NASA TECHNICAL MEMORANDUM

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10415_" 1_3,

FASTRAN-II - A FATIGUE CRACK GROWTH

STRUCTURAL ANALYSIS PROGRAM

J. C. Newman, Jr.

FEBRUARY 1992 (REVISED COPY)

,J

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Vi_inia 23665

(NASA-TM-I04159) FASTRAN-2: A

FATIGUE CRACK GROWTH STRUCTURAL

ANALYSIS PROGP.AM (NASA) I03 p

N92-_0964

Unclas

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G3/39 0109564

CONTENTS OF MANUAL

Page

I. SUMMARY ..................................................... ,,..... 1

II. THEORETICAL MANUAL ................................................. I

A. Introduction ..................................................... I

B. Analytical Crack-Closure Model ................................... 2

C. Additions and Modifications in FASTRAN-II ........................ 5

D. Effective Stress-lntensity Factor against Crack-Growth Rate ...... 11

E. Computer Systems ................................................. 15

F. References ....................................................... 16

Ill. USER GUIDE FOR FASTRAN-II .......................................... 18

A. Introduction ........................................ ,............. 18

B. Analysis (Input Data) File ....................................... 18

I. Problem Title ................................................. 18

2. Spectrum Filename and Time Limit .............................. 18

3. Material ...................................................... 18

4. Material Tensile Properties ................................... 18

5. Fatigue-Crack Growth Rate Option .............................. 20

6. Fatigue-Crack Growth Rate Equation and Fracture Properties .... 20

a. Constant-Amplitude Crack-Growth Rate Equation ............. 20

b. Fracture Criterion .......................... ,............. 21

7. Fatigue-Crack Growth Rate Table ............................... 22

8. Crack Growth Rates at Transition (NALP = I option only) ....... 23

9. Data Output Options ............. . ..................... 24

IO. Specimen Type and Loading . .................................. 25

11. Specimen and Crack Starter-Notch Dimensions 32

12. Final Crack Length Requested .................................. 33

13. Special Input for Various Crack Configurations ................ 33

14. Input Constant-Amplitude Loading to Initiate Crack from

Starter Notch (Precracking Stage) ............................ 34

15. Special Input for Proof Test or Constant Crack-Opening

Stress Concept 34

16. Input Primary Fatigue Loading ................................. 35

a. Constant- or Variable-Amplitude Loading ................... 35

b. TWIST or Mini-TWIST Flight-Load Sequence... ....... 36

c. FALSTAFF Flight-Load Sequence ............................. 38

d. Space Shuttle Flight Load Sequence ........................ 38

e. Gaussian Load Sequence .......... ,.......................... 38

f. Felix-28 Helicopter Flight Load Sequence .............. .... 39

g. Spectrum Read from List of Stress Points .................. 39

h. Spectrum Read from Flight-by-Flight Loading ............... 40

17. Input Variables for Load-Reduction Threshold Test ............. 40

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C. Spectrum Input Files ............................................. 41

I. List of Stress Points ......................................... 41

2. Flight-by-Flight Loading ...................................... 42

D. System of Units .................................................. 42

E. Error Messages ................................................... 43

IV. FASTRAN-II FLOW CHART .............................................. 46

V. FASTRAN-II ANALYSIS FILE--Quick Reference Guide .................... 48

VI. EXAMPLE PROBLEMS ................................................... 5]

A. Introduction

]. Center-Crack Tension under Constant-Amplitude Loading ......... 51

2. Through Cracks from Hole under Repeated Spike Loading ......... 53

3. Surface-Crack Tension under Constant-Amplitude Loading

(LFAST = ] Option) ........................................... 54

4. Corner Crack from Hole under User Specified Loading ........... 55

5. Corner Crack from Hole Under Space Shuttle Loading

(LFAST = 3 Option) ........................................... 55

6. Small Crack at Notch under Mini-TWIST Spectrum Loading

(NALP = ] Option) ......................

7. Compact Specimen under Stress-lntensity Factor'LoadinGiiiii_il 56

8. Center-Crack Tension under NFOPT = 8 Option ................... 57

g. Center-Crack Tension under NFOPT = 9 Option ................... 58

]0. Threshold Test (KTH = 3 Option) ............................... 58

B. Listing of Input Data Files for Examples ......................... 59

C. Listing of Output Data Files for Examples ........................ 65

VII. USER GUIDE FOR EFFECTIVE STRESS-INTENSITY FACTOR PROGRAM ........... 94

A. Introduction ..................................................... 94

B. Crack-Growth Rate Analysis (Input Date) File ..................... 94

I

2

3

4

5

6

7

Data Set Title ................................................ 94

Material ...................................................... 94

Specimen Type and System of Units Option ...................... 94

Material Tensile Properties ................................... 95

Crack Growth Rates at Transition (NALP = I option only) ....... 95

Data Points, Loading and Specimen Dimensions .................. 96

Stress-lntensity Factor Range and Rates ....................... 96

C. Interactive Input for Personal Computer .......................... 96

D. Example Input and Output Data Files .............................. 97

E. Error Messages ................................................... 100

II
FASTRAN-II

A Fatigue Crack Growth Structural Analysis Program

I. SUMMARY

FASTRAN-II is a life-prediction code based on the crack-closure concept

and is used to predict crack length against cycles from a specified initial

crack size to failure for many common crack configurations found in

structural components. FASTRAN-II is an update of the previous version

FASTRAN. The life-prediction method used in FASTRAN-II is built around an

analytical crack-closure model. The model is based on plasticity-induced

fatigue-crack closure and is used to calculate the stress level at which the

crack tip becomes fully open during cyclic loading. The applied cyclic

loads may be constant-amplitude, variable-amplitude or spectrum loading.

Several standardized flight-load spectra (TWIST, Mini-TWIST, FALSTAFF, and

Felix-28) and other load spectra (Space Shuttle and Gaussian) are included

as options, Spectrum loads may also be input by the user as either a list

of stress points or a flight-by-flight sequence. The program uses the

crack-closure concept to account for load-interaction effects. Tensile or

compressive loads can be applied. The program contains seventeen predefined

crack configurations; and the user can define one crack configuration. The

crack-opening stresses, as a function of load history and crack length, are

calculated from the model and the effective stress-intensity factor range,

used to correlate fatigue-crack growth rates, may be either elastic or

modified for plastic yielding at the crack tip. Ten example problems are

included with the user guide to demonstrate the many options in FASTRAN-II,

A computer program DKEFF was also developed to analyze laboratory

specimen data to obtain the effective stress-intensity factor against crackgrowth rate relations used by FASTRAN-II. A user guide for this program is

also included and discussed. (Computer programs are available from COSMIC.)

II. THEORETICAL MANUAL

A. Introduction

Fatigue tests on metallic materials have shown that fatigue cracks

remain closed during part of the load cycle under constant- and variableamplitude loading []]. The crack-closure concept has been used to correlate

crack growth rates under constant-amplitude loading and has been shown to be
a significant factor in causing load-interaction effects on crack growth

rates under variable-amplitude loading. Crack closure is caused by residual

plastic deformation remaining in the wake of an advancing crack. To develop

the rationale for predicting crack growth under general cyclic loading, a

mathematical model of crack closure was developed [2] and verified for

constant-amplitude, variable-amplitude and aircraft spectrum loading [3].

This user guide for FASTRAN-II is an update of a previous user guide

developed for FASTRAN and submitted to COSMIC [4].

FASTRAN-ii is a life-prediction code based on the crack-closure concept

[2,3]. The program calculates crack length against cycles from a user

specified initial crack size to failure for many common crack configurations

found in structural components. The life-prediction method uses an

analytical crack-closure model based on plasticity-induced fatigue crack

closure. The crack-closure model calculates the stress level at which the

fatigue crack tip becomes fully open during cyclic loading. The applied

cyclic loads may be constant-amplitude, variable-amplitude or spectrum

loading. The program uses the crack-closure concept to account for loadinteraction effects (acceleration and retardation). Either tensile or

compressive loads can be applied to a variety of crack configurations. The

current program has seventeen predefined crack configurations and the user

may define one other crack configuration by programming the stress-intensity

factor equation in a special subroutine. The crack-opening stresses as a

function of load history and crack length are calculated using either

contact stress-intensity factors [2,3] or crack-surface displacement

equations [5]; and the effective stress-intensity factor range (AKeff) may

be either elastic or modified for plastic yielding at the crack tip. Many

of the input and internal variables used in the program are defined in the

comment section at the beginning of the main program.

B. Analytical Crack-Closure Model

The analytical crack-closure model was developed for a central crack in

a finite-width specimen subjected to uniform applied stress [2]. The model

was later extended to through cracks emanating from a circular hole in a

finite-width specimen also subjected to uniform applied stress [5]. The

model was based on the Dugdale model [6], but modified to leave plastically

deformed material in the wake of the crack. The primary advantage in using

this model is that the plastic-zone size and crack-surface displacements are
obtained by superposition of two elastic problems--a crack in a plate

subjected to a remote uniform stress and a crack in a plate subjected to a

uniform stress acting over a segment of the crack surface.

Figure I shows a schematic of the model at maximum and minimum applied

stress. The model is composed of three regions: (1) a linear-elastic

region containing a circular hole with a fictitious crack of half-length

c'+ p, (2) a plastic region of length p, and (3) a residual plastic

deformation region along the crack surface. The physical crack is of length

c'- r, where r is the radius of the hole. The length of the compressive

plastic zone is _. Region I is treated as an elastic continuum. Regions 2

and 3 are composed of rigid-perfectly plastic (constant stress) bar elements

with a flow stress, %. The flow stress (%) is the average between the

yield stress and the ultimate strength of the material. This is a firstorder approximation for strain hardening. The shaded regions in Figure 1

indicate material that is in a plastic state. At any applied stress level,

the bar elements are either intact (in the plastic zone) or broken (residual

plastic deformation). The broken elements can carry compressive loads only,

and then only if they are in contact. At the maximum applied stress and

when the crack is fully open, the effects of state of stress on plastic-zone

size and displacements are approximately accounted for by using a constraint

factor, _. The constraint factor is used to elevate the tensile flow stress

for the intact elements in the plastic zone. The effective flow stress aa o

under simulated plane-stress conditions is oo (usual Dugdale model) and

under simulated plane-strain conditions is 3%. The value of 30o was

established from elastic-plastic finite-element analyses under plane-strain

conditions using an elastic-perfectly-plastic material (normal stress

elevation in the crack-tip region was about 2.7 from the analysis). For

sheet and plate material, fully plane-strain conditions may not be possible.

Irwin [7] suggested a modification to account for through-the-thickness

variation in stress state by introducing a constraint factor of a = 1.73 to

represent nominal plane-strain conditions. At the minimum applied stress,

some elements in the plastic zone and elements along the crack surface that

are in contact may yield in compression when the contact or compressive

stress reaches -%. The loss of constraint under compression was justified

on the grounds that when a crack closes the large stress gradient at the

crack tip is greatly reduced and a more uniform stress field is produced.

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C. Additions and Modifications in FASTRAN-II

FASTRAN-II has several additional options not available in the original

version of FASTRAN. The new program has the ability to model different

crack-growth properties in the crack depth (a) and crack length (c)

directions, for surface or corner cracks, and the properties may be given in

either equation or tabular form. To help the user input spectrum loadings

not included as special options, two additional methods have been added: a

list of stress points and a flight-by-flight input. Also, several

additional flight-load spectra, Space Shuttle, Gaussian, Felix-28, Inverted

FALSTAFF and clipped versions of TWIST and Mini-TWIST, were added to the

preexisting standard load spectra TWIST, Mini-TWIST, and FALSTAFF in the

original version. The ability to simulate proof testing on a center-crack

tension specimen was added. In addition to applied stresses, crack growth

can now be simulated under constant stress-intensity factor ranges for

center-crack and compact-tension specimens. Several new predefined crack

configurations were also added.

FASTRAN-II includes several modifications to the crack-closure model to

improve crack-growth rate correlations and to calculate crack-opening

stresses during variable-amplitude loading. These modifications will be

briefly discussed here.

Closure-Model Activation.-FASTRAN-II, like FASTRAN, is a fatigue crackgrowth program built around the analytical crack closure model. The closure

model calculates the crack-opening stress as a function of applied stress

history. During the crack-growth stage, the crack-opening stress is held

constant for a certain amount of crack extension, AC* (see ref. 2). The

stresses applied to the crack configuration are monitored during this stage

for the highest stress (SMAXH), the lowest stress (SMINB) before SMAXH, and

the lowest stress (SMINA) after SMAXH. Once the crack has grown the Ac*

increment, the closure model is activated and a new value of crack-opening

stress is calculated using SMINB, SMAXH, SMINA and ac*. However, during the

Ac*-growth stage, several other events can activate the model. The model is

automatically activated when the cycles needed to create Ac* exceed 1000

(the original model used 300). Also, stress excursions are monitored, and

if they exceed a preset value (such as large overloads or underloads), the

model is activated. The process is repeated until the configuration fails

or the desired crack length has been exceeded.
_rack-Lenqth Calculations.-The calculation of ac* the crack-growth

increment over which the crack-opening stress is held constant during

constant-amplitude, variable-amplitude or spectrum loading was changed. In

the original version, ac* was 5 percent of the monotonic plastic-zone size,

#. In the revised version, ac* is set at 20 percent of an approximate

cyclic plastic-zone size (_) as

ac* = 0.2 _ = 0.2 (p/4) (] - Rx)2 (I)

where p is the plastic-zone size using the current maximum stress and Rx is

the stress ratio using the current minimum and maximum applied stress. For

stress ratios less than zero, Rx is set equal to zero. Note that for a

stress ratio of zero, equation (I) reduces to 5 percent of the plastic-zone

size as used in the original version. This modification was made to improve

the crack-opening stress calculations under high stress ratio conditions.

Crack-0peninq Stresses.-The method used to calculate crack-opening

stresses (So) in the original version of FASTRAN assumed that the crackgrowth increments were small compared to crack length [2]. In proof testing

or severe spectrum loading, however, crack-growth increments may be large

compared to crack length. Therefore, the closure-model analysis in FASTRANII was modified. The contact stress-intensity factor equation for

calculating crack-opening stress (eq. (27) in ref. 2) was changed to

n

' = - _ [arcsin B2 arcsin BI]j So Smin j=1](2°j/=)

B k = [sin(_bk/2W)]/[sin(_c/2w)] for k = ] or 2.

(2)

where (3)

The prime denotes the modified crack-opening stress. The index j refers to

elements along the crack surface that are used to model plastic and

residual-plastic deformations, as shown in Figure ](b). The crack length,

c, is the sum of the hole radius plus all of the elements from j = 1] to n.

The dimension bk denotes an edge of an element. The width of an element is

b2 - bI. The specimen width (w) is measured from the centerline or edge.

The modification changed the upper limit of the summation in equation (2)

from n-1 to n to reflect the change in formulation from the crack length

before growth to actual or current crack length. Element ] is the first

intact element at the crack tip. Elements 1 to ]0 are always ahead of the

current crack tip and model the plastic zone, p. The intact elements in the

plastic zone may carry tensile stresses up to moo and compressive stresses
down to -ao. Elements 11 to n are located along the crack surface between

the initial notch and the current crack tip. In the physical model, the

crack was extended by the width of the n th element at the maximum applied

stress, see reference 2. The physical crack extension was ac*. However, in

the computation of the crack-opening stress in equation (2), the total width

of the nth element (ac*) was not used. The width used in equation (2) for

element n was the largest crack growth increment during the generation of

ac*. For low fatigue crack growth rates, the width of element n was

generally very small compared to ac*. But for proof testing, the width of

element n was equal to ac*, the crack extension during proof loading. At

the minimum applied stress, elements along the crack surface that are in

contact carry aj, the element contact stress, otherwise the element stress

l

is zero. For low fatigue crack growth rates, SO calculated from equation

(2) was nearly equal to So calculated from the previous equation developed

!

in reference 2. However, for large crack-growth increments, SO from

equation (2) is now a function of crack extension. See reference 8 for

further details on this modification and on the proof-test application.

Crack-Open!nq Stress Equations.-In reference 9, crack-opening stress

equations for constant-amplitude loading were developed from the analytical

crack-closure model (So) calculations. These equations were fit to the

results from the closure model and gave SO as a function of stress ratio

(R), maximum stress level (Smax/Oo) and a constraint factor (_). Herein, a

modification to correct these equations for large crack-growth extensions or

rates is presented [8]. In the closure model analysis, the nth element

always carry the compressive flow stress (-ao) during unloading. Thus, the

contribution of this element to the crack-opening stress can be estimated as

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follows. An approximate equation for SO under constant-amplitude loading

was derived from stress-intensity factors for a crack in an infinite plate

assuming that the contact stress distribution was linear. The assumed

contact stress distribution along the crack surfaces varied from the

compressive flow stress (-Oo) at the crack tip and to zero at the end of the

contact region. The modification to correct the crack-opening stress

equation is given by
J_

S° So + 0.3 % JAc/c /F for Smax/% < 0.6 (4)

where Ac is the crack-growth increment (or rate per one cycle) and c is the

current crack length. The boundary-correction factor, F, was added to these

equations to account for the influence of finite width. The crack-opening

stress SO is calculated from equations developed in reference 9. These

equations will be presented later. Comparisons with the model showed that

equation (4) was reasonably accurate for a wide range in constant-amplitude

loading conditions for Smax/% less than about 0.6. The difference between

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SO and SO was only significant (greater than a 2 percent effect on crackopening stresses) for growth rates greater than about 10.5 m/cycle. For

Smax/% > 0.6, the crack-opening stresses should be obtained from FASTRAN-II

by conducting an analysis of the test specimen.

The crack-opening stress equation developed in reference g will be

presented here for completeness. These equations were developed by fitting

to the calculated results from the closure model for a center-crack tension

specimen. Note that the crack-opening stress SO was calculated from

equation (2) where n is replaced by n-1 (original formulation). The

equation was given by

So/Sma x = A 0 + AIR + A2R2 + A3R3 for R _ 0 (5)

and So/Sma x = A 0 + AIR for R < 0 (6)

where R = Smin/Sma x, Sma x < 0.8%, Smi n > -%, So = Smin if So/Sma x is less

than R, and So/Smax = 0 if So/Sma x is negative. The A i coefficients are

functions of _ and Smax/Oo and are given by:

A0 = (0.825 0.34_ + 0.05_ 2) [cos(_SmaxF/2oo)] 1/_

A I = (0.415 O.07]e) SmaxF/oo

(7)

A2 = ] - A0 A] - A3

A3 = 2A0 + A] - ]

for e = ] to 3. Again, the boundary-correction factor, F, was added to

these equations to account for the influence of finite width on crackopening stresses. See reference 8 for a comparison between these equations

8
and the model calculations for a wide range in stress levels and stress

ratios. Equations (4)-(7) give approximate crack-opening stress equations

that agree fairly well with the results from the modified analytical crack

closure model. These equations are used to correlate fatigue crack-growth

rate data to obtain AKef f against crack-growth rate relations. (These

equations are programed in Subroutine SOEQN.)

Crack-Openinq Stresses for the Compact SDecimen.-Equations (5) and (6}

were originally developed for a through crack in a center-crack tension

specimen subjected to remote stress. Similar equations are not available

for the compact specimen. Because the compact specimen is used quite often

to obtain crack-growth rate data, crack-opening expressions are needed to

develop baseline AKeff-rate relations. A simple approximation can be made

to estimate crack-opening stresses. The approximation is based on matching

stress-intensity factors from the compact specimen to that for a center-

#

crack tension specimen. A stress Sma x is applied to a center-crack specimen

having the same crack length and width as the configuration of interest so

that the same stress-intensity factor is developed. Thus, the plastic-zone

size, the local crack surface displacements, the local residual plastic

deformations and the crack-opening stresses would be nearly the same in both

configurations. For example, the stress-intensity factor for the compacttension Specimen is

Kct = P/(wt) J_c Fct (8)

where P is the applied load and Fct is the boundary-correction factor. Note

that equation (8) is expressed as a stress P/(wt) times the square-root of

the crack length instead of in the usual form, P/(tJw). The stressintensity factor for the center-crack tension specimen is

Kcc t = S_a x J_-c Fcc t (g)

where Fcc t is the boundary-correction factor. Equating equations (8) and

(9) gives

S_a x = P/(wt) Fct/Fcc t (10)

Thus, S_a x replaces Sma x in equation (5) to estimate the crack-opening

stress for the compact-tension specimen (R _ 0). This approach gives crackopening stresses for the compact specimen that are a function of stress

g
ratio, stress level, and constraint. All other crack configurations in

FASTRAN-II are subjected to a remote applied stress and the original

procedure is expected to give accurate crack-opening stresses. Further

study is needed to evaluate the K-analogy procedure for other crack

configurations, especially the three-dimensional crack configurations.

Effective Stress-lntensity Factor Ranqe.-The effective stress-intensity

factor range developed by Elber [I] was based on linear-elastic analyses.

For high stress-intensity factors, such as those during proof testing or

near failure, plastic-zone sizes are no longer small compared to crack size

and linear-elastic analyses are inadequate. To modify the analysis to

account for plasticity, a portion of the Dugdale plastic-zone length (p),

like the well-known Irwin plastic-zone correction, was added to the current

crack length (c). Thus, the effective stress-intensity factor range

corrected for plasticity is

AKef f * (S - S_) J_-d F(dlw,d/r,...) (11)

!

where S is the maximum applied stress, SO is the crack-opening stress

developed by the previous cyclic load history, d = cx + _/4, where cx is the

current crack length (c) plus the crack-growth rate (dc/dN) per one cycle

and _ is the closure-corrected cyclic plastic-zone size. For large crackgrowth rates, the effective stress-intensity factor against rate relation is

used as a resistance curve [8], like a K R resistance curve, and the crack

driving force is calculated at the current crack length plus any crack

extension value. The boundary-correction factor F may be a function of

width, thickness, hole or notch radius and other configuration parameters.

The boundary-correction factor F is evaluated using the fictitious crack

length, d. The plasticity correction of "one-quarter of the cyclic plastic

zone" was developed in reference ]0 by equating an estimate of the cyclic Jintegral (AJ) to a plasticity-corrected cyclic stress-intensity factor

(AKp). (Note: The use of _/4 as the plasticity correction needs further

experimental and analytical verification.) The closure-corrected cyclic

plastic-zone size (_) is approximated by

= (p/n) (1 Reff)2 = (p/4) (! So/Smax ' )2 (12)

where p is the Dugdale plastic-zone size and Ref f is the ratio of crackopening stress to the maximum stress. The user has the option to select

either elastic or plasticity-corrected stress-intensity factors. If elastic

I0
stress-intensity factors are selected, then d = c in equation (11). With

either option, the baseline crack-growth rate data must be analyzed under

the same conditions, that is, using either elastic or plasticity-corrected

stress-intensity factors.

The plastic-zone length (p) for a center-crack tension specimen was

determined from an infinite periodic array of Dugdale models and was

modified (see ref. 5) for a finite-width specimen. The modified equation is

p = c((2w/_c) arcsin [sin(_c/2w) sec(_Sf/2_Oo)] -1) (13)

where f = 1 + 0.22(c/w) 2, e is the constraint factor [2], and oo is the flow

stress (average between yield stress and ultimate tensile strength). The

constraint factor elevates the flow stress at the crack tip due to the

three-dimensional stress state. For plane-stress conditions, e is assumed

to be unity and for plane strain, e = 3. For other crack configurations,

the plastic-zone size may be estimated by using the small-scale yielding

solution

p = _/8 (Kmax/_Oo)2 (14)

D. Effective Stress-lntensity Factor against Crack-Growth Rate

m

To make life predictions, AKef f (or AKeff) as a function of the crackgrowth rate must be obtained for the material of interest. Fatigue crackgrowth rate data should be obtained over the widest range in rates possible

(from threshold to fracture), especially if spectrum load predictions are

required. Data obtained on the crack configuration of interest would be

helpful but is not essential. The use of the plasticity-corrected stressintensity factor is only necessary if proof testing or severe loading (such

as low cycle fatigue conditions) are of interest. Most damage-tolerant life

calculations can be performed using the linear elastic stress-intensity

factor analysis with crack-closure modifications.

To obtain the baseline AKef f (or AKeff) against crack-growth rate

relations, the fatigue crack-growth rate data must be analyzed using the

equations previously presented. A computer program DKEFF (dkeff.for) was

also developed to help the user determine the effective stress-intensity

factor range against crack-growth rate relations. A user guide for this

program is given in Section VII.

I]
Under constant-amplitude loading, the only unknown in the equations is

the constraint factor, _. The constraint factor is determined by finding

(by trial-and-error) an _ value that will correlate the constant-amplitude

fatigue-crack-growth-rate data over a wide range in stress ratios, as shown

in references 3, 5, 8, 9, and 11. This correlation should produce a unique

relationship between AKef f (or AKeff) and crack-growth rate, as shown in

Figure 2. The AK-rate (linear-elastic fracture mechanics) data on 7075-T6

aluminum alloy, shown in Figure 2(a), were generated by C. M. Hudson []gJ

and E. P. Phillips (unpublished) at the NASA Langley Research Center. The

values of _ for the AKeff-rate correlation, shown in Figure 2(b), were

determined by trial-and-error. The variable-constraint (e) regime (1.8 >

> ].2) was associated with the transition region from flat- to slant-crack

growth (see ref. ]1).

In the large-crack-growth threshold regime for some materials, the

plasticity-induced closure model may not be able to collapse the threshold

(AK-rate) data onto a unique AKeff-rate relation because of other forms of

closure. Roughness- and oxide-induced closure (see for example, ref. 20)

appear to be more relevant in the threshold regime than plasticity-induced

closure. Further study, however, is needed to assess the interactions

between plasticity-, roughness- and oxide-induced closure in this regime.

If the plasticity-induced closure model is not able to give a unique AKef f

relation in the threshold regime, it is suggested that the high stress ratio

(R > 0.5) data be used to establish the AKef f relationship.

For variable-amplitude or spectrum load crack-growth predictions, the

constraint factor (e) should also be verified by some simple tests, such as

crack growth after a single-spike overload. Constraint Factors appear to be

more sensitive to crack-growth delays caused by single-spike overloads than

to crack growth under constant-amplitude loading at different stress ratios.

Higher values of constraint (e) will cause less load-interaction effects,

such as retardation or acceleration, than lower values of constraint. Thus,

spike-overload tests may be more useful in establishing values of _ than

constant-amplitude tests (see ref. 2]).

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E. Computer Systems

FASTRAN was designed to be a research program, rather than a production

program, due to the excessive computer time and cost involved in running the

program. However, in the last decade, major advancements made in computer

science and some modifications made in thecode aremaking implementation of

FASTRAN-II more practical. The computer time is directly related to the

number of cycles required to grow a crack from an initial size to failure.

Thus, crack-growth predictions under low stress levels will require more

time than under high stress levels for the same initial crack length. The

current version, FASTRAN-II, is written in FORTRAN-77 and executes on both

mainframe and personal computers using a Unix operating system.

A CONVEX-C2]O computer was used to generate the crack-growth

predictions made on the example problems included in this manual. The

CONVEX-C2]O uses the UNIX-based operating system CONVEX UNIX Version 7.].

If the source program is denoted as 'fastran2.f', the program can be

compiled on a CONVEX computer using the fc compiler statement:

fc -02 -o fastran.ex fastran2.f

The executable program is denoted as 'fastran.ex'. The executable program

runs about three times faster on a CONVEX computer under the -02 option

(scalar and vector optimization) than under normal compilation. The

IMPLICIT REAL*8 (A-H,O-Z)'Statement was used to make all operations double

precision; all intrinic functions are also used in double precision. The

program is executed interactively by

fastran.ex < infast &

where 'infast' is a file containing the analysis input data filename

'infile' on the first line and the output filename 'outfile' on the second

line. The program has internal timing statements using the SECONDS()

command to time a particular job and to terminate execution for each input

problem, if the user specified time limit is exceeded. The timing

statements can be removed if the SECONDS() command is not available.

On a Personal Computer (IBM PS/2 Model 80 with an 80387 math

coprocessor ) , the program 'fastran2.f' had to be subdivided into the main

program and severalsub-programs to compile using the Microsoft FORTRAN

compiler (Version 4.01). (The comment on all WRITE(6,...) statements can be

removed to activate some output to the screen. These write statements are

used to verify program execution. The complete output file is 'outfile'.)

The program excution times on the PC were about 60 times slower than those

using the optimized program on the CONVEX computer.

I5
F. References

]. Elber, W.: The Significance of Fatigue Crack Closure, Damaqe Tolerance

of Aircraft Structure-s, ASTM STP-486, 1971, pp. 230-242.

2. Newman, J. C., Jr.: A Crack-Closure Model for Predicting Fatigue Crack

Growth Under Aircraft Spectrum Loading, Methods and Models for Predictinq

Fatique Crack Growth Under Random Loadinq, J. B. Chang and C. M. Hudson,

eds., ASTM STP 748, ]98], pp. 53-84. (Also NASA TM-81941, Jan. 1981).

3. Newman, J. C., Jr.: Prediction of Fatigue Crack Growth Under VariableAmplitude and Spectrum Loading Using a Closure Model, Desiqn of Fatique

and FrBct_re Resistant Structures, P. R. Abelkis and C. M. Hudson, eds.,

ASTM STP 761, 1982, pp. 255-277. (Also NASA TM-81942, Jan. 1981).

4. Newman, J. C., Jr.: Fatigue Crack Growth Analysis of Structures

(FASTRAN)--A Closure Model, Computer Software Management and Information

Center (COSMIC), University of Georgia, Athens, Georgia, December 1984.

5. Newman, J. C., Jr.: A Nonlinear Fracture Mechanics Approach to the

Growth of Small Cracks, AGARD CP-328, 1983.

6. Dugdale, D. S.: Yielding of Steel Sheets Containing Slits, Journal of

Mechanics and Physics of Solids, Vol. 8, No. 2, 1960, pp. ]00-104.

7. Irwin, G. R.: Plastic Zone Near a Crack and Fracture Toughness,

Proceedings of the 7th Sagamore Conference, 1960, pp. IV.63-1V.77.

8. Newman, J. C., Jr.; Poe, C. C., Jr.; and Dawicke, D. S.: Proof Test and

Fatigue Crack Growth Modeling on 2024-T3 Aluminum Alloy, Fatigue 90,

Volume IV, H. Kitagawa and T. Tanaka, eds., Proceedings of the Fourth

International Conference on Fatigue and Fatigue Thresholds, 1990, pp.

2407-2416.

9. Newman, J. C., Jr.: A Crack-Opening Stress Equation for Fatigue Crack

Growth, International Journal of Fracture, Vol. 24, RI31-R]35, ]984.

IO. Newman, J. C., Jr.: Fracture Mechanics Parameters for Small Fatigue

Cracks, presented at the ASTM Symposium on Small-Crack Test Methods, San

Antonio, TX, November 14, ]990. (Also to appear in ASTM STP-]149,

1992.)

II. Newman, J. C., Jr.; Swain, M. H.; and Phillips, E. P.: An Assessment of

the Small-Crack Effect for 2024-T3 Aluminum Alloy, Small Fatique Cracks,

R. O. Ritchie and J. Lankford, eds., Metallurgical Society of AIME,

1986, pp. 427-452.

12. Newman, J. C., Jr.: Fracture Analysis of Various Cracked Configurations

in Sheet and Plate Materials, Properties Related to Fracture Touqhness,

ASTM STP 605, 1976, pp. 104-]23.

13. Newman, J. C., Jr. and Raju, I. S.: Prediction of Fatigue Crack-Growth

Patterns and Lives in Three-Dimensional Cracked Bodies, NASA TM-85787,

April I984.

16
14. deJonge, J. B.; Schutz, D.; Lowak, H and Schijve, J.: A Standardized

Load Sequence for Flight Simulation tests on Transport Aircraft Wing

Structures (TWIST), National Lucht-en Ruimtevaartlaboratium, NLR-TR

73029U, 1973.

15. Lowak, H.; deJonge, J. B.; Franz, J. and Schutz, D.: Mini-TWIST--A

Shortened Version of TWIST, Laboratorium Fur Betriebsfestigkeit,

Darmstadt, Report No. TB-146, 1979.

16. van Dijk, G. M. and deJonge, J. B.: Introduction to a Fighter Loading

Standard for Fatigue Evaluation--FALSTAFF, NLR MP 75017U, May 1975.

17. Huck, M.; Schutz, W.; Fischer, R.; and Kobler, H. G.: A Standard Random

Load Sequence of Gaussian Type Recommended for General Application in

Fatigue Testing, LBF Report No. 2909, IABG Report No. TF-570, 1976.

18. Edwards, P. R.; and Darts, J.: Standardised Fatigue Loading Sequences

for Helicopter Rotors (Helix and Felix), Part 2: Final Definition of

Helix and Felix, Royal Aircraft Establishment TR-84085, August 1984.

19. Hudson, C. M.: Effect of Stress Ratio on Fatigue-Crack growth in 7075-

T6 and 2024-T3 Aluminum-Alloy Specimens, NASA TN D-5390, National

Aeronautics and Space Administration, Washington, D.C., August 1969.

20. Ritchie, R. O. and Yu W.: Short Crack Effects in Fatigue, A Consequence

of Crack Tip Shielding, Small F_tique CrBcks, R. O. Ritchie and J.

Lankford, eds., Metallurgical Society of AIME, 1986, pp. 167-189.

21. Newman, J. C., Jr. and Dawicke, D. S.: Prediction of Fatigue-Crack

Growth in a High-Strength Aluminum Alloy under Variable-Amplitude

Loading, Advances in Fracture Re@earch, Pergamon Press, Vol. 2, 1989,

pp. 945-952.

17
Ill. USER GUIDE FOR FASTRAN-II

A. Introduction

A line-by-line explanation of the input data for FASTRAN-II is

presented herein. Each of the parameters used in the input data file is

defined and explained. The input data to the program is listed in Sections

B.I to B.17. The data are input in free format unless otherwise stated.

Several problems can be solved during each computer run by placing input

data files from Sections B.I to B.I7 in series. A flow chart of FASTRAN-II

is given in Section IV and a Quick Reference Guide for the Analysis (Input

Data) File is given in Section V. The input and output data files for ten

example problems are given and discussed in Section VI. These examples

problems cover many of the options available in FASTRAN-II.

B. Analysis (Input Data) File

I. Problem Title

READ TITLE(20)

FORMAT (20A4)

Any 80-character title describing the problem.

then the computer run is terminated.

If TITLE(1) = HALT,

2. Spectrum Filename and Time Limit

READ SPECTRA, TLIMIT

FORMAT (AIO,EIO.3)

SPECTRA is the filename of the spectrum loading file (used only for

NFOPT = 5, 8 and g). For other NFOPT values, create and use a

dummy file and filename, such as 'cstamp' (see example problem I).

TLIMIT is the time limit in CPU seconds allowed for the problem.

3. Material

READ MAT

FORMAT (20A4)

Any 80-character description of the material.

4. Material Tensile Yielding Properties

READ SYIELD, SULT, E, ETA, ALP, NALP, NEP, BETA

FORMAT (*)

SYIELD = Yield stress (0.2 percent offset)

SULT = Ultimate tensile strength

E = Elastic modulus

18
ETA= 0 for plane stress (normally used)

= Poisson's ratio for plane strain

ALP - Constraint factor

= I Plane-stress condition

= ].73 Irwin's plane-strain condition

- 3 Plane-strain condition

See references 3 and 9 for procedures to obtain the value of

ALP for a particular material, thickness, and test condition.

NALP = 0 Constraint factor (ALP) is constant as input

= I Constraint factor is variable (ALP is computed by the

program, see section 8, and refs. 8 and 11)

m

The crack-growth relation, AKef f (or AKef f) against

dc/dN or da/dN, must have been obtained using the same

constraint factor as specified by the NALP option.

Section VII presents a user guide for DKEFF, a program to

determine the effective stress-intensity factor against

crack-growth rate relation for either NALP option.

NEP - 0 Effective stress-intensity factor is elastic

=1 Effective stress-intensity factor is modified for plastic

yielding at the crack tip by adding one-quarter of the

cyclic plastic zone to the crack length (see eqn.(11)).

(Note: The plasticity corrected stress-intensity factor

in FASTRAN-II is different than that used in references 5

and 8. A closure corrected "cyclic" plastic zone size is

used instead of the maximum plastic zone size.)

=2 Effective stress-intensity factor is modified for plastic

yielding at the crack tip by adding one-quarter of the

monotonic plastic zone to the crack length (d = c + p/4)

in equation (]]), see reference 10.

BETA Constraint factor on compressive yielding; Compressive

yield stress = BETA*SFLOW (BETA is usually set equal to

unity; option has not been completely evaluated)

The flow stress, SFLOW, computed in the program is given by the

average between the yield stress and ultimate tensile strength.

KOPEN is used to select the method of calculating the crack-opening

stress, SO. For KOPEN = O, SO is computed from crack-surface

displacements. For KOPEN = I, SO is computed from contact stressintensity factors. Under constant-amplitude loading, very little

difference in SO is observed between KOPEN = 0 or I. But under

variable-amplitude loading, some differences are observed. KOPEN

is automatically set in the program. The current version of

FASTRAN-II has KOPEN = I. The user can make KOPEN = 0 to activate

the crack-surface displacement option. (The crack,opening stress

equations in the Theoretical Manual, equations (5)-(7), apply for

either value of KOPEN.)

19
5. Fatigue-Crack Growth Rate Option

READ IRATE

FORMAT(*)

IRATE= 1 da/dN is equal to dc/dN as a function of AKeff

= 2 da/dN and dc/dN have different AKeFf relations

The crack depth, a, is always measuredwith respect to plate

thickness (t) and crack length, c, is measured in the plate

width (w) direction.

Repeat sections. 6 to 7 IRATE times (J_= I to IRATE).

6. Fatigue-Crack Growth Rate Equation and Fracture Properties

READ CI(I,J), C2(I,J), C3(J), C4(J), C5(J), KF, M

FORMAT (*)

Cl(1,J) =

C2(I,J) =

C3(J) =

C4(J) =

C5(J) =

J=l

KF=

M =

Crack-growth coefficient for property J, C l

Crack-growth power for property J, C 2

Threshold constant, C 3

Threshold constant, C4

Cyclic fracture toughness or limiting value of maximum

stress-intensity factor, C 5

Properties in crack length c-direction

= 2 Properties in crack depth a-direction

Elastic-plastic fracture toughness, K F (see ref. ]2)

Fracture toughness parameter, 0 _ m _ ] (see ref. 12)

ConstaDt-Amplitude Crack-Growth Rate Equation [21

C2

dc/dN (or da/dN) = C l aKef f [l - (aKo/aKeff)2]/[1

where

aKo = C3 (I - C4Smin/Smax) = C3 (I C4R )

(Kmax/C5]2 (15)

(16)

Kma x = Sma x J_-c F

(17)

20
and

AKeff = (Smax - Sx) J=c x F (18)

f l

Sx = SO if SO is greater than Smin, otherwise Sx = Smi n

Smi n = Minimum applied stress

Sma x = Maximum applied stress

So Crack-opening stress

cx = Current crack length (c) plus rate (dc/dN) per cycle

F = Boundary-correction factor (F is computed in Subroutine BCF)

Kma x = Maximum stress-intensity factor

AKo = Effective stress-intensity factor threshold (eq. (16))

Note that equation (16) is slightly different than that used in

references 2 and 3. Caution should be exercised when using

thresholds. Information obtained on thresholds [5] may make many

of the thresholds invalid for small cracks.

Fracture Criterion (see ref. 12)

Failure occurs when Kma x _ Kie

Kle = _ K F (I - m Sn/Su)

where

1 for Sn < = _ Oy s

= Oys/Sn for Oy s < Sn < au

Sn = Net-section stress (computed from applied stress)

ay s = Yield stress (0.2 percent offset)

au = Ultimate tensile strength

Su = Net-section stress to produce plastic hinge

KF = Elastic-plastic fracture toughness

(19)

21
m = Fracture toughness parameter (0 _ m _ 1)

(m = 0 is for LEFM analysis; m - 1 is for very ductile

materials; see reference 12)

Fracture can be simulated in three different ways.

I. Using the crack-growth rate equation (eq. (15)), failure

occurs if Kma x > C5.

2. The Two-Parameter Fracture Criterion [12], KF and m, can also

be used. Both conditions: Kma x > C5 and Kma x > Kie are

checked. Failure occurs if any of the two conditions are

satisfied. [Set either C5 or KF to very large values (999.)

to deactivate.]

3. Failure occurs under any option if the plastic-zone size (#)

from the Dugdale model exceeds the remaining uncracked

ligament (or net section).

7. Fatigue-Crack Growth Rate Table (see ref. 5)

a. READ NTAB, NDKTH

FORMAT (*)

NTAB = 0 Program uses crack-growth rate equation (eq. (15))

- Value greater than one indicates number of data points

used to describe crack-growth rate data for tabular

input (table must contain at least two data points)

If NTAB > I then:

NDKTH - 0 dc/dN - f(AKeff) with no crack growth if AKeff< AKo

- 1 dc/dN = g(AKef f - AKo)

If IRATE - 1, da/dN and dc/dN have same AKef f relation

If IRATE - 2, da/dN and dc/dN have different tabular input for

AKef f relations

For NTAB > !, input table (I = 1 to NTAB)

b. READ DKETAB(I,J), CGRTAB(I,J)

FORMAT (*)

DKETAB(I,J) = Effective stress-intensity factor range (AKeff)

CGRTAB(I,I) = Crack-growth rate (dc/dN) at DKETAB(I,I)

CGRTAB(I,2) = Crack-growth rate (da/dN) at DKETAB(I,2)

22
Restriction: DKETAB(I,J) must not be equal to DKETAB(I+I,J),

no segmentswith infinite slope allowed.

The AKeff and dc/dN (or da/dN) relationships may be independent of

stress ratio (R) if the proper constraint factor (ALP) has been

used.

Thresholds:

The AKeff-thresholds are accounted for in the analysis by using

equation (16). A crack will not grow if AKef f is less than AK o.

Fracture:

Fracture can be simulated in two different ways.

1. Using the crack-growth rate table (section 7), the TwoParameter Fracture Criterion is used (see section 6.b).

Failure occurs if Kma x > Kie. [Note that C5 in the crackgrowth rate equation (eq. 15) is no__ttused.]

2. Failure occurs under any option if the plastic-zone size

exceeds the remaining uncracked ligament (or net section).

If IRATE = 2, repeat sectiQns 6 to 7, otherwise continue.

If IRATE = I, the crack-growth rate properties in the a- and cdirections are automatically set equal to each other, that is

CI(1,I) = C](I,2), C2(I,]) = C2(1,2), C3(I) = C3(2), C4(I) = C4(2)

and C5(I) =)C5(2. When the tabular input is used, CI(I,J) and

C2(I,J) (coefficient and power) are computed in Subroutine MATPROP

for linear segment I and property J and these values are used in

Subroutine RATE.

8. Crack Growth Rates at Transition (NALP = I option only)

READ RATE], ALP], BETA], RATE2, ALP2, BETA2

FORMAT (*)

If NALP = 0 No input is required, go to section 9.

If NALP = ] RATE] is the crack-growth rate near the start of

transition from flat-to-slant growth (ALP = ALP] and

BETA = BETA] for rates less than RATE]). RATE2 is the

crack growth rate near the end of the transition from

flat-to-slant growth (ALP = ALP2 and BETA = BETA2 for

rates greater than RATE2). For rates (RATES) between

RATE] and RATE2:

log(RATES) - log(RATE2) } ALP2) log(RATE]) -

ALP = ALP2 + (ALP] (20)

23
ALP - ALP] }

BETA = (BETA2 - BETAI) + BETAI

ALP2 ALPI

(21)

where ALP is calculated from equation (20).

These equations are in the main program under the VARIABLE

CONSTRAINT section. BETA is the constraint factor on compressive

yielding. The influence of BETA on crack-opening stresses has not

been studied. Thus, BETAI and BETA2 should be set equal to unity.

9. Data Output Options

READ NIPT, NPRT, LSTEP, NDKE, DCPR

FORMAT (*)

NIPT = 0 Internal print off

= Value greater than zero indicates output at a half-cycle

after "NIPT" crack-growth increments. Applied stress,

crack length, element sizes, element displacements, element

stresses, and maximum plastic-zone size are printed out at

LSTEP applied load increments during each half-cycle.

NPRT specifies frequency of crack-length-against-cycles output.

= Zero or negative number indicates crack-length-againstcycles output at specified crack-growth increment (DCPR)

One or greater, indicates crack-length-against-cycles

output at every "NPRT"th crack-growth (_c*) increment in

the closure model.

For either value of NPRT, the program prints the following"

total blocks (or flights) applied (NFLT), crack length (_1"or

crack length minus hole radius (C' - RAD), crack-depth-toplate-thickness (A/T) ratio, number of cycles (N), constraint

factor (ALP), crack-opening stress-intensity factor ratio

(KO/KMAX), either the stress-intensity factor range (DKC) or

effective stress-intensity factor range (DKEC) at C, and the

crack-growth rate (DC/DN). For surface and corner cracks, the

stress-intensity factor range (DKA) or effective stressintensity factor range (DKEA) at A and DA/DN are also printed.

LSTEP = Number of load steps from minimum to maximum load during

NIPT printout (usually set equal to one). Use values

greater than one to study how the crack surfaces open.

This option gives the contac_ stresses and stresses in the

plastic zone during cyclic loading.

NDKE = 0 Print out elastic stress-intensity factor ranges

- I Print out effective stress-intensity factor ranges

DCPR = Crack-growth increment at which crack-length-against-cycles

information are printed out

24
]0. Specimen Type and Loading (see Figs. 3 and 4)

READ NTYP, LTYP, LFAST, NS, NFOPT, INVERT, KCONST

FORMAT (*)

NTYP = Specimen type

- -11 Periodic array of symmetric through cracks at holes under S

- -10 Periodic array of symmetric through cracks at holes under

pin-loading (P), remote stress S=P/(2wt) and moment (_S)

--9

B -8

" -7

s -6

•" -5

= -4

s -3

•J -2

w -]

= 0

= ]

= 2

- 3

- 4

= 5

- 6

Corner crack at edge of semi-circular edge notch under S

Through crack at semi-circular edge notch under S

Surface crack at center of semi-circular edge notch under S

Two symmetric surface cracks at center of hole under S

One surface crack at center of hole under S

Two symmetric through cracks at a hole under S

One through crack at a hole under S

Two symmetric corner cracks at a hole under S

One corner crack at a hole under S

Surface crack under tension (S) and bending (Sb)

Center crack tension under S

Compact specimen (Input: S = P/(wt))

Single-edge crack tension under S

User specified crack configuration (see discussion below)

Through crack in pressurized cylinder (Input: S = p'radius/t)

Corner-crack in bar specimen (circular crack, a = c) under S

For surface and corner cracks, the crack-opening stress is

assumed to be constant along the crack front. Cracks are

grown in two directions--at the maximum depth point and at

the point of the intersection of the crack with the free

surface (see ref. 13). The analysis terminates if the limits

on the stress-intensity factor equations are exceeded. (NTYP

must be negative for cracks growing from a hole or notch.)

The user can define crack configuration of interest by using

NTYP = 4. The user must program the stress-intensity

boundary-correction factor equations for the desired crack

configuration in Subroutine SIF4 and re-compile the program.

The stress-intensity factor solution must be in the form

u

K = S J=c Fc for a through crack and

Ka = S J_-c Fa and Kc = S J=_ Fc for surface or corner

cracks. Fa and Fc are the correction factors at the crackdepth location (Fa= FA) and at the surface location (Fc= FC).

S is a characteristic stress used to defined the stressintensity factor and is input in sections 14 and 16. The

crack length c must be measured in the width (w) direction

and crack depth is measured in the thickness (t) direction.

The current configuration defined in Subroutine SIF4 is a

through crack in a bend bar, see Figure 4(p).

25
_a_

II II

O0 O0

l

1

--_ o

f0 0

V

0

i.__

0

r

C

v 0

I--

I

U_

0

Iv

r

26
M S

I I I I

I I

I I I I

I I I

t ,_G__I_ - _ I

I I I

I I I I

I

I_ 2w _1 2w ,i!: - 2w _

S

(a)

(b)

NTYP - -11

(P = M = 0)

NTYP= -10

(Sb :YS)

c'=r+c

A

S

w

(c)

cl'--

NTYP = -9

a

c

c

1

(d) NTYP =-8

27
(f)

2w -I

-*lc 

NTYP = -6

3-

2t

J_

2w -I

(g) NTYP =-5

(h) NTYP -- -4

--ft

_1_

.-_c_

(i) NTYP----3

(j) NTYP =-2

7-

t

_L

(k) NTYP = -1

(i) NTYP = 0

--ft

__L

ii_iiiiiiiiiiiii_iii_iiiiiiiiiiiiiii

:.:,:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.

:::::::::::::::::::::::::::::::::::::::::

iiiiiii_iiiiiiii-iiii_iiii_i_iii!ii

EE:_i:i:ii:!:i:_:_:.:'.i:_:_:i:!

::::::::::::::::::::::::::::::::::::::

iiiiiii_iiiii!ii!!i!iii!@!iiiiiii!ii

iiiii!i_iiii_._iii_!iiiii!i!_iiiii!iii

(m) NTYP = 1

Figure 4.- Crack configurations analyzed (continued).

Z8
TI L c _

2h_

(n) NTYP = 2

(hlw= 0.6)

S

T

2h

l

4

W

/

(o) NTYP - 3

(h/w = 2)

M wi ,_ (p) NTYP = 4

' M

c I Sb - 6Mtw 2

(Example of user defined configuration, NTYP = 4.)

p_ ©

(q) NTYP = 5

(S = p r / t)

(r) NTYP = 6

(a=c)

Figure 4.- Crack configurations analyzed (concluded).

29
LTYP = Type of loading (currently applies only for NTYP = O)

= 0 Remote tension (S)

= I Remote bending (input stress on outer surface, Sb)

= 2 Combined tension (S) and bending (_S)

(input S and _; bending stress on outer surface is _S)

LFAST = 0 Normal crack-closure model (S_ is computed from the

analytical crack-closure model for NTYP = I or -4 and

the opening stresses are assumed to apply for the other

crack configurations. NTYP = I is used for NTYP _ O;

and NTYP = -4 is used for all negative NTYP.)

=I

=2

=3

Uses an equivalent crack-opening stress, So (SOBAR),

for c _ Cmax and NFLT > 2*MAXSEQ. The Cma x value is

equal to the initial crack length, CI, plus ]0 times

the expected maximum cyclic plastic-zone size. MAXSEQ

equals the total number of blocks (or flights) in a

sequence to be repeated. NFLT is the block or flight

counter. The sequence to be repeated must be repeated

at least two times before the equivalent crack-opening

stress concept is activated. SOBAR is calculated in

the program by a damage-weighted average during the

second application of the full sequence as:

5o = (z Ac S_ Wdf)/(z ac Wdf ) where Wdf= ] - ASeff/AS

I

Linear cumulative damage calculations using So

equation for constant-amplitude loading [9] (equations

are programed in Subroutine SOEQN). The crack-opening

stress equations (4)-(6) must have been used to obtain

the effective stress-intensity factor range against

crack-growth rate relationship.

Uses a constant crack-opening stress concept for

precracking (Subroutine SOEQN) and for block (or

flight) loading (Subroutine SOFLY). In Subroutine

SOFLY, the highest and lowest stresses in the total

block (or flight) history are used with the constantamplitude crack-opening equations [9] to calculate the

"constant" crack-opening stress. Again, the crackopening stress equations must have been used to obtain

the effective stress-intensity factor range against

crack-growth rate relation. (Caution: This option has

not been completely evaluated. For short spectra, that

is, spectra that are repeated quite often, this option

may give good, inexpensive predictions if the proper

constraint factor has been used. For spectra, such as

TWIST or FALSTAFF, this is not a good option.)

30
-4 Sameas LFAST = 3 option, except that the constant

crack-opening stress for block (or flight) loading is

!

manually input as an So/Sma x ratio. The integer NRC

must be set equal to -1 and the DVALUE input is the

crack-opening stress ratio (see section 15).

NS = Number of elements used to define starter notch; elements are

of equal width. These elements are used if the notch surfaces

come into contact. The minimum value of NS is I for cracks

growing from a starter notch; and the minimum value of NS is 2

for cracks growing from a starter notch at a hole.

NFOPT = Block (or flight) loading option

= 0 Constant-amplitude loading

= I Blocks (or flights) are applied by user specified periodic

sequence (NSQ(1) = periodic sequence of block I)

=2 TWIST flight-load sequence [I4] (SMEAN = mean stress)

Options:

INVERT = 0 or ! normal TWIST loading

- 2 maximum stress clipped above Level II

= 3 maximum stress clipped above Level Ill

= 4 maximum stress clipped above Level IV

= 5 maximum stress clipped above Level V

(Level I is the highest stress in TWIST; Level II is

second highest; Level Ill is third highest; and etc.

Clipping above Level Ill means that a cycle in TWIST that

exceeds Level Ill is reduced to Level Ill but the cycle is

applied.)

= 3 Mini-TWIST flight-load sequence [I5] (SMEAN = mean stress)

INVERT options same as those for TWIST (see above)

--4 FALSTAFF flight-load sequence [16]

Options:

INVERT = 0 normal FALSTAFF sequence (SPEAK = highest

stress)

= I inverted FALSTAFF sequence (SPEAK = lowest

stress; mirror image of FALSTAFF about zero)

-5 Space Shuttle load sequence (SPEAK = highest stress)

Options:

INVERT = 0 full shuttle spectrum from "stsn" file

- I short shuttle spectrum from "stsn" file

(The short shuttle spectrum have stress ranges in the full

shuttle spectrum grouped together in take-off and landing

sequence, see file 'stsn'.)

= 6 Gaussian (Irregularity factor = 0.99) load sequence [17]

(SPEAK = highest stress; SMEAN = mean stress)

orIrregularity a mean stress factorof ofzero.0.99) is nearly an R = -] spectrum

31
7 Felix-28 flight-load sequence [18] (SPEAK= highest stress)

-8

The spectrum generator for Felix-28 is the same generator

or the full Felix, Helix and Helix-32 spectra. The data

statements in Subroutine HIXFIX need to be replaced or

modified for these other spectra, see reference 18.)

Spectrum read from tape as a list of stress points.

Input: Total number of stress points (Sma x and Smin) in

spectra; highest and lowest stress values in list;

INVERT = 0 stresses are max, min,..., max, min;

INVERT = I stresses are min, max,..., min, max;

if INVERT = I spectrum is reordered and written to

file 'spectra8' with INVERT I 0 input; re-run program

with spectra filename equal spectra8; SPEAK = highest

stress desired.

19 Spectrum read from tape as flight-by-flight sequence.

Input: Total number of flights in spectra; highest and

lowest stress values in total spectra;

INVERT = 0 stresses are max, min,..., max, min;

INVERT = 1 stresses are min, max,..., min, max;

if INVERT = 1 spectrum is reordered and written to

file 'spectra9' with INVERT : 0 input; re-run program

with spectra filename equal spectrag; SPEAK = highest

stress desired. For each flight, input flight

number, number of stress points in flight, and list

of stress values.

INVERT I Value between 0 and 5 to modify or select special feature

of spectrum loading (see notes under NFOPT options above)

KCONST : 0 Normal value to apply stress or load as external

loading

- 1 Apply stress-intensity factor for loading; input Kma x

and Kmi n instead of stress; only for NTYP = I or 2.

11. Specimen and Crack Starter-Notch Dimensions (see Fig. 3)

READ W, T, CI, AI, CN, AN, HN, RAD

FORMAT (*)

W = One-half width, w (or width for NTYP = 3, 2, -7, -8 and -9)

T = Thickness, t (one-half thickness for NTYP = -5, -6 or -7)

CI = Initial crack length, c_ (crack length plus hole or notch

radius, c i + r, for NTYP = -I to -11, otherwise c_ - ci)

AI = Initial crack depth, a i, for surface or corner crack (AI is

automatically set equal to T for a through crack)

32
CN = Starter notch length, cn (notch length plus hole or notch

radius, c i + r, for NTYP = -I to -11, otherwise c i = ci)

AN = Starter notch depth, an, for surface or corner crack (AN is

automatically set equal to T for a through crack)

HN = Starter notch height, hn

RAD = Radius (r) of circular hole located in center of specimen or

radius of semi-circular edge notch

(For NTYP > O, RAD is automatically set equal to zero.)

See Figures 3 and 4 for specimen nomenclature. A through crack of

length c (or c') is assumed when crack depth, a, is equal to plate

thickness, t, for a surface crack, corner crack, surface crack (or

corner crack) at a hole or notch configurations.

12. Final Crack Length Requested

READ CF

FORMAT (*)

CF - Final crack length, c_, desired

(analysis will terminate when crack length exceeds c_)

13. Special Input for Various Crack Configurations

a. If NTYP = 0 (with LTYP = 2) or NTYP = -10 then:

READ GAMMA

FORMAT (*)

GAMMA = Ratio of outer fiber bending stress to remote tensile

stress

b. If NTYP = -7, -8 and -9 then:

READ XKT, NBCF

FORMAT (*)

XKT = Elastic stress-concentration factor (see NBCF options)

NBCF = 0

I

2

3

Uniform stress (h/w = 2; r/w - 1/16; XKT= 3.17)

Uniform displacement (h/w = 1.5; r/w = 1/16; XKT= 3.15)

Uniform displacement (h/w 2; r/w = 1/16; XKT- 3.17)

Uniform displacement (h/w _ 3; r/w 1/8; XKT= 3.30)

Caution: These particular crack configurations should have the

stress-concentration factors (XKT) input as specified.

The program will accept the XKT value as input. The

stress-concentration factors are based on gross stress.

33
c. If NTYP = 5 then:

READ RADIUS

FORMAT (*)

RADIUS = Radius of pressurized cylinder

i

14. Input Constant-Amplitude Loading to Initiate Crack from Starter Notch

READ SMAX, SMIN

FORMAT (*)

The crack grows from an "initial notch size" (cn or CN) to the

"initial crack length" (ci or CI) under constant-amplitude loading.

The number of cycles are counted but cycles are reset to zero at

the end of the initiation stage (c = ci). When c = ci, the current

value of crack depth, a, may not be equal to ai. The current crack

depth is then reset to the input value of a i. The crack-opening

stress computed in the initiation stage is used at the start of the

Primary Fatigue Loading in the model. (If CN is set equal to CI,

then the crack configuration is not precracked. This input is

required but the values are not used in the program.)

]5. Special Input for Proof Test or Constant Crack-Opening Stress Concept

READ NRC, DVALUE

FORMAT (*)

NRC _ 0 DVALUE not used

-1 Proof test simulation for NTYP = I only, where DVALUE is

the amount of crack extension from proof loading (first

load application under primary fatigue loading). DVALUE

is determined from other sources, such as a resistancecurve analysis. [Note: The value of crack extension

during the proof stress can be computed from the program

by setting NRC = 0 and putting the desired proof stress as

the first load application under Primary Fatigue Loading,

see next section. The first cycle is assumed to be

applied under stroke control and the cracked body will not

fail unless the plastic zone in the model exceeds the net

section (see ref. 8).]

--1

DVALUE -

#

DVALUE is the crack-opening stress (So/Smax) ratio

manually input under the LFAST = 4 option and is

determined from other sources.

Input value for NRC option (see above)

34
16. Input Primary Fatigue Loading

a. Constant- or Variable-Amplitude Loading (NFOPT = 0 or I):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ =

MAXBLK =

LPRINT =

MAXLPR -

Total number of blocks (or flights) in the

sequence to be repeated

Number of different blocks (or flights, such

as Ist, 5th, IOth, etc.) that are used to

compose load history. Each block or flight

may be composed of various number of stress

levels (maximum and minimum stress) Each

stress level can have any number of cycles.

0 Print option, not used for NFOPT = 0 or i

0 Parameter not used for NFOPT = 0 or !

Line 2: READ SCALE

FORMAT (*)

SCALE = Scales stress levels in all blocks (or flights)

SMAXP(I,J) = SCALE * SMAXP(I,J)

SMINP(I,J) = SCALE * SMINP(I,J)

Line 3: READ NBLK, NSL(1), NSQ(1)

FORMAT (*)

NBLK = Specific block (or flight) number

(numbered consecutively and NBLK s 7)

NSL(1) = Number of stress levels to define block (or

flight) I

NSQ(1) = Periodic sequence of block (or flight) I,

block or flight I is repeated every "NSQ"

blocks or flights.

Line 4: READ SMAXP(I,J), SMINP(I,J), NCYCP(I,J)

FORMAT (*)

SMAXP(I,J) = Maximum applied stress of level J in block

(or flight) I

SMINP(I,J) = Minimum applied stress of level J in block

(or flight) I

NCYCP(I,J) - Number of cycles of level J in block (or

flight) I

Maximum number of different blocks (or flights) I

allowed is 5; maximum number of stress levels J allowed

is 1205.

Repeat line 4 for each stress level in block (or flight) I.

Repeat lines 3 Bnd 4 for each block (or fliqht) until MAXBLK is

completed. The blocks (or flights) must be ordered in increasing

periodic sequence (NSQ). In other words, BLOCK (or FLIGHT) 2 must

have a larger NSQ value than BLOCK (or FLIGHT) I, BLOCK (or FLIGHT)

3 must have a larger NSQ value than BLOCK (or FLIGHT) 2, etc.

35
Examples:

I. For constant-amplitude loading (NFOPT= 0), as shownin Figure 5(a):

MAXSEQ= MAXBLK= I LPRINT - 0 MAXLPR _ 0

NBLK = NSL(1) = NSQ(I) - I

SMAXP(I,I) = Maximum applied stress, Sma x

SMINP(I,I) - Minimum applied stress, Smi n

NCYCP(I,I) - May be set equal to any number of cycles (normally set

equal to 1000). Program runs slightly faster if NCYCP

is set equal to or greater than NMAX. NMAX is the

maximum number of cycles allowed before the "analytical"

closure model is exercised. NMAX is set to 1000 cycles

in the program.

2. For variable-amplitude loading (NFOPT = I), as shown in Figure 5(b):

MAXSEQ i 5 MAXBLK = 2 LPRINT = 0

NBLK _ I NSL(1) i 2 NSQ(1) = I

SMAXP(I, _ $I SMINP ,21- $3

NBLK - 2 NSL(2) = 4 NSQ(2) = 4

SMAXP(2,1) = Sl SMINP(2,1) = $2

SMAXP(2,2) SMAXP(2,3) i S4$I SMINPI_:_I SMINP =- $5$2

SMAXP(2,4) Sl SMINP(2,4) = $6

MAXLPR = 0

NCYCP( , = I

NCYCP(2,1) = 2

NCYCP(2,_I NCYCP(2, == II

NCYCP(2,4) = I

The unloading half-cycle, maximum to minimum stress, is used to set up the

input data, as indicated by the heavy lines in Figure 5(b).

b. TWIST [14] or MINI-TWIST [15] Flight-Load Sequence

(NFOPT - 2 or 3, respectively):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = 4000 MAXBLK = 10

LPRINT = 0 No internal spectrum information printed

] Print out flight numbers

2 Print out flight numbers, stress levels and

cycles (may result in large output file)

MAXLPR = Number of flights to be printed out

Line 2: READ SMEAN

FORMAT (*)

SMEAN = Mean stress for TWIST or MINI-TWIST spectrum

Flight sequences and stress levels are computed

in Subroutine TWIST; program scales stresses to

desired stresses using SMEAN.

36
Applied

stress

Sequence

Block 1

q A

_1_ Sequence

rl- Repeated

. 11 1l

ii i l

I Il I l

l

I

I

I

l

- I

v v i _._

Smax

Smin

Time

(a) Constant-amplitude loading

Applied

stress

Sequence of 5 blocks Sequence

Block 1 Block 1 Block 1 Block 2 Block 1 Repeated

$1 A $4

V-_II !I II _' II II _I

Time $6

(b) Variable-amplitude loading

Figure 5.- Typical constant- and variable-amplitude loading.

37
c. FALSTAFF [16] Flight-Load Sequence (NFOPT = 4):

Line 1: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = MAXBLK - 200

LPRINT = 0 No internal spectrum information printed

I Print out flight numbers

2 Print out flight numbers, stress levels and

cycles (may result in large output file)

MAXLPR - Number of flights to be printed out

Line 2: READ SPEAK

FORMAT (*)

SPEAK = Highest stress in FALSTAFF spectrum (level 32)

Flight sequences and stress levels are computed

in Subroutine FALSTAF; program scales stresses

to desired stresses using SPEAK.

d. Space Shuttle Flight-Load Sequence (NFOPT = 5):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = MAXBLK = MAXLPR = 2

LPRINT = 0

I

No internal spectrum information printed

Print out stress levels and cycles in either

full or short spectra.

Line 2: READ SPEAK

FORMAT (*)

SPEAK = Highest desired stress in Space Shuttle spectrum

Flight sequences and stress levels are computed

in Subroutine SHUTTLE from 'stsn' file; program

scales stresses to desired stresses using SPEAK.

e. Gaussian Load Sequence [17] (NFOPT = 6):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = MAXBLK = 839

LPRINT = 0 No internal spectrum information printed

] Print out part of stress levels and cycles

MAXLPR = 4195 (only part of the spectrum is printed)

Line 2: READ SPEAK, SMEAN

FORMAT (*)

SPEAK = Highest stress desired in Gaussian spectrum

SMEAN = Mean stress desired in Gaussian spectrum

38

._k
Flight sequences and stress levels are computed

in Subroutine GAUSS with an irregularity factor

of 0.99 (nealy R = -I); program scales spectrum

to desired mean and highest desired stress.

f. Felix-28 [18] Helicopter Flight-Load Sequence (NFOPT = 7):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = 140 MAXBLK = 12

LPRINT - 0

I

2

No internal spectrum information printed

Print out flight numbers and levels

Print out spectrum tables, stress levels and

cycles (may result in large output file)

MAXLPR = Number of flights to be printed out

Line 2: READ SPEAK

FORMAT (*)

SPEAK = Highest stress desired in Felix-28 spectrum

Flight sequences and stress levels are computed

in Subroutine HIXFIX; program scales spectrum to

desired stresses using SPEAK.

g. Spectrum Read from List of Stress Points (NFOPT _ 8):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

FORMAT (*)

MAXSEQ = MAXBLK = (NPOINTS-I)/2400 + ]

NPOINTS = total number of stress points in spectrum

LPRINT = 0

1

2

No internal spectrum information printed

Limited output of block numbers and cycles

Print out block numbers, stress levels and

cycles (may result in large output file)

MAXLPR - Number of blocks to be printed out

Line 2: READ SPEAK

FORMAT (*)

SPEAK = Highest stress desired

Sequences and unit stress levels are read from

SPECTRA filename; program scales stresses to

desired stresses using SPEAK.

39
h. Spectrum Readfrom Flight-by-Flight Loading (NFOPT= 9):

Line 1: READMAXSEQ,MAXBLK,LPRINT,MAXLPR

FORMAT(*)

MAXSEQ= MAXBLK= NFLIGHTS

NFLIGHTS= total numberof flights in spectrum

LPRINT= 0

I

2

No internal spectrum information printed

Limited output of flight numbers and cycles

Print out flight numbers, stress levels and

cycles (may result in large output file)

MAXLPR= Numberof flights to be printed out

Line 2: READ SPEAK

FORMAT(*)

SPEAK- Highest stress desired

Sequencesand unit stress levels are read from

SPECTRAfilename; program scales stresses to

desired stresses using SPEAK.

17. Input Variables for Load-Reduction Threshold Test

READ KTH, SMAXTH,RTH, CONST

FORMAT(*)

KTH- 0 No threshold test (normal program, applied stresses are

obtain from section 16)

--1 ASTMrecommendedpractice (CONST = dK/dc/K)

For a given value of CONST, the percentage change in K

is constant for equal increments of crack length. CONST

must be equal to or greater than -80 m -I or -2 in-I .

--2 Stress-intensity gradient procedure (CONST = dK/dc)

For a given value of CONST, the rate of change in _K is

constant for equal increments of crack length.

-3 Ten-percent load reduction procedure (CONST = dc)

For a given value of CONST, the load reduction is 10

percent of the previous load for equal increments of

crack length.

SMAXTH = Maximum stress level at start of load-reduction test

RTH = Stress ratio for threshold test

CONST _ Constant related to how the load is reduced during crack

extension

A threshold test is restricted to a center-crack tension (NTYP = I)

specimen, under applied remote stress (KCONST = O) and the normal

crack-closure model (LFAST = O) with NFOPT = O.

40
The current maximum applied stress level is given by

KTH - 1: Sma x - {K(ci)/K(c) EXP[ CONST (c ci)]) SMAXTH

KTH - 2: Sma x = [K(ci) + CONST (c ci)] SMAXTH

KTH - 3: Sma x - SMAXTH 0.9 Int[(c ci)/CONST]

where K(c) and K(ci) are the stress-intensity factors evaluated at

the current crack length, c, and at the initial crack length, c i,

respectively. The KTH = I and 2 methods are continuous functions and

the KTH - 3 method is a step function. The minimum stress Smi n is

RTH times Sma x.

Repeat sections B.I to B.17, to input another problem or inDut next

problem title as HALT to terminate computer run (see section B.I).

C. Spectrum Input Files

1. List of Stress Points (NFOPT = 8)

Line I: READ TITLE

FORMAT (20A4)

Any 80-character title describing the spectrum.

Line 2: READ NPOINTS, MAXST, MINST, INVERT, LFORMAT

FORMAT (*)

NPOINTS = Number of stress points in total spectrum

MAXST = Highest stress point in total spectrum

MINST = Lowest stress point in total spectrum

INVERT = Value indicating order of stresses

= 0 Smax 1, Smin],..., Smax n, Smin n

= I Smin I, Smax1,..., Smin n, Smax n

LFORMAT = Specified format for stress values

= ] FORMAT(2014)

= 2 FORMAT(1615)

= 3 FORMAT(]OI8)

Note: Stress values must conform to the specified

format field. Either 20, 16 or ]0 stress values

must be on all lines except the last line. The

last line may have any number less than the

specified format field.

Line 3: READ FORMAT (NSIG(I) (2014 ]615I=],NPOINTS) or I018)

NSIG(1) = stress expressed as an integer

41
If INVERT= ], the spectrum is read and the array NSIG is

reordered to correspond to the INVERT = 0 option. A new spectrum

file is generated named 'spectra8'. FASTRAN-II must then be run

using the new spectrum filename.

2. Flight-by-Flight Loading (NFOPT = 9)

Line 1: READ TITLE

FORMAT (20A4)

Any BO-character title describing the spectrum.

Line 2: READ NFLIGHT, MAXST, MINST, INVERT

FORMAT (*)

NFLIGHT = Number of flights in total spectrum

MAXST = Highest stress point in total spectrum

MINST = Lowest stress point in total spectrum

INVERT = Value indicating order of stresses in flights

= 0 Smax I, Smin],..., Smax n, Smin n

= ] Smin], Smax],..., Smin n, Smax n

Line 3: READ JFLT(M), NUM, (NSIG(1), I=I,NUM)

FORMAT (*)

JFLT(M) = Flight number for flight m

NUM = Number of stress points in flight m

NSIG(1) = stress expressed as an integer

Repeat Line 3, NFLIGHT times.

If INVERT = ], the spectrum is read and the array NSIG is

reordered to correspond to the INVERT = 0 option. A new spectrum

file is generated named 'spectrag' FASTRAN-II must then be run

using the new spectrum filename.

D. System of Units

All units used for input data must be consistent.

SI Units

For exampl e,

Length: m

Stress: MPa

AK (or K): MPa-m I/2

dc/dN (or da/dN): m/cycle

U.S. Customary

in.

ksi

ksi-inl/2

in/cycle

42
E. Error Messages _ ,_

FASTRAN-II has numerous error checks on the input data, some execution

errors and many terminating action messages that may be helpful to the user.

These error messages are listed and some are briefly discussed.

I. Data Input Errors:

a. ERROR: FINAL LENGTH CF < CI INITIAL CRACK LENGTH '

Value of CF, final crack length, must be greater than the

initial crack length, CI.

b. INPUT ERROR: IRATE OR NDKTH

One or more of the values may be out of bound.

c. INPUT ERROR: NEP OUT OF RANGE--NEP = 'value'

NEP must be either O, ] or 2.

d. ERROR: NOTCH LENGTH CN > CI INITIAL CRACK LENGTH

Value of CI, initial crack length, must be greater than or

equal to notch length, CN.

e. INPUT ERROR: CHECK CRACK-GROWTH RATE TABLE VALUES

(SIF AND RATES MUST BE IN ASCENDING ORDER)

f. PROOF TEST SIMULATION FOR NTYP=I (CCT) ONLY

g. THRESHOLD SIMULATION FOR NTYP=I (CCT) ONLY

WITH KCONST = LFAST - NFOPT = 0

h. INPUT ERROR: CHECK VALUE OF NRC OR LFAST

OR SULT MAY BE LESS THAN SYIELD

Input value of NRC is out of bound or NRC is not -I for

LFAST = 4. Input value of SULT may be less than SYIELD.

i. INPUT ERROR: ALPI MUST BE GREATER THAN ALP2

RATE] MUST BE LESS THAN RJ_TE2

J. INPUT ERROR: NTYP, LFAST, NFOPT, INVERT OR KCONST

Values are either out of bound or they are not correct

for KCONST = I. For KCONST =I, NTYP must be I or 2,

NFOPT must be less than or equal to ] and LFAST= O.

k. FATIGUE LOADING INPUT ERROR:

SMIN FOR PREVIOUS CYCLE IS GREATER THAN OR EQUAL TO SMAX

FOR CURRENT CYCLE NBLK = 'value at error'

LEVEL = 'value at error'

I. INPUT ERROR: NFOPT = 'value at error'

Value is out of bound.

m. INPUT ERROR: NALP MUST BE ZERO FOR LFAST=I OPTION

n. INPUT ERROR: CRACK LENGTH (CI) OR NOTCH LENGTH (CN)

IS LESS THAN HOLE RADIUS')

43
o. INPUT ERROR: NTYP - 'value' OR NBCF = 'value'

One or more of the values may be out of bound.

p. LIMITS ON BCF EQUATIONS FOR NTYP - 'value' EXCEEDED

q. INPUT ERROR: NFOPT = 0 AND MAXBLK > I

Value of MAXBLK must be I for NFOPT equal to O.

r. INPUT ERROR: CHECK LFORMAT

Value is out of bound.

s. INPUT ERROR: CHECK VALUE OF INVERT IN SPECTRUM FILE

Value of INVERT is not correct for spectrum input.

t. CURRENT SMAX LESS THAN OR EQUAL TO PREVIOUS SMIN IN

BLOCK 'value' NEAR STRESS POINT 'value'

u. CURRENT SMAX LESS THAN OR EQUAL TO PREVIOUS SMIN IN

FLIGHT 'value' NEAR CYCLE 'value'

v. ERROR: NUMBER OF STRESS POINTS EXCEED 2400 IN FLIGHT 'value'

Number of stress points for flight 'value' exceeds 2400.

Flight can be divided into smaller flights with less than 2400

stress points per flight.

w. NUMBER OF CYCLES FOR SPECTRUM EXCEEDED (P=5)

Gaussian spectrum can only be repeated five times. Parameter

P in Subroutine GAUSS may be set to larger value than 5 but

the repetition of the spectrum should be verified.

x. INPUT ERROR: NSQ MUST BE IN ASCENDING ORDER

WITH NSQ(1) EQUAL TO UNITY

2. Execution Errors

a. ERROR: PRIMARY LOADING BEING APPLIED FOR C < CI

Check analysis (input data) file.

b. ERROR: CXO ARRAY OUT OF BOUND NIC - 'value at error'

For crack growth increments (or rates) greater than I percent

of the crack length, the AKeff-rate curve is treated as a

"resistance curve" and a iterative technique is used to solve

for crack extension values. Maximum number of iterations

allowed is 100. Solution could not be found in I00

iterations. This error may also indicate specimen failure.

c. PLASTIC ZONE (RHO) COULD NOT BE FOUND AFTER 100 ITERATIONS

IN SUBROUTINE PLZ SMAX = 'value'

Solution for plastic-zone size (p) for crack emanating from a

hole (NTYP _ -4) cannot be found in I00 iterations. Error may

be in the applied loading or tensile properties.

d. ITERATIONS EXCEED 100

An iterative method is used to solve for intact and contact

stresses in the analytical crack-closure model. Program

terminates if a solution cannot be found in 100 iterations.

44
e. NUMBEROFELEMENTSEXCEEDMAXIMUMALLOWED'= value'

Maximum number of elements allowed equals 50.

Check input data or increase dimension on all

arrays in COMMON blocks containing 50.

3. Terminating Actions

a. CRACK LENGTH EXCEEDS INPUT VALUE FOR CF

Set CF = W (width or half-width) to deactivate option.

b. NUMBER OF CYCLES EXCEED 10 MILLION

Ten million is the limit on cycles that may be applied in a

load-reduction test. Limit may be increased by user by

changing statement IF(NTC.GT.I.0E+07) GOTO 992.

c. EFFECTIVE STRESS INTENSITY LESS THAN THRESHOLD

Effective stress-intensity factor range is less than the

effective threshold during the Fatigue Precracking stage.

Check input values of applied stress or crack lengths.

d. NUMBER OF CYCLES EXCEED 10 MILLION:

SO '= value' MAY BE GREATER THAN SMAX '= value'

CAUTION: CRACK MAY GROW AROUND OVERLOAD PLASTIC ZONES (IF ANY)

Number of cycles exceed IO million during fatigue precracking

or the primary fatigue loading stage. For variable-amplitude

loading, such as high-to-low block loading, cracks have been

shown to grow around overload plastic zones. The current

model will try to grow the crack straight through the overload

zone which may be physically impossible. This condition may

not be prevalent for spectrum loading where peak loads which

create the overload zones will be repeated. These peak loads

will tend to grow the crack through the overload zone.

e. BENDING STRESS-INTENSITY FACTOR IS NEGATIVE

Crack-growth analyses will terminate if the stress-intensity

factor range for a surface or corner crack becomes negative.

f. SPECIMEN FAILED DURING PRECRACKING OR C < CI

PRIMARY FATIGUE LOADING WAS ** NOT ** APPLIED

If configuration fails or cycles exceed 10 million during

fatigue precracking, the primary fatigue loading is printed

out but not applied.

g. When a crack configuration fails, the final crack length and

number of cycles at failure are printed out. In addition, the

terminating action will be also printed. There are seven

locations in the program where this terminating action may be

initiated. They are identified by the NFCODE. These failure

conditions are:

1. NFCODE - O:

2. NFCODE = 1:

3. NFCODE = 2:

4. NFCODE _ 3:

5. NFCODE = 4:

6. NFCODE = 5:

7. NFCODE = 6:

KMAX > C5

CRACK DRIVE > MATERIAL RESISTANCE

KMAX > C5

SMAX > O.99*SFLOW

KMAX > KIE (ELASTIC SIF AT FAILURE)

CRACK LENGTH EXCEEDS WIDTH

CRACK LENGTH PLUS PLASTIC ZONE EXCEEDS WIDTH

45
IV. FASTRAN-IIFLOW CHART

Material Properties ProblemandTitle Configuration Input i

Input

precracking

stresses

i

I Initialize Model

I

_<-_ Fatigue precracking (Cn < ci)? I

I SMINO - 0.0 i

I SMINO = SMINI Input

primary fatigue --<---_

stresses

i

Set c = c i and a = ai

Apply stresses _<--_

I

Monitor applied stresses

for SMINB, SMAXH, and SMINA

(see Appendix A, Ref. 2)

I

I Compute Ac* l

I

Compute crack growth

Aa and Ac from AKef f

I

Crack length

a=a+Aa

C=C+AC

Is AcT _ Ac* ? --> >

v

46
Closure

Model

v

I

Variable constraint? _>__->_ Compute a I

<, .<

Fracture

or -->__--->--_ StoP]

c > cf?

crack-opening Equivalent -> ___ >- crack-opening Compute

stress concept stress

and c > ci?

if SMINB < SNINO

I

I at SMAXH cr. -t o (c = ce ement + Ac*) !

I

I Lumpelements )

[

Contact analysis at SMINA

set SMINO = SMINA

I

Solve for contact

stresses, aj

I

Crack-opening stress ]

calcul ation

I

rst time ?I ' '

47
V. FASTRAN II ANALYSIS FILE--Quick Reference Guide

I. Problem Title

READ TITLE

FORMAT (20A4)

2. Spectrum Filename and Time Limit

READ SPECTRA, TLIMIT

FORMAT (AIO,EIO.3)

3. Material

READ MAT

FORMAT (20A4)

4. Material Tensile Yielding Properties

READ SYIELD, SULT, E, ETA, ALP, NALP, NEP, BETA

5. Fatigue-Crack Growth Rate Option

READ IRATE

Repea_ sections 6 to.Z IRATE times (J = ] to IRATE).

6. Fatigue-Crack Growth Rate Equation and Fracture Properties

READ C](I,J), C2(I,J), C3(J), C4(J), CS(J), KF, M

7. Fatigue-Crack Growth Rate Table (see ref. 5)

a. READ NTAB, NDKTH

If NTAB = O, go to 8, otherwise continue

b. READ DKETAB(I,J), CGRTAB(I,J)

Repeat 7b NTAB times.

8. Crack Growth Rates at Transition (NALP = I option only)

READ RATEI, ALP], BETAI, RATE2, ALP2, BETA2

g. Data Output Options

READ NIPT, NPRT, LSTEP, NDKE, DCPR

10. Specimen Type and Loading

READ NTYP, LTYP, LFAST, NS, NFOPT, INVERT, KCONST

11. Specimen and Crack Starter-Notch Dimensions

READ W, T, CI, AI, CN, AN, HN, RAD

48
12. Final Crack Length Requested

READ CF

13. Special Input for Various Crack configurations

a. If NTYP = 0 (with LTYP = 2) or NTYP = -lO then;

READ GAMMA

b. If NTYP - -7, -8 and -9 _hen:

READ XKT, NBCF

c. If NTYP = 5 then:

READ RADIUS

14. Input Constant-Amplitude Loading to Initiate Crack from Starter Notch

READ SMAX, SMIN

15. Special Input for Proof Test or Constant Crack-Opening Stress Concept

READ NRC, DVALUE

I

16. Input Primary Fatigue Loading

a. Constant- or Variable-Amplitude Loading (NFOPT = 0 or 1):

Line 1: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

LPRINT - HAXLPR = 0

Line 2: READ SCALE

Line 3: READ NBLK, NSL(1), NSQ(1)

Line 4: READ SMAXP(I,J), SMINP(I,J), NCYCP(I,J)

Repeat lines 3 and 4, MAXBLK times.

b. TWIST [14] or MINI-TWIST [15] Flight-Load Sequence

(NFOPT = 2 or 3, respectively):

Line ]: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ = 4000 MAXBLK = 10

LPRINT - O, 1, or 2

Line 2: READ SMEAN

c. FALSTAFF [16] Flight-Load Sequence (NFOPT - 4):

Line 1: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ = MAXBLK = 200

LPRINT - O, 1, or 2

49
Line 2: READ SPEAK

d. SpaceShuttle Load Sequence(NFOPTz 5):

(SPECTRA- stsn)

Line 1: READ MAXSEQ,MAXBLK,LPRINT,MAXLPR

MAXSEQ- MAXBLK- MAXLPR- 2

LPRINT- 0 or I

Line 2: READ SPEAK

e. Gaussian Load Sequence[17] (NFOPT = 6):

Line 1: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ _ MAXBLK - 839

LPRINT - 0 or I

MAXLPR = 4195

Line 2: READ SPEAK, SMEAN

f. Felix/28 [18] Helicopter Load Sequence (NFOPT - 7):

Line 1: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ = 140 MAXBLK _ 12

LPRINT _ O, 1, or 2

Line 2: READ SPEAK

g. Spectrum Read from List of Stress Points (NFOPT = 8):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ = MAXBLK _ (NPOINTS-I)/2400 + I

LPRINT = O, 1, or 2

Line 2: READ SPEAK

h. Spectrum Read from Flight-by-Flight Input (NFOPT _ 9):

Line I: READ MAXSEQ, MAXBLK, LPRINT, MAXLPR

MAXSEQ = MAXBLK _ NFLIGHTS

LPRINT = O, 1, or 2

Line 2: READ SPEAK

17. Input Variables for Load-Reduction Threshold Test

READ KTH, SMAXTH, RTH, CONST

5O
-i

-i

Vl. EXAMPLE PROBLEMS

A. Introduction

Ten examples problems have been selected to demonstrate the various

options in FASTRAN-II. These example problems may be used to verify the

operation of the code on other computer systems and to help the user. The

material crack-growth rate data in these examples are close to actual

properties for most of the materials considered. However, they should only

be used as examples and should not be used in damage-tolerance design

without verification. For damage-tolerant life calculations, actual crackgrowth properties using the crack-closure model analysis should be developed

from appropriate tests and analyses on the particular material and crack

configurations of interest.

Five computer files are required to run these example problems. They

are:

Filename

fastran2.f

examples

stsn

snfopt8

snfopt9

Description

Source Code (main and subroutines)

Analysis Input Data Files for Examples

Spectrum Loading for Space Shuttle (NFOPT = 5)

Sample Spectrum Loading for NFOPT = 8

Sample Spectrum Loading for NFOPT = 9

The analysis (input data) files for the ten examples are listed in Section

VI.B. A listing of the output data files are given in Section VI.C. In the

following sections, the ten example problems will be briefly described to

discuss what options are being tested and to describe the various output

data files and options. The ten examples were analyzed during one computer

run by placing the ten input data files in series with the word HALT at the

end of the last example. The word HALT maybe placed at the end of any other

input data file but the computer run will terminate after that example. The

total CPU time for all ten examples was 226 seconds on a CONVEX-C210

computer with scalar and vector optimization. The output files for the

first few examples will be described in more detail than the other examples.

1. Center-Crack Tension under ConstBn_-Amplitude Loadinq

The original analytical-crack closure model [2] was developed for a

modified Dugdale model of a center-crack tension specimen under plane-stress

conditions (constraint factor, e = ]). The first example shows results for

this specimen made of HT-80 steel under plane-stress conditions. The

specimen was fatigue precracked under a different stress level than the

primary fatigue loading. The precracking occurred from the saw-cut notch

51
length (CN) to the initial crack length (CI). The initial crack and notch

depths were completely through the thickness (AN= AI = T = thickness). A

simple power-law was used to express crack-growth rate as a function of

effective stress-intensity factor (C|, C2). No threshold stress-intensity

factor was used (C3 = C4 = 0.0). Linear-elastic facture mechanics was used

to describe the fracture toughness of the material (KF = 80 ksi-Jin and m =

0.0). The program options output line indicates that the normal closure

model was used (LFAST = O) and contact stress-intensity factors were used to

calculate crack opening values (KOPEN = ! is automatically set in the main

program). (Note: A user can change KOPEN in the main program and set KOPEN

= O. This will cause the program to calculate crack-opening values from

crack-surface displacement equations [5].) For LSTEP = 2, a detailed output

of contact stresses along the crack surface will be given at the minimum

applied stress (-10 ksi), at the mean applied stress (10 ksi), and at the

maximum applied stress (30 ksi) after every 150 (NIPT) crack-extension

increments (Ac*) in the model. Crack length against cycles results will be

printed out after every 0.02 inches (DCPR) of crack extension (NPRT s 0).

Other information given is KO/KMAX (ratio of opening to maximum stressintensity factor), elastic AK (DKC) at crack length c (NDKE = O) and crackgrowth rate (DC/DN). The primary fatigue loading (constant amplitude) was

input under NFOPT = 0 option. The primary fatigue loading was applied in

blocks of ]000 cycles with a periodic sequence (NSQ) of unity. SCALE was

unity, therefore, SMAX and SMIN are as input. When the NIPT option is

activated, the contact stresses are printed out. This output shows the

number of iterations required to obtain a solution for the contact stresses

(usually 10 to 20 iterations), applied stress level and current crack

length. (Note: The iterative technique uses the current contact stresses in

memory as the initial guess for the new solution. This accounts for the low

number of iterations at the minimum applied stress.) The contact-stress

output shows element number, element length (residual plastic deformation),

crack-opening displacement of elastic body (crack-opening displacement is

element LENGTH minus DISPLACEMENT), contact stress along crack surface

(element 11 to 23), stresses in plastic zone (elements i to 10) and width of

element. Element 1! is at the centerline of the crack, elements 11 to 23

are along the crack surface with element 23 (MELE = largest element number)

being the element created during the last crack extension and elements ! to

10 are used to model the plastic zone (see Fig. I). (MELE is the total

number of elements along the crack surface and in the plastic zone.)

52
Element ] is at the crack tip and element 10 is at the end of the plastic

zone. The current plastic-zone length is also printed out. The specimen

failed after 58,026 cycles at a crack length of ].767. NFCODE = 4 indicates

that failure was causedby the maximum stress-intensity factor exceeding the

elastic fracture toughness. The CPU time was 2.7 seconds.

2. Throuqh Cr_k$ from Hole under Repeated Spike Loadinq

The analytical-crack closure model was modified for cracks emanating

from a circular hole [5]. This model included the influence of the hole on

stress-intensity factors, crack-surface displacements, plastic-zone sizes

and crack-opening stresses.

The next example shows results for two symmetric through cracks

emanating from a circular hole (NYTP = -4) in a 7075-T65] aluminum alloy

plate subjected to repeated spike overloads. A single spike overload was

repeated every 5000 cycles after some amount of precracking. A constraint

factor of 2 was found from a correlation of constant-amplitude fatigue crack

growth rate data (like that shown in refs. 3 and g). A table look-up

procedure was used to relate effective stress intensity factor range (DKEFF)

to growth rate using seven data points (NTAB = 7). For AKef f values below

or above the extreme values given in the table, power-law extrapolations are

used using either the first two or last two data points, respectively. Note

that crack length in the width direction (c') is measured from the center of

the hole and the input of Cl, CN and CF must include the physical crack

length from the hole plus the hole radius (c + r). (The analysis will

terminate if the crack length input is less than the hole radius for NTYP <

O. The code has other input data checks to help protect the user on

improper input data, see Error Messages, Section III.E.) A AKeff-threshold

(C3) of I ksi-Jin was used for all stress ratios, R (C4 = 0.0). The TwoParameter Fracture Criterion (TPFC) []2] was used to characterize the

fracture behavior of this material (KF = 86.5 ksi-Jin and M = 0.7). The

output at every 50 (NPRT) analytical crack-growth increments shows the

physical crack length (c' r), the effective stress-intensity factor at

crack length c (DKEC) and other information as shown. The KO/KMAX ratio and

stress-intensity factors under the primary fatigue loading were based on the

highest and lowest levels in the applied sequence. LSTEP = I causes

detailed contact information to be printed out at the maximum and minimum

applied stresses after every 3000 (NIPT) crack-growth increments. Because

this configuration was a crack growing from a hole, element 11 (LENGTH = -

0.125) automatically became the negative of the radius of the hole. Element

53
12 simulated a sawcut (hn- 0.01) that was made at the edge of the hole to

act as a crack starter. Elements are automatically placed along any sawcut,

such as element 12, because they may come into contact under extreme

compressive loading. (NIPT may be set to a very large number, say ggg9, to

prevent printing of detailed information on contact stresses.) Again, the

specimen failed when KMAX was greater than the elastic stress-intensity

factor at failure (KIE) computed from the TPFC.

3. $qrf_ce-Crack Tension qnder Constant-Amplitude Loadinq (LFAST = .! Option)

The analytical-crack closure model was developed for two-dimensional

crack configurations. However, there was a need to calculate damagetolerant life for three-dimensional crack configurations, such as a surface

crack in a plate under tension. To use the model for 3D crack

configurations, the crack-opening stress levels computed from the 2D model

(under a constraint condition simulating 3D behavior) was assumed to apply

at all locations along the surface-crack front. For 3D crack

configurations, crack growth was also calculated at two locations: (I) crack

length c (or width) direction and (2) the crack depth a (or thickness)

direction. Thus, a surface crack may change shape as it grows.

This example shows the application of the model to the growth of a

surface crack in 221g-T851 aluminum alloy under constant-amplitude loading

for the LFAST = 1 option. Under constant-amplitude loading, the crackopening stress levels were found to stablize [2] after some amount of crack

growth. The LFAST = ] option causes the program to compute a "damageweighted average" crack-opening stress (denoted SOBAR). After the crack has

grown a predetermined amount (C > CMAX) and if the applied blocks are

greater than 2, then the program assumes that the opening stress is constant

at SOBAR. At this point, the analytical closure model is turned off and

crack-growth calculations are made very rapidly using the SOBAR value and

the stress history. The output shows when the "equivalent crack-opening

stress" concept is activated. In this example, the equivalent KO/KMAX value

of 0.534 was used beyond a crack length of 0.1455. The output shows the

stress-intensity factors and crack-growth rates at crack length (c) and at

crack depth (a). When the crack depth, a, reaches the plate thickness (a/t

= 1), then a through crack of length, c, is assumed in the analysis. The

specimen failed when the KMAX value exceeded the maximum stress-intensity

factor (C5) in the crack-growth rate equation (eq. 15). The CPU time was

less than I second whereas the full model (LFAST = O) would have required 14

54
seconds. For the LFAST= I option, crack-growth lives are usually less than

those under LFAST - 0 option. In this case, the life was 6 percent less.

4. Corner CrBck from Hole under User Specified Loadinq

Under variable-amplitude or spectrum loading, the user has the option

to define a repeating stress sequence. The user can define up to seven (7)

different block (or flight) histories. Each block (or flight) history may

have as many as 1200 stress level pairs (1200 values of SMAX; 1200 values of

SMIN) in any sequence and each stress level may have any number of cycles.

The repeating sequence is based on a periodic sequence number (NSQ) for each

block (or flight). This particular option was an early option in FASTRAN

and is useful for block (or flight) histories that have a small number of

stress levels and those that can be characterized by periodic sequence. If

the stress history is extensive, then NFOPT option 8 or 9 should be used.

The fourth example is a corner crack growing from an open hole in a

7075-T651 aluminum alloy plate subjected to a user-defined loading. The

user-defined sequence represents a transport-wing spectrum. The spectrum is

composed of five (5) different blocks (or flights). Each block is defined

with different set of stress levels and cycles. BLOCK I has a periodic

sequence number (NSQ) of unity and is repeated every time. BLOCK 2 has a

periodic sequence number (NSQ) of 5. Thus, every fifth block, such as 5,

15, 25,..., 95 are replaced by BLOCK 2. Every 10th block is replaced by

BLOCK 3 (NSQ = 10); the 50th block is replaced by BLOCK 4 (NSQ - 50); and

the 100th block is replaced by BLOCK 5 (NSQ = 100). Once the total sequence

has been applied, the sequence is repeated until the cracked component

fails. Note that SCALE has been used to scale the input spectrum to the

desired spectrum stress levels. Here the component failed because the crack

length plus plastic zone exceeded the component width.

5. Corner Crack from Hole under _pace Shuttle Loading (LFAST = 3 Option)

Example 5 shows an application of the model to predict crack growth

under a Space Shuttle load sequence (NFOPT - 5). This option requires that

the special file 'stsn' be read in as file SPECTRA. In this example, INVERT

was set equal to one which applies the short (reduced) shuttle spectrum. If

INVERT - O, the full shuttle spectrum, composed of 11,430 load reversals,

would have been applied. LPRINT was set equal to one so that the reduced

spectrum would be printed out. (Note: Using the full shuttle spectrum and

LPRINT _ I will print more information about the spectrum.) The LFAST = 3

option was also used in this case. This option uses the crack-opening

stress equations [9], developed for constant-amplitude loading, to calculate

55
a "constant" crack-opening stress using the maximum and minimum applied

stress in the total stress sequence. (Caution: This option is extremely

fast, low CPU seconds, but the use of this option is only useful for spectra

that are repeated quite often, such as short spectra. This option is not

recommended for general use and with spectra such as TWIST or FALSTAFF.)

6. Small Crack _t Notch qnder Mini-TWIST Spectrum Loadinq (NALP = I Op_iQn)

Example 6 illustrates the growth of small (micro-) cracks at a notch

in 7075-T6 aluminum alloy sheet under the Mini-TWIST loading. Two new

options are used in this example. First, the crack-growth rate properties

in the c and a directions differ. Second, the "variable-constraint" option

[11] is also used. In this option, the constraint factor (e) is equal to

ALP] (1.8) for rates lower than RATE] and is equal to ALP2 (1.2) for rates

greater than RATE2. For example, spike overloads applied during the MiniTWIST sequence may cause a loss of constraint by causing large scale

yielding at the crack tip. The baseline constant-amplitude crack-growth

rate data must have been analyzed using this same option to obtain the AKef f

against crack-growth rate relations. A change in constraint has been

associated with the transition from flat-to-slant crack growth. Note that

for variable-amplitude or spectrum loading the stress-intensity factors

(elastic or effective) that are printed out are based on the highest and

lowest stresses in the sequence and not on the individual stress values.

The crack-growth rates, on the other hand, are rates calculated from

individual stress values. Thus, DKC and DC/DN do not correspond. A zero

rate is printed out if the effective stress-intensity factor is less than

the threshold value (C3 = I MPa-Jm). For surface or corner cracks, DKA and

DA/DN have no meaning when A/T becomes unity (the crack becomes a through

crack) and values of zero are printed out for these quantities.

7. Compact Specimen under Stress-lntensitv Factor Loadino

Many laboratory tests are conducted under constant stress-intensity

factor loading conditions. This option (KCONST = I) was also added to

FASTRAN-II. However, the current version is restricted to only the centercrack and compact specimens (NTYP = I and 2, respectively) tested under the

NFOPT options of 0 or I (constant and variable amplitude, respectively).

This example shows results for an FVS0812 aluminum alloy compact

specimen tested under a repeated spike overload every 100,000 cycles after

some precracking at constant stress-intensity factor range. The large

number of cycles was selected so that the effects of only a single spike

overload could be studied. The precracking and primary fatigue loading is

56
input as Kmax and Kmi n instead of stress. During primary fatigue loading,

the KO/KMAX ratio is based on the "current" maximum stress-intensity factor

in the sequence (KMAX - 16.5 or 33 ksi-Jin). The NALP - ] option (variable

constraint) was also selected. This example shows the loss of constraint

during the spike overload, ALP - ].g versus 2.17 during constant-amplitude

loading. Note that load-interaction effects on constraint have not been

considered in FASTRAN-II. In the variable-constraint regime (2.4 > = >

1.9), a is computed as if constant-amplitude loading is being applied and,

therefore, a is a constant value after the spike overload. (Further study

is needed to model load-interaction effects on constraint.) The analysis

was terminated when the crack length exceeded the input value of CF.

8. Center-Crack Tension under NFOPT = B Optiqn

The next two examples were included to demonstrate the two new methods

of inputing spectrum load data. The spectrum loading selected is the first

twenty-two flights of the lower-wing structure for a fighter. Example 8

uses the list of stress points (NFOPT = 8) option. The spectrum contains

2,678 stress points with the highest stress point at 723 and the lowest

stress point at -g9. This option requires that the spectrum be read from a

file with INVERT = 0 (stresses are integer numbers input as maximum,

minimum, maximum, ..., minimum with a user-selected format). The userselected format may be either 2014, 1615 or I018 (LFORMAT - I, 2 or 3,

respectively). If stresses are input with a minimum stress as the first

stress point (INVERT - 1), then the program moves the first minimum stress

to the end of the spectrum, sets INVERT to zero, creates a new file named

'spectra8' and terminates. The original input spectrum file is not changed

or deleted. The user should then re-run FASTRAN-II with the filename

'spectraS'. (There is no limit on the number of stress points. The program

reads 2,400 stress points at a time, uses them to grow the crack and returns

for another set. Once the total number of stress points in the file has

been used, the file is rewound and the process is repeated.) This spectrum

was composed of 2 blocks, one block had 2400 stress points and the other had

278 stress points. The maximum desired stress (SPEAK) was input as 35 ksi.

Thus, all integer stress points have been scaled by 35/723 and the lowest

stress was about -4.79 ksi. In this example, LPRINT was ] and MAXLPR was

10. Thus, 10 blocks of spectrum information were printed out as the stress

history was applied to the specimen. The specimen failed in 25,325 cycles

when the maximum applied stress-intensity factor exceeded Kie, the elastic

stress-intensity factor at failure [12].

57
9. ¢_nter-qr_ck Tension under NFOPT = 9 Option

Example g uses the same flight-load history as used in the last

example. However, the stresses are read in flight-by-flight sequence. As

previously mentioned, the spectra had 22 flights with the highest stress

point at 723 and the lowest at -gg. Each flight has a flight number and a

specified number of stress points in the flight. The flight number, number

of stress points and stress values are read in free format. Again, the

INVERT options (0 and I) still apply. If stresses are input with a minimum

stress as the first stress point (INVERT = 1), then the program moves the

first minimum stress to the end of the spectrum (last stress point in last

flight), moves the first minimum stress of the other flights to the end of

the preceding flights, sets INVERT to zero, creates a new file named

'spectrag' and terminates. The original input spectrum file is not changed

or deleted. The user should then re-run FASTRAN-II with the filename

'spectrag'. In this example, LPRINT was I and MAXLPR was 5. Thus, spectrum

information was printed out on at least five flights. (Information on seven

flights were printed because the program automatically uses seven flights at

a time to growth the crack.) The other differences between the output for

examples 8 and g is that example g prints out effective stress-intensity

factors (DKEC) instead of elastic stress-intensity factors (DKC), see

SectionIII.B.g (NDKE), and the crack-length-against-cycles information is

printed out in equal crack extension increments (DCPR = 0.04 in.). The

specimen failed at the same number of cycles as example 8.

10. Threshold Test (KTH = 3 Option)

The last example is a simulated large-crack threshold test (KTH

option). A threshold test is restricted to a center-crPck tension specimen

under remote applied stress using the normal closure model (LFAST = 0).

There are three types of load-shedding tests that can be simulated. This

example uses option KTH = 3 which reduces the applied stress in 10 percent

steps after each crack-growth increment of Ac (= CONST). In this example,

CONST equals 0.0078 inches. After a precracking stage, the load-shedding

procedure is applied. In threshold tests, KO/KMAX and stress-intensity

factors are based on the current maximum applied stress and stress ratio.

The maximum applied stress is also printed out, in addition to the usual

output. Note that with each stress reduction, there is a corresponding rise

in the crack-opening stress level and a reduction in the effective stressintensity factor. The program terminates when the number of cycles exceed

10 million.

5B
/

JT -

: L_ _

B. Listing of Input Data Files for Examples

SAMPLE 1: Center-Crack Tension under Constant-Amplitude Loading

200.0

STEEL

]19. 30000. 0.0 1.00 0 0

cstamp

HT-80

115.

I

4.O000E- 10

0 0

150 -5

I 0

4.0

4.0

25.0

0

1

1.0

1

30.0

0

3.00 O. O. 999. 80.

2 0 0.02

0 2 0 0 0

0.4 0.8 0.4

0.0

0.0

1 0 0

1 1

-10.0 1000

0.0 0.0 0.0

0.7 0.4 0.0 0.0

SAMPLE 2: Through Cracks from Hole under Repeated Spike Loading

cstamp 200.0

7075-T651 i

85.0 10400.0 0.0 2.0 0 0 1.0

2.860 1.0 .0 999.0 86.5 0.7

0

.25 0.135 .25 0.01

77.0

1

3. 2000E-08

7 0

1.0 3.199E-08

4.0 1.679E-06

5.0 5.000E-06

10.0 2.000E-05

12.0 3.500E-05

19.0 2.000E-04

25.0 I .O00E-03

3000 50 I 1 0.02

-4 0 0 1 1 0

2.00 0.25 0.145

2:00

15.0 0.0

0 0.0

2 2 0 0

].0

1 1 1

20.0 0.0 1

2 1 2

15.0 O.0 5000

O 0.0 0.0 0.0

.125

:L

_t 59
S_PLE 3: Surface-Crack Tension

cstamp 200.0

2219-T851

45.0 65.0 10000.0

I

1.2000E-07 3.12 2.7

0 0

50 5 I 0 0.02

0 0 I I 0 0

4.0 0.3 .136

4.0

0.0 0.0

0 0.0

1 1 0 0

1.0

1 1 1

30.0 15.0 1000

0 0.0 0.0 0.0

SAMPLE 4: Corner Crack from Hole

cstamp 200.0

7075-T651

77.0 85.0 I0400.0

I

3.2000E-08 3.O0 O.

0 0

1000 20 I 0 0.02

-I 0 0 I I

2.00 0.25 0.145

2.00

0.0 0.0

0 0.0

100 5 0 0

15.00

I 6 1

1.2473 0.9000

1.3183 0.8301

1.4007 0.7814

1.3183 0.8301

I.2473 O.9000

I.0000 O.0903

2 10 5

1. 2473 O. 9000

1.3183 0.8301

1.4007 0.7814

1.4925 0.6989

1. 5806 O. 6323

1. 4925 O. 6989

1. 4007 O. 7814

1.3183 0.8301

1. 2473 O. 9000

1.0000 O. 0000

0

5

2

I

I

4

I

5

3

2

I

I

I

I

2

5

I

under Constant-Amplitude Loading (LFAST = I)

0.0 1.9 0 0 1.0

0.8 70. 70. .0

0

.030 .136 .030 0.0 0.0

under User Specified Loading

0.0 1.0 0 0 1.0

•0 999.0 86.5 0.7

0

.02 0.145 .02 0.0 .125

60
r,

3 11

1.2473

1.3183

1.4007

1.4925

1.5806

1.6796

1.4925

1.4007

1.3183

1.2473

1.0000 4 15

1.2473

1.3183

1.4007

1.4925

1.5806

1.6796

1.7742

1.8796

1.7742

1.5806

1.4925

1.4007

1.3183

1.2473

1.0000 5 17

1.2473

1.3183

1.4007

1.4925

1.5806

1.6796

1.7742

1.8796 1 9785 ] 7742 1 6796 1 5806 1 4925 1 4007

1.3183

1.2473

1.0000 0

10

0.90005

0.83013

O. 78142

0.69892

0.63231

O. 55911

O. 69891

0.78141

0.83012

0.90005

0.00001

50

O. 90005

0.83013

0.78142

O. 69893

0.63231

0.55911

0.45811

0.37421

0.458]1

0.63231

O. 69892

0.78141

0.83012

0.90005

0.00001

100

O. 90005

0.83013

0.78142

O. 69893

0.63232

O. 55911

0.45812

0.37421

O. 25812

0.45811

0.5591I

0.63231

O. 69893

0.78141

0.83012

O. 90005

0.00001

0.0 0.0 0.0

6]
SAMPLE 5: Corner Crack at Hole - Space Shuttle Spectrum (LFAST = 3)

stsn 200.0

7075-T7351

447. O0

]

9. 9999E- 10

6 0

1.0

2.0

7.0

12.0

20.0

30.0

3000 10

-1

.0350

• 035O

150.0

0

2

150.00

0

520.0 71100. .33 2.50 0 0 1.0

4.00 2.50 1.0 999• 80. 0.0

1.O00E-1!

2.000E-09

1.O00E-07

4.700E-07

0.250E-05

1.200E-05

1 0 0.001

0 3 2 5 1 0

.014 .013750 .001200

-41.4

0.0

2 I 2

0.0 0.0 0.0

.013750 .001200 0.0 .0075

SAMPLE 6: Small Crack at Notch under Mini-TWIST (NALP = 1 Option)

cstamp 200.0

7075-T6

0.520E+03 0.575E+03 6.960E+04 0.0

2

9. 999E- 11 3. O00E+O0 O. 8 O. 0

7 0

0.90 1.00E-11

1.35 1.20E-09

3.40 I.OOE-08 •

5.20 1.00E-07

11.9 I.OOE-06

18.8 1.00E-05

29.0 1.00E-04

9.999E-11 3.000E+O0 1.0 0.0

4 0

0.80

1.25

3.40

14.0

O.50E-06

4000 30

-7 0

0•0500

0.05

3.15

0.0

0

4000

90.0

0

1.00E-11

I.OOE-09

I .OOE-08

I.OOE-06

].8

I 0

0 2

0.00115

0.180E+01 1 00.IOOE+01

0.999E+04 0.500E+02 O.O00E+O0

0.999E+04 0.500E+02 O.O00E+O0

1.0 0.50E-05 1.2 1.0

0.01

3 0 0

0.0031840 0.0000025 0.0031840 0.0000025 0.0000025 .003175

1

0.0

0.0

10 0 0

0.0 0.0 0.0

62
SAMPLE 7: Compact Specimen under Stress-lntensity Factor Loading

200.0

0.660E+02

3.000E+O0

1.OOE-O9

1. OOE-O8

1.00E-07

1.00E-06

1.00E-05

4.40E-05

1.00E-03

2.4

1 1

0 1

0.250

1.65

0 0

1.280E+04 0.0 0.240E+01 1 0 0.100E+01

0.0 0.0 0.999E+04 0.400E+02 O.O00E+O0

1.0 6.000E-04 1.90 1.0

0.02

1 0 I

0.510 0.250 0.48 0.250

cstamp

FVS0812

O.570E+02

I

9.999E-11

7 0

0.65

0.94

1.4

3.1

7.0

11.3

22.0

4.000E-06

2000 I0

2 0

2.0

0.650

16.5

0 .00

2 2

1.0

1 1

33.0

2 1

16.5

0 0.0

1

3.30 1

2

1.65 100000

O.O 0.0

SAMPLE 8: Center-Crack Tension

snfopt8 200.0

2024-T3

52.0 72.0 10400.

I

2.8700E-09 4.07 O.

0 0

3000 3 1 0 O. 04

1 O 0 1 8

12.0 .05 .2

12.0

20.0 2.0

0 0.0

2 2 1 10

35.0

0 0.0 0.0

O.O0

under NFOPT = 8 Option

0.0 1.00 0

O. 99g. 400.

O 0

.05 .1 .05

1.0

1.0

.0

0.0

0.0

0.0

63
SAMPLE 9: Center-Crack Tension under NFOPT - 9 Option

snfopt9 200.0

2024-T3

52.0 72.0

]

2.8700E-09 4.07

0 0

3000 -20 ]

] 0 0

]2.0 .05

12.0

20.0 2.0

0 0.0

22 22 I

35.0

0 0.0

10400.

O.

] 0.04

l 9 0

.2

0.0 1.00 0

O. 999. 400.

0

.05 .I .O5

5

0.0 0.0

SAMPLE lO: Threshold Test (KTH = 3

cstamp 200.

2618-T651

60.0 60.0

1

4.0000E-09 3.50

0 0

3000 30 ]

] 0 0

12.0 0.1

0.70

7.0 0.7

0 0.0

I l 0

1.0

I I ]

7.0 0.7

3 7.0

HALT

10000.

O.

0 O.Ol

1 0 0

0.45

0.0

O.

0

O.l

0

1000

0.1 .0078

Option)

1.00

999.

0.4

0

80.

0.1

0

1.0

1.0

.0

1.0

O.

0.0

0.0

0.0

64
II

SAMPLE !: Center-Crack Tension under Constant-Amplitude Loading

SPECTRUM FILE : cstamp

MATERIAL PROPERTIES: HT-80

YIELD STRESS : 0.1150E+03

ELASTIC MODULUS = 0.3000E+05

PLANE STRESS SOLUTION:

TIME LIMIT : 200.0 SECONDS

STEEL

ULTIMATE STRENGTH = 0.1190E+03

CONSTRAINT FACTOR = O.IOOE+01

BETA = O.IOOE+01

DKEFF IS ELASTIC:

CRACK GROWTH CONSTANTS:

CI - 0.4000E-09 C2 = 3.000

FRACTURE TOUGHNESS PROPERTIES:

C3 - 0.00 C4 - 0.00 C5 = o.g9gE+03

KF = 0.800E+02 M : 0.00

CENTER CRACK TENSION: NTYP = I

SPECIMEN WIDTH - O.400E+OI THICKNESS : O.400E+O0

INITIAL CRACK LENGTH (CI) = O.80000E+O0 INITIAL CRACK DEPTH = O.40000E+O0

NOTCH LENGTH (CN) - O.70000E+O0 NOTCH DEPTH = O.40000E+O0

NOTCH HEIGHT - O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = 2

FINAL CRACK LENGTH = (CF) REQUESTED = 0.40000E+01

PROGRAM OPTIONS: LFAST = 0

ERR = 0.234E+01 PDC = 0.20

DCPR - O.200E-OI

KOPEN = ] KCONST = 0 INVERT = 0 LSTEP - 2

NMAX :I000 NIPT - 150 NPRT - -5 NDKE - 0

PRECRACKING FROM STARTER NOTCH:

MAX STRESS = 0.2500E+02 MIN STRESS - O,O000E+O0

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

0 0.7200E+00 1.0000 2299 1.00 0.384 38.:37 0.528E-05 0.00 O.O00E+O0

0 0.7400E+00 1.0000 6826 _1.00 0.450 38._94 0.392E-05 " 0.00 O.O00E+O0

0 0.7600E+00 1.0000 12258 1.00 0.475 39.51 0.357E-05 0.00 O.O00E+O0

0 0.7800E+00 1.0000 17929 I.O0 0.485 40.07 0.351E-05 0.00 O.O00E+O0

0 O.8000E+O0 1.0000 24237 1.00 0.490 40.64 0.357E-05 0.00 O.O00E+O0

65
BLOCK(OR FLIGHT) LOADING: NFOPT = 0 LPRINT = 0

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = I

MAXLPR =

= I

BLOCK I NUMBER OF STRESS LEVELS = I NSQ = I

LEVEL SMAX SMIN CYCLES

I 0.3000E+02 -0.I000E+02 1000

SCALE = 1.00

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

2 0

4 0

5 0

7 0

10 0

12 0

13 0

16 0

17 0

19 0

21 0

22 0

24 0

26 0

27 0

29 0

30 0

32 0

33 0

34 0

36 0

37 0

38 0

39 0

40 0

42 0

43 0

44 0

44 0

45 0

46 0

47

48

49

50

51

5]

52

53

53

54

55

55

.8200E+00 1.0000 1545

.8400E+00 1.0000 3240

.8600E+00 1.0000 5160

.8800E+00 1.0000 7157

.9000E+O0 1.0000 9181

.9200E+00 1.0000 11200

.9400E+00 1.0000 13172

.9600E+00 1.0000 15096

.9800E+00 1.0000 16969

.1000E+01 1.0000 18795

.I020E+OI 1.0000 20560

.I040E+01 1.0000 22273

.I060E+01 1.0000 23930

.I080E+01 1.0000 25539

.1100E+01 1.0000 27087

.1120E+01 1.0000 28586

.II40E+OI 1.0000 30037

.1160E+01 1.0000 31443

.1180E+01 1.0000 32806

.1200E+01 1.0000 34113

.1220E+01 1.0000 35373

.1240E+01 1.0000 36612

.1260E+01 1.0000 37805

.1280E+01 1.0000 38957

.1300E+011.O000 40075

1320E+01 1.0000 41156

1340E+01 1.0000 42201

1360E+01 1.0000 43214

1380E+01 1.0000 44198

1400E+OI 1.0000 45146

1420E+OI 1.0000 46064

0.1440E+01 1.0000 46953

0.1460E+01 1.0000 47815

0.1480E+011.O000 48648

0.1500E+01 1.0000 49455

0.1520E+01 1.0000 50230

0.1540E+01 1.0000 50981

0.1560E+01 1.0000 51708

0.1580E+01 1.0000 52410

0.1600E+01 1.0000 53092

0.1620E+01 1.0000 53758

0.1640E+01 1.0000 54403

0.1660E+01 1.0000 55025

1.00

1.00

1.00

1.00

1.0o

1.00

1.00

1 O0

I O0

I oo

l O0

I O0

I O0

I O0

I O0

I O0

l O0

I O0

I O0

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.352 65.91 0.131E-04

0.398 66.80 0.I09E-04

0.421 67.68 0.I01E-04

0.433 68.55 0.992E-05

0.440 69.43 0.990E-05

0.446 70.31 0.I00E-04

0.447 71.16 0.I02E-04

0.451 72.04 0.105E-04

0.450 72.89 0.I09E-04

0.451 73.74 0.111E-04

0.452 74.59 0.115E-04

0.453 75.44 0.118E-04

0.453 76.29 0.123E-04

0.453 77.15 0.126E-04

0.452 78.00 0.131E-04

0.452 78.87 0.137E-04

0.453 79.69 0.139E-04

0.454 80.56 0.144E-04

0.453 81.38 0.149E-04

0.450 82.26 0.157E-04

0.450 83.09 0.161E-04

0.452 83.93 0.164E-04

0.451 84.82 0.171E-04

0.450 85.67 0.177E-04

0.450 86.52 0.182E-04

0.449 87.38 0.189E-04

0.448 88.24 0.195E-04

0.449 89.11 0.200E-04

0.448 89.91 0.205E-04

0.446 90.79 0.214E-04

0.446 91.68 0.222E-04

0.445 92.58 0.229E-04

0.444 93.40 0.236E-04

0.444 94.32 0.245E-04

0.443 95.15 0.251E-04

0.440 96.09 0.264E-04

0.440 96.93 0.269E-04

0.438 97.90 0.282E-04

0.438 98.76 0.289E-04

0.437 99.63 0.298E-04

0.439 100.50 0.302E-04

0.437 101.53 0.317E-04

0.435 102.43 0.329E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

66
CONVERGENCE IN

APPLIEDSTRESS

ELEMENT LENGTH

11 O.O0000E+O0

12 O.O0000E+O0

13 0.21583E-03

14 0.39472E-03

15 0.53862E-03

16 0.62370E-03

17 0.66698E-03

18 0.69641E-03

19 0.71348E-03

20 0.71947E-03

21 0.72559E-03

22 0.73176E-03

23 0.73805E-03

] 0.72764E-03

2 0.72276E-03

3 0.71441E-03

4 0.69072E-03

5 0.62833E-03

6 0.51392E-03

7 0.38240E-03

8 0.24910E-03

9 0.11944E-03

10 0.72254E-05

PLASTIC ZONE =

CONVERGENCE IN

APPLIED STRESS

ELEMENT LENGTH

]1 O.O0000E+O0

12 O.O0000E+O0

13 0.21583E-03

14 0.39472E-03

15 0.53862E-03

16 0.62370E-03

17 0.66698E-03

18 0.69641E-03

19 0.71348E-03

20 0.71947E-03

2] 0.72559E-03

22 0.73176E-03

23 0.73795E-03

] 0.72764E-03

2 0.72276E-03

3 0.71441E-03

4 0.69072E-03

5 0.62833E-03

6 0.51392E-03

7 0.38240E-03

8 0.24910E-03

9 0.]1944E-03

10 0.72254E-05

PLASTIC ZONE =

2 ITERATIONS

= -0.I000E+02

DISPLACEMENT

-0.64029E-06

0.80951E-06

0.21668E-03

0.39549E-03

0.5392]E-03

0.62323E-03

0.66558E-03

0.69528E-03

0.71313E-03

0.71988E-03

0.72568E-03

0.73186E-03

0.73795E-03

0.72786E-03

0.72289E-03

0.71457E-03

0.69097E-03

0.62862E-03

0.51396E-03

0.38246E-03

0.24914E-03

0.11946E-03

0.72443E-05

0.1918E+00

19 ITERATIONS

= O.IO00E+02

DISPLACEMENT

0.14191E-02

0.13667E-02

0.12941E-02

0.11807E-02

0.99137E-03

0.85420E-03

0.78040E-03

0.73387E-03

0.71752E-03

0.71997E-03

0.72620E-03

0.73207E-03

0.73779E-03

0.72689E-03

0.72237E-03

0.71438E-03

0.69071E-03

0.62834E-03

0.51393E-03

0.38241E-03

0.24911E-03

0.11945E-03

0.72321E-05

0.1918E+00

CRACK LENGTH

STRESS

-6.561

-1.116

-]2.165

-15.361

-19.142

-26.98!

-34.556

-43.739

-53.454

-65.637

-67.708

-79.308

-117.000

-115.208

-116.272

-116.526

-]16.592

-86.809

-29.664

-4.222

11.134

22.406

32.066

CRACK LENGTH

STRESS

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-8.490

-23.569

-38.176

-78.008

-76.020

-74.342

-77.326

-78.311

-49.568

6.141

30.025

44.350

54.576

63.150

0.16772E+01

WIDTH CENTER0.35000E+00 LINE

0.35000E+00

0. I4949E+00

0.37225E+00

0.21732E+00

0.95395E-01

0.51257E-01

0.44736E-01

0.91738E-02

0.92688E-02

0.93569E-02

0.94185E-02

0.95153E-02 CRACK

0.19176E-02 TIP

0.]9176E-02

0.38352E-02

0.76704E-02

0.11506E-01

0.17258E-01

0.23011E-01

0.28764E-01

0.38352E-0!

0.57528E-01

END OF PLASTIC ZONE

0.1

0

0

0

0

0

0

0

0

0

0

0

0

END

CENTERLINE

CRACK

TIP

6772E+01

WIDTH

0.35000E+00

0 35000E+00

0 14949E+00

0 37225E+00

0 21732E+00

0 95395E-01

0 51257E-01

0.44736E-01

0 91738E-02

0 92688E-02

0 93569E-02

94185E-02

95153E-02

19176E-02

19176E-02

38352E-02

76704E-02

11506E-01

17258E-01

23011E-0!

28764E-01

38352E-01

57528E-01

OF PLASTIC ZONE

67
CONVERGENCE IN 2

APPLIEDSTRESS=

ELEMENT

11

12

13

14

15

16

17

18

19

20

21

22

23

1

2

3

4

5

6

7

8

9

10

LENGTH

O.O0000E+O0

O.O0000E+O0

0.21583E:03

0.39472E-03

0.53862E-03

0.62370E-03

0.66698E-03

0.69641E-03

0.71348E-03

0.71947E-03

0.72559E-03

0.73176E-03

0.73795E-03

0.72764E-03

0 72276E-03

0 71441E-03

0 69072E-03

0 62833E-03

0 51392E-03

0 38240E-03

0 24910E-03

0 11944E-03

0 72254E-05

ITERATIONS

0.3000E+02

DISPLACEMENT

0.40000E-02

0.38313E-02

0.35956E-02

0.32218E-02

0.25713E-02

0.20546E-02

0.17317E-02

0.I4680E-02

0.12878E-02

0.]2176E-02

0.I1400E-02

0.]0522E-02

0.94598E-03

0.85312E-03

0.82353E-03

0.78518E-03

O.71995E-03

0.62833E-03

0.51392E-03

0.38240E-03

0.24910E-03

O.11944E-03

0.72254E-05

PLASTIC ZONE = o.IgI8E+O0

CRACK LENGTH = 0.16772E+01

STRESS

0.000

0.000

0.000

0.000

0 000

0 000

0 000

0 000

0 000

0 000

0 000

0.000

0.000

I]7.000

]17.000

]17.000

117.000

117.000

117.000

117.000

I]7.000

117.000

117.000

WIDTH

0.35000E+00

0.35000E+00

0.14949E+00

0.37225E+00

0.21732E+00

0.95395E-01

0.51257E-0]

0.44736E-01

0.91738E-02

o.g2688E-02

0.93569E-02

o.g4]85E-02

0.95]53E-02 CRACK

0.19176E-02 TIP

0.]9176E-02

0.38352E-02

0.76704E-02

0.11506E-01

0.17258E-01

0.23011E-01

0.28764E-01

0.38352E-01

0.57528E-01

END OF PLASTIC ZONE

CENTERLINE

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

56 0.]680E+01 1.0000 55626

57 0.1700E+01 1.0000 56209

57 0.1720E+01 ].0000 56772

58 0.1740E+01 1.0000 57317

1.00 0.434 ]03.34 0.339E-04

1.00 0.433 ]04.26 0.350E-04

1.00 0.431 105.]9 0.362E-04

1.00 0.431 106.13 0.373E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH = ].7670 CRACK DEPTH = 0.4000 TOTAL CYCLES = 58026

NFCODE = 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME = 2.7 SECONDS

SAMPLE 2: Through Cracks from Hole under Repeated Spike Loading

SPECTRUM FILE - cstamp TIME LIMIT - 200.0 SECONDS

MATERIAL PROPERTIES: 7075-T651

YIELD STRESS = 0.7700E+02 ULTIMATE STRENGTH = 0.8500E+02

ELASTIC MODULUS = 0.1040E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.200E+01

BETA = 0.I00E+01

68
DKEFFIS ELASTIC:

CRACKGROWTHRATESFROMTABLELOOKUP(NDKTH= 0):

DKEFF RATE

0.I000E+01 0.3199E-07

0.4000E+01 0.1679E-05

0.5000E+01 0.5000E-05

0.I000E+02 0.2000E-04

0.1200E+02 0.3500E-04

0.1900E+02 0.2000E-03

0.2500E+02 0.I000E-02

THRESHOLDCONSTANTS: C3 = 1.00 C4 - 0.00

FRACTURE TOUGHNESS PROPERTIES: KF = 0.865E+02 M = 0.70

TWO SYMMETRIC THROUGH CRACKS AT HOLE: NTYP = -4

SPECIMEN WIDTH = O.200E+OI THICKNESS _ 0.250E+00

INITIAL CRACK LENGTH (CI) = 0.14500E+00 INITIAL CRACK DEPTH = 0.25000E+00

NOTCH LENGTH (CN) = 0.13500E+00 NOTCH DEPTH = 0.25000E+00

NOTCH HEIGHT - O.IOOE-OI ELEMENTS ON STARTER NOTCH (NS) _ 2

FINAL CRACK LENGTH (CF) REQUESTED = 0.20000E+01

HOLE RADIUS (RAD) E 0.12500E+00

PROGRAM OPTIONS: LFAST = 0 KOPEN = ] KCONST _ 0 INVERT - 0 LSTEP _ I

ERR = 0.162E+01 PDC = 0.20 NMAX =1000 NIPT = 3000 NPRT = 50 NDKE _ I

PRECRACKING FROM STARTER NOTCH:

MAX STRESS - 0.1500E+02 MIN STRESS _ O.O000E+O0

BLOCK C'-RAD A/T CYCLES ALP KO/KMAX DKEC DC/DN

0 0.]270E-01 1.0000 388 2.00 0.315 5.73 0.657E-05

0 0.1580E-0] 1.0000 838 2.00 0.329 6.02 0.725E-05

0 0.]924E-01 1.0000 1288 2.00 0.335 6.33 0.802E-05

DKEA DA/DN

0.00 O.O00E+O0

0.00 O.OOOE+O0

0.00 O.O00E+O0

BLOCK (OR FLIGHT) LOADING: NFOPT - I LPRINT _ 0

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) _ 2

MAXLPR _ 0

2

BLOCK 1 NUMBER OF STRESS LEVELS = I NSQ = I SCALE = 1.00

LEVEL SMAX SMIN CYCLES

I 0.2000E+02 O.O000E+O0 I

BLOCK 2 NUMBER OF STRESS LEVELS = ] NSQ = 2 SCALE = 1.00

LEVEL SMAX SMIN CYCLES

1 0.1500E+02 O.O000E+O0 5000

NOTE: KO/KMAX AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

BLOCK C'-RAD A/T CYCLES ALP KO/KMAX DKEC DC/DN DKEA DA/DN

2 0.2013E-0] 1.0000 16 2.00 0.270

2 0.2405E-01 1.0000 505 2.00 0.258

2 0.2819E-01 ].0000 955 2.00 0.256

2 0.3263E-01 1.0000 1405 2.00 0.257

2 0.3732E-01 1.0000 1855 2.00 0.258

2 0.4224E-01 1.0000 2305 2.00 0.260

6.]6 0.760E-05

6.63 0.878E-05

6.91 0.955E-05

7.13 0.102E-04

7.31 0.107E-04

7.47 0.112E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

69
2 0.4737E-01 1.0000

2 0.5269E-01 1.0000

2 0.5820E-01 1.0000

2 0.6390E-01 1.0000

2 0.6976E-01 1.0000

4 0.7591E-01 1.0000

4 0.8247E-01 1.0000

4 0.8903E-01 1.0000

4 0.9559E-01 1.0000

4 0.1023E+00 1.0000

4 0.I093E+00 1.0000

4 0.1164E+00 1.0000

4 0.1237E+00 i.0000

4 0.1312E+00 1.0000

4 0.1389E+00 1.0000

4 0.1468E+00 1.0000

6 0.1551E+00 1.0000

6 0.1640E+00 1.0000

6 0.1726E+00 1.0000

6 0.1814E+00 1.0000

6 0.1906E+00 ].0000

0.2001E+00 1.0000

0.2101E+00 1.0000

6

6

6 0.2206E+00

6 0.2315E+00

6 0.2430E+00

6 0.2544E+00

6 0.2657E+00

8 0.2779E+00

8 0.2900E+00

8 0.3028E+00

8 0.3163E+00

8 0.3306E+00

8 0.3449E+00

8 0.3590E+00

8 0.3739E+00

8 0.3899E+00

8 0.4070E+00

8 0.4242E+00

8 0.4408E+00

8 0.4587E+00

10 0.4781E+00

10 0.4977E+00

10 0.5191E+00

10 0.5395E+00

10 0.5609E+00

10 0.5842E+00

10 0.6094E+00

10 0.6370E+00

10 0.6629E+00

10 0.6892E+00

10 0.7182E+00

10 0.7502E+00

10 0.7861E+00

1.0000

1.oooo

1.0000

1.0000

1.0000

1.0000

1.0000

1.oooo

1.0000

].0000

].0000

1.0000

1.0000

1.0000

1.0000

1.0000

1 0000

1 0000

1 0000

1 0000

1 0000

I 0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

I.OOOO

1.0000

2755

3205

3655

4105

4555

5010

5521

5987

6437

6887

7337

7787

8237

8687

9137

9587

10035

10547

11003

11453

11903

12353

12803

13253

13703

14153

14583

14988

15447

15847

16247

16647

17047

17426

17776

18126

18476

18826

19156

19456

19756

20051

20382

20674

20932

21182

21432

21682

21931

22147

22347

22547

22747

22947

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

0.260

0.26]

0.261

0.26]

0.262

0.262

0.266

0.264

0.263

0.263

0.263

0.263

0.263

0.264

0.264

0.263

0.223

0.269

0.264

0.263

0.264

0.263

0.264

0.263

0.264

0.263

0.262

0.264

0.270

0.265

0.264

0.264

0.264

0.266

0.264

0.264

0.265

0.265

0.267

0.265

0.265

0.302

0.268

0.264

0.267

0.267

0.267

0.266

0.273

0.268

0.268

0.267

0.268

0.268

7.62 0.]16E-04

7.76 0.120E-04

7.90 0.125E-04

8.02 0.129E-04

8.14 0.132E-04

8.25 0.136E-04

8.31 0,138E-04

8.46 0.143E-04

8.60 0.148E-04

8.72 0.152E:04

8.84 0.156E-04

8.95 0.160E-04

9.07 0.165E-04

9.18 0.169E-04

9.31 0.173E-04

9.45 0.178E-04

10.36 0.223E-04

9.60 0.184E-04

9.83 0.193E-04

9.98 0.]99E-04

10.11 0.207E-04

10.27 0.217E-04

10.40 0.226E-04

10.59 0.238E-04

10.73 0.248E-04

10.91 0.261E-04

]1.10 0.276E-04

11.23 0.286E-04

I].27 0.288E-04

i].55 0.311E-04

I].75 0.328E-04

11.96 0.346E-04

12.15 0.367E-04

12.32 0.386E-04

12.55 0.415E-04

12.77 0.443E-04

12.98 0.471E-04

13.22 0.505E-04

13.40 0.532E-04

13.67 0.574E-04

13.93 0.617E-04

13.]4 0.494E-04

14.41 0.700E-04

]4.82 0.779E-04

]5.03 0.822E-04

15.34 0.888E-04

15.68 0.966E-04

16.06 0.I06E-03

16.26 0.111E-03

16.80 0.125E-03

17.20 0.137E-03

17.67 0.152E-03

18.]7 0.169E-03

18.75 0.190E-03

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

70
CONVERGENCE IN

APPLIEDSTRESS

ELEMENT LENGTH

11 -0.12500E+00

12 -0.I0000E-01

13 0.24429E-04

14 0.47516E-04

15 0.82233E-04

16 0.11551E-03

17 0.13841E-03

18 0.15296E-03

19 0.16238E-03

20 0.16879E-03

21 0.17184E-03

22 0.17351E-03

23 0.17460E-03

24 0.17498E-03

25 0.17553E-03

26 0.17545E-03

27 0.17532E-03

28 0.17535E-03

I 0.17389E-03

2 0.17327E-03

3 0.17225E-03

4 0.16969E-03

5 0.16409E-03

6 0.15203E-03

7 0.12529E-03

8 0o86756E-04

9 0.47740E-04

10 0.I0388E-04

PLASTIC ZONE -

2 ITERATIONS

= 0.1500E+02

DISPLACEMENT

0.32001E-02

0.31769E-02

0.31587E-02

0.30420E-02

0.26859E-02

0.21697E-02

0.17011E-02

0.13446E-02

0.I0625E-02

0.82044E-03

0.63324E-03

0.51855E-03

0.43337E-03

0.38271E-03

0.35922E-03

0.33328E-03

0.30791E-03

0.27901E-03

0.25437E-03

0.24655E-03

0.23634E-03

0.21881E-03

0.19389E-03

0.16231E-03

0.12529E-03

0.86756E-04

0.47740E-04

0.I0388E-04

0.I251E-01

CRACK LENGTH

STRESS

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

162.000

162.000

162.000

162.000

162.000

162.000

162.000

162.000

162.000

162.000

0.94685E+00

WIDTH CENTER0.12500E+00 LINE

O.]O000E-01

0.58931E-01

0.21447E+00

0.22244E+00

0.I3894E+00

0.72762E-01

0.41574E-01

0.26205E-01

0.17840E-01

0.78680E-02

0.43891E-02

0.29872E-02

0.67788E-03

0.82884E-03

0.68413E-03

0.62813E-03

0.62782E-03 CRACK

0.12513E-03 TIP

0.12513E-03

0.25026E-03

0.50053E-03

0.75079E-03

0.11262E-02

0.15016E-02

0.18770E-02

0.25026E-02

0.37539E-02

END OF PLASTIC ZONE

CONVERGENCE IN

APPLIED STRESS

ELEMENT LENGTH

11 -0.12500E+00

12 -0.10000E-O]

13 0.24429E-04

14 0.47516E-04

15 0.82233E-04

16 0.I1551E-03

17 0.13841E-03

18 0.15296E-03

19 0.16238E-03

20 0.16879E-03

21 0.17184E-03

22 0.17351E-03

23 0.17467E-03

24 0.17553E-03

25 0.17545E-03

26 0.17532E-03

27 0.17535E-03

28 0.25437E-03

1 0.25464E-03

2 0.24681E-03

3 0.23659E-03

4 0.21904E-03

11 ITERATIONS

= O.O000E+O0

DISPLACEMENT

0.65898E-04

0.66479E-04

0.66945E-04

0.70125E-04

0.82714E-04

0.11538E-03

0.13806E-03

0.15251E-03

0.16207E-03

0.16884E-03

0.17227E-03

0.17411E-03

0.17496E-03

0.17542E-03

0.17508E-03

0 17517E-03

0 17537E-03

0 17544E-03

0 17399E-03

0 17336E-03

0 17235E-03

0 16980E-03

CRACK LENGTH

STRESS

0.000

0.000

0.000

0.000

0.000

-0.929

-1.950

-3.442

-5.647

-9.475

-14.805

-21.423

-30.375

-43.874

-48.356

-48.520

-57.22]

-81.000

-81.000

-81.000

-81.000

-81.000

0.94749E+00

WIDTH CENTER0.12500E+00 LINE

0.I0000E-01

0.58931E-01

0.21447E+00

0.22244E+00

0.13894E+00

0.72762E-01

0.41574E-01

0.26205E-01

0.17840E-01

0.78680E-02

0.43891E-02

0.36651E-02

0.82884E-03

0.68413E-03

0.62813E-03

0.62782E-03

0.63537E-03 CRACK

0.12526E-03 TIP

0.12526E-03

0.25051E-03

0.50103E-03

71
5 0.19410E-03 0.1642]E-03 -81.000

6 0.16248E-03 0.]5216E-03 -81.000

7 0.]2543E-03 0.12537E-03 -40.361

8 0.86848E-04 0.86813E-04 14.134

9 O.477gOE-04 0.47784E-04 38.998

10 0.10399E-04 O.]0395E-04 59.202

PLASTIC ZONE = 0.]253E-01

0.75154E-03

0.11273E-02

0.15031E-02

0.18789E-02

0.25051E-02

0.37577E-02

END OF PLASTIC ZONE

BLOCK C'-RAD A/T CYCLES ALP KO/KMAX DKEC DC/DN DKEA DA/DN

10 0.8231E+00 1.0000 23130

10 0.8593E+00 1.0000 23280

10 0.9037E+00 1.0000 23430

10 0.9519E+00 1.0000 23562

10 0.9981E+00 1.0000 23662

10 0.I059E+01 1.0000 23762

10 0.1142E+01 1.0000 23857

10 0.1216E+01 1.0000 23916

2 O0

2 O0

2 O0

2 O0

2 O0

2 O0

2 O0

2 O0

0.272 19.22 0.214E-03 0.00 O.O00E+O0

0.269 19.94 0.265E-03 0.00 O.O00E+O0

0.270 20.70 0.331E-03 0.00 O.O00E+O0

0.276 21.35 0.396E-03 0.00 O.O00E+O0

0.274 22.38 0.522E-03 0.00 O.O00E+O0

0.275 23.61 0.714E-03 0.00 O.O00E+O0

0.272 25.74 0.119E-02 0.00 O.O00E+O0

0.284 27.07 0.159E-02 0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH 1 1.4405 CRACK DEPTH = 0.2500 TOTAL CYCLES = 23963

NFCODE = 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME = 60.5 SECONDS

SAMPLE 3: Surface-Crack Tension under Constant-Amplitude Loading (LFAST = I)

SPECTRUM FILE = cstamp TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 2219-T851

YIELD STRESS = 0.4500E+02 ULTIMATE STRENGTH = 0.6500E+02

ELASTIC MODULUS = 0.I000E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.190E+01

BETA = 0.I00E+01

DKEFF IS ELASTIC:

CRACK GROWTH CONSTANTS:

C] = 0.I200E-06 C2 = 3.120

FRACTURE TOUGHNESS PROPERTIES:

C3 = 2.70 C4 = 0.80 C5 = 0.700E+02

KF = 0.700E+02 M = 0.00

SURFACE CRACK TENSION: NTYP = 0 AND LTYP = 0

SPECIMEN WIDTH = 0.400E+01 THICKNESS = O.300E+O0

INITIAL CRACK LENGTH (CI) = 0.]3600E+00 INITIAL CRACK DEPTH = 0.30000E-01

NOTCH LENGTH (CN) = 0.13600E+00 NOTCH DEPTH = 0.30000E-01

NOTCH HEIGHT = O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = 1

FINAL CRACK LENGTH (CF) REQUESTED = 0.40000E+01

72
PROGRAMOPTIONS: LFAST= ] KOPEN= ] KCONST= 0 INVERT= 0 LSTEP= ]

ERR= 0.110E+01 PDC= 0.20 NMAX=1000 NIPT = 50 NPRT= 5 NDKE= 0

BLOCK(ORFLIGHT)LOADING:NFOPT= 0 LPRINT= 0

TOTALNUMBEROFBLOCKS(OR) FLIGHTSTO BEREPEATED=

NUMBEROFDIFFERENTBLOCKS(ORFLIGHTS)= 1

MAXLPR = 0

1

BLOCK ] NUMBER OF STRESS LEVELS = I NSQ = 1 SCALE - 1.00

LEVEL SMAX SMIN CYCLES

I 0.3000E+02 0.1500E+02 1000

EQUIVALENT cRAcK-OPENING STRESS CONCEPT IF CRACK LENGTH >= 0.1455

AND TOTAL BLOCKS (OR FLIGHTS) > 2

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

] 0.]360E+00 0.1000

] 0.1370E+00 0.1352

I 0.1379E+00 0.1567

2 0.1389E+00 0.1735

2 0.1399E+00 0.]881

2 0.1409E+00 0.2008

2 0.1419E+00 0.2124

2 0.1429E+00 0.2232

2 0.1439E+00 0.2333

2 0.1450E+00 0.2427

] 1.90 0.500

617 1.90 0.523

930 1.90 0.529

]145 1.90 0.532

]313 1.90 0.532

1450 1.90 0.533

]568 1.90 0.533

1672 1.90 0.534

]765 1.90 0.534

]849 1.90 0.533

2 40 O.O00E+O0

3 03 0.227E-05

3 48 0.371E-05

3 8] 0.508E-05

4 07 0.646E-05

4 29 0.773E-05

4 49 0.909E-05

4 67 0.103E-04

4 83 0.]15E-04

4 97 0.129E-04

4.90 0.121E-04

5.43 0.]89E-04

5.77 0.220E-04

5.98 0.245E-04

6.]5 0.268E-04

6.28 0.285E-04

6.39 0.303E-04

6.49 0.316E-04

6.58 0.330E-04

6.66 0.345E-04

CONVERGENCE IN 2

APPLIED STRESS =

ELEMENT

11

12

13

14

15

16

17

18

19

20

21

22

i

2

3

4

5

6

7

8

9

10

LENGTH

O.O0000E+O0

0.16650E-03

0.16999E-03

0.17208E-03

0.17370E-03

0.17442E-03

0 ]7468E-03

0 17496E-03

0 17521E-03

0 17545E-03

0 17572E-03

0 17595E-03

0.17527E-03

0.]7430E-03

0.]7238E-03

0.]6681E-03

0.15225E-03

0.12719E-03

0.97948E-04

0.67645E-04

0.37100E-04

0.80019E-05

ITERATIONS

0.3000E+02

DISPLACEMENT

0.80751E-03

0 36146E-03

0 31940E-03

0 28878E-03

0 26131E-03

0 24681E-03

0 24161E-03

0 23604E-03

0 23018E-03

0 22401E-03

0.21721E-03

0.20951E-03

0.20062E-03

0.]9432E-03

0.]8613E-03

0.]7211E-03

0.15225E-03

0.12719E-03

0.97948E-04

0.67645E-04

0.37100E-04

0.80019E-05

PLASTIC ZONE = 0.1620E-01

CRACK LENGTH = 0.14557E+00

STRESS

0.000

0.000

0.000

0.000

0.000

0.000

0 000

0 000

0 000

0 000

0 000

0 000

104 500

104 500

104 500

104 500

104.500

104.500

104.500

104.500

104.500

104.500

WIDTH

0.13600E+00

0.36752E-02

0.17958E-02

0.16232E-02

0.I0276E-02

0.20261E-03

0.20757E-03

0.21312E-03

0.20611E-03

0.20727E-03

0.21133E-03

0.20298E-03 CRACK

0.]6199E-03 TIP

0.16199E-03

0.32399E-03

0.64798E-03

0.97197E-03

0.14580E-02

0.19439E-02

0.24299E-02

0.32399E-02

0.48598E-02

END OF PLASTIC ZONE

CENTERLINE

73
CONVERGENCE IN

APPLIEDSTRESS

ELEMENT

11

12

13

14

15

16

17

18

19

20

21

22

I

2

3

4

5

6

7

8

9

10

LENGTH

O.O0000E+O0

0.16650E-03

0.17098E-03

0.17370E-03

0.17442E-03

0 17468E-03

0 17496E-03

0 17521E-03

0 17545E-03

0 17572E-03

0 17595E-03

0.20062E-03

0.20090E-03

0.19460E-03

0.18639E-03

0.17235E-03

0.15247E-03

0.12737E-03

0.98086E-04

0.67740E-04

0.37152E-04

0.80130E-05

PLASTIC ZONE =

12 ITERATIONS

- 0.1500E+02

DISPLACEMENT

0.42052E-03

0.21834E-03

0 19755E-03

0 18389E-03

0 18029E-03

0 17916E-03

0 17808E-03

0 17709E-03

0 17628E-03

0 17576E-03

0 17586E-03

0 17618E-03

0 17551E-03

0 17454E-03

0 17262E-03

0 16705E-03

0 15251E-03

0 12741E-03

0.98107E-04

0.67740E-04

0.37150E-04

0.80112E-05

0.1622E-01

CRACK LENGTH

STRESS

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-0.815

-23.132

-55.000

-55.000

-55.000

-55.000

-55.000

-23.909

21.323

40.592

52.015

60.084

66.915

0.14578E+00

WIDTH CENTER0.13600E+00 LINE

0.36752E-02

0.34191E-02

0.I0276E-02

0.20261E-03

0.20757E-03

0.21312E-03

0.20611E-03

0.20727E-03

0.21133E-03

0.20298E-03

0.20371E-03 CRACK

0.16222E-03 TIP

0.16222E-03

0.32444E-03

0.64889E-03

0.97333E-03

0.14600E-02

0.19467E-02

0.24333E-02

0.32444E-02

0.48667E-02

END OF PLASTIC ZONE

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

2 0.1460E+00 0.2516 1926 1.90 0.536 5.11 0.138E-04 6.73 0.351E-04

2 0.1470E+00 0.2603 1998 1.90 0.536 5.23 0.150E-04 6.79 0.362E-04

EQUIVALENT CRACK-OPENING SIF RATIO: KO/KMAX = 0.534

4 0.1821E+00 0.4472 3163

4 0.2281E+00 0.6049 3747

5 0.2868E+00 0.7625 4106

5 0.3620E+00 0.9279 4335

5 0.4615E+00 1.0000 4471

5 0.5928E+00 1.0000 4563

5 0.7727E+00 1.0000 4634

5 0.1036E+01 1.0000 4683

1 90

1 90

1 90

l 90

1 90

1 90

1 90

1 90

0.534 7.59 0.523E-04 8.06 0.637E-04

0.534 9.58 0.113E-03 9.30 0.103E-03

0.534 11.79 0.229E-03 10.76 0.168E-03

0.534 14.40 0.460E-03 12.45 0.276E-03

0.534 18.21 0.I09E-02 0.00 O.O00E+O0

0.534 20.74 0.184E-02 0.00 O.O00E+O0

0.534 23.91 0.347E-02 0.00 O.O00E+O0

0.534 28.20 0.863E-02 0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH - 1.3667 CRACK DEPTH = 0.3000 TOTAL CYCLES = 4701

NFCODE = O: KMAX > C5

CPU TIME = 0.8 SECONDS

74
SAMPLE 4: Corner Crack from Hole under User Specified Loading

SPECTRUM FILE - cstamp TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 7075-T651

YIELD STRESS - 0.7700E+02 ULTIMATE STRENGTH = 0.8500E+02

ELASTIC MODULUS - 0.I040E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.I00E+01

BETA = 0.I00E+01

DKEFF IS ELASTIC:

CRACK GROWTH CONSTANTS:

CI - 0.3200E-07 C2 - 3.000

FRACTURE TOUGHNESS PROPERTIES:

C3 = 0.00 C4 = 0.00 C5 = 0.9ggE+03

KF - 0.865E+02 M = 0.70

ONE CORNER CRACK AT HOLE: NTYP = -I

SPECIMEN WIDTH = 0.200E+01 THICKNESS = 0.250E+00

INITIAL CRACK LENGTH (CI) = 0.14500E+00 INITIAL CRACK DEPTH = O.20000E-O]

NOTCH LENGTH (CN) = 0.14500E+00 NOTCH DEPTH = O.20000E-OI

NOTCH HEIGHT - O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = 2

FINAL CRACK LENGTH (CF) REQUESTED - O.20000E+O]

HOLE RADIUS (RAD) = 0.12500E+00

PROGRAM OPTIONS: LFAST = 0 KOPEN = I KCONST = 0 INVERT = 0 LSTEP = I

ERR - 0.162E+01 PDC = 0.20 NMAX =IO00 NIPT = 1000 NPRT = 20 NDKE = 0

BLOCK (OR FLIGHT) LOADING: NFOPT - I LPRINT = 0 MAXLPR =

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED = I00

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = 5

0

BLOCK I NUMBER OF STRESS LEVELS = 6 NSQ = I

LEVEL SMAX SMIN CYCLES

I 0.1871E+02 0.1350E+02 5

2 o. Ig77E+02 0.1245E+02 2

3 0.2101E+02 0.I172E+02 ]

4 0.1977E+02 0.1245E+02 I

5 0.1871E+02 0.1350E+02 4

6 0.1500E+02 0.I354E+01 ]

SCALE = 15.00

BLOCK 2 NUMBER OF STRESS LEVELS = I0 NSQ = 5

LEVEL SMAX SMIN CYCLES

I 0.1871E+02 0.1350E+02 5

2 o. Ig77E+02 0.1245E+02 3

3 0.2101E+02 0.1172E+02 2

4 0.2239E+02 0.1048E+02 I

5 0.2371E+02 o.g485E+01 ]

6 0.2239E+02 O.104BE+02 1

7 0.2101E+02 0.1172E+02 I

8 0.1977E+02 0.1245E+02 2

9 0.1871E+02 0.1350E+02 5

IO 0.1500E+02 O.O000E+O0 I

SCALE = 15.00

75
BLOCK

BLOCK

BLOCK

NUMBER OF STRESS

LEVEL SMAX

1 0.1871E+02

2 0.1977E+02

3 0.2101E+02

4 0.2239E+02

5 0.2371E+02

6 0.2519E+02

7 0.2239E+02

8 0.2101E+02

9 0.1977E+02

10 0.1871E+02

11 0.1500E+02

NUMBER OF STRESS

LEVEL SMAX

I 0.1871E+02

2 0.1977E+02

3 0.2101E+02

4 0.2239E+02

5 0.2371E+02

6 0.2519E+02

7 0.2661E+02

8 0.2819E+02

9 0.2661E+02

10 0.2371E+02

11 0.2239E+02

12 0.2101E+02

13 0.1977E+02

]4 0.1871E+02

15 0.1500E+02

NUMBER OF STRESS

LEVEL SMAX

I 0.1871E+02

2 0.1977E+02

3 0.2101E+02

4 0.2239E+02

5 0.2371E+02

6 0.2519E+02

7 0.266]E+02

8 0.2819E+02

9 0.2968E+02

10 0.2661E+02

11 0.2519E+02

12 0.2371E+02

13 0.2239E+02

14 0.2101E+02

15 0.1977E+02

]6 0.1871E+02

17 0.1500E+02

LEVELS = 11

SMIN

0.1350E+02

0.1245E+02

0.1172E+02

0.I048E+02

0.9485E+01

0.8386E+01

0.I048E+02

0.1172E+02

0.1245E+02

0.1350E+02

O.O000E+O0

LEVELS = 15

SMIN

0.1350E+02

0.1245E+02

0.1172E+02

0.1048E+02

0.9485E+01

0.8386E+01

0.6872E+01

0.5613E+01

0.6872E+01

0.9485E+01

0.I048E+02

0.1172E+02

0.1245E+02

0.I350E+02

O.O000E+O0

LEVELS = 17

SMIN

0.1350E+02

0.1245E+02

0.I172E+02

0.I048E+02

0.9485E+01

0.8386E+01

0.6872E+01

0.5613E+01

0.3872E+01

0.6872E+01

0.8386E+01

0.9485E+01

0.I048E+02

0.I172E+02

0.1245E+02

0.1350E+02

O.O000E+O0

NSQ

NSQ

NSQ

= I0

CYCLES

5

3

2

2

1

I

I

I

2

5

1

= 50

CYCLES

5

3

2

3

1

I

I

1

I

I

2

1

2

5

I

= I00

CYCLES

5

3

2

3

2

I

2

I

2

I

I

I

3

1

2

5

I

SCALE

SCALE

SCALE -

- 15.00

= 15.00

15.00

76
NOTE: KO/KMAX AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

BLOCK C'-RAD A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

I 0.2001E-01 0.0801 ] 1.00 0.000

1116 0.452]E-0I 0.2334 17820 1.00 0.333

2146 0.7355E-01 0.4147 34273 1.00 0.473

3112 0.1096E+00 0.6349 49716 1.00 0.491

4028 0.1489E+00 0.8468 64346 1.00 0.491

4863 0.]951E+00 1.0000 77695 1.00 0.344

5707 0.2507E+00 1.0000 91186 1.00 0.504

6525 0.3136E+00 I.O000 104252 1.00 0.361

7388 0.3842E+00 1.0000 118037 1.00 0.498

8222 0.4680E+00 1.0000 ]31374 1.00 0.486

9038 0.5793E+00 1.0000 144408 1.00 0.482

9695 0.6981E+00 1.0000 154910 1.00 0.462

10345 0.8649E+00 1.0000 165294 1.00 0.297

10877 0.]116E+01 1.0000 173797 1.00 0.367

]0.97 0.I06E-04

15.58 0.192E-05

18.40 0.772E-06

20.93 0.156E-05

23.38 0.110E-05

26.19 0.113E-04

27.55 0.563E-05

29.11 0.22IE-04

30.87 0.443E-05

32.95 0.123E-04

35.69 0.167E-04

38.66 0.878E-05

43.07 0.241E-03

50.92 0.831E-04

13.11 0.181E-04

18.34 0.184E-05

21.65 0.126E-05

23.90 0.232E-05

25.33 0.140E-05

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH = 1.3067 CRACK DEPTH = 0.2500 TOTAL CYCLES = 175153

NFCODE = 6: CRACK LENGTH PLUS PLASTIC ZONE EXCEEDS WIDTH

CPU TIME = 18.8 SECONDS

SAMPLE 5: Corner Crack at Hole - Space Shuttle Spectrum (LFAST = 3)

SPECTRUM FILE = stsn TiME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 7075-T7351

YIELD STRESS = 0.4470E+03 ULTIMATE STRENGTH= 0.5200E+03

ELASTIC MODULUS = 0.7IIOE+05

PLANE STRAIN SOLUTION: CONSTRAINT FACTOR = 0.250E+01

BETA = 0.I00E+01 POISSONS RATIO = 0.330E+00

DKEFF IS ELASTIC:

CRACK GROWTH RATES FROM TABLE LOOKUP (NDKTH = 0):

DKEFF RATE

O.IO00E+OI

0.2000E+01

O.7000E+OI

0.1200E+02

0.2000E+02

0.3000E+02

0.1000E-I0

0.2000E:OB

O.IO00E-06

0.4700E-06

0.2500E-05

0.1200E-04

THRESHOLD CONSTANTS: C3 = 2.50 C4 = 1.00

FRACTURE TOUGHNESS PROPERTIES: KF = 0.800E+02 M = 0.00

77
ONE CORNER CRACK AT HOLE: NTYP = -1

SPECIMEN WIDTH - 0.350E-01 THICKNESS - 0.140E-01

INITIAL CRACK LENGTH (Cl) = 0.13750E-01 INITIAL CRACK DEPTH = 0.12000E-02

NOTCH LENGTH (CN) - 0.13750E-01 NOTCH DEPTH = 0.12000E-02

NOTCH HEIGHT - O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) - 2

FINAL CRACK LENGTH (CF) REQUESTED = 0.35000E-01

HOLE RADIUS (RAD) = 0.75000E-02

PROGRAM OPTIONS: LFAST = 3 KOPEN = I KCONST = 0 INVERT = I LSTEP - !

ERR - 0.967E+01 PDC - 0.20 NMAX -I000 NIPT - 3000 NPRT - !0 NDKE - 0

BLOCK (OR FLIGHT) LOADING: NFOPT = 5 LPRINT - !

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO B_ REPEATED -

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) 2

MAXLPR - 2

2

CONSTANT CRACK-OPENING CONCEPT: KO/KMAX = 0.283

SHORT SHUTTLE SPECTRUM: HIGHEST STRESS = 0.]500E+03

NOTE: KO/KMAX AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

NLR GENERATED STS TEST SEQUENCE

11430 LOADING REVERSALS -30 => X <= 100 (= ! FLIGHT)

BLOCK SMAX SMIN CYCLES TOTAL CYCLES

1 0.7500E+02 O.O000E+O0 ] 1

1 0.6750E+02 0.7500E+01 6 7

1 0.3000E+02 -0.3000E+02 11 18

1 0.6000E+02 0.1500E+02 523 541

1 0.6750E+02 0.7500E+01 813 1354

I 0.6750E+02 0.3000E+02 2000 3354

I 0.6750E+02 0.3000E+02 ggg 4353

! 0.6750E+02 -0.4500E+02 1 4354

1 0.1275E+03 0.7500E+01 1 4355

] 0.1050E+03 0.7500E+01 1 4356

] 0.]050E+03 0.3000E+02 ] 4357

1 0.3750E+02 -0.3750E+02 7 4364

2 0.8250E+02 0.2250E+02 14 14

2 0.I050E+03 0.6000E+02 88 102

2 0.6750E+02 0.7500E+01 542 644

2 0.1500E+03 -0.4500E+02 1 645

2 0.1350E+03 0.6750E+02 230 875

2 0.7500E+02 O.O000E+O0 2 877

2 0.I125E+03 0.5250E+02 4 881

2 0.1125E+03 -0.3000E+02 I 882

2 0.3000E+02 -0.3000E+02 37 919

2 0.1275E+03 0.8250E+02 29 948

2 0.1275E+03 0.3000E+02 1 949

2 0.6750E+02 0.3000E+02 400 1349

78
BLOCK C'-RAD A/T CYCLESALP KO/KMAXDKC DC/DN

]I 0.6267E-02 0.1438 31000 2.50

25 0.6471E-02 0.2964 70165 2.50

37 0.7169E-02 0.4971 106413 2.50

48 0.8392E-02 0.6984 136538 2.50

57 O.]O00E-01 0.8852 161495 2.50

63 0.1199E-01 1.0000 181340 2.50

69 0.1450E-01 1.0000 195437 2.50

74 0.1761E-01 1.0000 210526 2.50

77 0.2248E-01 1.0000 221397 2.50

DKA DA/DN

0.283

0.283

0 283

0

0

0

0

0

0

22.02 0.593E-08

30.13 0.]58E-07

283 36.07 0.533E-06

283 4].84 0.439E-07

283 49.84 0.757E-07

283 54.41 0.995E-07

283 62.24 0.147E-06

283 86.05 0.371E-06

12.44 0.364E-09 32.65 0.203E-07

40.08 0.385E-07

42.54 0.463E-07

44.27 0.I04E-05

46.64 0.616E-07

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH m 0.0350 CRACK DEPTH = 0.0140 TOTAL CYCLES = 222378

NFCODE = 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME = 0.4 SECONDS

SAMPLE 6: Small Crack at Notch under Mini-TWIST (NALP = I Option)

SPECTRUM FILE = cstamp TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 7075-T6

YIELD STRESS = 0.5200E+03 ULTIMATE STRENGTH = 0.5750E+03

ELASTIC MODULUS = 0.6960E+05

PLANE-STRAIN-TO-PLANE-STRESS SOLUTION:

(ALP = ALP] TO ALP2 AND BETA = BETA] TO BETA2)

DKEFF IS ELASTIC:

CRACK GROWTH RATES (dc/dN) FROM TABLE LOOKUP (NDKTH = 0):

DKEFF RATE

0.9000E+O0 O.]O00E-IO

0.1350E+01 0.1200E-08

0.3400E+0] O.]O00E-07

0.5200E+0] O.]O00E-06

0.1190E+02 O.]O00E-05

0.1880E+02 0.I000E-04

0.2900E+02 0.I000E-03

THRESHOLD CONSTANTS: C3 = 0.80 C4 = 0.00

FRACTURE TOUGHNESS PROPERTIES: KF = 0.500E+02 M = 0.00

CRACK GROWTH RATES (da/dN) FROM TABLE LOOKUP (NDKTH = 0):

DKEFF RATE

O.8000E+O0 0.I000E-I0

0.1250E+01 0.I000E-08

0.3400E+01 0.I000E-07

0.]400E+02 0.1000E-05

THRESHOLD CONSTANTS: C3 = 1.00 C4 = 0.00

79
FRACTURE TOUGHNESS PROPERTIES: KF = 0.500E+02 M - 0.00

RATE AT START OF TRANSITION = 0.5000E-06 WITH ALPI = 1.80 AND BETAI = 1.00

RATE AT END OF TRANSITION - 0.5000E-05 WITH ALP2 = 1.20 AND BETA2 - 1.00

SURFACE CRACK AT cENTER OF SEMI-CIRCULAR EDGE NOTCH: NTYP = -7

SPECIMEN WIDTH = 0.500E-01 THICKNESS = 0.115E-02

INITIAL CRACK LENGTH (CI) = 0.31840E-02 INITIAL CRACK DEPTH = 0.25000E-05

NOTCH LENGTH (CN) = 0.31840E-02 NOTCH DEPTH - 0.25000E-05

NOTCH HEIGHT = 0.250E-05 ELEMENTS ON STARTER NOTCH (NS) = 2

FINAL CRACK LENGTH (CF) REQUESTED = 0.50000E-01

HOLE RADIUS (RAD) = 0.31750E-02

KT = 0.315E+01 NBCF OPTION = I

PROGRAM OPTIONS: LFAST = 0 KOPEN - I KCONST = 0 INVERT = 0 LSTEP = i

ERR = 0.I09E+02 PDC = 0.20 NMAX =1000 NIPT = 4000 NPRT = 30 NDKE = 0

BLOCK (OR FLIGHT) LOADING: NFOPT = 3 LPRINT = 0 MAXLPR =

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED = 4000

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = 10

MINI-TWIST FLIGHT-LOAD SEQUENCE: MEAN STRESS - 0.9000E+02

NOTE: KO/KMAX AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

BLOCK C'-RAD A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

1 0.9000E-05 0.0022 2 1.80 -0.231

1967 0.9084E-05 0.0041 29042 1.80 -0.163

3818 0.1001E-04 0.0067 59042 1.80 -0.016

5756 0.1203E-04 0.0102 89042 1.80 -0.022

7615 0.1713E-04 0.0167 118059 1.80 0.185

9316 0.3189E-04 0.0316 144312 1.80 0.218

10720 0.6726E-04 0.0641 164960 1.80 0.260

1.27 O.O00E+O0

2.06 O.O00E+O0

2.87 O.O00E+O0

3.60 O.O00E+O0

4.62 O.O00E+O0

6.23 0.171E-08

8.68 0.262E-08

2.51 0.119E-08

3.08 O.O00E+O0

3.53 O.O00E+O0

3.94 O.O00E+O0

4.73 O.O00E+O0

6.30 0.175E-08

8.92 0.278E-08

11564 0.1267E-03 0.1183 179860 1.80 0.133 11.55 0.213E-07 12.06 0.169E-07

12193 0.2064E-03 0.1879 190616 1.80 0.224 13.93 0.910E-08 15.01 0.111E-07

12831 0.3085E-03 0.2783 199373 1.80 0.267 16.20 0.I06E-07 18.19 0.151E-07

13184 0.4165E-03 0.3702 204576 1.80 0.257 17.93 0.670E-07 21.15 0.536E-07

13384 0.5263E-03 0.4683 207999 1.80 0.218 19.35 0.101E-06 23.87 0.794E-07

13594 0.6318E-03 0.5706 210912 1.80 0.216 20.65 0.364E-08 26.57 0.649E-08

13708 0.8368E-03 0.7841 213099 1.75 0.284 22.92 0.856E-08 32.41 0.248E-07

14138 0.I010E-02 1.0000 218348 1.55 0.342 37.30 0.127E-06

14211 0.1118E-02 1.0000 219369 1.63 0.340 38.16 0.134E-06

14240 0.1212E-02 1.0000 219744 1.61 0.099 38.83 0.474E-06

14273 0.1289E-02 1.0000 220142 1.56 0.284 39.35 0.249E-06

14289 0.1378E-02 1.0000 220455 1.80 0.318 39.93 0.348E-08

14317 0.1467E-02 1.0000 220792 1.78 0.282 40.47 0.631E-06

14324 0.1570E-02 1.0000 221047 1.79 0.297 41.07 0.301E-06

14334 0.1685E-02 1.0000 221325 1.67 0.296 41.72 0.319E-06

14365 0.1803E-02 1.0000 221784 1.37 0.293 42.35 0.176E,06

14391 0.1914E-02 1.0000 222221 1.76 0.332 42.94 0.226E-06

14415 0.2029E-02 1.0000 222652 1.67 0.312 43.54 0.325E-06

14437 0.2169E-02 1.0000 223073 1.55 0.334 44.25 0.558E-06

14474 0.2289E-02 1.0000 223563 1.80 0.314 44.86 0.128E-06

14504 0.2435E-02 1.0000 224033 1.56 0.318 45.58 0.172E-06

14543 0.2596E-02 1.0000 224527 1.54 0.331 46.38 0.I03E-06

14567 0.2742E-02 1.0000 225048 1.41 0.220 47.09 0.811E-06

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

80
14598

14626

14678

14683

14761

14796

14852

14856

15168

15260

15353

15446

15524

15530

0.2878E-02 1.0000

0.3040E-02 1.0000

0.3188E-02 1.0000

0.3445E-02 1.0000

0.3755E-02 1.0000

0.4039E-02 1.0000

0.4371E-02 1.0000

0.4969E-02 1.0000

0.5812E-02 1.0000

0.6338E-02 1.0000

0.6914E-02 1.0000

0.7630E-02 1.0000

0.8249E-02 1.0000

0.8852E-02 1.0000

225515 1.40

225993 1.38

226572 1.76

226878 1.41

228024 1.20

228786 1.41

229659 1.59

230134 1.27

235466 1.20

237398 1.20

238649 1.20

240055 1.43

241139 1.20

241433 1.33

0.337 47.76 0.271E-06

0.295 48.55 0.493E-06

0.310 49.29 0.168E-06

0.293 50.55 0.912E-08

0.378 52.06 0.175E-06

0.288 53.51 0.702E-06

0.209 55.19 0.183E-06

0.311 58.25 0.131E-05

0.413 62.67 0.140E-06

0.401 65.53 0.766E-06

0.354 68.77 0.576E-06

0.346 72.96 0.290E-06

0.357 76.64 0.789E-06

0.312 80.40 0.147E-05

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH - 0.0126 CRACK DEPTH = 0.0012 TOTAL CYCLES = 24]725

NFCODE = 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME = 72.6 SECONDS

SAMPLE 7: Compact Specimen under Stress-lntensity Factor Loading

SPECTRUM FILE = cstamp TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: FVS0812

YIELD STRESS = 0.5700E+02 ULTIMATE STRENGTH = 0.6600E+02

ELASTIC MODULUS = 0.1280E+05

PLANE-STRAIN-TO-PLANE-STRESS SOLUTION:

(ALP = ALPI TO ALP2 AND BETA = BETAI TO BETA2)

DKEFF IS ELASTIC:

CRACK GROWTH RATES FROM TABLE LOOKUP (NDKTH = 0):

DKEFF RATE

0.6500E+00 0.I000E-08

0.9400E+00 0.I000E-07

0.1400E+01 0.I000E-06

0.3100E+01 0.I000E-05

0.7000E+01 0.I000E-04

0.I130E+02 0.4400E-04

0.2200E+02 0.I000E-02

THRESHOLD CONSTANTS: C3 = 0.00 C4 = 0.00

FRACTURE TOUGHNESS PROPERTIES: KF = 0.400E+02 M = 0.00

RATE AT START OF TRANSITION = 0.4000E-05 WITH ALPI = 2.40 AND BETAI = 1.00

RATE AT END OF TRANSITION = 0.6000E-03 WITH ALP2 = 1.90 AND BETA2 = 1.00

81
COMPACT SPECIMEN: NTYP - 2

SPECIMEN WIDTH = 0.200E+01 THICKNESS = 0.250E+00

INITIAL CRACK LENGTH (CI) = 0.51000E+00 INITIAL CRACK DEPTH = 0.25000E+00

NOTCH LENGTH (CN) = 0.48000E+00 NOTCH DEPTH = 0.25000E+00

NOTCH HEIGHT = O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = I

FINAL CRACK LENGTH (CF) REQUESTED = 0.65000E+00

PROGRAM OPTIONS: LFAST = 0 KOPEN = I KCONST = 1 INVERT = 0 LSTEP = 1

ERR = 0.123E+01 PDC = 0.20 NMAX =1000 NIPT = 2000 NPRT = 10 NDKE = ]

PRECRACKING FROM STARTER NOTCH:

MAX SIF = 0.1650E+02 MIN SIF = 0.1650E+01

BLOCK C A/T CYCLES ALP KO/KMAX DKEC DC/DN DKEA DA/DN

0

0

0

0

0

0

0

0

0

0.4834E+00 1.0000

0.4866E+00 l.O000

0.4900E+00 1 0000

0 4932E+00 I

0 4964E+00 ]

0 4995E+00 I

0 5026E+00 ]

0 5056E+00 I

0 5087E+00 1

0000

0000

0000

0000

0000

0000

BLOCK (OR FLIGHT) LOADING:

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED =

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = 2

46 2.17 0.280 11.87 0.555E-04 0.00 O.O00E+O0

114 2.17 0.318 11.25 0.434E-04 0.00 O.O00E+O0

193 2.17 0.330 11.05 0.411E-04 0.00 O.O00E+O0

273 2.17 0.337 10.94 0.398E-04 0.00 O.O00E+O0

353 2.17 0.341 10.87 0.391E-04 0.00 O.O00E+O0

433 2.17 0.343 10.84 0.387E-04 0.00 O.O00E+O0

513 2.17 0.344 10.83 0.386E-04 0.00 O.O00E+O0

593 2.17 0.345 10.81 0.384E-04 0.00 O.O00E+O0

673 2.]7 0.345 10.81 0.383E-04 0.00 O.O00E+O0

NFOPT = I LPRINT = 0 MAXLPR = 0

BLOCK I NUMBER OF SIF LEVELS = I NSQ = I SCALE =

LEVEL KMAX KMIN CYCLES

1 0.3300E+02 0.3300E+01 I

1.00

BLOCK 2 NUMBER OF SIF LEVELS = I NSQ = 2 SCALE =

LEVEL KMAX KMIN CYCLES

1 0.1650E+02 0.1650E+01 IO0000

1.00

NOTE: KO/KMAX BASED ON "CURRENT" APPLIED KMAX IN SEQUENCE

BLOCK C A/T CYCLES ALP KO/KMAX DKEC DC/DN DKEA DA/DN

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

0 5125E+00 1.0000

0 5151E+00 1.0000

0 5176E+00 I.O000

0 5201E+00 1.0000

0 5226E+00 1.0000

0 5251E+00 1.0000

0.5275E+00 1.0000

0.5300E+00 1.0000

0.5325E+00 1.0000

0.5350E+00 1.0000

0.5375E+00 1.0000

0.5400E+00 1.0000

0.5426E+00 1.0000

0.5452E+00 1.0000

0.5478E+00 1.0000

I

133

372

814

1523

2364

3079

3592

3951

4210

4404

4548

4660

4761

4856

1.90

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

0.I87

0.510

0.610

0.682

0.722

0.721

0.696

0.651

0.606

0.559

0.516

0.474

0.432

0.419

0.412

26 84 0.254E-02

8 09 0.156E-04

6 44 0.791E-05

5 24 0.441E-05

4 59 0.303E-05

4 60 0.305E-05

5 01 0.388E-05

5 76 0.576E-05

6 51 0.814E-05

7 28 0.113E-04

7 98 0.150E-04

8 68 0.195E-04

9 37 0.246E-04

9 59 0.265E-04

9 71 0.275E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

82
2 0.5503E+00

2 0.5529E+00

2 0.5555E+00

2 0.5582E+60

2 0.5608E+00

2 0.5633E+00

2 0.5659E+00

2 0.5684E+00

2 0.5711E+_0

2 0.5737E+00

2 0.5764E+00

2 0.5791E+00

2 0.5818E+00

2 0.5845E+00

2 0.5873E+00

2 0.5900E+00

2 0.5928E+00

2 0.5956E+00

2 0.5983E+00

2 0.6009E+00

2 0.6035E+00

2 0o6062E+00

2 0.6087E+00

2 0.6112E+00

2 0.6136E+00

2 0.6161E+00

2 0.6186E+00

2 0.6211E+00

2 0.6236E+00

2 0.6262E+00

2 0.6287E+00

2 0.6312E+00

2 0.6337E+00

2 0.6363E+00

2 0.6388E+00

2 0.6413E+00

2 0.6439E+00

2 0.6464E+00

2 0.6489E+00

1.0000

1.0000

1 0000

1 0000

1 0000

1 0000

1 0000

1 0000

1o0000

1.0000

1.0000

1.0000

1 0000

I 0000

1 0000

1 0000

1 0000

1 0000

1 0000

1 0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

4946

5036

5126

5216

5300

5380

5460

5540

5620

5700

5780

5860

5940

6020

6100

6180

6260

6340

6417

6492

6567

6642

6713

6783

6853

6923

6993

7063

7133

7203

7273

7343

7413

7483

7553

7623

7693

7763

7833

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2.17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2 17

2.17

0.406

0.400

0.397

0.392

0.390

0.385

0.382

0.379

0.376

0.375

0.373

0.371

0.370

0.369

0.368

0.366

0.365

0.364

0.362

0.361

0.36]

0.359

0.362

0.362

0.361

0.360

0.360

0.359

0.359

0.358

0.358

0 358

0 357

0 357

0 357

0 357

0 357

0 356

0 356

9.80 0.283E-04 0.00 O.O00E+O0

9.90 0.292E-04 0.00 O.O00E+O0

9.95 0.297E-04 0.00 O.O00E+O0

10.04 0.305E-04 0.00 O.O00E+O0

10.06 0.307E-04 0.00 O.O00E+O0

10.16 0.316E-04 0.00 O.O00E+O0

10.20 0.321E-04 0.00 O.O00E+O0

10.25 0.325E-04 0.00 O.O00E+O0

10.29 0.329E-04 0.00 O.O00E+O0

10.32 0.332E-04 0.00 O.O00E+O0

10.35 0.335E-04 0.00 O.O00E+O0

10.37 0.337E-04 0.00 O.O00E+O0

10.40 0.340E-04 0.00 O.O00E+O0

10.42 0.342E-04 0.00 O.O00E+O0

10.43 0.344E-04 0.00 O.O00E+O0

10.46 0.347E-04 0.00 O.O00E+O0

10.47 0.348E-04 0.00 O.O00E+O0

10.49 0.349E-04 0.00 O.O00E+O0

10.52 0.353E-04 0.00 O.O00E+O0

10.54 0.355E-04 0.00 O.O00E+O0

10.55 0.356E-04 0.00 O.O00E+O0

10.57 0.358E-04 0.00 O.O00E+O0

]0.52 0.353E-04 0.00 O.O00E+O0

10.53 0.354E-04 0.00 O.O00E+O0

10.55 0.356E-04 0.00 O.O00E+O0

10.56 0.356E-04 0.00 O.O00E+O0

10.56 0.357E-04 0.00 O.O00E+O0

10.58 0.359E-04 0.00 O.O00E+O0

10.58 0.359E-04 0.00 O.O00E+O0

10.59 0.359E-04 0.00 O.O00E+O0

10.59 0.360E-04 0.00 O.O00E+O0

10.59 0.360E-04 0.00 O.O00E+O0

10.60 0.362E-04 0.00 O.O00E+O0

10.60 0.361E-04 0.00 O.O00E+O0

10.61 0.363E-04 0.00 O.O00E+O0

10.61 0.362E-04 0.00 O.O00E+O0

10.62 0.363E-04 0.00 O.O00E+O0

10.63 0.364E-04 0.00 O.O00E+O0

10.63 0.364E-04 0.00 O.O00E+O0

CRACK LENGTH EXCEEDS INPUT VALUE FOR CF

CPU TIME = 9.5 SECONDS

SAMPLE 8: Center-Crack Tension under NFOPT = 8 Option

SPECTRUM FILE - snfopt8 TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 2024-T3

YIELD STRESS = 0.5200E+02 ULTIMATE STRENGTH = 0.7200E+02

ELASTIC MODULUS = 0.I040E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.I00E+01

BETA = 0.I00E+01

83
DKEFFIS ELASTIC:

CRACKGROWTHCONSTANTS:

CI - 0.2870E-08 C2 - 4.070

FRACTURETOUGHNESSPROPERTIES:

C3 = 0.00 C4 = 0.00 C5 = 0.999E+03

KF = 0.400E+03 M = 1.00

CENTER CRACK TENsIoN: NTYP - I

SPECIMEN WIDTH = 0.120E+02 THICKNESS = O.500E-OI

INITIAL CRACK LENGTH (CI) = O.20000E+O0 INITIAL CRACK DEPTH = 0.50000E-01

NOTCH LENGTH (CN) = O.]O000E+O0 NOTCH DEPTH - 0.50000E-01

NOTCH HEIGHT = O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = I

FINAL CRACK LENGTH (CF) REQUESTED = 0.12000E+02

PROGRAM OPTIONS: LFAST - 0 KOPEN = ] KCONST = 0 INVERT = 0 LSTEP = I

ERR = 0.124E+01 PDC = 0.20 NMAX =]000 NIPT = 3000 NPRT = 3 NDKE = 0

PRECRACKING FROM STARTER NOTCH:

MAX STRESS = 0.2000E+02 MIN STRESS = O.2000E+O)

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

0 0.1022E+00 1.0000 93

0 0.I044E+00 1.0000 316

0 0.1067E+00 1.0000 647

0 0.I090E+00 1.0000 1050

0 0.1114E+00 1.0000 1495

0 0.1138E+00 1.0000 1967

0 0.1163E+00 1.0000 2454

0 0.1188E+00 1.0000 2948

0 0.1214E+00 1.0000 3441

0 0.1241E+00 1.0000 3934

0 0.1268E+00 1.0000 4422

0 0.1296E+00 1.0000 4904

0 0.1324E+00 1.0000 5378

0 0.1353E+00 1.0000 5845

0 0.1382E+00 1.0000 6303

0 0.1412E+00 1.0000 6752

0 0.1443E+00 1.0000 7194

0 0.1474E+00 1.0000 7628

0 0.1506E+00 1.0000 8052

0 0.1539E+00 1.0000 8468

0 0.1573E+00 1.0000 8876

0 0.1607E+00 1.0000 9276

0 0.1642E+00 1.0000 9668

0 0.1678E+00 1.0000 10052

0 0.1715E+00 1.0000 10428

0 0.1752E+00 1.0000 10796

0 0.1790E+00 1.0000 11156

0 0.1829E+00 1.0000 11509

0 0.1869E+00 1.0000 i]854

0 0.1910E+00 1.0000 12192

0 0.1952E+00 1.0000 12521

0 0.1994E+00 1.0000 12845

1.00 0.262

1.00 0.374

1.00 0.425

1.00 0.450

1.00 0.463

1.00 0.472

1.00 0.478

1.00 0.482

1.00 0.485

1.00 0.487

1.00 0.489

1.00 0.490

1.00 0.491

1.00 0.491

1.00 0.491

1.00 0.492

1.00 0.492

1.00 0.494

1.00 0.493

1.00 0.493

1.00 0.493

1.00 0.494

1.00 0.494

1.00 0.494

1.00 0.494

1.00 0.495

1.00 0.495

1.00 0.495

1.00 0.495

1.00 0.495

1.00 0.495

1.00 0.495

10.20 0.161E-04

10.31 0.858E-05

10.42 0.635E-05

10.53 0.555E-05

10.65 0.523E-05

10.76 0.513E-05

10.88 0.509E-05

11.00 0.517E-05

11.]2 0.529E-05

11.24 0.542E-05

11.36 0.560E-05

11.48 0.579E-05

11.61 0.602E-05

11.73 0.628E-05

11.86 0.652E-05

11.99 0.678E-05

12.12 0.707E-05

12.25 0.731E-05

12.38 0.768E-05

12.52 0.800E-05

12.65 0.836E-05

12.79 0.871E-05

12.93 0.910E-05

13.07 0.948E-05

13.21 0.992E-05

13.36 0.103E-04

13.50 0.I07E-04

13.65 0.112E-04

13.80 0.117E-04

13.95 0.123E-04

14.10 0.128E-04

14.25 0.134E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

84
BLOCK(OR FLIGHT) LOADING: NFOPT = 8 LPRINT = I

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED =

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = 2

MAXLPR = 10

2

SPECTRUM LOAD SEQUENCE (NFOPT - 8): MAX STRESS = 0.3500E+02

NOTE: KO/KMAX AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

SPECTRUM: LIST OF STRESS POINTS

BLOCK ACCUMULATIVE

POINTS CYCLES

1 2400 1200

2 2678 1339

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

I 0.2040E+00 1.0000

1 0.2161E+00 1.0000

I 0.2312E+00 1.0000

209 1.00 0.321 31.86 0.761E-04

562 1.00 0.086 32.79 0.233E-03

1158 I.O0 0.285 33.92 0.614E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

BLOCK ACCUMULATIVE

POINTS CYCLES

I 2400 1200

2 2678 1339

3 0.2470E+00 1.0000

4 0.2660E+00 1.0000

1634 1.00 0.355 35.06 0.310E-04

2596 I.O0 0.408 36.39 0.417E-04

0.00 O.O00E+O0

0.00 O.O00E+O0

BLOCK ACCUMULATIVE

POINTS CYCLES

I 2400 1200

2 2678 1339

5 0.2802E+00 1.0000 32]9 1.00 0.259 37.34 0.253E-03 0.00 O.O00E+O0

BLOCK ACCUMULATIVE

POINTS CYCLES

I 2400 1200

2 2678 1339

7 0.2989E+00 I.O000

7 0.3180E+00 1.0000

4623 1.00 0.392 38.57 0.550E-04

5166 I.O0 0.423 39.79 0.134E-03

0.00 O.O00E+O0

0.00 O.O00E+O0

BLOCK ACCUMULATIVE

POINTS CYCLES

1 2400 1200

2 2678 1339

9 0.3452E+00 1.0000

11 0.3679E+00 1.0000

12 0.3947E+00 1.0000

13 0.4211E+00 1.0000

15 0.4509E+00 1.0000

15 0.4827E+00 1.0000

17 0,5144E+00 1.0000

17 0.5475E+00 1.0000

19 0.5906E+00 1.0000

21 0.6278E+00 1.0000

21 0.6703E+00 I.O000

22 0.7190E+00 I.O000

23 0.7708E+00 1.0000

24 0.8167E+00 1.0000

6271 1.00 0.396 41.46 0.460E-03

6848 1.00 0.208 42.80 0.145E-03

8023 I.O0 0.440 44.34 0.584E-03

8870 1.00 0.344 45.80 0.703E-04

9382 1.00 0.215 47.40 0.115E-02

10206 I.O0 0.277 49.05 0.857E-03

10993 1.00 0.294 50.64 0.679E-03

11865 !.00 0.450 52.25 0.235E-03

12819 1.00 0.448 54.28 0.217E-03

13405 1.00 0.447 55.97 0.463E-03

14337 1.00 0.495 57.85 0.821E-03

14694 1.00 0.213 59.93 0.619E-02

15834 1.00 0.501 62.07 0.927E-03

16057 1.00 0.226 63.92 0.110E-01

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0 O00.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

85
25 0.8656E+00 1.0000

27 0.9244E+00 1.0000

27 0.9732E+00 1.0000

27 0.I026E+01 1.0000

29 0.1103E+01 1.0000

29 0.1176E+01 1.0000

30 0.1267E+01 1.0000

31 0.1351E+01 1.0000

31 0.1447E+01 1.0000

32 0.1563E+01 1.0000

33 0.1655E+01 1.0000

33 0.1756E+01 1.0000

33 0.1856E+01 1.0000

35 0.1974E+01 1.0000

35 0.2113E+01 1.0000

35 0.2225E+01 1.0000

35 0.2343E+01 1.0000

36 0.2564E+01 1.0000

37 0.2697E+01 1.0000

37 0.2889E+01 1.0000

37 0.3035E+01 1.0000

37 0.3226E+01 1.0000

37 0.3392E+01 1.0000

37 0.3602E+01 1.0000

37 0.3773E+01 1.0000

16849 1.00 0 302 65.83 0.538E-03

17552 ].00 0 498 68.05 0.]56E-03

18006 1.00 0 455 69.85 0.135E-02

18518 1.00 0 211 71.75 0.436E-02

19023 I.O0 0 457 74.44 0.312E-03

19663 1.00 0 213 76.94 0.154E-01

20050 1.00 0.338 79.91 0.891E-02

20784 1.00 0.504 82.59 0.278E-03

21022 1.00 0.467 85.58 0.282E-03

21413 1.00 0.20] 89.11 0.457E-01

22002 1.00 0 263 91.79 0.239E-02

22371 1.00 0

22544 1.00 0

22874 1.00 0

23331 1.00 0

23598 1.00 0

23796 1.00 0

24091 1.00 0

463 94.67 0.897E-02

339 97.48 0.903E-02

455 100 71 0.4IIE-03

521 104 39 0.832E-02

203 107 48 0.297E-01

527 110 48 0.137E-02

439 115 99 0.130E-01

24222 1.00 0.431 119 42 0.I03E-01

24527 1.00 0.499 124 06 0.151E-01

24701 1.00 0.483 127 71 0.134E-01

24832 1.00 0.137 132 52 0.275E-01

24936 1.00 0.450 136.53 0.705E-02

25083 1.00 0.487 ]41.43 0.178E-01

25220 1.00 0.243 ]45.97 0.104E-02

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH _ 3.9745 CRACK DEPTH I 0.0500 TOTAL CYCLES = 25325

NFCODE - 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME I 9.2 SECONDS

SAMPLE 9: Center-Crack Tension under NFOPT _ 9 Option

SPECTRUM FILE _ snfopt9 TIME LIMIT = 200.0 SECONDS

MATERIAL PROPERTIES: 2024-T3

YIELD STRESS = 0.5200E+02 ULTIMATE STRENGTH _ 0.7200E+02

ELASTIC MODULUS _ 0.I040E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.100E+01

BETA _ 0.I00E+01

DKEFF IS ELASTIC:

CRACK GROWTH CONSTANTS:

CI _ 0.2870E-08 C2 = 4.070

FRACTURE TOUGHNESS PROPERTIES:

C3 = 0.00 C4 = 0.00 C5 = 0.999E+03

KF = 0.400E+03 M = 1.00

86
CENTERCRACKTENSION:NTYP- 1

SPECIMENWIDTH= 0.120E+02 THICKNESS= 0.500E-01

INITIAL CRACKLENGTH(CI) = O.20000E+O0 INITIAL CRACKDEPTH= 0.50000E-01

NOTCHLENGTH(CN) = O.IO000E+O0 NOTCHDEPTH= O.50000E-O]

NOTCHHEIGHT= O.O00E+O0 ELEMENTSONSTARTERNOTCH(NS) = I

FINAL CRACKLENGTH(CF) REQUESTED= 0.12000E+02

PROGRAMOPTIONS: LFAST= 0

ERR= 0.124E+01 PDC= 0.20

DCPR= 0.400E-01

KOPEN= I KCONST= 0 INVERT= 0 LSTEP= I

NMAX=1000 NIPT = 3000 NPRT=-20 NDKE= I

PRECRACKINGFROMSTARTERNOTCH:

MAXSTRESS= 0.2000E+02 MIN STRESS= 0.2000E+01

BLOCK C A/T CYCLES

0 O.I400E+O0 1.0000 6572

0 0.1800E+00 1.0000 11245

ALP KO/KMAX

1.00 0.492

1.00 0.495

DKEC DC/DN

6.72 0.668E-05

7.58 0.109E-04

DKEA DA/DN

0.00 O.O00E+O0

0.00 O.O00E+O0

BLOCK (OR FLIGHT) LOADING: NFOPT = 9 LPRINT = I

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED =

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = 22

MAXLPR = 5

22

SPECTRUM LOAD SEQUENCE (NFOPT = 9): MAX STRESS = 0.3500E+02

NOTE: KO/KMAX'AND SIF BASED ON HIGHEST AND LOWEST APPLIED STRESS IN SEQUENCE

FLIGHT-BY-FLIGHT SPECTRUM

FLIGHT POINTS CYCLES

I 104 52

2 138 69

3 104 52

4 104 52

5 ]8 9

6 110 55

7 160 80

BLOCK C A/T

22

53

87

109

132

152

167

184

200

212

221

236

242

255

264

273

279

287

294

0.2400E+00 1 0000

0.2800E+00 1 0000

0.3200E+00 I 0000

0.3600E+00 I 0000

O.4000E+O0 ] 0000

0.4400E+00 1 0000

0.4800E+00 I 0000

0.5200E+00 I 0000

0.5600E+00 1 0000

O.6000E+O0 I 0000

0.6400E+00 I 0000

0.6800E+00 I 0000

0.7200E+00 1.0000

0.7600E+00 I.O000

0.8000E+O0 I.O000

0.8400E+00 ].0000

0.8800E+00 I.O000

0.9200E+00 1.0000

0.9600E+00 1.0000

CYCLES

1281

3215

5281

666]

8053

9276

10183

11205

12162

]2918

13489

14365

14716

15512

16055

16560

16972

17472

17853

ALP

I .00

1.00

1.00

1.00

1.00

I O0

I O0

l O0

l O0

l O0

1 O0

I O0

I O0

1 O0

1 O0

I O0

1 O0

1.00

1.00

KO/KMAX

0.097

0.259

0.427

0.432

0.226

0.440

0.277

0.446

0.45]

0.449

0.303

0.288

0.439

0.501

0.226

0.482

0.447

0.498

0.455

DKEC DC/DN

15.41 0.196E-03

16.40 0.253E-03

1].74 0.647E-04

15.85 0.220E-03

10.35 0.388E-04

17.65 0.341E-03

22.14 0.857E-03

]6.26 0.244E-03

]8.87 0.447E-03

16.]6 0.238E-03

24.33 0.126E-02

29.85 0.289E-02

20.46 0.622E-03

22.56 0.927E-03

41.43 O.IIOE-01

11.99 0.708E-04

28.07 0.226E-02

14.57 0.156E-03

24.76 0.135E-02

DKEA DA/DN

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

87
299 0.I000E+01 1.0000

306 0.1040E+01 1.0000

308 0.I080E+01 1.0000

315 0.1120E+01 1.0000

323 0.1160E+01 1.0000

326 0.1200E+01 1.0000

329 0.1240E+01 1.0000

331 0.1280E+01 1.0000

335 0.1320E+01 1.0000

342 0.1360E+01 1.0000

342 0.1400E+01 1.0000

345 0.1440E+01 1.0000

347 0.1480E+01 1.0000

352 0.1526E+01 1.0000

353 0.1572E+01 1.0000

360 0.]617E+01 1.0000

363 0.1665E+01 1.0000

367 0.1713E+01 1.0000

368 0.1760E+01 1.0000

370 0.1808E+01 1.0000

371 0.1856E+01 1.0000

373 0.1903E+01 1.0000

375 0.1964E+01 1.0000

378 0.2026E+01 1.0000

382 0.2087E+01 1.0000

385 0.2152E+01 1.0000

388 0.2217E+01 1.0000

390 0.2283E+01 1.0000

391 0.2348E+01 1.0000

393 0.2418E+01 1.0000

393 0.2488E+01 1.0000

396 0.2574E+01 1.0000

397 0.2660E+01 1.0000

400 0.2746E+01 1.0000

403 0.2832E+01 1.0000

404 0.2924E+01 1.0000

406 0.3016E+01 1.0000

407 0.3107E+01 1.0000

408 0.3210E+01 1.0000

409 0.3312E+01 1.0000

410 0.3414E+01 1.0000

412 0.3517E+01 1.0000

412 0.3619E+01 1.0000

413 0.3722E+01 1.0000

18216 1.00 0.499 25.58 0.155E-02

18602 1.00 0.436 22.18 0.865E-03

18754 1.00 0.457 17.27 0.312E-03

19077 1.00 0.504 26.86 0.189E-02

19661 1.00 0.213 44.96 0.154E-01

19838 1.00 0.446 29.27 0.268E-02

20026 1.00 0.434 20.46 0.623E-03

20146 1.00 0.462 21.61 0.778E-03

20372 1.00 0.504 16.78 0.278E-03

20785 1.00 0.281 52.83 0.297E-01

20866 1.00 0.326 24.51 0.130E-02

21005 1.00 0.467 16.84 0.282E-03

21124 1.00 0.498 24.81 0.137E-02

21412 1.00 0.201 58.73 0.457E-01

21501 1.00 0.406 20.09 0.579E-03

21857 1.00 0.496 35 39 0.581E-02

22069 1.00 0.499 19

22325 1.00 0.224 22

22377 1.00 0 409 34

22499 1.00 0

22543 1.00 0

22704 1.00 0

22847 1.00 0

22976 1.00 0

23231 1.00 0

23423 1.00 0

23597 1.00 0

23694 1.00 0

492 34

339 39

474 39

455 18

526 21

521 38

491 22

203 52

396 37

61 0.525E-03

55 0.925E-03

01 0.493E-02

26 0.509E-02

45 0.903E-02

12 0.875E-02

47 0.411E-03

O0 0.693E-03

64 0.832E-02

83 0.975E-03

B5 0.297E-01

34 0.722E-02

23801 1.00 0 492 41.78 O.II4E-01

23878 1.00 0 301 63.65 0.636E-01

23907 1.00 0.439 43.12 0.130E-01

24093 1.00 0.152 34.85 0.544E-02

24157 1.00 0.431 40.75 0.103E-01

24312 1.00 0.523 28.83 0.253E-02

24461 1.00 0.499 44.71 0.151E-01

24544 1.00 0.467 21.56 0.773E-03

24680 1.00 0.483 43.44 0.134E-01

24757 1.00 0.526 24.65 0.133E-02

24830 1.00 0.137 51.83 0.275E-01

24891 1.00 0.431 48.44 0.209E-01

24957 1.00 0.542 52.41 0.290E-01

25033 1.00 0.487 46.52 0.178E-01

25087 1.00 0.386 47.46 0.193E-01

25169 1.00 0.467 46.63 0.180E-01

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

SPECIMEN FAILED:

CRACK LENGTH = 3.9745 CRACK DEPTH = 0.0500 TOTAL CYCLES I 25325

NFCODE - 4: KMAX > KIE (ELASTIC SIF AT FAILURE)

CPU TIME _ 9.1 SECONDS

88
SAMPLE10: Threshold Test (KTH = 3 Option)

SPECTRUM FILE - cstamp TIME LIMIT - 200.0 SECONDS

MATERIAL PROPERTIES: 2618-T651

YIELD STRESS = 0.6000E+02 ULTIMATE STRENGTH = 0.6000E+02

ELASTIC MODULUS= O.IO00E+05

PLANE STRESS SOLUTION: CONSTRAINT FACTOR = 0.I00E+01

BETA = 0.I00E+01

DKEFF IS ELASTIC:

CRACK GROWTH CONSTANTS:

CI = 0.4000E-08 C2 - 3.500

FRACTURE TOUGHNESS PROPERTIES:

C3 = 0.00 C4 = 0.00 C5 = 0.999E+03

KF = 0.800E+02 M = 0.00

CENTER CRACK TENSION: NTYP = I

SPECIMEN WIDTH = 0.120E+02 THICKNESS = O.IOOE+O0

INITIAL CRACK LENGTH (CI) = 0.45000E+00 INITIAL CRACK DEPTH = O.IO000E+O0

NOTCH LENGTH (CN) = O.40000E+O0 NOTCH DEPTH = O.IO000E+O0

NOTCH HEIGHT = O.O00E+O0 ELEMENTS ON STARTER NOTCH (NS) = 1

FINAL CRACK LENGTH (CF) REQUESTED = O.700OOE+O0

PROGRAM OPTIONS: LFAST = 0 KOPEN = I KCONST = 0 INVERT = 0 LSTEP = I

ERR = 0.120E+01 PDC = 0.20 NMAX =1000 NIPT = 3000 NPRT = 30 NDKE = 0

PRECRACKING FROM STARTER NOTCH:

MAX STRESS = O.7000E+OI MIN STRESS = O.7000E+O0

BLOCK C A/T CYCLES ALP KO/KMAX

0 0.4104E+00 1.0000 16270 I.O0 0.51I

0 0.4210E+00 1.0000 39947 1.00 0.522

0 0.4319E+00 ].0000 64040 1.00 0.524

0 0.443]E+00 ].0000 87907 1.00 0.525

DKC DC/DN DKA DA/DN

7.16 0.462E,06

7.25 0.447E-06

7.34 0.461E-06

7.44 0.478E-06

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

BLOCK (OR FLIGHT) LOADING: NFOPT = 0 LPRINT = 0

TOTAL NUMBER OF BLOCKS (OR) FLIGHTS TO BE REPEATED =

NUMBER OF DIFFERENT BLOCKS (OR FLIGHTS) = !

MAXLPR =

I

BLOCK I NUMBER OF STRESS LEVELS = I NSQ = I

LEVEL SMAX SMIN CYCLES

I 0.7000E+01 O.7000E+O0 1000

SCALE = 1.00

LOAD REDUCTION SCHEME: METHOD 3

SMAXTH = 0.700E+01 RTH = O.IOOE+O0 CONST = 0.780E-02

NOTE: KO/KMAX AND SIF BASED ON APPLIED STRESS, SMAX, AND R

BLOCK C A/T CYCLES ALP KO/KMAX DKC DC/DN DKA DA/DN

SMAX = 7.00

2 0.4503E+00 1.0000 1585 1.00 0.635

SMAX = 7.00

20 0.4585E+00 1.0000 ]9363 1.00 0.530

7.50 0.196E-06

7.57 0.213E-06

0.00 O.O00E+O0

0.00 O.O00E+O0

89
SMAX = 6.30

25 0.4590E+00 1.0000 24034 1.00 0.696

SHAX = 6.30

40 0.4600E+00 1.0000 39307 1.00 0.665

SHAX = 6.30

50 0.4621E+00 ]40000 49493 1.00 0.568

SMAX = 6.30

65 0.4662E+00 1.0000 64862 1.00 0.543

SHAX = 5.67

72 0.4666E+00 1.0000 71581 1.00 0.717

SMAX = 5.67

94 0.4675E+00 1.0000 93009 1.00 0.674

SMAX - 5.67

107 0.4692E+00 1.0000 106034 1.00 0.578

SMAX = 5.67

124 0.4726E+00 1.0000 123399 1.00 0.551

SMAX = 5.67

129 0.4736E+00 1.0000 128914 1.00 0.548

SMAX = 5.10

138 0.4740E+00 1.0000 137082 1.00 0.721

SMAX = 5.10

163 0.4747E+00 1.0000 162994 1.00 0.676

SMAX = 5.10

179 0.4761E+00 1.0000 178288 1.00 0.579

SMAX = 5.10

199 0.4789E+00 1.0000 198695 1.00 0.553

SMAX = 5.10

218 0.4815E+00 !.0000 217604 1.00 0.548

SMAX = 4.59

227 0.4818E+00 1.0000 226797 1.00 0.719

SMAX = 4.59

256 0.4823E+00 1.0000 255270 1.00 0.675

SMAX = 4.59

273 0.4835E+00 1.0000 272835 1.00 0.580

SMAX= 4.59

297 0.4858E+00 1.0000 296669 1.00 0.554

SMAX = 4.59

330 0.4892E+00 1.0000 329550 1.00 0.544

SMAX = 4.13

341 0.4895E+00 1.0000 340166 1.00 0.719

SMAX = 4.13

376 0.4899E+00 1.0000 375164 1.00 0.680

SMAX = 4.13

396 0.4909E+00 1.0000 395696 1.00 0.580

SMAX = 4.13

424 0.4927E+00 1.0000 423217 1.00 0.554

SMAX = 4.13

473 0.4965E+00 ].0000 472433 ].00 0.545

SMAX = 4.13

481 0.4970E+00 1.0000 480256 1.00 0.544

SMAX = 3.72

493 0.4972E+00 1.0000 492264 1.00 0.717

SMAX = 3.72

532 0.4976E+00 1.0000 531297 1.00 0.676

SMAX = 3.72

556 0.4983E+00 1.0000 555171 1.00 0.580

SMAX = 3.72

588 0.4999E+00 1.0000 587017 1.00 0.555

6.81 0.742E-07

6.82 0.104E-06

6.84 0.256E-06

6.87 0.133E-06

6.18 0.412E-07

6.19 0.673E-07

6.20 0.167E-06

6.22 0.211E-06

6.23 0.907E-07

5.61 0.276E-07

5.61 0.469E-07

5.62 0.118E-06

5.64 0.147E-06

5.65 0.644E-07

5.09 0.203E-07

5.09 0.339E-07

5.10 0.829E-07

5.11 0.I03E-06

5.13 0.477E-07

4.62 0.144E-07

4.62 0.227E-07

4.62 0.589E-07

4.63 0.736E-07

4.65 0.799E-07

4.65 0.337E-07

4.19 0.I05E-07

4.19 0.169E-07

4.19 0.420E-07

4.20 0.517E-07

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 O.000E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 O.000E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.O00E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 O.O00E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

90
SMAX= 3.72

644 0.5030E+00 ].0000 643967 1.00 0.545

SMAX = 3.72

676 0.5047E+00 1.0000 675831 1.00 0.543

SMAX = 3.35

690 0.5049E+00 1.0000 689644 1.00 0.717

SMAX = 3.35

735 0.5052E+00 1.0000 734721 1.00 0.676

SMAX = 3 35

763 0.5058E+00 1.0000 762195 ].00 0.579

SMAX = 3 35

800 0.5071E+00 1.0000 799121 I 00 0.555

SMAX = 3 35

866 0.5097E+00 1.0000 865380 1 00 0.547

SMAX = 3 35

939 0.5125E+00 1.0000 938977 I 00 0.543

SMAX = 3 01

956 0.5127E+00 1.0000 955047 i 00 0.717

SMAX = 3 01

1007 0.5129E+00 ].0000 ]006985 1 00 0.676

SMAX = 3 01

1039 0.5134E+00 ].0000 1038452 I 00 0.578

SMAX = 3 01

1082 0.5145E+00 1.0000 ]081417 I 00 0.556

SMAX = 3.01

1159 0.5166E+00 ].0000 1158656 ] 00 0.548

SMAX = 3.01

1294 0.5203E+00 ].0000 1293331 ] 00 0.543

SMAX = 2.7]

1312 0.5204E+00 ] 0000 ]311956 I 00 0.7]7

SMAX = 2.71

1374 0.5206E+00 I 0000 13733]9 ] 00 0.677

SMAX = 2.71

1410 0.521]E+00 ] 0000 1409886 ] 00 0.581

SMAX = 2.71

]460 0.5219E+00 I 0000 1459620 ] 00 0.556

SMAX = 2.71

1550 0.5236E+00 I 0000 1549173 I 00 0.547

SMAX = 2.71

1722 0.527]E+00 I 0000 1721103 1.00 0.545

SMAX = 2.71

]772 0.5281E+00 1.0000 1771787 1.00 0.545

SMAX = 2.44

]794 0.5282E+00 1.0000 1793724 1.00 0.7]9

SMAX = 2.44

1865 0.5283E+00 1.0000 1864692 1.00 0.675

SMAX = 2.44

1908 0.5287E+00 ].0000 1907194 1.00 0.581

SMAX = 2.44

1966 0.5294E+00 1.0000 1965254 1.00 0.558

SMAX = 2.44

2071 0.5308E+00 1.0000 2070353 1.00 0.549

SMAX = 2.44

2273 0.5336E+00 ].0000 2272061 1.00 0.548

SMAX = 2.44

2435 0.5359E+00 1.0000 2434246 ].00 0.545

SMAX = 2.20

246] 0.5360E+00 ].0000 2460219 1.00 0.721

4.21 0.563E-07

4.22 0.242E-07

3.80 0.749E-08

3.80 0.120E-07

3.80 0.300E-07

3.81 0.367E-07

3.82 0.394E-07

3.83 0 172E-07

3.45 0 530E-08

3.45 0 854E-08

3.45 0 214E-07

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.O00E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

3.45 0 258E-07 0.00 0.000E+00

3.46 0 277E-07

3.47 0.122E-07

3 12 0.375E-08

3 13 0.596E-08

3 ]3 0.149E-07

3 ]3 0.182E-07

3 ]3 0.197E-07

3 14 0.203E-07

3 15 0.853E-08

2.83 0.261E-08

2.83 0.433E-08

2.83 0.I05E-07

2.84 0.128E-07

2.84 0.137E-07

2.85 0.140E-07

2.85 0.604E-08

2.57 0.181E-08

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+O0

0.00 0.000E+00

0.00 0.O00E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

91
SHAX = 2.20

2546 0.5361E+00 1.0000 2545561 1.00 0.679

SMAX = 2.20

2596 0.5364E+00 1.0000 2595533 1.00 0.580

SMAX = 2.20

2664 0.5370E+00 1,0000 2663069 1.00 0.558

SMAX = 2.2O

2786 0.5381E+00 1.0000 2785229 1.00 0.551

SHAX = 2.20

3022 0.5404E+00 1.0000 3021697 1.00 0.549

SMAX = 2.20

3348 0.5436E+00 I 0000 3347553 1.00 0.546

SMAX = 1.98

3378 0.5437E+00 1 0000 3377740 1.00 0.721

SMAX = 1.98

3476 0.5438E+00 ] 0000 3475028 1.00 0.676

SMAX = .98

3533 0.5441E+00 1 0000 3532265 1.00 0.581

SMAX - .98

3611 0.5445E+00 I 0000 3610661 1.00 0.557

SMAX = 1.98

3752 0.5455E+00 I 0000 3751033 1.00 0.548

SMAX = .98

4024 0.5474E+00 1.0000 4023910 1.00 0.548

SMAX - 1.98

4565 0.5512E+00 1.0000 4564451 1.00 0.546

SMAX = 1.98

4600 0.5514E+00 1.0000 4599161 1.00 0.547

SMAX = 1.78

4635 0.5515E+00 1,0000 4634641 1.00 0.722

SMAX = 1.78

4751 0.5516E+00 1.0000 4750169 1.00 0.678

SMAX = 1.78

4818 0.5518E+00 1.0000 4817320 1.00 0.583

SMAX - 1,78

4909 0.5522E+00 1,0000 4908497 1.00 0.558

SMAX = 1.78

5073 0.5529E+00 ],0000 5072714 1.00 0.550

SMAX = 1.78

5393 0.5545E+00 1.0000 5392968 1.00 0.550

SMAX = 1.78

6029 0.5576E+00 1.0000 6028876 1.00 0.549

SMAX = 1.78

6357 0.5592E+00 1.0000 6356577 1.00 0.548

SMAX - 1.60

6399 0.5593E+00 1.0000 6398688 1.00 0.725

SMAX = 1.60

6536 0.5594E+00 1.0000 6535808 1.00 0.679

SMAX = 1.60

6615 0.5595E+00 1.0000 6614322 1.00 0.584

SMAX - 1.60

6722 0.5598E+00 1.0000 6721288 1.00 0.558

SMAX = 1.60

6914 0.5605E+00 1.0000 6913013 1.00 0.551

SMAX = 1.60

7290 0.5618E+00 1.0000 7289119 1.00 0.551

SMAX = 1.60

8037 0.5643E+00 1.0000 8036807 1.00 0.551

2.57 0.296E-08

2.57 0.754E-08

2.57 0.907E-08

2.57 0.960E-08

2.58 0.983E-08

2.59 0.425E-08

2.33 0.128E-08

2.33 0.215E-08

2.33 0.530E-08

2.33 0.647E-08

2.33 0.693E-08

2.34 0.702E-08

2.34 0.717E-08

2.34 0.299E-08

2.110.890E-09

2.11 0.150E-08

2.11 0.371E-08

2.11 0.455E-08

2.11 0.485E-08

2.12 0.488E-08

2.12 0.498E-08

2.13 0.209E-08

1.91 0.614E-09

0.00 O.000E+00

0.00 O.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+O0

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 O.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

1.91 0.105E-08 0.00 O.O00E+O0

1.91 0.261E-08

1.91 0.321E-08

1.92 0.340E-08

1.92 0.343E-08

1.92 0.346E-08

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

0.00 0.000E+00

92
SMAX- 1.60

8817 0.5670E+00 1.0000 8816028 1.00 0.550

SMAX = 1.44

8867 0.5671E+00 1.0000 8866675 1.00 0.728

SMAX - 1.44

9035 0.5671E+00 1.0000 9034217 1.00 0.686

SMAX - 1.44

9127 0.5673E+00 1.0000 9126504 1.00 0.584

SMAX - 1.44

9252 0.5675E+00 1.0000 9251880 1.00 0.560

SMAX - 1.44

9478 0.5680E+00 1.0000 9477158 1.00 0.553

SMAX - 1.44

9921 0.5691E+00 1.0000 9920340 1.00 0.554

1.93 0.146E-08

1.73 0.417E-09

1.73 0.692E-09

1.73 0.184E-08

1.73 0.224E-08

1.74 0.237E-08

1.74 0.237E-08

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

0.00 O.O00E+O0

NUMBER OF CYCLES EXCEED 10 MILLION

CPU TIME - 42.1 SECONDS

93
VII. USER GUIDE FOR EFFECTIVE STRESS-INTENSITY FACTOR PROGRAM

A. Introduction

A computer program DKEFF (dkeff.for) was also developed to help the

user determine the AKeff (or AKeff) against crack-growth rate relations for

a given material and thickness using either center-crack or compact

specimens. The program is a FORTRAN code designed to be executed

interactively on a personal computer. A line-by-line explanation of the

crack-growth rate input data file and interactive terminal input is

presented herein. The parameters used in the input data file and for

terminal input are defined and explained in Sections VII.B and VII.C,

respectively. The input data are in free format unless otherwise stated.

Several data sets can be analyzed for the same specimen type during each

computer run by placing input data sets in series. An example input and

output data file are given in Section VII.D. A list of error messages are

also given and discussed.

B. Crack-Growth Rate Analysis (Input Data) File

i. Data Set Title

READ TITLE(20)

FORMAT (20A4)

Any 80-character title describing the data set.

2. Materlal

READ MAT

FORMAT (20A4)

Any 80-character description of the material.

3. Specimen Type and System of Units Option

READ NTYP, LUNIT

NTYP = Specimen Type

- ! Center-crack tension specimen

= 2 Compact tension specimen

LUNIT = 0 Input and output data have same units

= I Input in English and output in SI units

= 2 Input in SI and output in English units

All units used for input data must be consistent.

SI Units

For example,

U.S. Customary

Length: m in.

Stress: MPa ksi

_K (or K): MPa-m I/2 ksi-in 1/2

dc/dN (or da/dN): m/cycle in/cycle

94
4. Material Tensile Yielding Properties

READ SYIELD, SULT, E, NEP, NALP, ALP

SYIELD = Yield stress (0.2 percent offset)

SULT = Ultimate tensile strength

E - Elastic modulus

NEP = 0 Effective stress-intensity factor is elastic

=I Effective stress-intensity factor is modified for

yielding at the crack tip. One-quarter of cyclic

plastic zone is added to crack length (see eqn.(]1)).

(Note: The plasticity corrected stress-intensity factor

in FASTRAN-II is different than that used in references

5 and 8. A closure corrected "cyclic" plastic zone size

is used instead of the maximum plastic zone size.)

=2 Effective stress-intensity factor is modified for

yielding at the crack tip. One-quarter of monotonic

plastic zone is added to crack length.

NALP = 0 Constraint factor (ALP) is constant as input

= I Constraint factor is variable (ALP is computed by the

program, see Section III.B.8).

FASTRAN-II must use the crack-growth rate relation, AKef f

(or AKeff) against dc/dN or da/dN, that is obtained with

the specified NALP option.

ALP = Constraint factor

= ] Plane-stress condition

= 1.73 Irwin's plane-strain condition

= 3 Plane-strain condition

See references 3 and 9 for procedures to obtain the value of ALP

for a particular material, thickness, and test condition. The

flow stress, SFLOW, computed in the program is the average of the

yield stress and ultimate tensile strength.

5. Crack Growth Rates at Transition (NALP = I option only)

READ RATE], ALP], RATE2, ALP2

If NALP = 0

If NALP = ]

No input is required, go to section 6.

RATEI is the crack-growth rate near the start of

transition from flat-to-slant growth (ALP = ALPI

for rates less than RATE]). RATE2 is the crack

growth rate near the end of the transition from

flat-to-slant growth (ALP = ALP2 for rates greater

than RATE2). For rates (RATES) between RATEI and

RATE2 a linear relation is used, see equation (20).

95
6. Data Points, Loading and Specimen Dimensions

READ MTAB, R, SNAX, g, T

MTAB = Number of _K-rate data points to be analyzed

R = Stress ratio (Smin/Smax)

SMAX - Remote applied stress on center-crack tension specimen

or P/(WT) for compact tension specimen (P - applied load)

g - Specimen half-width for center-crack tension specimen

or specimen width for compact tension specimen

T - Specimen thickness

7. Stress-Intensity Factor Range and Rates

READ OK(1), RATE(1)

DK(1) = Elastic stress-intensity factor range (AK = Kma x- Kmin)

RATE(1) = Corresponding crack-growth rate (dc/dN)

_epeat _tep 7 MTAB times. To analyze another data set on the same

specimen type (NTYP), repeat sections 6 and 7. Repeat until all

data sets are input. Enter '0 0 0 00' at the end of the last data

set to terminate analysis.

C. Interactive Input on Personal Computer

(computer response is in bold letters)

1. ENTER DATA FILE NAME :

READ DATFILE

FORMAT(A8)

Crack-growth rate input data filename

2. ENTER OUTPUT FILE NAME :

READ OUTFILE

FORMAT(A8)

Output filename

3. SHORT FILE: DKEFF-DATA ? YES- 1

READ NSIG

NSIG - 0

NO-O

Long output file

Echos all input data and prints out effective

stress-intensity factor, crack-growth rate, crackopening stress, maximum applied stress, constraint

factor and crack length.

96
Short output file

Miminal output of effective stress-intensity

factor, crack-growth rate, crack-opening stress,

maximum applied stress, constraint factor and crack

length.

4. DELTA-K(O) DELTA-KEFF(Z) OR DELTA-KBAREFF(2) ?

READ NDKEP

NDKEP = 0

1

Elastic stress-intensity factor ranges printed out

Effective (elastic) stress-intensity factor

ranges printed out

Effective (elastic-plastic) stress-intensity factor

ranges printed out (NEP = ) or 2).

5. ERROR: NEP - 0 FOR ELASTIC-PLASTIC ANALYSIS

INPUT DESIRED VALUE OF NEP (0, 1, OR 2)?

READ(5,*) NEP

For NDKEP = 2, NEP was not set to ] or 2 in the input data

file. User can override and set NEP = O, ) or 2. An elasticplastic analysis will be conducted using either NEP = ] or 2.

Program will terminate if NEP is input as zero.

If NEP - I or 2, and NALP = O, program goes to 6.

If NEP- I or 2, and NALP = I, program goes to 7.

6. IS VALUE OF ALP OK ? YES-1 NO-O

READ NQ

If NQ = O, .... CORRECT VALUES IN INPUT DATA FILE ....

and program terminates.

If NQ = ], program continues and completes the analysis.

7. ARE RATE AND ALP VALUES OK ? YES-1 NO=O

READ NQ

If NQ = O, .... CORRECT VALUES IN INPUT DATA FILE ....

and program terminates.

If NQ = I, program continues and completes the analysis.

D. Example Input and Output Data Files

I. Input File.-An example crack-growth rate input data file is

contained in the file 'ctn.dat' The input data file was given in English

units but the output is requested in SI units (LUNIT = I). The variable

constraint option (NALP = I) was also selected. The values of RATEI and

RATE2 were selected to correspond closely with the transition region from

flat-to-slant crack growth. The values of constraint have been selected by

trial-and-error to correlate crack-growth rate dataat differenct stress

ratio (R) conditions. Two data sets on compact specimens are included, one

97
at a stress ratio of 0.5 and the other at R = 0.1.

as foll ows:

Compact Tension Specimens

7075-T651

2 1

77. 85. 10400. 0

2.0E-05 1.9 1.0E-3

13 0.5 1.0 3.0

4.69 2.29E-06

4.84 2.58E-06

5.02 2.94E-06

5.22 3.30E-06

5.45 3.74E-06

5.74 4.26E-06

6.07 4.93E-06

6.35 5.50E-06

6.67 6.14E-06

7.08 6.94E-06

7.58 8.17E-06

7.89 9.15E-06

8.26 1.05E-05

12 0.1 0.7 3.0

5.43 6.71E-07

5.84 1.72E-06

6.37 3.03E-06

7.41 4.78E-06

9.37 6.82E-06

11.66 1.27E-05

12.79 1.65E-05

13.73 2.07E-05

14.84 2.47E-05

16.02 3.32E-05

17.48 4.16E-05

19.21 6.39E-05

1 1.9

1.2

0.25

0.25

The input data file is

2. Output File.-The data in the example input data file was given in

English units and the output file is given in SI units (LUNIT = I). The

long output file option (NSIG - O) was selected using terminal input (see

VII.C.3). The long output file gives the input AK-rate data and information

on loading and specimen dimensions. Because the compact specimen was being

analyzed, a stress level (Smax') was calculated for a center-crack tension

specimen that would give the same elastic stress-intensity factor as that

applied to the compact specimen. This stress level is used to calculate an

equivalent crack-opening stress level (So/Smax) for the compact specimen

(Smax' - RSF * Smax). The corresponding output data file is as follows:

Compact Tension Specimens

MATERIAL PROPERTIES : 7075-T651

COMPACT SPECIMEN

INPUT IN ENGLISH AND OUTPUT IN SI UNITS

YIELD STRESS - .5309E+03 ULTIMATE STRENGTH = .5861E+03

ELASTIC MODULUS - .7171E+05 MPa

98
PLANE-STRAIN-TO-PLANE-STRESS SOLUTION:

(ALP = ALP] TO ALP2)

RATEI = .5080E-06 m/cycle ALPI =

RATE2 = .2540E-04 m/cycle ALP2 =

NTAB - 13

R _m .50 Smax = .6895E+01MPa

DELTA-K RATE

MPa-m^O.5 m/cycle

.5154E+01

.5319E+01

.5517E+01

.5737E+01

.5990E+01

.6308E+01

.667]E+01

.6979E+01

.7330E+01

.7781E+01

.8330E+0]

.8671E+01

.9078E+01

.5817E-07

.6553E-07

.7468E-07

.8382E-07

.9500E-07

.1082E-06

.1252E-06

.1397E-06

.]560E-06

.1763E-06

.2075E-06

.2324E-06

.2667E-06

DKEFF ELASTIC:

DKEFF RATE

MPa-m^O.5 m/cycle

So/Smax'

.4499E+0] .5817E-07

.4640E+01 .6553E-07

.4811E+0] .7468E-07

.5000E+O! .8382E-07

.521gE+O] .9500E-07

.5495E+01 .1082E-06

.5808E+01 .1252E-06

.6075E+01 .1397E-06

.6380E+01 .1560E-06

.6771E+01 .1763E-06

.7246E+0] .2075E-06

.7540E+01 .2324E-06

.7889E+01 .2667E-06

.5636E+00

.5638E+00

5640E+00

5642E+00

5644E+00

5645E+00

5647E+00

5648E+00

.5648E+00

.5649E+00

.5651E+00

.5652E+00

.5655E+00

.190E+01

.120E+01

w = .7620E-01 m

RSF

.5426E+01

.5459E+01

.5501E+01

.5549E+01

.5607E+01

.5683E+01

.5774E+01

.5854E+01

.5948E+01

.6072E+01

.6229E+01

.6327E+01

.6447E+01

t = .6350E-02 m

Smax' ALP c

MPa m

.3741E+02

.3764E+02

.3793E+02

.3826E+02

.3866E+02

.3918E+02

.3981E+02

.4036E+02

.4101E+02

.4187E+02

.4295E+02

.4363E+02

.4445E+02

.1900E+01

.1900E+01

.1900E+O]

1900E+0]

1900E+01

1900E+01

1900E+01

1900E+01

1900E+01

.1900E+01

.1900E+01

.1900E+O]

.]900E+01

.2177E-01

.2269E-01

.2377E-01

.2493E-0]

.2621E-01

.2775E-01

.2939E-01

.3069E-01

.3209E-01

.3375E-01

.3558E-01

.3663E-01

.377gE-01

NTAB = ]2

R I¢ .10 Smax = .4826E+01MPa

DELTA-K RATE

MPa-m^0.5 m/cycle

.5968E+01

.6418E+01

.7001E+01

.8144E+0]

.]030E+02

.1281E+02

.1406E+02

.1509E+02

.1704E-07

.4369E-07

.7696E-07

.1214E-06

.1732E-06

.3226E-06

.419]E-06

.5258E-06

w = .7620E-0] m

RSF

.5349E+0]

.5414E+01

.5509E+0!

.5721E+01

.6183E+0]

.6773E+01

.7070E+0!

.7318E+01

t = .6350E-02 m

99
.1631E+02

.]761E+02

.]92]E+02

.2111E+02

DKEFF ELASTIC:

DKEFF

MPa-m^O.5

4145E+0]

4438E+0]

4826E+01

5606E+01

7097E+01

8817E+01

9665E+01

1035E+02

1110E+02

118]E+02

1275E+02

.1369E+02

.6274E-06

.8433E-06

.1057E-05

.1623E-05

RATE

m/cycl e

.7610E+01

.7918E+0]

.8294E+01

.8734E+01

So/Smax' Smax'

MPa

1704E-07 .3748E+00 .2582E+02

4369E-07 .3777E+00 .2613E+02

7696E-07 .3796E+00 .2659E+02

1214E-06 .3804E+00 .2761E+02

1732E-06 .3798E+00 .2984E+02

3226E-06 .3807E+00 .3269E+02

4191E-06 .3812E+00 .3412E+02

5258E-06 .3825E+00 .3532E+02

6274E-06 .3872E+00 .3673E+02

8433E-06 .3961E+00 .3821E+02

1057E-05 .4025E+00 .4003E+02

1623E-05 .4164E+00 .4215E+02

ALP

1900E+01

1900E+01

1900E+01

1900E+01

1900E+01

IgOOE+01

1900E+01

1894E+01

1862E+0]

1809E+0]

1769E+01

1692E+O]

C

m

.1935E-01

.2143E-01

.2398E-01

.2846E-01

.3508E-01

.4052E-01

.4259E-01

.4409E-01

.4564E-01

.4710E-01

.4866E-01

.5025E-01

E. Error Messages

The program DKEFF (dkeff.for) has several error checks on the input

data and execution errors that may be helpful to the user. These error

messages are listed and some are briefly discussed.

I. INPUT ERROR: NEP OUT OF RANGE--NEP = 'value'

NEP must be either O, I or 2.

2. INPUT ERROR: NTYP MUST BE I OR 2

Only center-crack tension and compact specimens are used

to obtain baseline crack-growth rate data.

3. INPUT ERROR: MTAB= 'value' > MAXN= 200'

Maximum number of AK and dc/dN pairs are limited to MAXN.

MAXN is currently set at 200.

4. INPUT ERROR: CHECK LUNIT VALUE

5. INPUT ERROR: RATE] MUST BE LESS THAN RATE2 AND

ALP] MUST BE GREATER THAN ALP2

6. CAUTION: SMAX > 0.6 SFLOW SHOULD RUN FASTRAN-II

TO OBTAIN CRACK-OPENING STRESSES

7. WARNING: SMAX > 0.8 SFLOW !tI MUST RUN FASTRAN-II

TO OBTAIN CRACK-OPENING STRESSES

8. ITERATIONS EXCEED 100 IN SUBROUTINE FNC

Crack length cannot be found from input AK and Smax.

Check input data, loading and specimen dimensions.

100

Form Approved

REPORT DOCUMENTATION PAGE OMBNo.oTo_-ores

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Div,'. H*qhw_v. Su,te 1204, A¢IiPKJKOn. VA 22202-4302, and to the Office of M&nageme_t and Budget. P&p_t'work RffduCl:lOn Proiect (070 -0188). w&shmgto . OC 20503.

1. AGENCY USE ONLY (Leave btank) 12. REPORT DATE'

[February 1992

4_=TITLE AND SUI_TITLE

FASTRA.N-II - A Fatigue Crack Growth S_ructura] Analysis Program

6. AUTHOR(S)

ames C. Newman, Jr.

7'. PERFORMING ORGANIZATION NAME(S) AND AOORESSiES)

qASA Langley Research Center

tampton. VA 23665-5225

9. SPONSORING/MONITORINGAGENCYNAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

_Vashington, DC 20546-0001

3. REPORT TYPE AND DATES COVERED

Technical Memorandum

$. FUNDII_G NUMBERS

505-63-50-04

mlml

8. PERFORMING ORGANIZATION

REPORT NUMBER

10. SPON SO'RING / MONITORING

AGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES

NASA TM-104159

12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited

Subject Category 39

13. ABSTRACT (Maximum 200words)

Fastran-II is a life-prediction code based on the crack-closure concept and is used to predict crack length against cycles

from a specified Initialcrack size to failure for many common crack configurations found in structural componments.

FASTRAN-II is an update of the previous version FASTRAN. The life-predictionmethod used in FASTRAN-II is built around

an analytical crack-closure model. The model is based on plasticity-inducedfatigue-crack closure and is used to calculate

the stress level at which the crack tip becomes fully open dudng cyclic loading. The applied cyclic loads may be

constant-amplitude, variable-amplitude or spectrum loading. Several standardized flight-load spectra (TWIST, Mini-TWIST,

FASTAFF, and Felix-28) and other load spectra (Space Shuttle and Gaussian) are included as options. Spectrum loads

may also be input by the user as either a list of stress pointsor a flight-by-flight sequence. The program uses the

crack-closure concept to account for Io_d-interectlon effects. Tensile or compressive loads can be applied. The program

contains seventeen predeflned crack configuratiorm;and the user can define one crack configuration. The crack-opening

stresses, as a function of load history and crack length, are calculated from the model and the effective stress-intensity

factor range, used to correlate fatigue-cra_,-'tkgrowth rates, may be either elastic or modified for plastic yielding at the crack

tip. Ten example problems are includedwith the user guide to demonstrate the many options in FASTRAN-II.

A computer program DKEFF was also developed to analyze laboratory spedman data to obtain the effective stress-intensity

factor agains crack-growth rate relations used by FASTRAN-II. A user guide forthis program is also includedand

d!_,_,j=_s_,,__. IComp_er _rame are ay__!!_,__,from COSMIC.}

14. SUBJECT T|RMS

FASTRAN; Fatigue; Cracks; Fracture; Crack pmpogation; Plasticity;Strass-intansltyfactor

17. SECURITY CLASSIFICATION tO. SECURITY CLASSIFICATION 19.

OF REPORT OF THIS PAGE

I I._..._l,ui

NSN 7540-01-280-$500

! lrv, lme_=PlbsH ..

15. NUMBER OF PAGES

1031

16.._I_oE COOE

SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF ABSTRACT

Standard Form 2_)B (Rev. 2-89)

Pr_ttll_,_ by AN¢_I ¢*td Z3q*IB

2911-102