loading

Geo Panel: Analysis > FATIGUE > Fatigue Load (FT_LOAD)

Reference number> 2

Associated event >1

Associated load case > -1

4

3

2

1 5

6

t

(psi)x

Loading

Figure 5-2. Loading History

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Reference number> 3

Associated event >2



Reference number > 4

Associated event >2




Associated event >3Associated load case > 0



Associated event >3


You may review all your input so far by using list command FT_EVENTLIS(Analysis > FATIGUE > FATIGUE LIST > Events). You may modify/correct any

loading specification by repeating command FT_LOAD(Analysis > FATIGUE >Fatigue Load), or may delete a loading by using command FT_LOADDEL(Analysis > FATIGUE > FATIGUE DELETION > Loads).

Caution: Deleting an event erases all loadings which were associated to that

event, i.e., all parameters which are otherwise listed for that event [using

command FT_EVENTLIS(AnalysIs > FATIGUE > FATIGUE LIST >

Events)] will be deleted.

Defining a Location

Define a location for fatigue calculation.

Geo Panel: Analysis > FATIGUE > Fatigue Location (FT_LOC)

Reference number for fatigue location > 1

Associated node label > 1

For this example Node label is irrelevant. It is useful for problems for

which the stress conditions are available in the database from initial

COSMOSM processing phase calculations.

Stress concentration in X dir > 1.0

Stress concentration in Y dir > 1.0In

dex

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Stress concentration in Z dir > 1.0

Stress concentration factors are considered to be unity in all

directions.

You may review your inputs by using list command FT_LOCLIST (Analysis >

FATIGUE > FATIGUE LIST > Locations) (use default for all prompts). Dashedlines under headings X, Y, Z, and CS are due to the irrelevance of nodal coordinates

in this example.

Specifying the Stress

Specify the stress conditions for the defined loadings (as are specified in Figure

5-2) and location.

Geo Panel: Analysis > FATIGUE > Apply Stress (FT_STREAD)

Location label > 1

Fatigue loading label >2

Item number > Actual stresses

Normal stress in X direction > 50000Normal stress in Y direction > 0.0

Normal stress in Z direction > 0.0

Shear stress TAU_XY > 0.0

Shear stress TAU_XZ > 0.0

Shear stress TAU_YZ > 0.0

(For location 1, loading 2, define SX = 50,000. All other stress components are zero).


Location label > 1

Fatigue loading label >4


Normal stress in X direction > -30000

Normal stress in Y direction > 0.0






Location label > 1

Fatigue loading label >6In

dex

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Normal stress in X direction > -20000

Normal stress in Y direction > 0.0





To review the stress conditions specified by command FT_STREAD (Analysis >

FATIGUE > Apply Stress) use list command FT_STLIST(Analysis > FATIGUE >

FATIGUE LIST > Stress Conditions).Geo Panel: Analysis > FATIGUE > FATIGUE LIST > Stress Conditions

(FT_STLIST)

First location > 1

Last location > 1

Increment > 1

Loading label > All loadings


You may use command FT_STREAD (Analysis > FATIGUE > Apply Stress) to

modify any stored stress condition (stress may have been stored by using command

FT_STREAD (Analysis > FATIGUE > Apply Stress) or pre-stored in the data base

in the processing phase) for any combination of location and loading. Command

FT_STDEL (Analysis > FATIGUE > FATIGUE DELETION > Stresses) may be

used to delete stress conditions stored using command FT_STREAD(Analysis >FATIGUE > Apply Stress).

Specifying the Fatigue Curve

Specify the fatigue design curve (S-N curve) using command FT_CURDEF(Analysis > FATIGUE > Property Curve).

Geo Panel: Analysis > FATIGUE > Property Curve (FT_CURDEF)

Curve/prop item number > 1Default implies that the user is going to specify points on the S-N

curve

Stress ratio R > -1

Min. over Max. Stress. Default -1 implies a fully reversible stress

cyclingInd

ex

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Cycles at point 1 > 2000

Stress at point 1 > 80E3







To input the fifth point, the command must be re-issued.


Curve/prop item number > 1

Stress ratio R > -1



Cycles at point 6 >

To review your inputs for

the fatigue design curve you

may use the commands

ACTXYPRE(Display >XY PLOTS > Activate

Pre-Proc) and XYPLOT(Display> XY PLOTS >

Plot Curves). The above

five points define a fatigue

design curve according to

Figure 5-3.

Geo Panel: Display > XY PLOTS > Activate Pre-Proc (ACTXYPRE)

Graph number > 1

Curve type > Fatigue

Curve type > SN

Curve number > 1

Graph color > 12

Graph line style > Solid

Graph symbol style > Circle

Geo Panel: Display > XY PLOTS > Plot Curves (XYPLOT)

Cycle

20

60

40

103

104

105

106

107

108

x

xx

0

80

x

xalt

(ksi)

Figure 5-3. Defined Fatigue DesignCurve (S-N Curve)

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Running the Fatigue Analysis

Through steps 1 to 7, all necessary data are stored in the database. In order to make

a fatigue calculation at a location, first we have to activate that location.

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Geo Panel: Control > ACTIVATE > Set Entity (ACTSET)

Set label > Loc

Location for fatigue calculation > 1

(This is necessary because the program is designed in such a way that the user may

define more than one location).

To run the fatigue problem, use command R_FATIGUE(Analysis > FATIGUE >Run Fatigue Analysis).

When the analysis is complete, the program will return to the GEOSTAR menu.

The resulting outputs are stored in the output file with extension .FTG. The resultsmay be reviewed either with the editor, using the command EDIT(File > Edit...), orusing the list command FTGLIST(Results > LIST > Fatigue Usage Factor).

Geo Panel: Results > LIST > Fatigue Usage Factor (FTGLIST)

Location label > 1

Lists the latest results for location 1 (see Figure 5-4.):

Table Table 5-1. List of Results

Interpretation of the Results

A cumulative fatigue usage factor of 0.074 means that 7.4% of the life of the

component (structure) is used up by this combination of events. It is interesting to

note that even though the number of the loading combinations is 9 for this example,

there are only three sets which produce partial factors. There are basically two

reasons for this. First, loadings with identical stress conditions produce zeroalternating stress intensities. Secondly, updating the alternating stress intensity list

every time one of the partial factors is evaluated (Analysis Procedure For Cyclic

Loading section), results in the elimination of some of the remaining sets in the list

(e.g., the combination of loadings 2 and 6 makes no contribution).

Loading

(EVN)

Loading

(EVN)

Cycles Alternate

Stress

Partial

FactorUsed Allowed

2

4

3

1

2

2

4

6

4

2

3

2

800

1200

300

0.1240E+05

0.1245E+06

0.1000E+09

4000

25000

15000

0.64510E-01

0.96357E-02

0.30000E-05

Cumulative Fatigue Usage Factor = 0.741483E-01 Total Solution Time = 3 seconds

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Table 5-2. Results [Reference 1]

For more complicated problems, often an initial finite element structural analysis is

necessary in order to find the stress conditions. The following example illustrates

the use of COSMOSM. Modules such as STAR, NSTAR, or HSTAR, to evaluate

the stress conditions needed for fatigue analysis.

Fatigue Caused by Pressure Loading Example

(This example is based on using the processing phase module, STAR, in the Basic

System.)

A circular nozzle is under a varying internal pressure. Figure 5-4 shows the model

and the internal pressure cycle. In addition to a normal pressure build up which isexpected to occur 10,000 times during the service life, the component is expected to

experience an abnormal condition which occurs only 2000 times with the

characteristics illustrated in Figure 5-4.

Figure 5-4. Geometry and Load Cycle

Theoretical COSMOSM

Fatigue usage factor at

location 17.4% 7.4%

x

x

x

x x

Loadingx

3

2

1 4

0.02 0.22

Event 1

(10,000 cyc )

Event 2

(2000 cyc)

0.1 0.20.12

t

P (psi)

20,000

10,000P

2"

4"

2"

2"3"

5"

2"

1"

1"

CL

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To evaluate the usage factor (percentage of life used up) at different locations, first

the stress conditions at extreme points within a cycle will be evaluated. Therefore,

we need only to evaluate the stress field for the two stress levels 10,000 and 20,000

psi.

Given

E = 30 x 106 psi

= 0.3

The processing phase consists of the following three steps.

Structural Modeling and Stress Analysis

Type the following cryptic input in the command window.

GEO> VIEW,0,0,1,0,

GEO> PLANE,Z,0,1,

GEO> GRIDON,,,1,1,10,10,,

GEO> SCALE,0,

GEO> PT,1,8,0,0,GEO> PT,2,6,0,0,

GEO> PT,3,3,0,0,

GEO> PT,4,3,3,0,

GEO> PT,5,3,6,0,

GEO> PT,6,3,8,0,

GEO> PT,7,3,10,0,

GEO> PT,8,4,10,0,

GEO> PT,9,4,8,0,

GEO> PT,10,5,6,0,

GEO> PT,11,5,3,0,GEO> PT,12,6,2,0,

GEO> PT,13,8,2,0,

GEO> PT,14,6,3,0,

GEO> CRLINE,1,1,2,

GEO> CRLINE,2,2,3,

GEO> CRLINE,3,3,4,

GEO> CRLINE,4,4,5,

GEO> CRLINE,5,5,6,

GEO> CRLINE,6,6,7,

GEO> CRLINE,7,7,8,GEO> CRLINE,8,8,9,

GEO> CRLINE,9,9,10,

GEO> CRLINE,10,10,11,

GEO> CRPCIRCLE,11,14,12,1,-90,1,


GEO > CRLINE,13,13,1

GEO> CRBRK,11,11,1,2,0,

GEO> SF2CR,1,1,12,0,

GEO> ACTMARK,SF,

GEO> SF2CR,2,2,11,0,GEO> SF2CR,3,3,14,0,

GEO> SF2CR,4,4,10,0,


GEO> SF2CR,5,5,9,0,

GEO> SF2CR,6,6,8,0,

GEO> EGROUP,1,PLANE2D,0,1,1,0,0,0,0,0,

GEO> MPROP,1,EX,3.D7,

GEO> MPROP,1,NUXY,.3,

GEO> MPROP,1,DENS,.0003,

GEO> M_SF,1,3,1,4,4,4,1,1,GEO> M_SF,5,6,1,4,4,4,1,1,

GEO> M_SF,4,4,1,4,6,4,1,1,

GEO> NMERGE,1,160,1,0.0001,0,1,0,

GEO> DCR,1,UY,0,2,1,,

GEO> ACTSET,LC,1

GEO> PEL,33,10000.0,1,36,1,

GEO> PEL,81,10000.0,1,86,1,

GEO> PEL,49,10000.0,1,52,1,

GEO> PEL,65,10000.0,1,68,1,

GEO> ACTSET,LC,2GEO> PEL,33,20000.0,1,36,1,

GEO> PEL,81,20000.0,1,86,1,

GEO> PEL,49,20000.0,1,52,1,

GEO> PEL,65,20000.0,1,68,1,

GEO> A_STATIC,N,0,0,1e-06,1e+10,0,0,0,

GEO> R_STATIC

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At this stage you have the option of analyzing the fatigue problem at one location

(node) at a time or at all nodes at once. The first option provides you with more

details of the calculation. However, the all nodes option relieves you from the

painstaking task of studying each location one by one and provides you with a

graphical display of the fatigue life. These options are outlined in the following:

Fatigue Analysis (Normal Procedure,

i.e., One Location at a Time)

1. It is recommended to first identify the areas of stress concentration. This may bedone, quantitatively, by studying the stress distribution inside the structure using

commands in the Results menu. From the displayed contours it is apparent that

the tip of the nozzle, at node 105, experiences the highest level of stress.









Associated event > 1


Zero stresses at all locations





Scale factor > 1





Scale factor > 1In

dex

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Geo Panel: Analysis > FATIGUE > Fatigue Location (FT_LOC)

Reference number for fatigue location > 1

Associated node label > 105

Stress concentration in X dir > 1.0

Stress concentration in Y dir > 1.0

Stress concentration in Z dir > 1.0



Stress ratio R > -1











Stress ratio R > -1



Geo Panel: Analysis > OUTPUT OPTIONS > Set Print Options

(PRINT_OPS)

Displacement print flag >Yes

...

Output flag > Append

Geo Panel:Control > ACTIVATE > Set Entity (ACTSET)

Set label > Loc

Location for fatigue calculation > 1

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Geo Panel: Analysis FATIGUE R F ti A l i (R FATIGUE)

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Geo Panel: Analysis > FATIGUE > Run Fatigue Analysis (R_FATIGUE)

Perform fatigue calculations

Geo Panel: Results > LIST > Fatigue Usage Factor (FTGLIST)

List fatigue results for location 1

2. You may review the output file by using the EDIT (File > Edit...) command.

Geo Panel: File > Edit a File (EDIT)

Review all results

3. You may continue by defining new locations.

There are other alternative ways of solving a linear static problem such as the

one considered above which may significantly reduce the computational time.

One alternative to the above is running the static problem for only one load

case (e.g., load case 1) and then associating fatigue loadings 2 and 3 with that

load case with a scale factor of 1 and 2.0, respectively.

4. This results in the modification of structural modeling part of the problem as

follows (type the cryptic input in the command window):

GEO>ACTSET,LC,1

GEO>PEL,33,10000.0,1,36,1,

GEO>PEL,81,10000.0,1,86,1,

GEO>PEL,49,10000.0,1,52,1,

GEO>PEL,65,10000.0,1,68,1,

GEO>A_STATIC,N,0,0,1e-06,1e+10,0,0,0,

GEO>R_STATIC

and of Fatigue Analysis part of the problem:.

.

.

GEO>FT_LOAD,2,1,1,1

GEO>FT_LOAD,3,2,1,2

.

.

.

5. The above modification is applicable only to the linearly elastic materials for

which the stress (response) is linearly proportional to the applied load.In

dex

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Fatigue Analysis (All-Nodes Option)

Type the following cryptic input in the command window.

GEO>FT_EVENT,1,10000

GEO>FT_EVENT,2,2000

GEO>FT_LOAD,1,1,0

GEO>FT_LOAD,2,1,1,1

GEO>FT_LOAD,3,2,2,1

GEO>FT_LOAD,4,2,0

GEO>FT_CURDEF,1,,2000,80E3,4000,60E3,10000,30E3,100000,5E3

GEO>FT_CURDEF,1,1000000,1E3

GEO> ACTSET,LOC,0

Activate all-nodes option

GEO> R_FATIGUE

Perform fatigue calculations

GEO> FTGLIST,0

List the fatigue results at all nodes

1. You may review the output file by using the EDIT (File > Edit...) command, or

issuing the SYSTEM command and using your favorite editor.

Geo Panel: File > Edit... (EDIT)

Review results

2. At this step, you may utilize the graphic capability of GEOSTAR to displaycolor plots of the fatigue life and identify fatigue critical regions. Fatigue curves

can be plotted using the XYPLOT (Display > XY PLOTS > Plot Curves)

command.

Geo Panel: Results > PLOT > Fatigue (ACTFTG)

Fatigue Caused by Thermal Loading Example

(Using the processing phase modules: HSTAR and NSTAR.)

Index

Index


Assume that the nozzle of Figure 5-5 Nozzle Geometry and a Section

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Assume that the nozzle of

example 2 is exposed to a fluid

heat up condition which is

expected to occur 5000 times

during its service life. Onecomplete cycle of this heat up

condition is shown in Figure 5-6.

Given

E = 30*106 psi

= 0.3 = 8.0E-6 in/in/F

Kx = 0.1 BTU/in hr F

h = 1.0 BTU/in2 hr F

(outside surface)

h = 5.0 BTU/in2 hr F

(inside surface)

Tref = 60 FTmax = 300 F

Creating the Model

Geometry

Structural modeling (similar to thethermal loading example); type the

following cryptic input in the

command window.

GEO> TITLE, FT3A: FATIGUE OF A NOZZEL WITH CYCLIC INTERNAL FLUIDTEMP.

GEO> PLANE,Z,0,1,

GEO> VIEW,0,0,1,0,

GEO> CRSPOLY,1,8,0,0,

L,8,2,0,

L,6,2,0,

A,5,3,0,

x

2"

4"

2"

2"3"

5"

2"

1"

1"

CL

2 3

Figure 5-5. Nozzle Geometry and a Section

Time HR

300

150

0.02 0.1 0.20.12

60

0

Temperatur

e

Figure 5-6. Temperature Variation Cycle

Index

Index


L,5,6,0,

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, , , ,

L,4,8,0,

L,4,10,0,

L,3,10,0

L,3,8,0,L,3,6,0,

L,3,3,0,

L,3,0,0,

L,6,0,0,

L,8,0,0,

GEO> SCALE,0,

GEO> CRBRK,3,3,1,2,0,GEO> SF2CR,1,13,2,0,

GEO> SF2CR,2,12,3,0,

GEO> SF2CR,3,11,14,0,

GEO> SF2CR,4,10,4,0,

GEO> SF2CR,5,9,5,0,

GEO> SF2CR,6,8,6,0,

GEO> EGROUP,1,PLANE2D,0,1,1,0,0,0,0,GEO> M_SF,1,3,1,4,4,4,1,1,

GEO> M_SF,5,6,1,4,4,4,1,1,

GEO> M_SF,4,4,1,4,6,4,1,1,

GEO> DCR,12,UY,0,13,1,,

Specifying and Running the Thermal Analysis

Thermal analysis. Using the HSTAR Module, the nodal temperatures of the

structure will be evaluated at time intervals of 0.01 hours. Type the following

cryptic input in the command window.

GEO> CURDEF,TIME,1,1,0.000000E+00,60.0000,0.200000E-01,300.000,

GEO> CURDEF,TIME,1,3,0.100000,300.000,0.120000,60.0000,

GEO> CURDEF,TIME,1,5,0.200000,60.0000,

GEO> MPROP,1,ALPX,0.800000E-05,GEO> MPROP,1,DENS,0.300000E-03,

GEO> MPROP,1,C,40.0000,

GEO> MPROP,1,KX,0.100000,

GEO> ACTSET,TC,0,

GEO> ACTSET,TP,0,Index

Index


GEO> CEL,4,1.,60.,2,16,4,0,

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GEO> CEL,13,1.,60.,3,16,1,0,

GEO> CEL,29,1.,60.,3,32,1,0,

GEO> CEL,45,1.,60.,3,48,1,0,

GEO> CEL,99,1.,60.,3,104,1,0,

GEO> CEL,61,1.,60.,3,64,1,0,

GEO> CEL,77,1.,60.,3,80,1,0,

GEO> CEL,65,5.,1.,1,68,1,1,

GEO> CEL,49,5.,1.,1,52,1,1,

GEO> CEL,81,5.,1.,1,86,1,1,

GEO> CEL,33,5.,1.,1,36,1,1,

GEO> NMERGE,1,160,1,0.0001,0,1,0,

GEO> CRMERGE,1,20,1,0.0001,1,1,0,

GEO> TIMES,0.000000E+00,0.200000,0.100000E-01,

GEO> TUNIF,60.0000,

GEO> TOFFSET,273.000,

GEO> INITIAL,TEMP,1,159,1,60.,

GEO> A_THERMAL,T,0.100000E-02,5,1,20,0,

GEO> R_THERMAL

Specifying and Running the Stress Analysis

Stress analysis. Type the following cryptic input in the command window.

GEO> MPROP,1,EX,30E6,NUXY,.3,

GEO> TREF,60,

GEO> A_NONLIN,S,1,1,20,0.001,0,T,0,0,

GEO> PRINT_OPT,1,0,0,1,0,1,0,0,0,1,

GEO> NL_PLOT,2,20,2,0,

GEO> R_NONLINEAR

Stresses are available for postprocessing (fatigue analysis or stress plot) from

the nonlinear module only for those time steps which are specified by com-

mandNL_PLOT (Analysis > NONLINEAR > Plot Options).

Specifying and Running the Fatigue Analysis

Fatigue analysis. Type the following cryptic input in the command window.In

dex

Index


GEO> FT_EVENT,1,5000,

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GEO> FT_LOAD,1,1,2,1,









GEO> FT_LOAD,10,1,20,1,

GEO> FT_LOC,1,130,1,1,1,

GEO> FT_CURDEF,1,-1,2000,80E3,4000,40E3,10000,10E3,100000,3.E3,

GEO> ACTSET,LOC,1,

GEO> R_FATIGUE

If you wish to perform fatigue analysis based on a simplified elastic-plastic

formulation then continue with:

Running the Analysis Based on the Elastic-Plastic Formulation

Analysis based on elastic-plastic formulation. Type the following cryptic input in

the command window.

GEO> FT_LOC,2,51,1,1,1,

GEO> FT_LOC,3,71,1,1,1,

GEO> FT_SEC,1,2,3,25,-1,1,GEO> ACTSET,LOC,2,

GEO> FT_CURDEF,30,2.,.8,

GEO> FT_CURDEF,20,60,.5E5,200,.15E5,400,.8E4,

GEO> R_FATIGUE

You may review all the results by using either EDIT (File > Edit...) or FTGLIST

(Results > LIST > Fatigue Usage Factor) commands.

Index

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Index

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6 A Brief Theoretical Backgroundfor Simplified Elastic-Plastic

Formulation

Introduction

This chapter contains additional details on subjects mentioned in earlier chapters.

Section Orientation in Junctions

For sections in the nozzle to shell junctions,

rational planes of bending should be

approximated such that they are perpendicularto the mid-plane and have the same angle

between the section and the surface on both

sides. This is done by forming an isosceles

triangle which has the section as its base and

the mid-plane as its altitude as shown in

Figure 6-1. One side of the triangle is tangent

to the fillet at the point of interest.

LC

Figure 6-1. Defining a Sectionin an Irregular Area

Index

Index

Chapter 6 A Brief Theoretical Background for Simplified Elastic-Plastic Formulation

Stress Linearization

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For plane strain, plane stress and

axisymmetric structures, stresslinearization is performed along a

section path. A section is defined by

nodes N1 and Nn according to Figure

6-2. The program interpolates n-2

equally spaced (integration) points along

the path between nodes N1 and Nn. For

each point, structural elements are

searched to identify an element whichcontains that point. Once the element is

identified, the stresses at that point are

interpolated linearly from the element

corner nodes. These stresses are denoted

as actual stresses.

The equivalent linearized stresses are found along a section by evaluating the

membrane and bending stresses according to one of the following two methods.

Cartesian Formulation (Approximation)

This formulation is applicable to plane stress or plane strain problems or to the

axisymmetric structures where the radial dependency could be ignored. For

axisymmetric structures, radial dependency is due to the fact that for axisymmetric

structures, there is more material at a greater radius than at a smaller one. Thisoption is implemented into the calculation if the input quantity curvature_radius

in command FT_SEC (Analysis > FATIGUE > Fatigue Section) is set to zero (see

the Axisymmetric Formulation section for the definition of curvature_radius).

The membrane components of the stresses are evaluated by integrating along the

section path according to:

N2

N1

LC

N3

Nn-1

Nn

N4

Figure 6-2. A Section with n EquallySpaced Integration PointsAlong its Path

Index

Index


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(6-1)

where, superscripts m and a

correspond to membrane and actual

stresses, respectively, l is the lengthof the section, x and y are the local

coordinates along and perpendicular

to the section path (Figure 6-3) with

an origin at the mid-wall, and X, Y

and Z are the global Cartesian

coordinates. Membrane stresses are

considered to be constant along the

section.

The bending component of stresses

at position x along the section path

(Figure 6-3) is calculated according

to:

(6-2)

where superscript b corresponds to the bending stress.

The linearized stress at any point along the section is the sum of membrane and

bending stresses.

LC

N1Y

X

y

x,N

n

/2/2

Figure 6-3. Definition of the SectionLocal Coordinates

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Index


Axisymmetric Formulation

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In this case the axisymmetric features

of the model is more properly

implemented in the formulation, foraxisymmetric structures, than the

Cartesian case. This option is

considered in fatigue calculation if a

non-zero value is assigned to the

input quantity curvature_radius in

command FT_SEC (Analysis >

FATIGUE > Fatigue Section).

Curvature_radius corresponds to the

radius of curvature of the average

mid-wall centerline in the xy plane

as represented by in Figure 6-4.A large value (or -1) for

curvature_radius corresponds to

straight walls (e.g., cylinder or cone).

In order to find the linearized

stresses, it is desired to obtain applied

forces and moments along the

section. Figure 6-5 represents a free-

body diagram of the section. A right-

handed local coordinate system, x, y,

and z is established on the section

with the origin at the mid-wall (sameas in Figures 6-3 or 6-4). FN and FT

correspond to the inplane normal and

shear forces on the section in y and x

directions and MZ is the bending

moment.

The three inplane forces and moment

on the section over a small sector in the hoop direction are defined as:

L

N1

Y

X

C

yx

x

Nm

Figure 6-4. Curvature-Radiusof the Mid-Wall

MZ

L

Y

X

C

FTMZ

FT

x,

FN

Figure 6-5. Applied Forces and MomentsAlong the Section

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(6-3)

where, ya and xy

a are the actual stresses along the section in local coordinates, X

is the global coordinate (or equivalently the radius) of point being integrated along

the section and xfin local coordinates is the offset of the sector neutral axis from

the center line.

From FN in the above equation, the average normal membrane stress in the y

direction is computed once it is divided by the sector area (Xc is the global

coordinate of the mid-section).

(6-4)

The MZ contribution to the normal bending stress in the y direction is computed by

applying the familiar relationship s = Mz (x-xf)/I with I as moment of inertia of the

sector.

(6-5)

An average membrane shear stress (xy component) is computed by dividing FT

[in Equation (6-3)] by the sector area, assuming that xy bending shear stress has

insignificant contribution (since the shear stress distribution is assumed to be

parabolic and equal to zero at the two free surface ends).

Index

Index


(6 6)

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(6-6)

Average membrane stress in the x direction is computed by averaging the actualstresses along the section according to:

(6-7)

The bending stress in the x direction (thickness direction) is ignored if in command

FT_SEC(Analysis > FATIGUE > Fatigue Section) the corresponding flag is set toone, otherwise it is approximated at the two ends as the difference of the actual and

membrane stresses.

The hoop membrane and bending stresses are calculated by considering a small

sector () in the XY plane. By integrating the total normal force on the sector and

averaging it over the corresponding area, the average membrane stress (in hoop

direction) is computed according to:

(6-8)

For straight walls where , Equation (6-8) reduces to the familiar form.

The hoop bending stress is calculated by evaluating the applied bending moment on

the sector. Once the bending moment is evaluated, the hoop bending stress is found

analogous to that of the y direction bending stress.

(6-9)

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Index


As mentioned before, the linearized stress at any point along the section is the sum

of membrane and bending stresses.

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Simplified Elastic-Plastic Option

For a desired location (e.g., at one end of a section), each alternating stress intensity

(evaluated according to Chapter 2) is increased by a factor Ke (Reference 1). This

factor is a function of the equivalent linearized alternating stress intensity L as

well as the design stress m interpolated from the Sm-T curve. The equivalent

linearized alternating stress intensity is evaluated analogously to the alternatingstress intensity, based on the linear stresses, not actual stresses. Factor Ke is defined

for different ranges ofL as follows:

(6-10)

(6-11)

(6-12)

where, M and N are the elastic-plastic material parameters (input quantities on

FT_CURDEF (Analysis > FATIGUE > Property Curve) command).

References

1. ASME Boiler and Pressure Vessel Code, Edition 1983, Section III, Division 1,

Subsection NB.

2. Kroenke, W. C., Addicott, G. W. and Hinton, B. M., Interpretation of Finite

Element Stresses According to ASME Section III, Paper 75-PVP-63,ASME

Second National Congress on Pressure Vessels and Piping, June 1975.

3. Kroenke, W. C., Classification of Finite Element Stresses According to ASME

Section III Stress Categories, Pressure Vessels and Piping, Analysis and

Computer, ASME, June 1974.

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Index


4. Gordon, J. L., Outcur: An Automated Evaluation of Two-Dimensional Finite

Element Stresses According to ASME Section III Stress Requirements, Paper

76 WA/PVP 16 D b 1976

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76-WA/PVP-16, December 1976.

5. ASME Boiler and Pressure Vessel Code, Edition 1989, Section III, Division 1,Appendices.

Index

Index

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COSMOSM Advanced Modules I-1

Index

A

Analysis Options 4-2, 5-2Apply Stress 4-2, 5-2, 5-7, 5-8ASME Boiler and Pressure

Vessel Code 1-2, 6-7, 6-8

B

Bending Stress 6-3, 6-5, 6-6

D

Design Stress 6-7

Eelastic-plastic formulation 1-2,4-2, 5-21, 6-1

Event Cycle 4-2, 5-1, 5-5, 5-14Events 1-1, 4-2, 4-3, 5-1, 5-2, 5-3,

5-5, 5-6, 5-11

F

Fatigue 1-1, 1-2, 4-2, 4-3, 5-1, 5-2,

5-3, 5-5, 5-6, 5-7, 5-8, 5-9, 5-10, 5-11, 5-12, 5-14, 5-15, 5-16, 5-17, 5-18, 5-20, 5-21, 6-2, 6-4, 6-6, 6-7

Fatigue Analysis 1-1, 4-2, 5-1, 5-2, 5-10, 5-11, 5-12, 5-14, 5-16, 5-17, 5-20, 5-21

fatigue curve 5-8

fatigue design curve 5-8, 5-9fatigue life 1-2, 5-14, 5-17fatigue loading 4-2, 5-7Fatigue Location 4-2, 5-2, 5-3, 5-

6, 5-15

fatigue properties 4-2Fatigue Section 4-2, 5-3, 6-2, 6-4,

6-6

fatigue usage factor 1-1, 1-2, 5-2,5-11, 5-12, 5-16, 5-21

L

Linearized Stress 6-3, 6-7load history 5-4Loads 1-1, 4-3, 5-3, 5-6Locations 4-2, 4-3, 5-2, 5-3, 5-7,

5-13, 5-14, 5-16

M

Material Property 5-2Membrane Stress 6-5, 6-6

P

Plot Curves 5-9, 5-17Plot Options 5-1, 5-20Print Options 5-2, 5-15Property Curves 4-2, 4-3, 5-2, 5-3

S

Sections 4-2, 4-3, 5-3, 6-1Set Entity 5-2, 5-11, 5-15Sm-T Curve 5-3, 6-7S-N curve 5-2, 5-8, 5-9Stress Conditions 1-1, 4-2, 5-2, 5-

6, 5-7, 5-8, 5-11, 5-12, 5-13

Stresses 1-1, 4-3, 5-1, 5-2, 5-3, 5-5, 5-7, 5-8, 5-14, 5-20, 6-2, 6-3, 6-4, 6-5, 6-6, 6-7, 6-8

U

usage factor1-1, 1-2, 5-2, 5-11, 5-

12, 5-13, 5-16, 5-21

Index

Index

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