[American Institute of Aeronautics and Astronautics 45th AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

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Stress RangeMean StressAfter Cold-Working

Mean StressAfter Cold-Working

Stress Range

Cyclic Stress Strain LoopAway from Hole

Cyclic Stress Strain Loopat Edge of Hole

0

0

σ

ε

σ

ε

Residual StressesDue to Cold-Working

Figure 1. Effect of Residual Stresses from Cold-working

on Stress Ratio.

Probabilistic Risk Assessment of Joints with Cold-worked Holes

William T. Fujimoto* Advanced Structural Technology, Inc., Milford, CT, 06460

A methodology for performing the probabilistic fatigue analyses of joints with cold-worked holes is described. Probabilistic fatigue analyses are necessary in determining the susceptibility of cold-worked holes to cracking, or in performing damage tolerance analyses, because of the influence of many difficult-to-quantify variables on both the pattern of cracks which nucleate and on the nucleation and crack growth lives. A computer program for performing the probabilistic fatigue analyses of multi-layer joints with cold-worked holes is described. Examples are presented to illustrate how the methodology can predict failure rates and lives for cold-worked holes.

I. Introduction

Cold-working of fastener holes is used extensively to improve the life of aircraft. If properly exploited, cold-

working, either by itself, or in conjunction with other life enhancement techniques such as interference fit fasteners, can serve as the basis for a new generation of ultra long life, low maintenance aircraft if incorporated into new designs. Cold-working can also be used for restoring structural Integrity to non-cold-worked members experiencing premature failures.

Cold-working achieves the increase in the fatigue life of a hole by inducing beneficial residual compressive stresses at the edge of the hole. The residual stresses are induced by cold-expansion of the hole with an oversized

mandrel which has sufficient interference to plastically yield the material near the hole edge. The plastic yielding stretches the material in the region surrounding the hole, so that upon removal of the mandrel, the elastic material surrounding the plastically deformed region compresses it, inducing residual compressive hoop stresses in the vicinity of the hole edge, and residual tensile stresses away from the hole.

Since the residual stresses are permanent and remain unchanged unless reyielding of the plastically deformed region occurs, they cause a shift in the cyclic fatigue behavior at the hole edge by shifting the stress ratio R downward, as shown schematically in Figure 1. This “beneficial” shift in the stress ratio, in turn, increases the fatigue life of the hole. The residual tensile stresses away from the hole also cause a “detrimental” shift in the R-ratio by raising the stress ratio away from the hole in the residual tension zone. However, the

stresses away from the hole are much lower than the stresses at the hole edge, leading to a much lower rate of fatigue damage accumulation away from the hole. * President, [email protected], (203)-878-8327

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1743

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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While the general principles by which cold-working can extend life are understood, exploiting the full potential of cold-working has been held back by a lack of a full understanding of the failure mechanisms in cold-worked holes. The life increase which is possible in multi-layer aircraft joints is a function of many parameters, i.e., stress spectra, fastener system, % mandrel interference, edge distance, material properties, fastener clamp-up, restraint of secondary bending, surface finish and coating of the joint members, etc. Depending on how these parameters interact, the actual life improvement factor for a joint with cold-worked holes versus one without cold-working may vary from little or no increase, to an increase of an order of magnitude or more.

In this paper, the use of probabilistic fatigue analyses, as a tool for quantifying the effects of these variabilities and uncertainties upon reliability, is described. This approach can account explicitly for each of the major mechanisms affecting the crack formation and crack growth lives, including relaxation of the residual stress field and the progressive growth of a system of cracks. Thus, probabilistic fatigue analyses are an important and necessary part of an analytical testing capability for joints with cold-worked holes.

II. Key Factors Influencing Life

Key factors influencing life include the following:

• 3D nature of residual stress field due to mandrel pull-through • Multi-stage crack growth involving a progressively spreading system of cracks • Multiple potential initial crack sites, and the effect of site upon the multi-stage crack growth path. • Relaxation of the residual stress field due to overloads/underloads, or due to cyclic reyielding

from crack growth. • Interaction of the hole with adjacent structural elements for multi-layer joints.

A detailed discussion of the first three factors follows. For discussion of the remaining two factors, the reader is referred to ref. [i ]. A. Through-the-Thickness Variations in Residual Stresses due to the Directional Nature of Mandrel Pull-through.

The through-the thickness variations exist due to the directional nature of the cold-working process. In the split-sleeve cold-working method of Fatigue Technology, Inc.(FTI), the most popular method for airframe construction, an oversize tapered mandrel is pulled through the sheet by a pneumatic tool, as shown in Figure 1. A lubricated expandable split-sleeve is inserted in the hole prior to cold-working to prevent direct contact between the hole and the sliding mandrel. Upon pulling the mandrel through the hole, the hole is reamed to the final hole dimension. The directional process induces different residual stresses at the entrance surface, the mid-plane and at the exit surface.

Through-the-thickness variations in the residual stress field occur because the plate undergoes out-of-plane deformation during the mandrel pull-through process, as shown schematically in Figure 3. The out-of-plane rotation of the plate occurs because of the movement of the center of pressure of the mandrel contact pressure through the plate during pull-through.

Figure 4 from Forques, Bernard and Bui-Quocii illustrates the variation through the thickness of the residual hoop stresses for a cold-worked 7475-T7351 aluminum plate with a W/D of 5 and a 4% mandrel interference level. The residual stresses are shown at three stations through the thickness: along the mandrel entry surface, at mid-section, and along the mandrel exit surface. In obtaining these stresses, Forgues, Bernard and Bui-Quoc used a 3D axisymmetric ABAQUS finite element model. As Figure 2 shows, the predicted residual tangential stresses at the entry surface are significantly lower in magnitude than at the mid-plane and at the exit surface. These results point to a significant

Figure 2. Split-Sleeve Cold-Working.


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reduction in the residual stresses at the entry surface, a fact independently confirmed by ANSYS 3D and slice synthesis analyses summarized in this paper.

An understanding of the mechanisms causing this behavior was obtained using a slice synthesis model of mandrel pull-throughiii. In the slice synthesis model, mandrel pull-through was modeled by idealizing the plate as a stack-up of slices. The residual stresses at any slice were found by combining solutions for the uniform elastoplastic expansion of a hole and the elastic solution for out-of-pane rotation of a plate with a circular hole with moments applied to the hole periphery. Determination of the contributions of the planar and out-of-plane components on the residual stresses at any station in the thickness required iterative solution of a non-linear system of equations. From the slice synthesis model, it was determined that during the entrance of the mandrel tapered ramp into the hole, the moment resulting from the eccentricity causes an “opening” up of the hole, reducing the effective interference at the entry surface. Because of the tapered entry ramp, the loss of interference at the entry surface is not negated during the passage of the mandrel through the hole.

The net result of the out-of-plane rotation is that the entry surface residual stresses are significantly lower than they would be for uniform expansion of the hole, while the residual stresses in the interior and the exit surface are less affected by the rotation, approaching the values expected for uniform expansion of the hole. B. Initial Cracking at Hole Edge vs. Away from Hole and Multi-Stage Crack Growth. Most analytic simulations of crack

growth from a cold-worked hole assume cracking at the edge of the hole. However, test data on cold-worked 2014-T651 aluminum low load transfer lap shear joint specimens which were tested in fatigue spectrum loading with a peak far field stress of 29.7 ksi, then examined after a crack had formed and grown, indicate that cracking can occur away from the holeiv, see Figure 5. In Figure 5, the fracture surface indicates that the primary crack nucleated away from the hole in the residual tension zone. Furthermore, the indications of secondary cracking at the hole edge are indicative of a multi-stage crack growth pattern where the secondary cracks coalesce with the primary crack, leading to a single through-the-thickness crack. This secondary cracking indicates a relaxation of the residual stress field, leading to a loss of the residual compressive stress at the edge of the hole and the subsequent formation of secondary cracking. This relaxation can stem from two mechanisms. The primary mechanism is the inability of the cracked material to transmit the residual tension stresses needed to equilibrate the residual compression stresses, leading to a monotonic decay of the residual stress field as the crack growsv. The second mechanism is cyclic reyielding of the net section residual stress field due to crack tip plasticity.

T

T

(a) Mandrel Flat in Contact with Hole.

(b) Mandrel Ramp in Contact with Hole.

h0

h

h0

h

Figure 3. Out-of-Plane Plate Rotation During Mandrel Pull-through.

Figure 4. Analytically Predicted Residual Stress Distribution in Cold-Worked 7475-T7351 Aluminum Plate with a W/D of 5 and a 4% Mandrel Interference Level . Reproduced from 3-D axisymmetric numerical analysis and experimental study of the fastener hole cold-working process in Computer Methods and Experimental Measurements for Surface Treatment Effects, edited by M.H. Aliabadi and C.A. Brebbia, Computational Mechanics Publications, Southampton, UK, 1993, ISBN 1-56252-150-0.


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Figure 6 summarizes the possible scenarios for multi-stage stage cracking following crack formation at the hole edge or away from the hole. Whether the crack nucleates at the hole edge or away from the hole is a function of the geometry, i.e., e/D, and the interference level, as well as the degree of secondary bending and the localized peaking of bearing

stresses due to fastener tilting. In either case, formation of the primary crack is followed by formation of a secondary crack and the subsequent coalescence of the cracks into a single crack system. Since different crack growth lives result from whether the crack forms at the hole edge or away from the hole, correctly identifying the most probable initial crack location and the subsequent form of damage progression is a vital part of performing a damage tolerance analyses.

III. Stress Field Synthesis

In order to perform a probabilistic analysis of a cold-worked hole, the stress field around the hole must be determined. The stress field causing formation and growth of the fatigue crack was determined by stress field synthesisi. In stress field synthesis, Figure 7, the resultant stress field is synthesized by superimposing the residual stresses resulting from cold-working and the elastically-determined applied stresses. The stresses due to the applied loading are obtained by idealizing the plate as a stack-up of infinitesimally thin slices , and by forming the stress field at any thickness location by combining the two-dimensional solutions for an open hole subjected to far field tension and an loaded hole. The bypass stress and

the bearing stress for each slice are determined from a multi-layer joint analysis to account for the effect of adjacent layers and fasteners on the stress state surrounding the hole. The residual stresses due to cold-working were determined from ANSYS three-dimensional finite element models. The results of the finite element analyses were used to obtain correction factors for transforming a closed-from 2D

solutioniii into 3D residual stress solutions. To capture the effect of the mandrel on the stresses, the mandrel pull-through process was explicitly modeled. In order to simplify the modeling process, the rectangular plate was approximated by a circular annulus whose inner radius was equal to the hole diameter, and whose outer diameter was equal to the plate width W. By reducing the geometry to an equivalent circular annulus, axi-

symmetric models, in which only half of the net section is explicitly modeled, could be used.

Figure 5. Experimentally Determined Fracture Surfaces for 2014-T651 Aluminum Low Load Transfer Lap Shear Joint Specimensiv.

Direction of MandrelPull-Through

Primary Crack

Secondary Crack

Resultant Crack

Direction of MandrelPull-Through

Primary Crack

Secondary Crack

Resultant Crack

Primary Crack Nucleationat Hole Edge

Primary Crack NucleationAway From Hole

Figure 6. Possible Multi-Stage Cracking Scenarios.


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Figure 8 shows a typical axi-symmetric model. The flared split sleeve was explicitly modeled in the model shown (although it was later removed in subsequent analyses to reduce the model size). The mandrel was modeled as a rigid surface with sharp corners† at the intersections of the mandrel entry and exit ramps with the mandrel flat. Mandrel pull-through was simulated by incrementally advancing the mandrel as a rigid body through the hole. Frictionless contact is assumed during pull-through.

Typical residual stresses predicted by the model for the entry, exit and mid-plane are shown in Figure 8 for a 0.10 in. thick by 1.0 in. wide 7075-T74 Al plate with an 0.25 in. dia. hole which has been cold-worked to 3% interference, followed by a 10% post-ream. In determining these stresses, the “kill element” feature of ANSYS was used to remove the elements to simulate the post-ream, while retaining the residual plastic strains resulting from cold-working. Figure 9 shows the same through-the-thickness distribution of the residual stresses as shown in Figure 2: residual tangential stresses at the entry surface which are significantly lower in magnitude than at the mid-plane and at the exit surface. These residual stresses are stored in multi-dimensional tables so that correction factors to convert a two-dimensional closed form solution to three-dimensions could be generated during the probabilistic analysis‡.

† Later models had rounded corners to eliminate the singularity caused by sharp corners. ‡ A closed-form solution was used as the fundamental solution for the probabilistic analysis to permit the effects of variations in yield stress and interference level to be simulated.

Stress Due to Bypass Stress

Stress Due toFastener Bearing

Residual StressDue to Cold-working

Freebodyof Hole

Figure 7. Stress Field Synthesis of Three-Dimensional Stress State.

Figure 8. Axisymmetric ANSYS model of Work-Piece with Sleeve and Rigid Mandrel.


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IV. Probabilistic Fatigue Analyses

Because of the possibility of multi-modal failures, Monte Carlo simulationvi was used for the probabilistic fatigue analyses so that separate failure rate curves could be generated for each mode. Figure 10 summarizes the architecture of ProbabilityContour4, a module developed for automating the probabilistic fatigue analyses. The input to the module consists of the identifications for the plate and mandrel materials, the interference level, the hole diameter D, the plate width W, the and the stress spectrum(currently the module is limited to constant amplitude loading with a specified stress ratio R). With this information, the module extracts the mean material properties from the material database, along with the parameters for the assumed stochastic distributions for the interference level, the yield stress, and the fatigue strength.

The module first calls the stress field synthesis subroutine StressFieldSynthesizer to form the elastically determined net section distribution of stresses and strains resulting from the applied loading§. The stress field synthesis program is linked to the multi-layer joint analysis program, so that the secondary bending and the fastener bearing peaking gradient can be used in synthesizing a three-dimensional stress gradient. Relaxation of the residual stress field from overloads or underloads is accounted for using the method of strain invarianceiii.

Once the mean applied stress distribution has been determined, Monte Carlo simulation is used to determine the most probable location on the net section for initial crack formation. 100 trials are used for each location on the net section. In the Monte Carlo simulation, the mandrel interference level, the material elastic properties and the yield stress, and the applied load severity factors (the factor by which the mean value is multiplied) are treated as stochastic variables with a Gaussian distribution and allowed to vary. Values of the parameters which correspond to the probability distributions assumed for the variables are generated by an inverse transform methodvi from random numbers generated by a random number generator.

For each trial, the residual stress field corresponding to the mandrel inference and yield strength is obtained from the cold-worked analysis routine. The steady stress level, and the vibratory stress amplitude for each maneuver in the usage spectrum is obtained by multiplying the mean applied stresses by the severity level for the trial and superimposing the applied stresses on the residual stress state.

In performing the Monte Carlo simulations, the stress ratio transformation equation is treated as a stochastic variable. Normally, when a stress ratio transformation such as the linear Goodman is used, it is treated as a deterministic quantity, and is assumed to be given by:

σσσ

vib Rvib

steady

TU

q

F

g =−=

−LNMOQP

1

1

(1)

§ The stresses and strains in the x-direction(perpendicular to the net section) and in the y-direction(parallel to the net section) are synthesized from elastic stress field solutions for a hole in a finite width strip under bypass loading and bearing loading, respectively. These solutions are stored in multi-dimensional tables and read in during the start of the module.

Residual Stress, 3% Interference, Isotropic Hardening, W/D = 4After 10% Ream Operation

-100000

-80000

-60000

-40000

-20000

0

20000

40000

0.1250 0.2500 0.3750 0.5000

Radial Distance (in)

Tang

entia

l Res

idua

l Str

ess

~psi

ANSYS-InletANSYS-MidplaneANSYS-Exit

Rigid mandrel has 0.125 inch fillets.

D=0.25 W=1.0 T=0.10Material: 7050-T74 Aluminum

Figure 9. ANSYS 3D Residual Tangential Stresses in W/D=4 Plate Cold-worked to 3% Interference Level, Followed by 10% Ream.


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where the exponent q is normally assumed to be 1. However, for a cold-worked hole, the likelihood of a crack forming away from the hole or at the hole edge is highly dependant upon the form of the equation for finding the equivalent stress.

Figure 11, which shows a constant life diagram for both tensile and compressive mean stresses, schematically illustrates why this is so. As Figure 11 shows, there are two different transformation equations involved, depending on whether the mean stress is tensile (away from hole) or is compressive(at the edge of the hole). If the mean stress is tensile, the use of eq. (1) above is appropriate, with q ranging from 1 to 2**, depending on the material. If the mean stress is compressive, this transformation equation is physically incorrect, since it implies an unbounded increase in the allowable vibratory stress amplitude with increasingly negative mean stress. The exact form of the transformation equation is not known, due to a scarcity of SN test data for negative mean stresses; however, it can be assumed to be bounded by the cross-hatched region shown in the constant life diagram in Figure 11 for the negative mean stress regime. In other words, an

upper bound estimate is given by the unmodified Goodman transformation equation, while a lower bound estimate can be obtained by assuming once the mean stress becomes negative, the effect is the same as a zero mean stress.

The corresponding stress ratio transformation equation for compressive mean stresses becomes:

σσ

φσvib Rvib

steady

TUF

g =−=

−1

1 (2)

where φ varies between 0 and 1. In the Monte Carlo simulation, φ was assumed to follow an uniform distribution, where any value between 0 and 1 was equally likely††.

The crack formation life corresponding to the equivalent stress amplitude is then found by a safe life fatigue analysis. As the lives are determined for each trial, they are written to an array. Once all the trials have been completed, the failure rate curves for each location are generated using the median rank regression methodvii.

** It should be noted that the probability of crack formation is extremely sensitive to the distribution assumed for q, with higher or lower probabilities of crack formation away from the hole possible by varying the value of q. †† This was due to the lack of any test data which would indicate a definitive probable distribution for the values. It should be noted that the use of φ=0 is favored by the author, based on his experience as a consultant evaluating lap joint fatigue test data from testing of cold-worked hole specimens..

Specify Materials,Interference, D, W, MaxStress, Stress Ratio R

Load Material Propertiesfrom Data Base & Generate StochasticDistribution Parameters for Residual

Stress & Applied Load Variation

Multi-Layer Joint Analysis to DetermineBypass Stress,Bearing Stress, Bearing

Peaking at Hole

Vary StressTransformation

Equation Parameters

Find Equivalent Stressfrom Goodman Eq.

Determine FatigueLife using Universal

Fatigue Machine

Place Life in UnsortedVector

Loop Through Net Section Location(1=HoleEdge--Exit Surface, 2=Away from Hole),3=

Hole Edge--Entry Surface

Next Location

Form Failure RateUsing Median Rank

Regression

Compare Failure RateCurves to Identify Most

Probable Crack Site

Cold-Worked

HoleAnalysis

SynthesizeResultant Stress

Field(BeforeReyielding)

Adjust ResidualStresses for

Reyielding due toOverload/Underload

Monte Carlo Loop(Vary % Mandrel

Interference, Yield Stress,Load Severity, Goodman

Parameter)

If EntrySurface, Adjust

ResidualStress

3

21

Mandrel Pull-thru

3% Interference, 20 ksi Peak Stress, R=0

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1000 10000 100000 1000000 10000000

Life ~ Cycles

Prob

abili

ty o

f Cra

ck N

ucle

atio

n

Hole Edge, Exit SurfaceAway From Hole, Exit SurfaceHole Edge, Entry Surface

Figure 10. Flowchart for Probabilistic Fatigue Analysis to Identify Most Probable Initial Crack Location and Time to Initial Cracking.


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V. Examples

An example of the failure rate curves generated using this module are shown in Figure 12 for the exit surface hole edge, the exit surface away from the hole, and the entry surface hole edge‡‡ for a 7050-T74 aluminum member which is cold-worked to an interference level of 3%, then subjected to constant amplitude cycling at a far field peak stress of 20 ksi and a stress ratio R of 0.

In generating the curves, the yield stress, the interference level, and the fatigue strength were assumed to be governed by Gaussian distributions with a coefficient of variation(COV) of 0.10. The assumption of a Gaussian distribution with a COV of 0.10 was arbitrary, in the absence of more specific data, and was selected primarily to illustrate the effects of the variabilities of the parameters on life. In fact, since ProbabilityContour4 uses the two and three-parameter Weibull distributions as the native internal distributions, the Gaussian distributions were converted into equivalent Weibull distributions. Thus, virtually any probability distribution which can be

idealized by a two or three-parameter Weibull distribution can be handled. From the Monte Carlo simulation, different combinations of yield stress and interference level satisfying the

probability functions were formed for each trial. For each combination of yield stress and interference level, the two-dimensional residual stress distribution was determined. The stresses at the entry, the exit and away from the hole, as obtained from the two-dimensional analysis, were multiplied by the three-dimensional correction factors from the finite element analyses. Because the two-dimensional residual stresses reflected the variability of yield stress and interference level, the synthesized three-dimensional residual stresses likewise reflected the variability.

Failure rate curves such as shown in Figure 12 can be used to to identify the most probable initial cracking location. To compare the relative likelihood of cracking at a location other than hole edge at the entry surface, the median life of the hole edge on the entry surface can be divided by the mean life of the hole edge on the exit surface, as schematically illustrated in Figure 13, to serve as a measure of the likelihood of cracking occurring at the hole edge on the exit surface. A ratio greater than 1 is indicative of cracking initially occurring at the hole edge at the exit surface, while a ratio less than 1 is indicative of cracking initially occurring at the hole edge on the entry surface, where the residual stresses have been reduced by rotation of the work-piece during mandrel pull-through. Similarly, the median life of the hole edge on the entry surface can be divided by the mean life of the exit surface away from the hole edge to serve as a measure of the likelihood of cracking occurring at the exit surface away from the hole edge.

Figure 14 shows a carpet plot summarizing the non-dimensional likelihood of cracking on the exit surface versus peak stress and interference for a 7050-T74 aluminum plate subjected to constant amplitude cycling at R=0. Figure 15 shows a carpet plot summarizing the non-dimensional likelihood of cracking on the exit surface versus peak stress and stress ratio R for the same plate cold-worked to an interference level of 3%.

‡‡ The failure rate was not generated for the entry surface, away from the hole, as this location would have significantly lower residual tension stresses than for the exit surface. Hence, for an unloaded hole, in the absence of secondary bending stresses, the probability of initial cracking here prior to initial cracking on the exit surface away from the hole would be very low.

σσσ

vib Rvib

steady

TU

q

F

g =− =

−LNMOQP

1

1

0 1.0

Negative Mean Stress Positive Mean Stressσ vib

σ s tea d y

tuF

q=1.0

q=2.0

σσ

φσvib Rvib

steady

TUF

g =− =−

1

1

φ = 1

φ = 0

Cross-Hatched Area Designates Probable Range of Isolife Contour

Figure 11. Variabilities in the Constant Life Diagram to Determine the Equivalent R=-1Strain(or Stress) for Tensile & Compressive Mean Stresses.


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The results summarized in Figures 14 and 15 indicate that, regardless of stress level, interference, or load magnitude, the hole edge on the entry surface is predicted to be the most likely location for initial cracking for specimens with unloaded holes, with little likelihood of cracking occurring first on the exit surface away from the hole edge§§.

Furthermore, on the exit surface, it can be seen that the likelihood of cracking occurring first on the exit surface away from the hole is always less than at the hole edge, although it approaches the likelihood of cracking occurring at the hole edge for higher interference levels and stress levels. However, it should be noted that these results are only valid for specimens with normal e/D ratio’s and no secondary bending. Specimens with “shy” e/D ratio’s are more susceptible to cracking away from the hole, and thus may behave differently.

§§ Even though the overall likelihood of initial cracking away from the hole edge was small, during the Monte Carlo simulations, numerous instances occurred where initial cracking was predicted to occur away from the hole. Hence, it must be emphasized that cracking away from the hole can occur on unloaded cold-worked holes, although the rate of occurrence, relative to the population, can be expected to be low.

3% Interference, 20 ksi Peak Stress, R=0

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1000 10000 100000 1000000 10000000

Life ~ Cycles

Prob

abili

ty o

f Cra

ck N

ucle

atio

n

Hole Edge, Exit SurfaceAway From Hole, Exit SurfaceHole Edge, Entry Surface


Exit Surface

Entry Surface

Figure 12. Failure Rate Curves Generated Using ProbabilityContour4.

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1000 10000 100000 1000000 10000000

Hr.

Prob

abili

ty o

f Fai

lure

F(i)

Edge of HoleResidual Tension Zone

Location 2

Location 1

Probability of Cracking of Hole Edge For 50% Probability of Cracking Away From Hole

50% Probability of Cracking Nucleating Away From Hole

Figure 13. Finding the Probability of Cracking at Secondary Crack Locations Relative to the Probability of Cracking at the Primary Crack Site.


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Since Figures 14 and 15 predict cracking to occur at the hole edge for the lower stress levels and stress ratio’s close to 0 or less, findings which are in agreement with the test results of Cook and Holdwayviii. In their testing, Cook and Holdway subjected 7050-T76 aluminum unloaded hole cold-worked specimens to constant amplitude fatigue testing at R=0.1 and a peak stress of 26.7 ksi. Of the sixteen specimens tested in fatigue, 62% experienced initial cracking at the hole at either the entry or the exit surface, while the remainder of the specimens either experienced cracking at the grips, or experienced a combination of cracking at the hole and the grips. In no case did cracking nucleate away from the hole in the residual tension zone. This finding is consistent with the probability contour plots of Figures 14 and 15, which predicts that cracking at the hole edge is significantly more probable as cracking in the residual tension zone for the combination of R-ratio and peak stress.

R=0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3 4 5

% Interference

Med

ian

Life

Ent

ry S

urfa

ce H

ole

Edge

/Med

ian

Life

at

Loca

tion

20ksi_HoleEdge,Exit30 ksi_HoleEdge,Exit40 Ksi_Hole Edge,Exit20kis_AwayFromHole,Exit30ksi_AwayFromHole,Exit40ksi_AwayFromHole,Exit


Exit Surface

Entry Surface

Figure 14. Likelihood of Cracking on the Exit Surface versus Cracking on the Entry Surface at the Hole Edge versus Peak Stress and Interference for a 7050-T74 Al Plate Subjected to Constant Amplitude Cycling at R=0.

0

0.1

0.2

0.3

0.4

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0.6

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0.8

-0.25 0 0.25 0.5

Stress Ratio R

Med

ian

Life

Ent

ry S

urfa

ce H

ole

Edge

/Med

ian

Life

at L

ocat

ion

20ksi_HoleEdge,Exit30 ksi_HoleEdge,Exit40 Ksi_Hole Edge,Exit20kis_AwayFromHole,Exit30ksi_AwayFromHole,Exit40ksi_AwayFromHole,Exit


Exit Surface

Entry Surface

Figure 15. Likelihood of cracking on the Exit Surface versus Cracking on the Entry Surface at the Hole Edge versus Peak Stress and Stress Ratio R for a 7050-T74 Al Plate Cold-worked to 3% Interference and Subjected to Constant Amplitude Cycling at R=0.


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Because ProbabilityContour4 can also generate, as part of the Monte Carlo fatigue analysis to identify the most probable location for initial cracking, the failure rate curves relating the probability of cracking to service life, the module can also be used to generate, for a specified probability level, the SN curves for cold-worked holes. Such curves can be used in a spectrum fatigue analysis for determining the life of a member for a specified reliability level.

The SN curves shown above in Figure 16 were generated using 50% probability lives from the failure rate curves at each of the locations. The curves show that the crack formation life of the hole at the entry side is predicted to be roughly half the life at the exit side due to the reduction of residual stresses. It should be noted that SN curves corresponding to a specified failure rate, i.e., 2% probability of failure, can be generated by this technique simply by finding the lives corresponding to the specified failure rate. SN curves generated in this manner account for the variability in the cold-working process and in loads severity, variability in the accuracy of the equation used

to transform the stress range into a form compatible with conventional SN or ε-N life curves, in addition to the variability of fatigue strength of the material. Thus statistical protection against variability can be built into the fatigue analysis, as protection against unexpected premature failures.

VI. Conclusions

A probabilistic fatigue analysis methodology for cold-worked holes has been developed. The methodology explicitly accounts for the three-dimensional nature of the residual stresses, as well as for the three dimensional nature of the stresses induced by the applied loading. The methodology accounts for the effects of variability of the mandrel interference level, the yield stress, the Goodman stress ratio transformation, loading spectrum, and the fatigue strength of the material on life.

The probabilistic fatigue analyses can be used, as part of virtual fatigue testing, to identify the most probable initial crack formation location, and to also track the probability of secondary cracking occurring at other locations. By performing a probabilistic analysis, rather than a deterministic analysis to identify the initial cracking location, the likelihood of initial cracking away from the hole can be estimated. This information, in turn, can be used to assess the risk of premature fatigue failures due to failure modes which are different than those used for establishing the expected life. Because each candidate failure mode can be explicitly examined, the probabilistic fatigue analyses can be used to provide a specified level of protection against premature fatigue failure, unlike a deterministic fatigue analysis, which can only provide protection against a single mode of failure. Thus the application of probabilistic fatigue analyses is an important and necessary step to implementing an analytical testing capability for joints with cold-worked holes.

Acknowledgments The work described in this paper was performed under a Phase I SBIR grant from the Air Force Research Laboratory

with Dr. Eric Tuegel as the Program Manager. The contribution of Mr. Antonio Portelinha in performing the finite element modeling of mandrel pull-through is also gratefully acknowledged.

3% Interference, R=0

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~ ks

i

Hole Edge, ExitAway From Hole, ExitHole Edge, EntryNon-Coldworked


Exit Surface

Entry Surface

Non-Coldworked

Hole Edge--Exit(Based on 2D Residual Stress

Hole Edge--Entry

Figure 16. 50% Probability SN Curves for Potential Initial Crack Locations for a 7050-T74 Al Plate Cold-worked to 3% Interference and Subjected to Constant Amplitude Fatigue.


12

References

i W.T. Fujimoto, “Modeling Multi-Stage Failure Mechanisms in Joints with Cold-Worked Holes”, Paper Presented to ASIP 2002 Conference, Savannah, GA, Dec. 2002. ii S.A. Forgues, M. Bernard & T. Bui-Quoc, “3D Axisymmetric Numerical analysis and Experimental Study of Fastener Hole Cold-working Process”, Computer Methods and Experimental Measurements for Surface Treatments, Ed. M.H. Aliabadi and C.A. Brebbias, Computational Mechanics Publications, 1993. iii W. Fujimoto, “Progress toward Developing a 2nd Generation Crack Growth Prediction Methodology for Joints with Cold-Worked Holes”, 2003 Air Force ASIP Conference , Savannah, GA, Dec., 2003. iv Mann,J.Y., Revill, G.W., and Lupson, W.F., “Improving the Fatigue Performance of Thick Aluminum Alloy Bolted Joints by Hole Cold-Expansion and the Use of Interference-Fit Steel Bushes”, ARL Structures Note 486, Aeronautical Research Laboratories, Melbourne, Australia, 1983. v A.T. Ozdemir & L. Edwards,”Relaxation of Residual Stresses at Cold-worked Fastener Holes Due to Fatigue Loading”, Fatigue Fract. Engnrg. Mater. Structure, Vol. 20, No. 10, 1997. vi S. Mahadevan, “Monte Carlo Simulation”, in Reliability-Based Mechanical Design, T.A. Cruse, Ed., Marcel Dekker, 1997. vii Dr. R.B. Abernethy, the New Weibull Handbook. viii R. Cook and P. Holdway, “Residual stresses Induced by Hole Cold Expansion”, Computer Methods and Experimental Measurements for Surface Treatments, Ed. M.H. Aliabadi and C.A. Brebbias, Computational Mechanics Publications, 1993.

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