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Evaluation of fatigue crack growth software for use on cracks in Evaluation of fatigue crack growth software for use on cracks in

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Joshua Marc Williams

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EVALUATION OF FATIGUE CRACK GROWTH SOFTWARE

FOR USE ON CRACKS IN COMPLEX

GEOMETRIES

By


A Thesis Submitted to the Faculty of Mississippi State University

in Partial Fulfillment of the Requirements for the Degree of Master of Science

in Aerospace Engineering in the Department of Aerospace Engineering

Mississippi State, Mississippi

May 2009

EVALUATION OF FATIGUE CRACK GROWTH SOFTWARE

FOR USE ON CRACKS IN COMPLEX

GEOMETRIES

By


Approved: __________________________________ __________________________________ Gregory D. Olsen Anthony J. Vizzini Assistant Professor Bill and Carolyn Cobb Chair and of Aerospace Engineering Department Head of Aerospace (Director of Thesis) Engineering (Committee Member) __________________________________ __________________________________ Steven R. Daniewicz Pasquale Cinnella Professor of Mechanical Engineering Professor of Aerospace Engineering (Committee Member) (Graduate Coordinator)

__________________________________ Sara A. Rajala Dean of the Bagley College of Engineering

Name: Joshua Marc Williams

Date of Degree: May 2, 2009

Institution: Mississippi State University

Major Field: Aerospace Engineering

Major Professor: Dr. Gregory Olsen

Title of Study: EVALUATION OF FATIGUE CRACK GROWTH SOFTWARE FOR USE ON CRACKS IN COMPLEX GEOMETRIES

Pages in Study: 81

Candidate for Degree of Master of Science

Fatigue-crack growth data for the lower arm of the Apache helicopter’s scissor

assembly is presented from an Army alternate source qualification test. The lower arm

model is imported to finite element analysis software to obtain the stress state at a crack

location. The stress state and geometry are used in seven fatigue-crack growth cases in

NASGRO and AFGROW, with an additional four cases discussed briefly. The results

from the fatigue-crack growth routines are compared to the fatigue-crack growth data

from the Army’s test. One case reproduces the crack growth data prior to breakthrough.

Some cases are shown to be more applicable to this configuration than others are. The

process of performing fatigue life estimates is discussed. Suggestions are made on the

viability of this approach and possible future avenues for development.

ii

DEDICATION

This work is dedicated to my family: my wife, Brandi, for her constant love and

support; my son, Aiden, for his smiles and coos; for my parents, for their unending

encouragement; and my siblings, for sharing this last bit of college with me.

iii

ACKNOWLEDGEMENTS

I would like to thank Dr. Greg Olsen for taking me under his wing and helping me

through this process. I want to thank Dr. Robert Vaughan and his team from the U.S.

Army’s Aviation Engineering Directorate, for providing the project and its relevant data.

I also want to thank Dr. James Newman Jr. and Dr. Steven Daniewicz for their lessons on

fracture mechanics and fatigue, which helped me understand the mechanics of the project

I was facing. Finally, I’d like to express gratitude for the Department of Aerospace

Engineering, the Bagley College of Engineering, and Mississippi State University for

providing me with a curriculum that has left me ready to enter the workforce.

TABLE OF CONTENTS

DEDICATION.................................................................................................................... ii

ACKNOWLEDGEMENTS............................................................................................... iii

LIST OF TABLES...............................................................................................................v

LIST OF FIGURES ........................................................................................................... vi

CHAPTER

I. INTRODUCTION .......................................................................................1

II. STRESS ANALYSIS.................................................................................11

III. FATIGUE ANALYSIS..............................................................................21

Analysis Using NASGRO..........................................................................24 Analysis Using AFGROW.........................................................................35

IV. CONCLUDING REMARKS.....................................................................45

REFERENCES ..................................................................................................................51

APPENDIX

A FILE FORMATS FOR STRESS DATA. ..................................................53

B STRESS DATA FILES..............................................................................57

iv

v

LIST OF TABLES

TABLE

1 NASGRO Case CC05 – Corner Crack in Plate, Top Portion. .........................29

2 NASGRO Case CC05 – Corner Crack in Plate, Entire Height........................29

3 NASGRO Case CC02 – Corner Crack at Hole in Plate. .................................32

4 NASGRO Case CC02, corrected .....................................................................33

5 NASGRO Case CC06 – Corner Crack at Hole in Plate, Weight Function......34

6 AFGROW Single Corner Crack at Hole..........................................................37

7 AFGROW Single Corner Crack at Hole, corrected.........................................38

8 AFGROW Internal Axial Crack in Thick Pipe, Weight Function...................40

9 AFGROW Axial Through Crack in Thick Pipe, Weight Function .................41

10 AFGROW Advanced Model............................................................................43

11 Case Summary .................................................................................................44

vi

LIST OF FIGURES

FIGURE

1 Apache Scissor Assembly2. ...............................................................................3

2 Scissor Assembly Test Stand5............................................................................5

3 Typical Crack Location, Sets F and G5..............................................................6

4 Set G Crack5.......................................................................................................7

5 Set F Cracks5......................................................................................................7

6 Measured Crack Lengths vs. Cycles..................................................................8

7 Fracture Initiation Site, Set F Crack 14 ..............................................................9

8 Boundary Conditions and Load .......................................................................14

9 Meshed Geometry............................................................................................14

10 Local Coordinate System and Crack Plane......................................................16

11 Path Configuration, Top Portion......................................................................17

12 Path Configuration, Entire Height ...................................................................17

13 Stress Normal to the Crack (Z direction).........................................................18

14 Tensile Stress Normal to the Crack (Z direction) ............................................19

15 Shear Stress in Crack Plane (XY direction).....................................................20

16 S-N Curve for Scissor Assembly4....................................................................22

17 NASGRO Case SC04 - Axial Surface Crack in Pipe16....................................25

vii

18 NASGRO Case TC07 – Axial Through Crack in Pipe16 .................................26

19 NASGRO Case CC01 – Corner Crack in Plate16 ............................................27

20 NASGRO Case CC05 – Corner Crack in Plate, Weight Function16 ...............28

21 NASGRO Case CC05 – Layout.......................................................................29

22 NASGRO Case CC02 – Corner Crack at Hole in Plate16................................31

23 NASGRO Case CC02, corrected – Crack Growth ..........................................32

24 NASGRO Case CC06 – Corner Crack at Hole in Plate, Weight Function16...34

25 NASGRO Case BE03 – Two Cracks at Hole in Plate, BEM Solution16 .........35

26 AFGROW Single Corner Crack at Hole..........................................................37

27 AFGROW Single Corner Crack at Hole, corrected - Crack Growth...............38

28 AFGROW Internal Axial Crack in Thick Pipe, Weight Function...................39

29 AFGROW Axial Through Crack in Thick Pipe, Weight Function .................40

30 AFGROW Axial Through Crack in Thick Pipe – Crack Growth....................42

1

CHAPTER I

INTRODUCTION

Fatigue-related failures are a threat to many systems, especially in aircraft, where

the structures encounter a wide spectrum of loadings. Ignoring the effects of fatigue can

lead to disasters, such as those experienced by the de Havilland Comets in the 1950s.1 As

such, fatigue must be considered in the design and analysis of aircraft components, and

requirements for fatigue life are often part of a component specification.

Helicopters have many components that are especially susceptible to fatigue-

related failures due to the nature of their loading. As a rotor blade spins, it encounters a

cyclic loading effect. As the blade moves into the wind, it experiences a higher relative

velocity, and therefore has more lift and drag acting on it. As the blade makes a half

rotation, it moves with the wind and has a lower relative velocity, decreasing the lift and

drag. As the blade continues another half rotation, it experiences the increased loading

again, causing a cyclic loading that is linked to the rotation of the rotor. These varying

cyclic loads are then transferred throughout the rotor assembly, making components of it

especially prone to fatigue-related failure. Furthermore, due to the presence of multiple

blades, some components will encounter multiple cycles per rev, leading some

components to experience fatigue lives on the order of 108 cycles.

One component of the AH-64 Apache helicopter that experiences such regular

cyclic loading is the scissor assembly, shown in

2

Figure 1. The scissor assembly is a critical component of the main rotor assembly

of the Apache. Its function is to transfer the torque from the rotating swashplate to the

main rotor hub. Its scissoring allows the swashplate to move, controlling the collective

and cyclic pitch of the rotor blades. It is composed of an upper drive arm, a lower drive

arm, and a bolt assembly. The drive arms are manufactured from forged 7075-T73

aluminum, and the bolt assembly is made of a heat-treated steel alloy. There are two

scissors in the main rotor assembly: the load bearing scissor assembly and the secondary

scissor assembly. They are identical except in placement; the secondary assembly is a

redundant component intended to take over should the load bearing scissor fail.2

3

Figure 1 Apache Scissor Assembly2

During a rotation of the main rotor, the scissor assembly will see a cyclic load.

For component testing purposes, the Army’s Aviation Engineering Directorate (AED)

has estimated the loading to be a steady +849.6 N in the direction of rotation with an

oscillatory component of +/- 3576.4 N at 3-5 Hz for the purpose of fatigue life testing.3

During actual flight, the scissors will see much more complicated loading conditions as

the helicopter maneuvers.

The Army has to verify the integrity of the parts they receive from a commercial

vendor. When a new vendor begins to manufacture the parts, the Army conducts an

Alternate Source Qualification Test (ASQT) to ensure that the parts meet the Army’s

Lower Arm

Upper Arm

4

requirements. A new vendor was beginning production of the Apache scissor assemblies,

so an ASQT was performed on the Scissor Assembly.

The ASQT for the scissor assembly consisted of two phases. The first phase used

assemblies from a qualified vendor to verify the experimental setup. The second phase

tested the assemblies from the new vendor. The test phase consisted of a visual

inspection of the assembly, a destructive fatigue test, followed by material testing. Four

specimens from the new vendor were tested. Upon receipt, they were inspected,

disassembled and inspected, and reassembled. They were then connected to a test stand

(Figure 2) that applied the test loading at 3-5 Hz. The load was applied from a calibrated

load cell on a simulated swash plate to the spherical bearing of the lower arm. The upper

arm was mounted on a simulated lower shoe of the rotor hub. The part was considered

failed once a crack was detected, and the test was run until the part fractured or the runout

of 3 million cycles was reached.

5

Figure 2 Scissor Assembly Test Stand5

6

Several cracks appeared throughout the course of the testing, but two cracks in

particular were noticed due to their similarity and slow crack growth. These two load

sets, F and G, had cracks initiate and propagate in the same area, shown with dye in

Figure 3, at the corner of the lower arm’s bushing hole and the main lug bushing. Set G

generated one crack, with its tip shown in marker in Figure 4, while set F generated two

cracks each on opposite sides of the bushing hole as shown in Figure 5. In both sets, the

cracks propagated outward from the bushing hole and were observed after breaking

through to the outside surface.4

Figure 3 Typical Crack Location, Sets F and G5

7

Figure 4 Set G Crack5

Figure 5 Set F Cracks5

8

Once detected, the crack lengths for both of these sets were approximated by

visual comparison to a ruler. Because the arm was still on the fixture so the test could

continue, these crack lengths were measured from the exterior corner down, rather than

from the initiation site. Measurements were taken at roughly every 1200 cycles for set F

and every 4000 cycles for set G. Several estimates to the crack length were made and

recorded, along with the cycle number when the observation was made. Plotting this

data, seen in Figure 6, shows linear crack growth for about the first 18mm. Beyond that,

the crack spreads into the U-cut and begins to turn and drift down the length of the arm.

Estimates of crack length were made for set G after this point, but not for set F.5

0

20

40

60

80

100

0 10000 20000 30000 40000 50000 60000

Cycles after Crack Detection

Cra

ck L

engt

h (m

m)

Set GSet F

Figure 6 Measured Crack Lengths vs. Cycles

9

After the fatigue test was completed, the assembly was removed from the fixture

and sent to the US Army Aviation and Missile Research Development and Engineering

Center (AMRDEC) for material testing and analysis. AMRDEC found that both set F

and set G failed due to fatigue, and that in both cases the fatigue crack was initiated by

fretting at the chamfer edge of the bushing hole. Figure 7 is a photograph of the surface

of crack 1 of set F, with the arrow pointing to the corner of the bushing hole where the

crack initiated. AMRDEC’s photo clearly shows fatigue beach marks, giving an

approximation of the shape of the crack at various points in its life.4

Figure 7 Fracture Initiation Site, Set F Crack 14

The linear growth and steady rate for the crack-growth data for sets F and G led

AED to consider the possibility of using this data to test the usefulness of fatigue crack-

10

growth software. For this purpose, AED provided 3D models of the lower arm of the

scissor assembly as well as the results from their testing.

The objective for this research is to consider possible approaches to modeling this

fatigue crack growth and to report on the findings. To this end, the part must be analyzed

to obtain the stress state at the crack location. Then the stress state and geometry must be

considered in implementing fatigue-crack-growth routines to reproduce the crack growth

observed on the test stand.

The stress analysis performed on the lower arm of the scissor assembly is detailed

in Chapter II. The stress state is determined in the entire lower arm as well as in the

crack plane. The information on the stresses in the lower arm is used to run the fatigue

analysis as discussed in Chapter III. Several case geometries are used to analyze the

crack growth in the lower arm. Chapter IV provides a brief recap of this approach to

estimating fatigue life. The results of the fatigue analysis performed are discussed, as

well as other potential approaches to the problem.

11

CHAPTER II

STRESS ANALYSIS

To provide meaningful inputs to any fatigue software, the stress acting on the

crack must be known. The stress state where the crack formed during testing is not

among the information that AED included in their report, as it had no direct impact on the

ASQT. However, with the 3D model, material properties, loads, boundary conditions,

and a finite elements software package, the stress state can be determined.

The finite element method involves dividing a volume into smaller parts called

elements. The elements are composed of nodes, which are the points in the model where

the problem is solved. These nodes are linked together based on the geometry of the

model. The nodes also have known conditions applied to them, such as boundary

conditions, loads, and material properties. The information is compiled into a system of

equations in matrix form. The matrix is then solved, providing some primary result,

which can be used to derive other results. In the case of structural problems, the solution

usually yields the displacements of each node, which can be used to determine the stress

from the stress-strain relationship.

One such finite elements software package is ANSYS. ANSYS was first released

in 1971 and has undergone many revisions in its lifetime.6 For this project, ANSYS

Academic Teaching Advanced, Release 11.0 is used. This version is limited to academic

use and has a 256K node limit, but it is sufficient for the static analysis required.

12

The model of the lower arm of the scissor assembly was provided by AED.

Because no ANSYS model had been developed, several formats were provided so that

one would be more likely to import successfully. Pro/E, IGES, parasolid and Solidworks

models were all sent, but only the parasolid model imported. While the parasolid model

would not render except as a collection of lines, it did produce an intact volume inside

ANSYS, which none of the other models were capable of. Hence, the parasolid model

was used as the base model in ANSYS.

Due to the complex nature of the lower arm’s geometry, the mesh elements

should be able to map freely to the volume, rather than having a structured mesh.

Additionally, the elements must be suitable for structural uses. ANSYS’s SOLID187 and

SOLID92 are both acceptable based on those requirements; for this project, SOLID187 is

used. It is a 10-node tetrahedral element that is useful for modeling irregular meshes,

such as CAD models, and meets the requirements for an accurate stress analysis of the

lower arm.7

The lower arm is manufactured from forged Aluminum 7075-T73. Al 7075-T73

has a Young’s Modulus, E, of 72.0 GPa and a Poisson’s Ratio, �, of 0.33.8 Because the

load applied should produce stresses well below the yield stress of 435 MPa, it is

acceptable to model the material with a linear-elastic isotropic behavior. While

SOLID187 is able to perform more advanced material models, it is unnecessary for this

simulation. Because the lower arm is modeled as having linear-elastic isotropic behavior,

no additional material properties are required by ANSYS for this simulation.

The boundary conditions for this simulation are relatively simple, consisting of

one load and one boundary. The most critical stress state in the area of interest occurs

13

when the maximum tensile stress occurs in the crack region. That region is under tension

during positive loading. The actual load is applied from the test fixture load cell, which

applies 4426 N via a linkage to a spherical bearing in the lower arm. To simulate the

load, a radial pressure is applied to the model over the lower face of the cylindrical

opening for the bearing, marked as “pressure” in Figure 8. The equivalent pressure

necessary to produce a resultant force of 4426 N is 7.931MPa. The lower arm in AED’s

test was pinned to the upper arm, which was pinned to a fixed part of the test stand. To

simplify the finite element analysis, the lower arm is assumed to be fixed in all three

dimensions at the pin where it connects to the upper arm, as shown by the label “fixed” in

Figure 8.

With the material properties and boundary conditions described, the model has to

be meshed. The model is meshed using ANSYS’s automeshing routine. The element

size is set at 10 mm, but in the area of interest, the mesh is refined to 2 mm, as can be

seen in Figure 9. Since SOLID187 is quadratic and has midside nodes, that gives about

16 nodes horizontally through the thickness where the crack is located and approximately

18 nodes vertically along the bushing hole before the cut where the crack begins to turn.

In total, 89788 nodes and 57268 elements are generated.

14

Figure 8 Boundary Conditions and Load

Figure 9 Meshed Geometry

Fixed

Pressure

15

Due to the irregular geometry and the presence of small faces in the model, a few

regions of the mesh produced elements that were outside ANSYS’s aspect ratio limits.

However, only 85 elements (0.15%) had this problem, and none of the offending

elements are in the vicinity of the crack location.

The meshed model is evaluated by ANSYS, providing the stress state in the entire

lower arm. However, the data required for the fatigue analysis consists primarily of the

stress normal to the crack in the crack plane. While no data was provided on the exact

location of the crack plane in the Army’s testing, observation of photographs (see Figure

5) from the test allows for an approximate location to be determined. A section cut is

made across the lower arm along the assumed crack plane, seen in Figure 10, which

permits the direct observation of the stress state inside.

To output stresses normal to the crack, the global Cartesian coordinate system is

not acceptable, because the crack plane is not oriented along its axes. So a local

coordinate system is created, with its X-Y plane lying in the crack plane and with its

origin at the outside corner of the crack plane, as shown in Figure 10. Outputting stresses

in this coordinate system allows for an observation of the stress state in the crack plane in

a form that is readily applicable to the crack growth.

16

Figure 10 Local Coordinate System and Crack Plane

The stress state in the plane also has to be exported from ANSYS for use by the

fatigue crack growth software. While ANSYS can output the stress on the surface, it

does so in a non-uniform fashion. Building paths on the surface allows the stress state to

be exported in a regular manner. One path is built along the top portion where the linear

crack growth occurred (Figure 11), and another path is built along the entire side of the

cross section (Figure 12). For both paths, each portion is divided into 10 sections, and

the portions are laid out so that the width of the model is also split into 10 sections. This

gives an 11x11 grid of data points for the top portion output, and an 11x21 grid for the

entire height output. Because the paths generated contain some portions that do not lie on

the desired grid, the unnecessary portions are removed using Excel (see Appendix B).

Additional paths are created in a similar manner as needed for additional data output.

17

Figure 11 Path Configuration, Top Portion

Figure 12 Path Configuration, Entire Height

18

The data from the path is exported as a file containing a brief header, followed by

two or more columns of data. The first column contains the distance from the beginning

of the path, and the following columns contain the data requested at those points. The

fatigue crack growth programs only require the stress normal to the crack plane, so that is

the data included in the file. The format is explained in greater detail in Appendix A.

When observing the tensile stresses normal to the crack plane in Figure 14, it is

not obvious that the crack should initiate at the interior corner. However, as AMRDEC

observed, fretting damage occurred near that boundary, creating a defect which causes

higher local stresses than this undamaged model shows. Once the crack was present, a

local stress concentration developed at its tip and followed it as it propagated.

While the top portion is in tension, the bottom portion is in compression, as seen

in Figure 13. When the load reverses, the opposite will be true. The maximum tensile

stress observed anywhere on the model is 188 MPa, which is well below the yield stress

of 435 MPa, so the assumption that the material was linear-elastic is valid.

Figure 13 Stress Normal to the Crack (Z direction)

19

Figure 14 Tensile Stress Normal to the Crack (Z direction)

20

One final observation can be made from the finite element analysis. Plotting the

shear stress in the crack plane, as seen in Figure 15, gives important information. While

the normal stress is the driving factor for Mode I fracture, the in-plane shear is the driving

factor for Mode II fracture. Cracks tend to propagate in the direction of minimum shear,

causing cracks to turn when they encounter shear.9 When the crack propagated into the

area where the U-cut is, it experienced higher amounts of shear and turned down the axial

length of the lower arm, which is supported by the presence of shear in that region of the

crack plane.

Figure 15 Shear Stress in Crack Plane (XY direction)

21

CHAPTER III

FATIGUE ANALYSIS

With the stress acting on the crack plane known, the next step in modeling the

crack growth is to combine the geometry and the loading and use that information to run

a fatigue crack-growth model. Fatigue analysis is another of the many areas of

engineering that has had a significant improvement in its model fidelity since the

personal computer became widespread.

The first major fatigue model was developed in 1870 by Wöhler. His work on

train axles indicated that the stress range applied to a structure was what governed its life.

His study led to the development of the S-N curve, which plots the stress range applied to

a part against its lifetime, in cycles (usually on a logarithmic scale). The S-N curve is

still used as a first-order estimate of fatigue life, as AED does in Figure 16.1, 4

22

Figure 16 S-N Curve for Scissor Assembly4

Subsequent developments improved on Wöhler’s work. Goodman developed an

equation to account for the effects of a nonzero mean stress, since Wöhler’s work only

included completely reversed stresses, where the stress amplitude is evenly distributed

between tensile and compressive stresses.1 Miner popularized a method of including the

damage from different stress amplitudes by assuming that the damage was linear and

cumulative. Both of these developments allowed for more types of cyclic loadings to be

analyzed. To describe the more complex loadings, R is used to denote the ratio of the

minimum stress, �min, to the maximum stress, �max.1

The next major advance in fatigue analysis came in 1961, when Paris proposed

that the fatigue crack growth rate, da/dN, was proportional to the change in stress intensity

factor, �K, at the crack tip.10 This allowed fatigue to be looked at in two separate

regimes: the first being the crack’s initiation, and the second being the crack’s

propagation (a third regime occurs as the crack growth becomes unstable). Previous

23

models had encompassed both initiation and growth. While Paris’ Law does not improve

the probabilistic model for crack initiation, it gives a good non-probabilistic approach to

the crack’s growth after it has started. However, for this approach to work, the stress

intensity factor on the crack has to be known at each cycle, and the crack length (and

possibly the loading) change after each cycle. To implement Paris’ approach, the use of

computational methods is almost necessary, as the integration of da/dN does not have a

closed-form solution.

Paris’ Law improved the field of fatigue mechanics significantly, but other

developments expanded it further. Paris’ original work did not include R effects. Walker

and Forman developed equations implementing R effects into the Paris equation,

expanding its usefulness.1 Elber described the importance of crack-closure on the crack

growth. Due to the plastic wake behind the crack tip extending the material, the crack

closes during part of the load cycle, retarding the crack growth.11 Again, this crack

closure has to be calculated or approximated at each step of the loading, making a

computational approach almost a necessity in performing calculations.

Fortunately, there are computer programs available that will perform the

calculations necessary to include these more advanced fatigue models. Two major ones

are NASGRO and AFGROW. Both have multiple geometric cases, a built-in material

library, options for loading, and relatively straightforward user interfaces as well as

several additional features. Both are considered in this study.

Before the specifics of each program are detailed, the concepts used in

approaching the problem must be explained. Because neither program has the exact

geometry of the lower arm, an approximate geometry must be used. Paris’ Law states

24

that the crack growth rate, da/dN, is linked to the change in stress intensity factor, �K.

Because the stress intensity factor is determined by the loading, which is known, and by

the geometry, selecting the correct geometry is essential.

The ideal approach would be to have a solution for the stress intensity factor of

the crack as a function of the crack length. No such solution exists for the lower arm, and

finding one would involve far more than is necessary for a fatigue-life estimate. Some

work has been done in using finite element analysis run in an iterative manner to solve

fatigue problems, but this approach is extremely computationally expensive.12

Because no K solution exists and making a solution would be excessive for a

single geometry, an approximation of the K solution must be used. Because K is linked

to the model’s geometry, selecting a simpler, similar geometry with a known K solution

can approximate the K solution for the actual geometry.

Analysis Using NASGRO

NASGRO was developed at NASA’s Johnson Space Center and had its first

release in 1986. Several revisions occurred, leading to NASGRO version 3.0, the first

Windows version, which was released via the internet in 1998. NASGRO 3.0 received a

handful of updates; then, in 2000, the responsibility for further development was given to

Southwest Research Institute and the NASGRO Consortium, leading to releases 4 and 5.

The current version of NASGRO is 5.2, but for this research, version 4.0 is used.

NASGRO has 51 of geometries to choose from in version 4.0. They are

organized into the type of crack or type of solution: through crack, corner crack,

embedded crack, surface crack, standard specimen, data table, polynomial series, and

25

boundary element. Because the geometry of the crack plane is complex, several cases are

reasonable to consider.

The crack in consideration grows from the bushing hole, which has a geometry

very similar to a thick-walled pipe. NASGRO case SC04, seen in Figure 17, deals with

surface cracks inside a hollow cylinder. Unfortunately, case SC04 is set up for a pressure

vessel with axial loading and has no input for the stresses across the cross section or

remote bending stress. This shortcoming makes it unsuitable for use analyzing the lower

arm.

Figure 17 NASGRO Case SC04 - Axial Surface Crack in Pipe16

26

Once the crack has propagated through the thickness, it can be assumed to be a

through crack. NASGRO case TC07 is a longitudinal crack in a cylinder as shown in

Figure 18, but, like SC04, it is limited to applying a pressure. Since it has no capacity for

the loading the lower arm encounters, it is also not a suitable case.

Figure 18 NASGRO Case TC07 – Axial Through Crack in Pipe16

27

NASGRO has six different corner crack geometries. NASGRO case CC01 is a

corner crack in a rectangular plate, as seen in Figure 19. The case requires the

assumption that the stress state on the crack plane can be approximated by a corner crack

with a far-field stress condition, which is not ideal, as it will not redistribute the stress

properly as the crack grows. Case CC01 does not allow for an arbitrary variation of the

stress state, but reduces it to a longitudinal stress and bending in two directions. By

approximating the stress at the crack plane into those three components, case CC01 could

be used. However, more appropriate cases exist in NASGRO.

Figure 19 NASGRO Case CC01 – Corner Crack in Plate16

28

Another corner crack case, CC05 seen in Figure 20, is a weight function based

solution and allows for an arbitrary stress state in the cross-section. The geometry is

identical to case CC01, only rather than breaking the stress down into three components,

it allows for an arbitrary stress distribution to be input from a file. This allows thee stress

data produced from the output from the ANSYS path to be used directly. The file format

for inputting the stresses is discussed in Appendix A.

Figure 20 NASGRO Case CC05 – Corner Crack in Plate, Weight Function16

While case CC05 is limited by the fact that its geometry does not match the lower

arm properly, it is approached two ways. The first, “top portion,” approach involves

using only the top left part of the cross-section, where the linear crack growth occurred,

as the boundary (Figure 21, left). The second, “entire height,” approach uses the left part

of the cross-section over the entire height (right). The input stresses come from the

respective ANSYS paths (see Figures 11 and 12). For both approaches, the crack does

not propagate correctly. For the top-only approach summarized in Table 1, the crack

barely grows at all for a 4 mm initial crack. If a 6 mm initial crack is input, the crack

29

propagates very quickly in the vertical (c) direction, but barely at all in the horizontal.

This is due in part to the nearby boundary, which does not actually exist, exerting too

much influence on the crack’s K. The second approach provides no crack growth at all,

even for a large initial crack, as seen in Table 2.

Figure 21 NASGRO Case CC05 – Layout

Table 1 NASGRO Case CC05 – Corner Crack in Plate, Top Portion

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)

Final Crack Size (mm)

Final Cycle Count

X y (a) t: 15.97 a0: 4.0 af: 4.00000

Y x (c) W: 18.29 c0: 4.0 cf: 4.06362 3000000

Stress File Scale Factors a0: 6.0 af: 6.64733

YX_TR.dat 1.0 -0.61608 c0: 6.0 cf: 18.3656 2529820

Table 2 NASGRO Case CC05 – Corner Crack in Plate, Entire Height

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X y (a) t: 15.97 a0: 4.0 af: 4.00000

Y x (c) W: 60.96 c0: 4.0 cf: 4.00000 3000000

Stress File Scale Factors a0: 6.0 af: 6.00000

YXt_TR.dat 1.0 -0.61608 c0: 6.0 cf: 6.00000 3000000

30

Case CC05 has been superseded in newer versions of NASGRO by case CC09,

which has the same geometry and stress input, but has an improved model.13 However,

the approach of using a rectangular plane as a portion of the cross-section is not likely to

improve significantly by the presence of an improved model. Limiting the rectangle to

part of the geometry does not allow the fatigue model to calculate the correct K. K is a

function of the geometry and the loading, both of which are abridged in this approach.

Case CC02 is the closest NASGRO case to the actual geometry of the lower arm

at the crack location. It has a corner crack initiating from a hole in a plate, acted on by

three stresses: a remote axial stress, a remote bending stress, and a pin-applied stress, as

shown in Figure 22. The stress acting at the crack plane of the lower arm is actually due

to bending caused by the load applied at the opposite end. This makes the stress

components in case CC02 simple to resolve. The distance from the load application to

the crack plane is 162 mm, giving bending moments of 718.3 and -422.5 N-m. While the

actual load on the lower arm causes a combination of bending and torsion, case CC02

only accounts for the bending. However, as the torsion does not significantly affect the

bending stress or the Mode I crack growth, it can be neglected.

31

Figure 22 NASGRO Case CC02 – Corner Crack at Hole in Plate16

Case CC02 provides reasonable results for the crack growth in the cross-section

as seen in Table 3. However, case CC02 assumes that the bending moment creates a

stress distributed over the entire cross-section. Figure 13 from the stress analysis

indicates that only one side of the cross-section is carrying the bending stress. To correct

for this, case CC02 is run again, but this time the effective width, W, used in the

calculation of the bending stress is halved, which doubles the bending stress. The results

of this correction are shown in Table 4 and Figure 23. Because the initial defect arose

from fretting damage, it is not possible to say exactly when it started or how large it was,

but for a 0.5 mm crack, in about 350000 cycles, perforation occurs, which is a reasonable

estimate. NASGRO release 4.0 does not have the capability to continue its crack-growth

analysis after break-through, and there is no NASGRO geometry appropriate to continue

the analysis after it occurs.

32

Table 3 NASGRO Case CC02 – Corner Crack at Hole in Plate

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size (mm)


Final Cycle Count

X x (c) B-D/2: 15.97 c0: 2.0 cf: 15.9732

Y y (a) t: 60.96 a0: 2.0 af: 14.1458 2898000

S1: 13.494 W: 85.94 c0: 4.0 cf: 15.9730

(MPa) -8.313 D: 54.00 a0: 4.0 af: 14.1455 2142000

S0 & S2: 0 B: 42.97 c0: 6.0 cf: 15.9709

a0: 6.0 af: 14.1449 1679000

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0 50000 100000 150000 200000 250000 300000 350000 400000

N (cycles)

Leng

th (

mm

)

ac

Figure 23 NASGRO Case CC02, corrected – Crack Growth

33

Table 4 NASGRO Case CC02, corrected

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X x (c) B-D/2: 15.97 c0: 0.5 cf: 16.0481

Y y (a) t: 60.96 a0: 0.5 af: 14.2033 348709

S1: 26.988 W: 85.94 c0: 1.0 cf: 16.0113

(MPa) -16.626 D: 54.00 a0: 1.0 af: 14.1781 246709

S0 & S2: 0 B: 42.97 c0: 2.0 cf: 16.0077

a0: 2.0 af: 14.1757 184709

c0: 4.0 cf: 15.9847

a0: 4.0 af: 14.1598 136709

c0: 6.0 cf: 16.0438

a0: 6.0 af: 14.2010 107709

Case CC06 has the same geometry as case CC02, only like case CC05, it is a

weight function solution that accepts an arbitrary stress, using the same type of file.

However, the coordinate system shown in Figure 24 does not function when used as

described. Converting X to the distance from the edge of the hole to the edge of the

plate, x/(B-D/2), appears to be the proper arrangement for the coordinate system. Like

case CC05, this case does not show any crack growth at all. The case results are

summarized in Table 5.

34

Figure 24 NASGRO Case CC06 – Corner Crack at Hole in Plate, Weight Function16

Table 5 NASGRO Case CC06 – Corner Crack at Hole in Plate, Weight Function

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X x (c) B-D/2: 15.97 c0: 4.0 cf: 4.00000 3000000

Y y (a) t: 60.96 a0: 4.0 af: 4.00000

Stress File XYt_TR.dat W: 85.94 c0: 6.0 cf: 6.00000 3000000

1.0 D: 54.00 a0: 6.0 af: 6.00000 Scale Factors -0.61608 B: 42.97

While case CC05 had a geometry that was not similar enough to the lower arm,

case CC06 is a good match. The fact that NASGRO considers the crack to be below the

threshold for crack growth indicates that there may be issues with the way the weight

function is used in this research. Furthermore, newer revisions of NASGRO have

replaced this case with an updated version, CC10.13

One more case exists in NASGRO that merits discussion. Case BE03 has two

cracks emanating from a hole in a plate: one a through-thickness crack, and the other a

35

corner crack. BE03 is a boundary-element case, which uses the boundary-element

method to solve for the stress intensity factor, rather than using an empirical formula or a

weight function. This makes the solution more time consuming, taking minutes to solve

rather than seconds. It does not allow for any stress variation in either direction, which

limits its usefulness for the crack in the lower arm. Furthermore, its solution is only valid

when the through-thickness crack is present, and there is no such crack in the lower arm.

Figure 25 NASGRO Case BE03 – Two Cracks at Hole in Plate, BEM Solution16

Analysis Using AFGROW

Like NASGRO, AFGROW has had a long development history. Its earliest

version was a BASIC program written for the Air Force in the 1980s called ASDGRO.

ASDGRO was added to, resulting in MODGRO, which underwent several revisions

including a rewrite from BASIC to Turbo Pascal and C. James Harter started working on

36

MODGRO in 1987 and has led its development since. In 1994, the code was renamed

AFGROW, and a Windows 95 version was developed in 1996. The Windows version

has been improved several times, implementing additional capabilities such as multiple

cracks and Microsoft COM implementation.14

Like NASGRO version 3, AFGROW is released publicly on the internet.

However, the current version, 4.0012.15, is the last version that is being released by the

Air Force Research Laboratory. Future work will be performed by LexTech, Inc.15

While AFGROW does not have the all of the same cases as NASGRO, it does

have a very wide selection of case geometries. It has standard solutions based on known

K solutions, as either equations or interpolated tables. It also has a set of weight function

based solutions, which allow for varying stress states in the part. Three configurations in

AFGROW reflect the geometry where the crack formed in the lower arm. The first is the

same geometry as NASGRO cases CC02 and CC06. The others are weight function

based solutions for edge and through cracks in thick pipes.

As with the NASGRO cases above, the crack in the lower arm can be modeled as

a corner crack initiating at a hole in a plate. AFGROW’s case is titled “Single Corner

Crack at Hole”, seen in Figure 26, and geometrically it is identical to NASGRO’s. The

approach to determining the bending stress is also the same, although the results, shown

in Table 6, are not. Internal differences in the programs account for this change, with the

most probable difference being the material models.

37

Figure 26 AFGROW Single Corner Crack at Hole

Table 6 AFGROW Single Corner Crack at Hole

Standard CS

AFGROW CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X x (c) B-D/2: 15.97 c0: 2.0 cf: 8.8831

Y y (a) T: 60.96 a0: 2.0 af: 9.1245 3000000

�bending: (MPa) 13.494 W: 85.94 c0: 4.0 cf: 13.180

R: -0.61608 D: 54.00 a0: 4.0 af: 12.951 3000000

c0: 6.0 cf: 15.806

a0: 6.0 af: 15.000 3000000

This solution is limited in the same way that NASGRO case CC02 was limited.

AFGROW believes the bending stress to be evenly distributed when it is not. Correcting

for this in the same manner (doubling the bending stress) provides the results in Table 7.

The crack growth can be seen in Figure 27. Even with the stress corrected, the

AFGROW version grows significantly slower than the NASGRO version, slow enough to

be outside the rate that occurred in the lower arm during testing.

38

Table 7 AFGROW Single Corner Crack at Hole, corrected

Standard CS

NASGRO CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X x (c) B-D/2: 15.97 c0: 0.5 cf: 15.9680

Y y (a) T: 60.96 a0: 0.5 af: 15.0080 1217750

�bending: (MPa) 26.988 W: 85.94 c0: 1.0 cf: 15.9670

R: -0.61608 D: 54.00 a0: 1.0 af: 15.0070 978500

c0: 2.0 cf: 15.9670

a0: 2.0 af: 15.0060 766250

c0: 4.0 cf: 15.7970

a0: 4.0 af: 14.8670 559500

c0: 6.0 cf: 15.9660

a0: 6.0 af: 14.9760 446000

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0 200000 400000 600000 800000 1000000 1200000 1400000

N (cycles)

Leng

th (

mm

)

ac

Figure 27 AFGROW Single Corner Crack at Hole, corrected - Crack Growth

39

AFGROW includes several weight function based solutions for thick walled

pipes, which are closer to the actual geometry than any of the other available cases.

However, none of the available thick pipe cases include corner cracks. The closest case

is an “Internal Axial Crack in Thick Pipe”, shown in Figure 28. By assuming that the

cross section is mirrored at the top, half of the geometry can be used as an approximation

of the actual lower arm. This will underestimate the crack growth rate; without going

into detail, the K for a single edge crack is higher than the K for a double edge crack, and

this difference will increase as the crack length increases.9 Understanding this limitation,

the case can still be considered. However, results from this case, shown in Table 8, have

the same shortcoming that plagued NASGRO cases CC05 and CC06: the crack refuses to

grow.

Figure 28 AFGROW Internal Axial Crack in Thick Pipe, Weight Function

40

Table 8 AFGROW Internal Axial Crack in Thick Pipe, Weight Function

Standard CS

AFGROW CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X y (a) 15.97 a0: 2.0 af: 2.000

Y x (c) W/2: 60.96 c0: 2.0 cf: 2.000 3000000

R: -0.61608 Do: 85.94 a0: 4.0 af: 4.000

Stress File: Di: 54.00 c0: 4.0 cf: 4.000 3000000

Stress - Average, Top Edge.wfs a0: 6.0 af: 6.000

c0: 6.0 cf: 6.000 3000000

Although the internal axial crack did not provide useful results, another weight

function based thick-walled pipe case is worth consideration. The “Axial Through Crack

in Thick Pipe” case, shown in Figure 29, has to be set up like the internal axial crack, by

doubling the length and understanding that the crack growth rate will be slower than it

should be.

Figure 29 AFGROW Axial Through Crack in Thick Pipe, Weight Function

41

Using the same data file as the internal axial crack, but neglecting the stresses in

the X-direction (as the crack has already perforated), the axial through crack case

produces the results shown in Table 9. A plot of the crack growth, Figure 30, shows that

this propagation rate is far too low, as the test data (Figure 6) shows a growth of 6 mm in

just over 10000 cycles. While the geometries in this case and in the internal axial case

above are similar to the correct geometry, the difference in the K is too large for these

results to be useful, because these cases do not have a crack with a free edge.

Table 9 AFGROW Axial Through Crack in Thick Pipe, Weight Function

Standard CS

AFGROW CS

Geometry Dimensions

(mm)

Initial Crack Size

(mm)


Final Cycle Count

X y c0: 4.0 cf: 13.946 3000000

Y x (c) W/2: 60.96 c0: 5.0 cf: 17.602 3000000

R: -0.61608 Do: 85.94 c0: 6.0 cf: 19.975 3000000

Stress File: Di: 54.00 c0: 7.0 cf: 21.804 3000000

Stress - Average, Top Edge.wfs c0: 8.0 cf: 23.335 3000000

c0: 9.0 cf: 24.628 3000000

42

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 500000 1000000 1500000 2000000 2500000 3000000

N (cycles)

Leng

th (

mm

)

c

Figure 30 AFGROW Axial Through Crack in Thick Pipe – Crack Growth

One final case exists in AFGROW that warrants mention as well. AFGROW has

the capability of modeling the crack growth on some simple arbitrary geometries with no

additional plug-ins. Using this tool, called Advanced Model, a geometry can be created

to resemble the cross section of the lower arm. However, this tool is not capable of

completely reproducing the cross section, and the closest geometry simply creates the

geometry from the single corner crack at a hole. Because it is an arbitrary geometry, it

does not have the same K solution as the single corner crack case above, but it produces

similar results, which can be seen in Table 10. From the comparison, it is apparent that

the difference is most pronounced for small cracks and large cracks. However, the

43

Advanced Model component of AFGROW is useful for considering geometries that are

not as readily approximated by one of the other cases.

Table 10 AFGROW Advanced Model

Standard CS

NASGRO CS

Initial Crack Size

(mm)


Final Cycle Count

% Rel. to Std. Geo.

X x (c) c0: 0.5 cf: 15.8220

Y y (a) a0: 0.5 af: 15.8890 1737000 43%

�bending: (MPa) 26.988 c0: 1.0 cf: 15.8060

R: -0.61608 a0: 1.0 af: 15.8750 1126000 15%

c0: 2.0 cf: 15.7940

a0: 2.0 af: 15.8610 785750 3%

c0: 4.0 cf: 15.8180

a0: 4.0 af: 15.8640 511250 -9%

c0: 6.0 cf: 15.7650

a0: 6.0 af: 15.7830 388000 -13%

From the results of both NASGRO and AFGROW, one point to notice is that

none of the cases can properly encompass this geometry. Some cases, notably NASGRO

case CC02, provide acceptable estimates of the crack growth prior to break-through.

Even that case required modifications based on understanding of what was physically

occurring to begin giving reasonable results. Other cases provide results that may be

plausible if no data is available to compare to, like the unmodified NASGRO CC02, the

AFGROW “Single Corner Crack at Hole”, or AFGROW’s “Axial Through Crack in

Thick Pipe”. In addition, some cases provide results that are obviously not correct, such

as NASGRO CC05 and CC06, and AFGROW’s “Internal Axial Crack in Thick Pipe”. A

summary of the cases run is shown in Table 11.

44

Table 11 Case Summary

Sof

twar

e

Cas

e

Cra

ck G

rew

Val

idat

ed b

y B

each

Mar

ks

Val

idat

ed b

y C

ycle

Cou

nt

Comments

NASGRO CC05

(Weight) No No No 1 run grew, but had large aspect ratio and slow growth

NASGRO CC02 yes Yes Yes Did not meet cycle count expectation until bending stress was corrected

NASGRO CC06

(Weight) No No No

AFGROW SCCaH Yes Yes No Crack growth too slow, even after bending stress correction.

AFGROW IACiTP

(Weight) No No No

AFGROW ATCiTP (Weight) Yes N/A No

Crack growth too slow. Invalid approximation of K solution.

AFGROW Adv.

Model Yes Yes No Similar results to AFGROW SCCaH, but different solution method.

The other point to notice is that the weight function solutions (NASGRO CC05

and CC06, and AFGROW IACiTP and ATCiTP) did not predict this crack growth well,

despite the fact that they are the only ones that had the capacity to input arbitrary stress

states. The weight function solutions tended to calculate the K as being below the

threshold, but the cause of this shortcoming is still undetermined.

45

CHAPTER IV

CONCLUDING REMARKS

Fatigue life estimation is highly sensitive to small differences in a multitude of

factors. This sensitivity makes estimating fatigue life a challenging endeavor. Short of

physically testing an article, it is nearly impossible to predict the characteristics of the

failure that will occur with the exception of carefully controlled and highly studied

standard test articles. As the amount of information about the failure improves, so too

does the modeling fidelity, but it is unlikely that simulation will be able to replace testing

as a means of determining fatigue life.

If no information is known about the article except its geometry, loading, and

environment, several steps must be taken to attempt to predict the life of the article. First,

a decision must be made about where a defect is likely to produce a crack. Because crack

formation starts on a microstructural level, this is not a simple task. Notches, corners,

and boundaries all contribute to likely locations, as do chemical interactions with the

environment and mechanical interactions with other objects.

For the case of the scissor assembly, in the case of testing eight assemblies, three

different crack locations were noted (a fourth was due to an accident during the test)

while applying the same loads to the same geometry with the same constraints in the

same environment. Unless there is an obvious notch that would encourage crack

initiation, the best choice for estimating fatigue life is to select several sites and analyze

46

them independently. Complicating matters further, there is no formula or simple

engineering tool to identify likely sites, only engineering experience.

With a potential crack site selected, estimation of fatigue life improves. With a

known site, stress analysis of the component can be used to develop an idea of what loads

the site experiences. This step is the most straightforward of the analysis, as 3D finite

element analysis has become very common and quite powerful with the advances in

computer technology. With finite element tools, even very complex geometries and

loadings can be analyzed, giving a clear picture of the stress that would act on a crack

should one initiate. Still, using these tools without understanding the physics behind

them can cause errors due to the numerical approximations made.

Because the Army performed testing on the scissor assembly and identified the

crack locations, the analysis begins with this step. The lower arm was fed into ANSYS

as a solid model, and then it was meshed, loaded, constrained, and solved. The crack

plane identified by AED was put into the ANSYS model and used to output the stress

acting across it. This data was then exported into files to be imported into fatigue crack

growth software. The model was simplified from the actual test setup AED used, as the

upper arm and the bushing were not included. Because those components would have

added some flexibility to the lower arm at the bushing hole, the stress state output from

ANSYS is not an exact match to the true article, but it should be sufficient for a first-

order estimate.

The next step required for a fatigue life analysis is to consider the geometry and

the loading at the likely crack site and find a case in the fatigue crack growth software

that can be used to approximate the particular crack geometry and loading of the article.

47

Some simple geometries may translate exactly as one of the cases available, but many

will not. As with attempting to determine possible crack initiation points, deciding which

case to use depends largely on engineering judgment and experience. Understanding the

limits of the fatigue crack growth software, especially the limits caused by the differences

in the geometry and by the case itself, is critical in selecting the proper case.

This process was approached with the intention of trying a wide variety of

possible cases and evaluating them against the data collected by AED and AMRDEC.

Some of the cases considered in this study are examples of what approach not to take,

such as NASGRO’s case CC05, which was not a good match to the geometry at all.

Other cases considered appeared more promising, but they were still only approximations

of the actual geometry.

The final step is to set up and run the selected case and to ensure that the results

are meaningful. Setting up the selected case is usually simple enough, but there are a

number of choices available in fatigue crack growth software. After inputting the

dimensions, which is straightforward provided that the unit system is strictly adhered to,

options are available for material models, loading cycles, interaction models, residual

stresses, environmental effects, and many other variables. For simple analysis, many of

these factors are negligible, but even the simplest analysis requires a properly established

material model and loading. After the parameters have been set up and the model run,

the results must be evaluated. In the absence of test data to compare to, this final step is

also quite difficult. Again, experience and judgment are the key factors in checking the

results to determine whether they are reasonable.

48

Setting up the NASGRO and AFGROW cases for the crack in the lower arm was

not complicated because most of the advanced features of the software, such as crack

closure and variable loading, were neglected. The loading that the scissor assembly was

subject to was a very simple constant amplitude loading. The cracks were assumed to be

free of significant retardation or plasticity effects. The difference in the material models

likely had the largest impact on the results, but in both cases, 7075-T73 forged aluminum

was available in the built-in material libraries of the software. NASGRO uses the

NASGRO equation to calculate the fatigue crack growth rate, while AFGROW uses the

Harter-T method. Because AED provided test results, a basis was established to judge

the results of the cases that would not exist when attempting to predict a crack’s growth

without testing. Of course, some cases, such as NASGRO’s CC06 and AFGROW’s

internal axial crack in a thick pipe, provided obviously flawed data. Others provided data

that does not appear totally unreasonable at first, but it does not properly reflect the

physics, such as the uncorrected NASGRO CC02 and AFGROW single corner crack at a

hole, or the AFGROW axial through crack in a thick pipe. All of those gave crack

growth data that did not correlate with the test data. Only the correct geometry, correctly

loaded, provided properly meaningful results. Even then, the differences in software

packages, primarily in their material modeling, led to very different growth rates on an

identical geometry and load.

Of all the cases considered, the corrected NASGRO case CC02 provided the best

fatigue crack growth results for the crack in the lower arm of the scissors assembly.

While case CC02 does not allow for the continuation of the crack following break-

through and therefore cannot be directly compared to the Army’s test results, the number

49

of cycles required to cause break-through is a good estimate. For a 0.5 mm defect, the

corrected CC02 predicted 348709 cycles between crack initiation and break-through.

The two cracks AED detected were found at about 700000 and 920000 cycles at a length

of about 8 and 7 mm respectively. That would give 350000 to 570000 cycles for the

fretting to initiate the crack and for it to spread 7 or 8 mm externally, which is not an

unreasonable range.

Approaching a fatigue crack growth problem at a known location using finite

element analysis and crack growth software has some advantages. First, it is far less

expensive than full-scale testing. It also takes far less time than testing. When properly

set up, it can provide useful information on how a crack might grow if one initiates.

Furthermore, this approach can be very useful in determining how large a defect the

article can withstand without failing in a certain number of cycles.

However, this approach has some severe limitations. Without testing to find

likely crack locations, it is difficult to pick locations which are likely to have cracks

initiate. Even when the location is known, crack initiation has a lot of variability since it

occurs in the microstructure of the article. Because much of the article’s life will be

consumed in the crack’s initiation phase, this method is not very accurate for considering

initiation. Furthermore, this method is highly sensitive to small differences, which makes

using it as an estimation tool unsupported by experimental data a tenuous effort.

Improvements in computers have enhanced the field over the past years. In the

future, these tools will become more powerful and will be able to accurately predict crack

growth in complex geometries with greater success. At the cost of additional

development time and run time, the plug-in nature of these fatigue crack growth software

50

packages already allows for complex geometries to be analyzed in detail. This sort of

endeavor requires significant analysis of the article, though, as the stress intensity factor

must be determined with sufficient resolution to allow the fatigue software to accurately

predict the crack growth rate.

Other approaches have been made in using 3D finite element software to perform

3D fatigue crack growth analysis by using the finite elements to determine K in an

iterative method to predict crack growth. This approach is extremely computationally

intensive, although as the method improves it may become a useful complement to the

dedicated fatigue crack growth software.

Ultimately, the fatigue crack growth software currently available provides a

useful means of predicting crack growth and fatigue life beyond the simple S-N curve,

but it must be approached with care.

51

REFERENCES

1. Stephens, Fatemi, Stephens, and Fuchs, Metal Fatigue in Engineering, 2nd ed.,

Wiley, New York, 2001.

2. “Aviation Unit and Intermediate Maintenance Manual, Army Model AH-64A Helicopter,” TM 1-1520-238-23, United States Army, May 1994.

3. Fullerton, L., “Test Plan for Alternate Source Qualification Testing of the AH-64 Main Rotor Upper Controls Scissors Assemblies, Part Nos. 7-311511158-3 and -5,” AETP-FS-2005-001, Aviation Engineering Directorate, Redstone Arsenal, AL, Feb. 2005.

4. Thompson, Bryan, “Final Test Report for the New Source Testing of the Scissors Assembly,” Test Report 050098, Component Test and Surveillance Branch, Redstone Technical Test Center, Redstone Arsenal, AL, Feb. 2006.

5. Vaughan, Robert, Private communication, Sept. 2008.

6. Moaveni, Saeed, Finite Element Analysis: Theory and Application with ANSYS, 3rd ed., Pearson, Upper Saddle River, NJ, 2008.

7. ANSYS Help Files, “SOLID187,” 2007.

8. “Aluminum 7075-T73; 7075-T735x”, MATWEB. [http://www.matweb.com/search/DataSheet.aspx?MatGUID=6653b72914864cc0a0ff7adf5b720167&ckck=1. Accessed 8/10/2007]

9. Anderson, Ted L., Fracture Mechanics: Fundamentals and Applications, 3rd ed., CRC Press, 2005.

10. Paris, P. C. and Erdogan, F., “A Critical Analysis of Crack Propagation Laws,” Journal of Basic Engineering, 528-534, 1963.

11. Elber, W., “Fatigue Crack Closure Under Cyclic Tension,” Engineering Fracture Mechanics, 12, 37-45, 1970.

12. Hou, Goldstraw, Maan, and Knop, “An Evaluation of 3D Crack Growth Using ZENCRACK,” DSTO-TR-1158, Airframes and Engines Division, Aeronautical and Maritime Research Laboratory, Australian Defense Science and Technology Organization, Fisherman’s Bend, Australia, May 2001.

52

13. NASGRO Website, February 2009. [http://www.nasgro.swri.org/. Accessed 3/10/2009]

14. Harter, James A., AFGROW Users Guide and Technical Manual, AFRL-VA-WP-TR-2008-XXXX, Air Vehicles Directorate, WPAFB, OH, July 2008.

15. AFGROW Website, 2008. [http://afgrow.net/. Accessed 3/10/2009]

16. NASGRO Fracture Mechanics and Fatigue Crack Growth Analysis Software Reference Manual, Southwest Research Institute, May 2002.

53

APPENDIX A

FILE FORMATS FOR STRESS DATA

54

ANSYS .LIS File Format

The output from the paths set up in ANSYS is written to a file in the same manner

that ANSYS would output it to the screen. This makes the output more readable, but

increases the work that must be done prior to importing it into other programs. It consists

of columns of data. The first column, labeled ‘S’, marks the distance along the path that

the data point occurs at. Any additional columns contain ANSYS results that have been

mapped to that path. Since the only data required for NASGRO and AFGROW is the

stress in the Z direction, that is the only data that is output, in the column marked ‘S_Z’.

If no data is available for a given point on the path, it is left blank. The output units are

the units used in ANSYS. Every 46 lines, ANSYS puts header information over four

lines.

Sample structure:

PRINT ALONG PATH DEFINED BY LPATH COMMAND. DSYS= 0 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.0000 <value> <station> <value> <station> <value> <station> <value> <...> <...> ***** PATH VARIABLE SUMMARY ***** S S_Z <station> <value> <station> <value> <station> <value> <...> <...>

55

NASGRO .DAT File Format

NASGRO requires stresses to be input from a file for its weight function based

solutions. The filename is required to be DOS compatible, with a maximum of eight

characters prior to the extension. No effort is made for the readability of this file, as it is

strictly meant for input to NASGRO. Its first line contains two numbers, indicating the

number of points on the x- and y-axis, respectively. The next section contains a vertical

list of single numbers, indicating the normalized stations on the x-axis for which the

stress is input. Following that section is the third section, which contains two columns of

numbers. The first number is the normalized y-axis station, and the second number is the

stress at that y-axis station. Each group of these is for one of the x-axis stations listed in

the second section. The count at the top of the file must be correct, and the distribution

must be square. The units are either in MPa or ksi, depending on the units in NASGRO.

Sample Structure:

<# of X points> <# of Y points> <X1> <X2> <...> <Xn> <Y1> <S(x1,y1)> <Y2> <S(x1,y2)> <...> <S(x1,...)> <Y1> <S(x2,y1)> <Y2> <S(x2,y2)> <...> <S(x2,...)> <Y1> <S(xn,y1)> <Y2> <S(xn,y2)> <...> <S(xn,...)> <Ym> <S(xn,ym)>

56

AFGROW .wfs File Format

AFGROW can have the stress distribution input directly into a table in its user

interface, but it allows the numbers in that table to be read from and written to a file.

Like NASGRO’s .DAT file, no effort is made at readability. It contains a header whose

first line declares the unit system, indicated by a numeral. The next line indicates how

many distributions are in the file. The following section provides the details for the first

distribution. Its first line is the number of points in that distribution; the following lines

are the points, with the first value being the station (not normalized) and the second being

the stress. Subsequent distributions are included in the same manner. There is no

requirement that the distributions be the same length, but they are limited to 25 data

points each. Furthermore, only two distributions are used in the current version of

AFGROW, one for the x-direction and one for the y-direction.

Sample Structure:

UNITS=<unitType, 0 for English, 1 for Metric> <# of distributions> <# of points in distribution 1> <x1> <S(x1)> <x2> <S(x2)> <...> <S(...)> <xn> <S(xn)> <# of points in distribution 2> <y1> <S(y1)> <y2> <S(y2)> <...> <S(...)> <xm> <S(xm)>

57

APPENDIX B

STRESS DATA FILES

58

ANSYS Output: PRPATH Top Only.lis

This file contains the stress normal to the crack plane in the top of the cross-

section, with the path laid out as shown in Figure 11. Grayed out portions were removed

as they were the portions of the path that did not lie on the desired grid. Editing of the

file occurred in Excel.

PRINT ALONG PATH DEFINED BY LPATH COMMAND. DSYS= 0 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.0000 0.37836E+08 0.16836E-02 0.40648E+08 0.33672E-02 0.42771E+08 0.50508E-02 0.44704E+08 0.67344E-02 0.45460E+08 0.84180E-02 0.45481E+08 0.10102E-01 0.44836E+08 0.11785E-01 0.42203E+08 0.13469E-01 0.37811E+08 0.15152E-01 0.31328E+08 0.16836E-01 0.20604E+08 0.18538E-01 0.30245E+08 0.20241E-01 0.35481E+08 0.21943E-01 0.39309E+08 0.23646E-01 0.40175E+08 0.25348E-01 0.40250E+08 0.27050E-01 0.39226E+08 0.28753E-01 0.38241E+08 0.30455E-01 0.35967E+08 0.32158E-01 0.33219E+08 0.33860E-01 0.30358E+08 0.35681E-01 0.32411E+08 0.37501E-01 0.34191E+08 0.39322E-01 0.34870E+08 0.41143E-01 0.35521E+08 0.42963E-01 0.35006E+08 0.44784E-01 0.33372E+08 0.46605E-01 0.29917E+08 0.48425E-01 0.25004E+08

59

0.50246E-01 0.17545E+08 0.52067E-01 0.58617E+07 0.53903E-01 0.17104E+08 0.55740E-01 0.24004E+08 0.57576E-01 0.28075E+08 0.59413E-01 0.30306E+08 0.61249E-01 0.31391E+08 0.63086E-01 0.31682E+08 0.64922E-01 0.30855E+08 0.66759E-01 0.29266E+08 0.68596E-01 0.28070E+08 0.70432E-01 0.26731E+08 0.72252E-01 0.27678E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.74072E-01 0.28320E+08 0.75891E-01 0.29212E+08 0.77711E-01 0.29284E+08 0.79531E-01 0.28446E+08 0.81350E-01 0.26986E+08 0.83170E-01 0.23948E+08 0.84990E-01 0.20008E+08 0.86809E-01 0.13975E+08 0.88629E-01 0.60950E+07 0.90464E-01 0.13605E+08 0.92298E-01 0.19240E+08 0.94133E-01 0.22583E+08 0.95968E-01 0.24947E+08 0.97802E-01 0.25836E+08 0.99637E-01 0.25968E+08 0.10147 0.25753E+08 0.10331 0.25378E+08 0.10514 0.24729E+08 0.10698 0.24193E+08 0.10879 0.24430E+08 0.11061 0.24710E+08 0.11243 0.24729E+08 0.11425 0.24306E+08 0.11607 0.23610E+08 0.11789 0.22226E+08 0.11971 0.19875E+08 0.12153 0.16498E+08 0.12335 0.11523E+08 0.12516 0.40779E+07

60

0.12700 0.11325E+08 0.12883 0.16056E+08 0.13066 0.18960E+08 0.13249 0.20958E+08 0.13433 0.21987E+08 0.13616 0.22382E+08 0.13799 0.22517E+08 0.13983 0.22326E+08 0.14166 0.22282E+08 0.14349 0.22087E+08 0.14531 0.22081E+08 0.14713 0.21898E+08 0.14894 0.21626E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.15076 0.21031E+08 0.15258 0.20191E+08 0.15440 0.18852E+08 0.15622 0.16970E+08 0.15803 0.14158E+08 0.15985 0.10151E+08 0.16167 0.24139E+07 0.16350 0.10045E+08 0.16533 0.13738E+08 0.16716 0.16290E+08 0.16899 0.17779E+08 0.17082 0.18829E+08 0.17265 0.19534E+08 0.17449 0.19942E+08 0.17632 0.20158E+08 0.17815 0.20084E+08 0.17998 0.20144E+08 0.18179 0.19824E+08 0.18361 0.19636E+08 0.18543 0.19156E+08 0.18725 0.18384E+08 0.18906 0.17399E+08 0.19088 0.16324E+08 0.19270 0.14668E+08 0.19452 0.12432E+08 0.19633 0.91690E+07 0.19815 0.59997E+07 0.19998 0.91099E+07 0.20181 0.12104E+08

61


62


63

0.35663 0.40131E+07 0.35845 0.29484E+07 0.36027 0.26704E+07 0.36210 0.36305E+07 0.36391 0.17487E+07 0.36573 0.15131E+07 0.36755 0.16430E+07 0.36936 0.19014E+07 0.37118 0.19206E+07 0.37299 0.18002E+07 0.37481 0.17773E+07 0.37663 0.17414E+07 0.37844 0.20836E+07 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.38026 0.21519E+07

ANSYS Output: PRPATH - Full Height.lis

This file is the result of the output of the path shown in Figure 12. As in the

above section, the unneeded portions have been marked with gray. Note that blank

portions occur where the path went outside the boundary of the geometry, where there

was no data available. Editing occurred in Excel.

PRINT ALONG PATH DEFINED BY LPATH COMMAND. DSYS= 0 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.0000 0.37836E+08 0.30594E-02 0.42307E+08 0.61188E-02 0.45361E+08 0.91782E-02 0.45131E+08 0.12238E-01 0.41075E+08 0.15297E-01 0.30158E+08 0.18356E-01 0.21416E-01 0.24475E-01

64

0.27535E-01 0.30594E-01 0.33654E-01 0.36713E-01 0.39772E-01 0.42832E-01 0.45891E-01-0.30274E+08 0.48951E-01-0.41678E+08 0.52010E-01-0.45650E+08 0.55069E-01-0.45526E+08 0.58129E-01-0.43247E+08 0.61188E-01-0.35995E+08 0.61332E-01-0.35440E+08 0.61476E-01-0.34885E+08 0.61620E-01-0.34329E+08 0.61764E-01-0.33774E+08 0.61908E-01-0.33219E+08 0.62052E-01-0.32664E+08 0.62196E-01-0.32109E+08 0.62340E-01-0.31554E+08 0.62484E-01-0.30999E+08 0.62628E-01-0.30444E+08 0.65685E-01-0.33706E+08 0.68743E-01-0.35532E+08 0.71800E-01-0.35198E+08 0.74857E-01-0.31274E+08 0.77914E-01-0.22932E+08 0.80971E-01 0.84029E-01 0.87086E-01 0.90143E-01 0.93200E-01 0.96257E-01 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.99315E-01 0.10237 0.10543 0.10849 0.23052E+08 0.11154 0.31419E+08 0.11460 0.35312E+08 0.11766 0.34996E+08 0.12072 0.33836E+08 0.12377 0.30358E+08

65

0.12392 0.29611E+08 0.12406 0.29258E+08 0.12420 0.28944E+08 0.12435 0.28631E+08 0.12449 0.28317E+08 0.12464 0.28004E+08 0.12478 0.27690E+08 0.12492 0.27376E+08 0.12507 0.27063E+08 0.12521 0.26731E+08 0.12827 0.28297E+08 0.13132 0.29444E+08 0.13438 0.28594E+08 0.13743 0.25090E+08 0.14049 0.17996E+08 0.14354 0.56630E+07 0.14660 0.14965 0.15271 0.15577 0.15882 0.16188 0.16493 0.16799 -0.65402E+07 0.17104 -0.18319E+08 0.17410 -0.25020E+08 0.17715 -0.28487E+08 0.18021 -0.28832E+08 0.18326 -0.28412E+08 0.18632 -0.25885E+08 0.18646 -0.25700E+08 0.18661 -0.25516E+08 0.18675 -0.25331E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.18690 -0.25146E+08 0.18704 -0.24961E+08 0.18719 -0.24777E+08 0.18733 -0.24592E+08 0.18748 -0.24407E+08 0.18762 -0.24223E+08 0.18777 -0.24038E+08 0.19082 -0.24532E+08 0.19388 -0.24846E+08

66

0.19693 -0.23537E+08 0.19998 -0.20716E+08 0.20304 -0.14876E+08 0.20609 -0.32914E+07 0.20914 0.21220 0.21525 0.21830 0.22136 0.22441 0.22747 0.23052 0.32617E+07 0.23357 0.14820E+08 0.23663 0.20723E+08 0.23968 0.23672E+08 0.24273 0.24648E+08 0.24579 0.24646E+08 0.24884 0.24193E+08 0.24898 0.23973E+08 0.24913 0.23754E+08 0.24927 0.23535E+08 0.24942 0.23318E+08 0.24956 0.23166E+08 0.24970 0.22951E+08 0.24985 0.22735E+08 0.24999 0.22519E+08 0.25014 0.22303E+08 0.25028 0.22087E+08 0.25333 0.21924E+08 0.25638 0.21506E+08 0.25944 0.20249E+08 0.26249 0.17620E+08 0.26554 0.12520E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.26859 0.15880E+07 0.27165 0.27470 0.27775 0.28080 0.28385 0.28691 0.28996 0.29301 -0.15860E+07

67

0.29606 -0.12857E+08 0.29911 -0.17669E+08 0.30217 -0.20272E+08 0.30522 -0.21455E+08 0.30827 -0.22163E+08 0.31132 -0.22114E+08 0.31147 -0.21872E+08 0.31161 -0.21656E+08 0.31176 -0.21440E+08 0.31190 -0.21224E+08 0.31204 -0.21008E+08 0.31219 -0.20792E+08 0.31233 -0.20576E+08 0.31248 -0.20360E+08 0.31262 -0.20145E+08 0.31276 -0.19929E+08 0.31581 -0.19695E+08 0.31886 -0.18880E+08 0.32192 -0.17561E+08 0.32497 -0.15224E+08 0.32802 -0.11084E+08 0.33107 -0.56117E+07 0.33412 0.33717 0.34022 0.34327 0.34632 0.34937 0.35242 0.35548 0.57625E+07 0.35853 0.11340E+08 0.36158 0.15248E+08 0.36463 0.17427E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.36768 0.18851E+08 0.37073 0.19815E+08 0.37378 0.20144E+08 0.37392 0.20011E+08 0.37407 0.19794E+08 0.37421 0.19577E+08 0.37436 0.19360E+08 0.37450 0.19143E+08 0.37464 0.18926E+08

68

0.37479 0.18709E+08 0.37493 0.18488E+08 0.37508 0.18245E+08 0.37522 0.18002E+08 0.37827 0.17337E+08 0.38132 0.16314E+08 0.38437 0.15221E+08 0.38742 0.13111E+08 0.39047 0.99948E+07 0.39352 0.71363E+07 0.39657 0.95772E+07 0.39962 0.70451E+07 0.40267 0.25368E+07 0.40572 -0.12119E+06 0.40877 -0.24675E+07 0.41182 -0.69482E+07 0.41487 -0.97170E+07 0.41792 -0.71612E+07 0.42097 -0.10076E+08 0.42402 -0.13154E+08 0.42707 -0.15283E+08 0.43012 -0.16731E+08 0.43317 -0.17128E+08 0.43622 -0.17778E+08 0.43636 -0.17563E+08 0.43650 -0.17348E+08 0.43665 -0.17171E+08 0.43679 -0.16909E+08 0.43694 -0.16619E+08 0.43708 -0.16330E+08 0.43722 -0.16040E+08 0.43737 -0.15750E+08 0.43751 -0.15460E+08 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.43766 -0.15170E+08 0.44071 -0.14500E+08 0.44375 -0.13280E+08 0.44680 -0.12430E+08 0.44985 -0.10966E+08 0.45290 -0.89033E+07 0.45595 -0.73877E+07 0.45900 -0.80541E+07 0.46205 -0.56990E+07

69

0.46510 -0.28238E+07 0.46815 -0.23424E+06 0.47120 0.28361E+07 0.47424 0.58976E+07 0.47729 0.80969E+07 0.48034 0.72282E+07 0.48339 0.89067E+07 0.48644 0.10995E+08 0.48949 0.12663E+08 0.49254 0.13896E+08 0.49559 0.14662E+08 0.49864 0.15092E+08 0.49878 0.14731E+08 0.49893 0.14444E+08 0.49907 0.14214E+08 0.49922 0.13985E+08 0.49936 0.13755E+08 0.49951 0.13526E+08 0.49965 0.13297E+08 0.49980 0.13067E+08 0.49994 0.12721E+08 0.50009 0.12325E+08 0.50314 0.11206E+08 0.50618 0.10854E+08 0.50923 0.10144E+08 0.51228 0.87384E+07 0.51533 0.73367E+07 0.51838 0.62920E+07 0.52143 0.60579E+07 0.52447 0.47098E+07 0.52752 0.21379E+07 0.53057 -19149. 0.53362 -0.21870E+07 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.53667 -0.46895E+07 0.53972 -0.61135E+07 0.54277 -0.61844E+07 0.54581 -0.72523E+07 0.54886 -0.85398E+07 0.55191 -0.97555E+07 0.55496 -0.10575E+08 0.55801 -0.11297E+08 0.56106 -0.12307E+08

70

0.56120 -0.12040E+08 0.56134 -0.11773E+08 0.56149 -0.11506E+08 0.56163 -0.11239E+08 0.56178 -0.10972E+08 0.56192 -0.10397E+08 0.56206 -0.97857E+07 0.56221 -0.91748E+07 0.56235 -0.85639E+07 0.56250 -0.79530E+07 0.56554 -0.73535E+07 0.56859 -0.61866E+07 0.57164 -0.63584E+07 0.57469 -0.53231E+07 0.57774 -0.50438E+07 0.58078 -0.46454E+07 0.58383 -0.44547E+07 0.58688 -0.33190E+07 0.58993 -0.16735E+07 0.59298 -35080. 0.59603 0.16859E+07 0.59907 0.33267E+07 0.60212 0.45028E+07 0.60517 0.46882E+07 0.60822 0.46335E+07 0.61127 0.53652E+07 0.61431 0.55647E+07 0.61736 0.68100E+07 0.62041 0.74457E+07 0.62346 0.80471E+07 0.62360 0.74186E+07 0.62375 0.67900E+07 0.62389 0.61615E+07 ***** PATH VARIABLE SUMMARY ***** S S_Z 0.62403 0.55990E+07 0.62418 0.52777E+07 0.62432 0.49549E+07 0.62447 0.46305E+07 0.62461 0.43044E+07 0.62475 0.39767E+07 0.62490 0.36305E+07 0.62795 0.15420E+07 0.63099 0.16459E+07

71

0.63404 0.19170E+07 0.63709 0.17745E+07 0.64014 0.20520E+07 0.64319 0.21363E+07 0.64623 0.22185E+07 0.64928 0.16150E+07 0.65233 0.83140E+06 0.65538 19776. 0.65843 -0.85011E+06 0.66147 -0.16240E+07 0.66452 -0.23214E+07 0.66757 -0.21861E+07 0.67062 -0.20165E+07 0.67367 -0.16903E+07 0.67671 -0.18501E+07 0.67976 -0.18209E+07 0.68281 -0.16602E+07 0.68586 -0.37083E+07

NASGRO Input: YX_TR.dat

This input file for NASGRO contains the stress distribution in the top portion of

the cross section, but with the x- and y-axis swapped to match NASGRO’s geometry.

11 11 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 3.6305 0.1 8.0471 0.2 12.325 0.3 15.092 0.4 18.002 0.5 20.144

72

0.6 22.087 0.7 24.193 0.8 26.731 0.9 30.358 1 37.836 0.0 1.7487 0.1 6.6477 0.2 11.885 0.3 14.943 0.4 17.488 0.5 19.824 0.6 22.081 0.7 24.43 0.8 27.678 0.9 32.411 1 40.648 0.0 1.5131 0.1 7.1449 0.2 11.27 0.3 14.509 0.4 17.304 0.5 19.636 0.6 21.898 0.7 24.71 0.8 28.32 0.9 34.191 1 42.771 0.0 1.643 0.1 6.832 0.2 11.278 0.3 13.971 0.4 16.499 0.5 19.156 0.6 21.626 0.7 24.729 0.8 29.212 0.9 34.87 1 44.704 0.0 1.9014 0.1 6.0357 0.2 10.514 0.3 13.332 0.4 16.005 0.5 18.384 0.6 21.031 0.7 24.306

73

0.8 29.284 0.9 35.521 1 45.46 0.0 1.9206 0.1 5.5601 0.2 10.144 0.3 12.663 0.4 15.219 0.5 17.399 0.6 20.191 0.7 23.61 0.8 28.446 0.9 35.006 1 45.481 0.0 1.8002 0.1 6.3041 0.2 8.9742 0.3 11.85 0.4 14.094 0.5 16.324 0.6 18.852 0.7 22.226 0.8 26.986 0.9 33.372 1 44.836 0.0 1.7773 0.1 5.6312 0.2 8.3791 0.3 10.656 0.4 12.583 0.5 14.668 0.6 16.97 0.7 19.875 0.8 23.948 0.9 29.917 1 42.203 0.0 1.7414 0.1 4.8493 0.2 7.5934 0.3 9.2926 0.4 10.785 0.5 12.432 0.6 14.158 0.7 16.498 0.8 20.008 0.9 25.004

74

1 37.811 0.0 2.0836 0.1 4.8484 0.2 6.678 0.3 8.0429 0.4 8.8653 0.5 9.169 0.6 10.151 0.7 11.523 0.8 13.975 0.9 17.545 1 31.328 0.0 2.1519 0.1 4.6904 0.2 6.2697 0.3 7.2639 0.4 7.2755 0.5 5.9997 0.6 2.4139 0.7 4.0779 0.8 6.095 0.9 5.8617 1 20.604

NASGRO Input: YXt_TR.dat

This input file for NASGRO contains the stress distribution in the entire height of

the cross section, but with the x- and y-axis swapped to match NASGRO’s geometry.

11 11 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00 3.63

75

0.10 8.05 0.20 12.33 0.30 15.09 0.40 18.00 0.50 20.14 0.60 22.09 0.70 24.19 0.80 26.73 0.90 30.36 1.00 37.84 0.00 1.65 0.10 6.81 0.20 10.85 0.30 13.90 0.40 16.31 0.50 18.85 0.60 21.51 0.70 24.65 0.80 29.44 0.90 35.00 1.00 45.36 0.00 1.77 0.10 5.37 0.20 8.74 0.30 11.00 0.40 13.11 0.50 15.25 0.60 17.62 0.70 20.72 0.80 25.09 0.90 31.42 1.00 41.08 0.00 2.14 0.10 4.69 0.20 6.29 0.30 7.23 0.40 7.14 0.50 5.76 0.60 1.59 0.70 3.26 0.80 5.66 0.90 0.00 1.00 0.00 0.00 1.62 0.10 3.33 0.20 4.71

76

0.30 5.90 0.40 7.05 0.50 0.00 0.60 0.00 0.70 0.00 0.80 0.00 0.90 0.00 1.00 0.00 0.00 0.02 0.10 -0.04 0.20 -0.02 0.30 -0.23 0.40 -0.12 0.50 0.00 0.60 0.00 0.70 0.00 0.80 0.00 0.90 0.00 1.00 0.00 0.00 -1.62 0.10 -3.32 0.20 -4.69 0.30 -5.70 0.40 -6.95 0.50 0.00 0.60 0.00 0.70 0.00 0.80 0.00 0.90 0.00 1.00 0.00 0.00 -2.19 0.10 -4.65 0.20 -6.18 0.30 -7.39 0.40 -7.16 0.50 -5.61 0.60 -1.59 0.70 -3.29 0.80 -6.54 0.90 0.00 1.00 0.00 0.00 -1.69 0.10 -5.32 0.20 -8.54 0.30 -10.97 0.40 -13.15

77

0.50 -15.22 0.60 -17.67 0.70 -20.72 0.80 -25.02 0.90 -31.27 1.00 -41.68 0.00 -1.82 0.10 -6.19 0.20 -10.58 0.30 -13.28 0.40 -16.73 0.50 -18.88 0.60 -21.46 0.70 -24.85 0.80 -28.83 0.90 -35.53 1.00 -45.53 0.00 -3.71 0.10 -7.95 0.20 -12.31 0.30 -15.17 0.40 -17.78 0.50 -19.93 0.60 -22.11 0.70 -24.04 0.80 -25.89 0.90 -30.44 1.00 -36.00

NASGRO Input: XYt_TR.dat

This input file for NASGRO contains the stress distribution in the entire height of

the cross section with the standard x- and y-axis. It is identical to YXt_TR.dat except for

the axis reversal.

11 11 0.0 0.1 0.2 0.3 0.4

78

0.5 0.6 0.7 0.8 0.9 1.0 0.00 3.63 0.10 1.65 0.20 1.77 0.30 2.14 0.40 1.62 0.50 0.02 0.60 -1.62 0.70 -2.19 0.80 -1.69 0.90 -1.82 1.00 -3.71 0.00 8.05 0.10 6.81 0.20 5.37 0.30 4.69 0.40 3.33 0.50 -0.04 0.60 -3.32 0.70 -4.65 0.80 -5.32 0.90 -6.19 1.00 -7.95 0.00 12.33 0.10 10.85 0.20 8.74 0.30 6.29 0.40 4.71 0.50 -0.02 0.60 -4.69 0.70 -6.18 0.80 -8.54 0.90 -10.58 1.00 -12.31 0.00 15.09 0.10 13.90 0.20 11.00 0.30 7.23 0.40 5.90 0.50 -0.23 0.60 -5.70

79

0.70 -7.39 0.80 -10.97 0.90 -13.28 1.00 -15.17 0.00 18.00 0.10 16.31 0.20 13.11 0.30 7.14 0.40 7.05 0.50 -0.12 0.60 -6.95 0.70 -7.16 0.80 -13.15 0.90 -16.73 1.00 -17.78 0.00 20.14 0.10 18.85 0.20 15.25 0.30 5.76 0.40 0.00 0.50 0.00 0.60 0.00 0.70 -5.61 0.80 -15.22 0.90 -18.88 1.00 -19.93 0.00 22.09 0.10 21.51 0.20 17.62 0.30 1.59 0.40 0.00 0.50 0.00 0.60 0.00 0.70 -1.59 0.80 -17.67 0.90 -21.46 1.00 -22.11 0.00 24.19 0.10 24.65 0.20 20.72 0.30 3.26 0.40 0.00 0.50 0.00 0.60 0.00 0.70 -3.29 0.80 -20.72

80

0.90 -24.85 1.00 -24.04 0.00 26.73 0.10 29.44 0.20 25.09 0.30 5.66 0.40 0.00 0.50 0.00 0.60 0.00 0.70 -6.54 0.80 -25.02 0.90 -28.83 1.00 -25.89 0.00 30.36 0.10 35.00 0.20 31.42 0.30 0.00 0.40 0.00 0.50 0.00 0.60 0.00 0.70 0.00 0.80 -31.27 0.90 -35.53 1.00 -30.44 0.00 37.84 0.10 45.36 0.20 41.08 0.30 0.00 0.40 0.00 0.50 0.00 0.60 0.00 0.70 0.00 0.80 -41.68 0.90 -45.53 1.00 -36.00

AFGROW Input: Stress – Average, Top Edge.wfs

This AFGROW input file has the stresses averaged across the x-axis of the cross

section in the y-direction and the stress on the upper surface in the x-direction. Using the

stress on the surface of the bushing hole in the y-direction gives a stress state that is far

lower than the crack would experience.

81

UNITS=1 2 21 0 19.859 0.00305 20.274 0.0061 20.393 0.00914 19.627 0.01219 17.378 0.01524 12.983 0.01829 4.862 0.02134 6.091 0.02438 4.519 0.02743 2.006 0.03048 -0.078 0.03353 -2 0.03658 -4.456 0.03962 -6.132 0.04267 -4.955 0.04572 -13.058 0.04877 -17.387 0.05182 -19.671 0.05486 -20.333 0.05791 -20.336 0.06096 -19.575 11 0 3.6305 0.001597 8.0471 0.003194 12.325 0.004791 15.092 0.006388 18.002 0.007985 20.144 0.00958 22.087 0.0112 24.193 0.0128 26.731 0.0144 30.358 0.016 37.836

Evaluation of fatigue crack growth software for use on cracks in ...

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