Fatigue design of wire-wound pressure vessels using ASME-API 579 procedure

Engineering Failure Analysis 17 (2010) 748–759

Contents lists available at ScienceDirect

Engineering Failure Analysis

journal homepage: www.elsevier .com/locate /engfai lanal

Fatigue design of wire-wound pressure vessels usingASME-API 579 procedure

J.M. Alegre *, P.M. Bravo, I.I. CuestaStructural Integrity Group, Department of Civil Engineering, E.P.S. University of Burgos, C/Villadiego s/n, 09001 Burgos, Spain

a r t i c l e i n f o

Article history:Available online 2 September 2009

Keywords:ASME-API 579Wire-wound pressure vesselsFatigue lifeStress intensity factor

1350-6307/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.engfailanal.2009.08.008

* Corresponding author.E-mail address: [email protected] (J.M. Alegre).

a b s t r a c t

The wire winding of high pressure vessels is a technique usually applied to introduce initialcompressive stresses in the inner core of the vessel, with the aim to improve the fatigue lifeunder cyclic pressure conditions. In this work, the procedure followed to calculate thenumber of design cycles is presented, using the fracture mechanics approach and the struc-tural integrity concepts. In particular, the API 579-1/ASME FFS-1 procedure has been usedto analyse the structural integrity of the vessel through the crack propagation stage. Start-ing from a postulated internal semi-elliptical crack the number of design cycles is deter-mined, the flaw aspect ratio is updated and the structural integrity of the cracked vesselis evaluated using the Failure Assessment Diagram (FAD). Different propagation laws,which take into account for negative stress intensity ratio factors R = Kmin/Kmax < 0, arereviewed, because of their high influence on the fatigue life of wire-wound vessels. In addi-tion, this paper presents a number of useful expressions to calculate the stress intensityfactor (SIF) for internal semi-elliptical cracks in wire-wound pressure vessels, in order tocarry out the numerical integration of the number of cycles, updating the flaw aspect ratio,during the fatigue crack growth.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

For high pressure vessel design autofrettage techniques [1–3] or wire winding process [4] can be used, with the mainobjective to introduce initial residual stress in the inner core of the vessel. This internal compressive stress will improvethe fatigue life of the vessel, because it will reduce the mean stress level of the pressurization–depressurization fatigue cycles[5]. Nowadays, the most widespread technique used is the wire winding method, which makes it possible to reach higherinternal residual stress without yielding any part of the vessel during the wire winding process.

Basically, the winding method consists of a wire helically wound edge-to-edge in pretension in a number of turns andlayers around the outside of the inner cylinder. With each wound-wire layer becomes greater and greater compressive stres-ses in the cylinder, while the internal wire layers are slightly relieved as a consequence of the radial compressive stress gen-erated by the last wound layer. The final winding process is a vessel with internal compressive stress in the inner cylinderand a wire winding block under tensile stress. Under working pressure conditions the initial compressive stress in the cyl-inder is reduced and the winding tensile stress is increased, but the internal core of the vessel can remain under compressionif the initial compressive stresses are low enough. The compressive stress level that can be introduced in the vessel is limitedby the yield stress of the material under compression.

. All rights reserved.

http://dx.doi.org/10.1016/j.engfailanal.2009.08.008

mailto:[email protected]

http://www.sciencedirect.com/science/journal/13506307

http://www.elsevier.com/locate/engfailanal

J.M. Alegre et al. / Engineering Failure Analysis 17 (2010) 748–759 749

Obviously, the regulations governing these kinds of components must be strict enough to ensure the structural integrityof the high pressure vessel. In this sense, the ASME Code, article KD-9 [6], provides a number of requirements applicable tothe design of high pressure vessels consisting of an inner cylinder prestressed by a surrounding winding of at least ten layers.

The above mentioned requirements refer to the need for safety verification with respect to the plastic collapse pressure ofthe vessel, the value of the accumulated plastic strain damage, and its fatigue life under working pressure conditions. Thiswork is focused in this last requirement.

In order to calculate the number of design cycles based on crack propagation an initial crack size and an allowable finalcrack must be defined. The initial crack size to be used for the calculation of the crack propagation design cycles shall bebased on the nondestructive examination method to be used. In general, a surface crack not associated with a stress concen-tration can be assumed to be semi-elliptical with a ratio of depth to surface length of a/‘ = 1/3. However, the allowable finalcrack depth must be calculated using the Failure Assessment Diagram (FAD), available on the fitness for service procedures(FFS) as the API 579-1/ASME FFS-1, SINTAP, R6 or similar. This method to obtain the allowable final crack can provide a moreaccurate fatigue life design, however the numerical procedure is more time-consuming because of the aspect ratio, a/‘, shallbe updated as the crack size increases. The stress intensity factor (SIF), both at the deepest crack point and at the cornerpoint, must be used in order to update the aspect ratio of the crack thought its fatigue propagation. This continuous crackshape updating requires tedious double interpolations if typical tabular solutions for SIF calculation, such as the provided onthe mentioned codes, are used.

This work presents an application of the API 579-1/ASME FFS-1 procedure [7] to perform a fatigue analysis for wire-wound pressure vessels. A new set of analytical expressions are provided in order to calculate the SIF for a semi-ellipticalcrack, both in the deepest point and the surface points. These equations can be used to avoid the use of the tabular solutionsand, as a consequence, an automatic numerical integration of the propagation law can be effected.

2. Stress distribution in wire-wound vessels

The ASME code, Section VIII - Division 3, provides the analytical equations for the stress calculation in wire-wound vesselsunder linear-elastic conditions [6]. These equations provide the stress generated in wire and cylinder parts, during and afterwinding operation, at the central cross section. Other useful numerical methods, based on FE analysis, can be used to obtainthe stress distribution on the whole vessel, including the border effects [4].

The main dimensions for the problem considered in this study are shown in Fig. 1. It is assumed that the winding processis performed with a tensile stress in the wire Sw(x) and that this stress can be a function of the diameter coordinate x. Theradial and circumferential stress components, rr(x1) and rt(x1), for a cylinder coordinate x1, are defined by the followingexpressions:

ODifD

ID

1x

2x

Fig. 1. Nomenclature for wire-wound vessels.

750 J.M. Alegre et al. / Engineering Failure Analysis 17 (2010) 748–759

rrðx1Þ ¼ � 1� DI

x1

� �2" #Z Dw

Dif

x

x2 � D2I

SwðxÞ !

dx ð1Þ

rtðx1Þ ¼ � 1þ DI

x1

� �2" #Z Dw

Dif

x

x2 � D2I

SwðxÞ !

dx ð2Þ

where DI is the inside diameter, Dif is the interface diameter between the winding and cylinder, Dw is the outside diameter ofthe actual winding layer.

Moreover, the corresponding stress in the winding diameter coordinate x2 can be derived by:

rtðx2Þ ¼ Swðx2Þ � 1þ DI

x2

� �2" # Z Dw

x2

x

x2 � D2I

SwðxÞ !

dx ð3Þ

rrðx2Þ ¼ � 1� DI

x2

� �2" #Z Dw

x2

x

x2 � D2I

SwðxÞ !

dx ð4Þ

Subsequently, the above equations can be easily integrated for a constant tensile stress value in wire during the windingprocess, Sw(x), and they provide the stress distribution at all times of the winding operation. Once the winding process hasbeen concluded, the radial and circumferential stress components, both in the cylinder and the winding, can be obtainedreplacing Dw for DO in the above expressions, where DO represents the outside diameter when the winding process isfinished.

Under working pressure conditions, the radial and circumferential stress components in the cylinder can be calculated as

rTt ðx1Þ ¼ rtðx1Þ þ pw

ðDO=DIÞ2�1� D2

Ox2

1þ 1

� �rT

r ðx1Þ ¼ rrðx1Þ � pw


Ox2

1� 1

� � ð5Þ

And the corresponding stress distribution in the winding diameter as,

rTt ðx2Þ ¼ rtðx2Þ þ pw


Ox2

2þ 1

� �rT

r ðx2Þ ¼ rrðx2Þ � pw


Ox2

2� 1

� � ð6Þ

As an example, the Fig. 2 shows the stress distribution in a generic wire-wound vessel. It can be observed as the effect ofthe initial compressive stresses is to reduce the mean stress level of the circumferential stress component in the vessel.

3. Overview of API 579 for crack assessment

The API 579 procedure for evaluating cracks incorporates a Failure Assessment Diagram (FAD) methodology very similarto that in other documents, such as the British Energy R6 approach and the BS 7910 method. They provide a useful tool forthe determination of the allowable final crack in a fatigue crack growth analysis.

=

=

Fig. 2. Circumferential stress distribution for wire-wound vessels.


Three levels of assessment are provided in the API 579-1/ASME FFS-1 [7] for generating the FAD depending on the accu-racy required of the analysis and the information available to generate the diagram. In each case, the FAD is described by acontinuous curve and a cut-off value Lr

max. In all cases the FAD should be constructed from mean stress–strain data.The Level 1 assessment procedure is intended to provide a conservative analysis with a minimum amount of component

information. This level consists of a series of allowable flaw size curves that were generated using the Level 2 assessmentwith conservative input assumptions. Note that the API 579 Level 1 assessment of cracks is completely different than theBS 7910 Level 1 assessment.

The Level 2 assessment procedure in intended to provide a more detailed evaluation that produces results that are lessconservative than those from a Level 1 assessment. In this level assessment more detailed calculations are used in the eval-uation. And finally, the Level 3 assessment procedure is intended to provide the most detailed evaluation and the recom-mended analysis is based on numerical techniques such as the finite element method.

Fig. 3 illustrates the FAD concept. The FAD curve represents the predicted failure locus. If the assessment point falls withinthe curve, it is considered acceptable. The toughness ratio for the structure of interest is computed as:

Kr ¼ KPI þ /KSR

I

� �=Kmat ð7Þ

where KPI is applied stress intensity factor due to primary loads, KSR

I is the stress intensity factor due to secondary and resid-ual stress, / is a plasticity adjustment factor, and Kmat is the fracture toughness.

The load ratio in API 579 is defined as,

Lr ¼ rref =rYS ð8Þ

where rref is the reference stress and rYS is the yield strength. Eq. (8) is identical to the Lr definition in R6 and BS 7910. How-ever, API 579 proposes an alternative definition of the reference stress. Appendix C on the API 579 procedure contains anextensive library of the reference stress values for cracked bodies as well as stress intensity factor solutions [7].

The API 579 Level 2 uses the following FAD equation:

σ

σσ

=

σ

=

Fig. 3. Failure assessement diagram concept.


Kr ¼ ð1� 0:14L2r ½0:3þ 0:7 expð�0:65L6

r Þ� for Lr 6 Lmaxr

Kr ¼ 0 for Lr > Lmaxr

ð9Þ

which is the same as the R6 Option 1 FAD, as well as the one of the available Level 2 FAD expressions in BS 7910. This FAD hasa cut-off at Lmax

r , which is defined as

Lmaxr ¼ 1

21þ rTS

rYS

� �ð10Þ

where rTS is the tensile strength.In order to derive this diagram only the engineering values of the lower yield or 0.2% proof stress and the flow stress need

be known. The diagram provides a reasonable underestimate of the flaw tolerance of the component, but in some cases thedegree of underestimation may be excessive.

The API 579 Level 3 assessment is a more advance analysis. The available options for a Level 3 assessment include:

1. Method A: Level 2 assessment with user-generated partial safety factors or a probabilistic analysis.2. Method B: Material-specific FAD, similar to R6 Option 2.3. Method C: J-based FAD obtained from elastic–plastic finite-element analysis, similar to R6 Option 3.4. Method D: Ductile tearing assessment.5. Method E: Use a recognized assessment procedure, such as R6 or BS 7910.

The Level 3-B provides a Failure Assessment Diagram using the detailed stress–strain data. In this case the diagram is de-scribed by the next equations,

Kr ¼Eeref

LrrYSþ ðLrÞ3rYS

2Eeref

� �for 0:0 < Lr 6 Lmax

r

Kr ¼ 1:0 for Lr ¼ 0ð11Þ

where eref is the true strain obtained from the mean uniaxial tensile stress–strain curve at a true stress, Lr�rYS, and E is Young’smodulus, rYS is the lower yield or 0.2% proof stress. This diagram is suitable for all metals regardless of the stress–strainbehaviour.

It is recommended that the stress–strain curve be accurately defined for, at least, the next ratios of applied stress to yieldstress: r/rYS = 0.7, 0.8, 0.98, 1.0, 1.02, 1.1, 1.2 and intervals of 0.1 up to rTS.

Finally, the Level 3-C curve is based directly on the equivalence of the failure assessment curve to a J-integral analysis

Kr ¼ ðJe=JÞ1=2 for Lr 6 Lmaxr

Kr ¼ 0 for Lr > Lmaxr

ð12Þ

where Je and J are the values of the J-integral obtained for an elastic–plastic analysis and an elastic analysis respectively forthe same load (the load corresponding to the value, Lr).

As an example, Fig. 4 shows a scheme of the FAD curve of a high strength steel, obtained by means of the Level 3B of theAPI 579 procedure.

1000

1400

190

YS

TS

MPa

MPa

E GPa

σσ

===

ε σ σ σ1.000.900.860.800.760.730.520.39

Fig. 4. Failure Assessment Diagram obtained from stress–strain data material.


4. Numerical procedure to obtain the number of cycles for wire-wound vessels

The procedure presented here uses the fracture mechanics approach to describe the crack growth due to fatigue. Thenumber of cycles is predicted by integrating a crack growth rate law. Special features involved on the wire-wound vesseldesign are underlined. A scheme of the procedure is shown on Fig. 5.

4.1. Initial crack

An internal crack in the longitudinal direction of the vessel has been assumed in this study, as is shown in Fig. 6. The typ-ical initial aspect ratio a0/‘0 of 1/3 has been considered. An initial depth of a0 = 0.2 mm is postulated. Other initial crack shapeor sizes can be analyzed during the design stage of the vessel.

4.2. Calculation of the stress intensity factor range

The first step for the fatigue analysis is to calculate the stress intensity factor range at the deepest point and at the surfacepoints. In general, this requires the stress distribution in the wire-wound vessel to be previously calculated.

The residual stresses introduced by the wire winding process have been considered by calculating an equivalent negativestress intensity factor KI,res. The stresses of the internal pressure are considered separately, defining a minimum stress inten-sity factor KI(min) = 0 for P = 0, and KI(max) for the working pressure P = Pw.

As a result, a crack in a vessel without internal pressure has a stress intensity factor of KI(min) + KI,res, and a crack in thevessel under pressure is subjected to a stress intensity factor of KI(max) + KI,res. Then, the stress intensity factor range canbe obtained as:

DKI ¼ ðKIðmaxÞ þ KI;resÞ � ðKIðminÞ þ KI;resÞ ¼ KIðmaxÞ � KIðminÞ ð13Þ

And the stress intensity ratio factor,

Initial crack

0 0

0

/ 1/ 3

0.2

a

a mm

==

Calculation of the stress intensity factor range (Paragraph 6)

( ), (c)I IK a KΔ Δ

Stress analysis (uncracked geometry)

(Paragraph 2)

Structural Integrity Evaluation (FAD)

Number of cycles

[ ]0( ) ( )i m

I

aN

C f R K a

ΔΔ =⋅ ⋅ Δ

Updating the crack

0 0, ,a a a c c c= + Δ = + Δ

NOK

Crack growth control parameter

, ( . . 0.01 )a e g a mmΔ Δ =

Final allowable crack, Number of cycles,

,f fa c

ii

N N= Δ∑

Safe

Failure

win

ding

cylin

der

Fig. 5. Numerical procedure to calculate the number of fatigue cycles.

2= c

t

a

x

Inte

rnal

cyl

inde

r Fig. 6. Internal semi-elliptical crack in the longitudinal direction of the vessel.


R ¼ KIðminÞ þ KI;res

KIðmaxÞ þ KI;resð14Þ

Note that the effect of the wire winding gets to a displacement of the R – ratio to negative values, with no influence on thestress intensity range, DKI.

In this study, for wire-wound pressure vessels, the value of KI(min) = 0. The stress intensity factor values for KI,res and KI(max)

can be obtained using the numerical equations derived on the paragraph 6 of this work.

4.3. Structural integrity evaluation

The maximum value of the stress intensity factor can be found at the deepest point or at the surface points of the crack,depending on the stress levels of the vessel and the actual aspect ratio of the crack. Then,

Kr ¼max½KIðmaxÞða0Þ;KIðmaxÞðc0Þ�=Kmat ð15Þ

In order to obtain the load ratio Lr = rref/rYS the reference stress value rref must be calculated. In this sense, the API 579provides a solution for the present geometry, based on an adjustment of the stress distribution along the crack plane using afourth-order polynomial equation. The reference stress for this geometry can be found in the Appendix C on the ASME-API579 procedure [7].

After that, the structural integrity of the cracked vessel must be evaluated putting the corresponding point (Kr, Lr) on theFailure Assessment Diagram (FAD). If the evaluation point is located into the safety area the fatigue crack goes on growing.

4.4. Number of cycles for a crack advance Da

Once the stress intensity factor range at the deepest point and at the surface point is calculated, an incremental crackadvance Da for the deepest point is imposed (e.g. Da = 0.01 mm). The number of cycles DNi needed to produce this crackadvance is estimated using the propagation law as,

DNi ¼Da

C � ½f ðRÞ � DKIða0Þ�mð16Þ

where a general propagation law has been used, defined as

dadN¼ C � ½f ðRÞ � DKI�m ð17Þ

The Eq. (16) assumes a constant value for KI(a0) thought the integration step defined by the crack advance Da. The erroron the number of cycles obtained for this step depends on the integration step Da, and on the gradient of the stress intensityfactor in the interval Da.


4.5. Updating the crack

The next step is to calculate the new crack size for the next fatigue step simulation. In general, the stress intensity range atthe deepest point and at the surface point, DKI(a0) and DKI(c0), will be different. As a consequence, a different crack advanceis obtained for both positions after a number of cycles DNi. At the deepest point the crack advance is the fixed value Da, andthe crack advance at the surface point, Dc, can be obtained from:

DNi ¼Da

C � ½f ðRÞ � DKIða0Þ�m¼ Dc

C � ½f ðRÞ � DKIðc0Þ�mð18Þ

where,

Dc ¼ Da � DKIðc0ÞDKIða0Þ

� �m

ð19Þ

Note the crack aspect during fatigue crack propagation depends both on the geometry and the crack growth materialparameters. The value of a0 and c0 for the above equation must be updated by their actual values, a = a0 + Da andc = c0 + Dc, thought the crack propagation.

4.6. Final crack and number of design cycles

The procedure defined by the steps 1–5 is continuously repeated until the allowable final crack depth, predicted by FADcurve, is reached. The number of cycles accumulated, from the initial crack to the allowable final crack, defines the fatiguelife of the vessel. To ensure that the interval of crack depth used Da is sufficiently small, the calculation shall be repeatedusing intervals of decreasing size until no significant change in the calculated number of fatigue cycles is obtained.

This procedure allows both the initial aspect ratio of the crack or the initial crack size to be analyzed, and to determinetheir effect on the number of fatigue cycles.

5. Fatigue crack propagation law

For wire-wound vessel design the propagation law plays an important role in the number of design cycles calculated. Thecompressive stress distribution at the inner of the vessel, where the cracks initiates, leads to negative values of the stressintensity ratio factor, R = Kmin/Kmax < 0, that in some cases can be in the order of R 6 �2. As a consequence, the effect of R– ratio can be taken into account in the propagation law.

Many well-known models for the effect of R – ratio have been proposed in the last two decades since Paris [8] proposedtheir well-known crack growth law:

dadN¼ C � ðDKÞm ð20Þ

Elber [9] introduced the concepts of crack closure and the effective stress intensity factor range DKeff as the dominantdriving force for fatigue crack growth. The crack growth rate can be then expressed as:

dadN¼ C � ðDKeff Þm ð21Þ

As a consequence, many effective stress intensity factor range calculation models have been proposed in the literature[9–12]. In general, DKeff can be written as:

DKeff ¼ f ðR; . . .Þ � DK ð22Þ

where f(R, . . . , ) 6 1.0 for R < 0 which is dependent upon the R – ratio, material and geometry, etc.Walker [10] proposed an empirical model based on test data and curve fitting analysis. This model is frequently used to

account for the effects of the stress ratio

dadN¼ C � ½ð1� RÞpDK�m ð23Þ

Kujawski [11] expanded the Walker model for R < 0 by introducing a two-parameter driving force, defined by DK+ andKmax. The effective stress intensity factor range is then written as,

DKeff ¼ ðKmaxÞp � ðDKþÞ1�p ð24Þ

where DK+ is the positive part of the applied SIF range. This model assume that for R < 0, the negative part of DK does notcontribute to the crack growth.


To take into account for the crack closure effects, Huang [12] have recently proposed a different value of the parameter pfor R P 0 or R 6 0, in the form

DKeff ¼ ðKmaxÞp � ðDKþÞ1�p ¼ð1� RÞp � DK para 0 6 R 6 1

ð1� RÞ�1 � DK para � 1 6 R 6 0

(ð25Þ

where p is a constant depended of the material, but the exponent can taken as �1 for negative values of the R – ratio. Thismodel has been confirmed for many of engineering materials.

The ASME code [6] proposes a similar expression, but more conservative, taking the propagation law for R 6 0 as,

dadN¼ C

1:51:5� R

DK� �m

ð26Þ

The predicted number of cycles using different models can be significant. For example, considering an steel with a typicalParis law exponent of m = 3.0, and subjected to a load ratio of R = �1.0, the number of cycles predicted by the ASME code is1.73 times lower than the predicted by the Huang model one. And taking a value of R = �2.0 the difference between useASME or Huang model is increased to 2.12 times (approximately the half life predicted by the use of ASME code). If we alsoconsider that ASME code divides by a safety factor of two the calculated number of cycles, the fatigue life predicted can befour times lower than the corresponding one calculated by other models.

6. Method for determining stress intensity factor

Methods for calculating the stress intensity factors for several geometries in typical high pressure vessel are given in theAppendix D of the ASME code, or in Appendix C of API 579-1/ASME FFS-1.

In general, the commonly used method to calculate the stress intensity factor for internal semi-elliptical cracks in cylin-ders subjected to internal pressure, as the shown in Fig. 6, is based on the calculation of the stress distribution along thethickness, and then to fit it using a third-order polynomial equation as follows,

r ¼ A0 þ A1 � ðx=aÞ þ A2 � ðx=aÞ2 þ A3 � ðx=aÞ3 ð27Þ

where A0, A1, A2 and A3 are the coefficients of the polynomial equation, a is the crack depth, and x is referred as the distancethought the wall measured from the flawed surface, as is shown in Fig. 6.

If the distribution of stresses normal to the crack surface can be accurately represented by a single equation on the form ofEq. (27) over the entire range of crack depths of interest, an alternate method can be used to compute KI over this crackdepth. The stress distribution can be obtained as:

r ¼ A00 þ A01 � ðx=tÞ þ A02 � ðx=tÞ2 þ A03 � ðx=tÞ3 ð28Þ

For each value of a/t the values of A0i are converted to Ai values as,

A0 ¼ A00A1 ¼ A01 � ða=tÞA2 ¼ A02 � ða=tÞ2

A3 ¼ A03 � ða=tÞ3

ð29Þ

The stress intensity factor can be then calculated as:

KI ¼ffiffiffiffiffiffipaQ

r� ½G0ðA0 þ ApÞ þ G1A1aþ G2A2a2 þ G3A3a3� ð30Þ

where A0, A1, A2, A3 are the coefficients from Eq. (27), Ap is the internal pressure of the vessel if the pressure can acts on thecrack surfaces, G0, G1, G2, G3, are the free surface correction factors, and Q is the shape factor for an elliptical shape defined as,

Q ¼ 1þ 1:493ac

� �1:65for 0 6 a=c 6 1 ð31Þ

Each Gj has been obtained from the appropriate finite-element analysis by several authors, as Newman and Raju [13], andthey are usually provided in a tabulate form for different values of the crack shape, a/2c, and the crack size, a/t. Differentvalues must be used to compute the stress intensity factor at the deepest point or at the free surface. Usually the coefficientsnecessary to obtain the stress intensity factors at the deepest point and at the corner points are provided. These tabulatedGi-coefficients, or other similar, are the provided in the ASME code or in the API 579 procedure.


In this work the tabulated Gi-coefficients have been collected in closed form equations, allowing the numerical integra-tion of the propagation law to be more easily accomplished. The general expressions to be used for the calculation of thestress intensity factors are:

Gi¼0;1;2;3 ¼

Y0 þ 10 � ðY1 � Y0Þ a2c

for 0:0 < a

2c 6 0:1Y1 þ 10 � ðY2 � Y1Þ a

2c � 0:1

for 0:1 6 a2c 6 0:2

Y2 þ 10 � ðY3 � Y2Þ a2c � 0:2

for 0:2 6 a2c 6 0:3

Y3 þ 10 � ðY4 � Y3Þ a2c � 0:3

for 0:3 6 a2c 6 0:4

Y4 þ 10 � ðY5 � Y4Þ a2c � 0:4

for 0:4 6 a2c 6 0:5

8>>>>>><>>>>>>:

ð32Þ

where the value of the coefficients Yi can be obtained from Eqs. (33)–(36), for the crack surface points, and using the Eqs.(37)–(40) for the deepest point of the crack.

Go for KI;extremos

Y1 ¼ 0:657302 at

� �4 � 0:242102 at

� �3 þ 0:70961 at

� �2 þ 0:08712 at

� �þ 0:5452

Y2 ¼ 0:299912 at

� �4 � 0:049701 at

� �3 þ 0:64204 at

� �2 þ 0:07912 at

� �þ 0:7493

Y3 ¼ 0:135121 at

� �4 þ 0:005769 at

� �3 þ 0:59022 at

� �2 þ 0:06077 at

� �þ 0:9024

Y4 ¼ 0:051713 at

� �4 þ 0:006835 at

� �3 þ 0:54750 at

� �2 þ 0:03913 at

� �þ 1:0297

Y5 ¼ 0:001790 at

� �4 � 0:003425 at

� �3 þ 0:50321 at

� �2 þ 0:01709 at

� �þ 1:1406

8>>>>>>>>><>>>>>>>>>:

ð33Þ

G1 for KI;extremos

Y1 ¼ 0:320217 at

� �4 � 0:183483 at

� �3 þ 0:223074 at

� �2 þ 0:023451 at

� �þ 0:07259

Y2 ¼ 0:131695 at

� �4 � 0:046368 at

� �3 þ 0:150275 at

� �2 þ 0:065207 at

� �þ 0:10384

Y3 ¼ 0:056900 at

� �4 � 0:010124 at

� �3 þ 0:101266 at

� �2 þ 0:097197 at

� �þ 0:12799

Y4 ¼ 0:017104 at

� �4 þ 0:003203 at

� �3 þ 0:056165 at

� �2 þ 0:125117 at

� �þ 0:14841

Y5 ¼ 0:001342 at

� �4 � 0:001063 at

� �3 þ 0:016023 at

� �2 þ 0:149695 at

� �þ 0:16649

8>>>>>>>>><>>>>>>>>>:

ð34Þ

G2 for KI;extremos

Y1 ¼ 0:174216 at

� �4 � 0:102528 at

� �3 þ 0:103471 at

� �2 þ 0:011365 at

� �þ 0:02548

Y2 ¼ 0:075771 at

� �4 � 0:033684 at

� �3 þ 0:062586 at

� �2 þ 0:041126 at

� �þ 0:03444

Y3 ¼ 0:034858 at

� �4 � 0:013955 at

� �3 þ 0:033316 at

� �2 þ 0:064694 at

� �þ 0:04229

Y4 ¼ 0:010695 at

� �4 � 0:001067 at

� �3 þ 0:001861 at

� �2 þ 0:086597 at

� �þ 0:04949

Y5 ¼ �0:00109 at

� �4 þ 0:001749 at

� �3 � 0:025150 at

� �2 þ 0:105703 at

� �þ 0:05629

8>>>>>>>>><>>>>>>>>>:

ð35Þ

G3 for KI;extremos

Y1 ¼ 0:115045 at

� �4 � 0:074310 at

� �3 þ 0:063729 at

� �2 þ 0:006054 at

� �þ 0:01256

Y2 ¼ 0:049427 at

� �4 � 0:024738 at

� �3 þ 0:034269 at

� �2 þ 0:027114 at

� �þ 0:01578

Y3 ¼ 0:025587 at

� �4 � 0:015740 at

� �3 þ 0:016600 at

� �2 þ 0:043720 at

� �þ 0:01918

Y4 ¼ 0:003854 at

� �4 þ 0:004351 at

� �3 � 0:010507 at

� �2 þ 0:060288 at

� �þ 0:02259

Y5 ¼ 0:001255 at

� �4 � 0:002357 at

� �3 � 0:023994 at

� �2 þ 0:073333 at

� �þ 0:02609

8>>>>>>>>><>>>>>>>>>:

ð36Þ

G0 for KI;centro

Y0 ¼ 121:00898 at

� �4 � 134:73674 at

� �3 þ 53:8910 at

� �2 � 5:485392 at

� �þ 1:22121

Y1 ¼ �1:399094 at

� �4 � 0:182952 at

� �3 þ 2:15566 at

� �2 � 0:024252 at

� �þ 1:09650

Y2 ¼ �0:725037 at

� �4 þ 0:006716 at

� �3 þ 0:91244 at

� �2 þ 0:000082 at

� �þ 1:08561

Y3 ¼ �0:433628 at

� �4 þ 0:002685 at

� �3 þ 0:51853 at

� �2 þ 0:000203 at

� �þ 1:07271

Y4 ¼ �0:323452 at

� �4 � 0:080372 at

� �3 þ 0:48237 at

� �2 � 0:001379 at

� �þ 1:05634

Y5 ¼ �0:304319 at

� �4 � 0:005642 at

� �3 þ 0:306019 at

� �2 � 0:000499 at

� �þ 1:03659

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð37Þ


G1 for KI;centro

Y0 ¼ 41:80340 at

� �4 � 46:70392 at

� �3 þ 19:05639 at

� �2 � 2:075257þ 0:79655

Y1 ¼ �0:16052 at

� �4 � 0:300104 at

� �3 þ 0:775207 at

� �2 � 0:014967þ 0:66376

Y2 ¼ �0:14931 at

� �4 þ 0:001622 at

� �3 þ 0:287479 at

� �2 þ 0:000048þ 0:68260

Y3 ¼ �0:01597 at

� �4 þ 0:010961 at

� �3 þ 0:115395 at

� �2 þ 0:000616þ 0:70189

Y4 ¼ 0:037944 at

� �4 � 0:004449 at

� �3 þ 0:075087 at

� �2 � 0:000679þ 0:72142

Y5 ¼ �0:00653 at

� �4 þ 0:001969 at

� �3 þ 0:061521 at

� �2 þ 0:000476þ 0:74111

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð38Þ

G2 for KI;centro

Y0 ¼ 21:6358 at

� �4 � 24:24902 at

� �3 þ 10:07470 at

� �2 � 1:160662 at

� �þ 0:61850

Y1 ¼ �0:14253 at

� �4 þ 0:00275 at

� �3 þ 0:312871 at

� �2 þ 0:000325 at

� �þ 0:50779

Y2 ¼ �0:02125 at

� �4 � 0:01117 at

� �3 þ 0:141070 at

� �2 � 0:000684 at

� �þ 0:53099

Y3 ¼ 0:05948 at

� �4 þ 0:005773 at

� �3 þ 0:030759 at

� �2 þ 0:000634 at

� �þ 0:55559

Y4 ¼ 0:09914 at

� �4 þ 0:000476 at

� �3 � 0:003807 at

� �2 þ 0:000101 at

� �þ 0:58150

Y5 ¼ 0:04524 at

� �4 � 0:000179 at

� �3 þ 0:013343 at

� �2 � 0:000343 at

� �þ 0:60839

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð39Þ

G3 for KI;centro

Y0 ¼ 13:41269 at

� �4 � 15:073303 at

� �3 þ 6:366365 at

� �2 � 0:769575 at

� �þ 0:51684

Y1 ¼ �0:04979 at

� �4 � 0:003402 at

� �3 þ 0:182528 at

� �2 � 0:000325 at

� �þ 0:42459

Y2 ¼ 0:008699 at

� �4 þ 0:003878 at

� �3 þ 0:073959 at

� �2 þ 0:000512 at

� �þ 0:44800

Y3 ¼ 0:079331 at

� �4 � 0:000799 at

� �3 þ 0:004348 at

� �2 þ 0:000118 at

� �þ 0:47353

Y4 ¼ 0:105673 at

� �4 þ 0:004588 at

� �3 � 0:028071 at

� �2 þ 0:000395 at

� �þ 0:50062

Y5 ¼ 0:017883 at

� �4 þ 0:049532 at

� �3 � 0:022733 at

� �2 þ 0:002046 at

� �þ 0:52897

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð40Þ

7. Conclusions

In the present work the methodology used to obtain the number of design cycles for wire-wound vessels has been widelyexplained. The API 579 FFS assessment procedure has been used in order to obtain the allowable final crack.

Moreover, the aspect ratio of the crack is continuously updated during crack propagation by calculating the stress inten-sity factor at the deepest point and at the surface points. This procedure is more time-consuming but more accuracy for thecalculated number of fatigue cycles is expected. To facilitate the numerical integration of the propagation law, a set of equa-tions for the calculation of the stress intensity factors is provided, that avoids the tabulated values usually available in thedesign codes to be used.

Different propagation laws, that take into account the effect of negative R – ratios, have been reviewed. This is an impor-tant topic for wire winding vessel design, because R < �2 values can be easily reached during winding operation. The ASMEcode expression for the propagation law can be highly conservative when is compared to other accepted models available inthe literature. As a consequence, experimental fatigue crack growth tests at different R – ratios, to get a most accurate def-inition of the f(R) function, can be useful for wire-wound pressure vessels design.

Acknowledgements

The authors wish to record their thanks to JCyL (Project No. BU012A08) and NC-HYPERBARIC for sponsoring this researchProject.

References

[1] Gao XL. An exact elasto-plastic solution for an open-ended thick-walled cylinder of a strain-hardening material. Int J Pres Ves Pip 1992;52(1):129–44.[2] Majzoobi GH, Farrahi GH, Mahmoudi AH. A finite element simulation and an experimental study of autofrettage for strain hardened thick-walled

cylinders. Mater Sci Eng A 2003;359(1–2):326–31.[3] Alegre J, Bravo P, Preciado M. Design of an autofrettaged high-pressure vessel, considering the Bauschinger effect. J Proc Mech Eng. Proceedings of the

institution of mechanical engineers, part E 2006;220(1):7–16.[4] Alegre JM et al. Simulation procedure of high pressure vessels using the wire winding technique. Eng Fail Anal 2010;17(1):61–9.[5] Alegre JM, Bravo P, Preciado M. Fatigue behaviour of an autofrettaged high-pressure vessel for the food industry. Eng Fail Anal 2007;14(2):396–407.


[6] ASME boiler and pressure vessel code: alternative rules for construction of high pressure vessels ASME, S.V. Division 3; 2007.[7] ASME. API 579-1/ASME FFS-1, in fitness-for-service. Am Soc Mech Eng; 2007.[8] Paris PFE. A critical analysis of crack propagation laws. J Basic Eng. Trans Am Soc Mech Eng 1963:528–34.[9] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2(1):37–44.

[10] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T 3 and 7075-T 6 aluminium. Effects of environments and complexand complex load history on fatigue life. ASTM STP 462 1970;2:1–14.

[11] Kujawski D. A new ([Delta]K+Kmax)0.5 driving force parameter for crack growth in aluminum alloys. Int J Fatigue 2001;23(8):733–40.[12] Huang X, Moan T. Improved modeling of the effect of R-ratio on crack growth rate. Int J Fatigue 2007;29(4):591–602.[13] Newman JC, Raju IS. Stress intensity factor for internal and external surface cracks in cylindrical vessels. J Pres Ves Technol 1982;104:293–8.

Fatigue design of wire-wound pressure vessels using ASME-API 579 procedure

Documents

Transcript of Fatigue design of wire-wound pressure vessels using ASME-API 579 procedure