ap d

D-1

APPENDIX D Compendium of Reference Stress Solutions(Jan, 2000)

D.1 General

D.1.1 Overview

D.1.1.1 This appendix contains reference stress solutions for many crack geometries which are likely to occurin pressurized components. Reference stress solutions are used in the assessment of crack-likeflaws, see Section 9.

D.1.1.2 A summary of the reference stress solutions in this appendix is contained in Table C.1 of Appendix C.These reference stress solutions are recommended for most applications based on consideration ofaccuracy, range of applicability and convenience.

D.1.1.3 Reference stress solutions not included in this appendix may be obtained from publications (forexample, see references [D.14.1] and [D.14.2]) if the tabulated solutions correspond to thecomponent and crack geometry, and the loading condition. Otherwise, the reference stress shouldbe computed using a numerical approach such as the finite element method.

D.1.1.4 The reference stress solutions for plates can be used to approximate the solutions for cylinders andspheres by introducing a surface correction (Folias or bulging) factor. This is an approximation that issupported by experimental results.

D.1.1.5 An identifier has been assigned to each reference stress solution in this appendix (see Table C.1 ofAppendix C). This identifier is a set of alpha-numeric characters that uniquely identifies thecomponent geometry, crack geometry, and loading condition. The identifier can be used todetermine the associated stress intensity factor solution to be used in an assessment of crack likeflaws (see Section 9). For example, if a flat plate with a through-wall crack subject to a membranestress and/or bending stress is being evaluated, the reference stress solution is RPTC and theassociated stress intensity factor solution to be used is KPTC.

D.1.2 Symbol Definitions

D.1.2.1 The following symbols defined below are used in this appendix.

a = Crack depth parameter (mm:in),A = Cross-sectional area of the flaw (mm2:in2),Ao = Cross-sectional area of the component computed for the flaw length (mm

2:in2),c = Crack length parameter (mm:in),dn = Mean nozzle diameter (see Figure C.26) (mm:in),d1 = Distance from plate surface to the center of an embedded elliptical crack (see

Appendix C, Figure C.3) (mm:in),F = Net section axial force acting on a cylinder (N:lbs),M = Resultant net-section bending moment acting on a cylinder (N-mm:in-lbs),Ms = Surface correction factor for a surface crack,Mt = Surface correction factor for a through-wall crack,p = Pressure (MPa:psi),Pij = Primary stress component being evaluated,Pij m, = Equivalent primary membrane stress for a stress component,Pij b, = Equivalent primary bending stress for a stress component,

D-2 API RECOMMENDED PRACTICE 579 Jan, 2000_________________________________________________________________________________________________

Pl = Generalized loading parameter, such as applied stress, bending moment or pressure,Ply = Value of the generalized loading parameter evaluated for the component with a crack-

like flaw at the yield stress,Pm = Primary membrane stress component (MPa:psi),Pb = Through-Wall primary bending stress component (MPa:psi),P0 = Uniform coefficient for polynomial primary stress distribution (MPa:psi),P1 = Linear coefficient for polynomial primary stress distribution (MPa:psi),P2 = Quadratic coefficient for polynomial primary stress distribution (MPa:psi),P3 = Third order coefficient for polynomial primary stress distribution (MPa:psi),P4 = Fourth order coefficient for polynomial primary stress distribution (MPa:psi),P5 = Net-section primary bending stress about the x-axis (MPa:psi),P6 = Net-section primary bending stress about the y-axis (MPa:psi),t = Plate or shell thickness (mm:in),tn = Nozzle thickness (see Appendix C, Figure C.26) (mm:in),Ri = Cylinder inside radius (mm:in),Rm = Cylinder mean radius (mm:in),Ro = Cylinder, round bar, or bolt outside radius, as applicable (mm:in),Rth = Root Radius of a threaded bolt (mm:in),x = Radial local coordinate originating at the internal surface of the component,xg = Global coordinate for definition of net section bending moment about the x-axis,yg = Global coordinate for definition of net section bending moment about the y-axis,W = Distance from the center of the flaw to the free edge of the plate (mm:in), = Shell parameter used to determine the surface correction factors, = Half-angle of the crack (degrees), ref = Reference stress (MPa:psi), and

ys = Yield stress (MPa:psi), see Appendix F.

D.1.2.2 The above symbols are also defined for different component and crack geometrys in Appendix C,Figures C.1 through C.32.

D.2 Stress Analysis

D.2.1 Overview

D.2.1.1 A stress analysis using handbook or numerical techniques is required to compute the state of stressat the location of a crack. The stress distribution to be utilized in determining the stress intensityfactor is based on the component of stress normal to the crack face. The distribution may be linear(made up of membrane and/or bending distributions) or highly nonlinear based on the componentgeometry and loading conditions.

D.2.1.2 The stress distribution normal to the crack face resulting from primary loads should be determinedbased on service loading conditions and the uncracked component geometry. If the component issubject to different operating conditions, the stress distribution for each condition should be evaluatedand a separate fitness-for-service assessment should be performed.

D.2.1.3 In this appendix, the variable P is used for to signify that stress calculations and the resultingstress distributions used to determine the reference stress and the Lr ratio for the assessment of acrack-like flaw using the FAD (see Section 9) are categorized as primary stress (see Appendix B).

Jan, 2000 RECOMMENDED PRACTICE FOR FITNESS-FOR-SERVICE D-3_________________________________________________________________________________________________

The reference stress based on the secondary and residual stress distributions is required todetermine the plasticity interaction factor, , used in the assessment of crack-like flaws (see Section9). In this case, the variable variable P can be used to represent the primary and/or residual stress.

D.2.2 Stress Distributions

D.2.2.1 Overview The reference stress solutions in this appendix are formulated in terms of the coefficientsof a linear stress distribution (membrane and bending stress). Therefore, it is necessary to derivethese coefficients from the results obtained from a stress analysis.

D.2.2.2 General Stress Distribution A stress distribution through the wall thickness at the location of acrack-like flaw can be determined using an elastic solution or a numerical analysis technique such asthe finite element method. In some cases, the stress distribution normal to the crack face may behighly non-linear. Statically equivalent membrane and bending stress components can bedetermined from the general stress distribution using the following equations; the integration isperformed along a line assuming a unit width, see Appendix B.

Pt

P dxij m ijt

, z1 0 (D.1)P

tP t x dxij b ij

t

, FHGIKJz6 22 0 (D.2)

D.2.2.3 Fourth Order Polynomial Stress Distribution The fourth order polynomial stress distribution can beobtained by curve-fitting a general stress distribution to obtain the coefficients of the best-fit fourthorder polynomial. The equivalent membrane and bending stress distributions for use in the referencestress solutions in this appendix can be obtained directly from the coefficients of this polynomial.

a. The general form of the fourth order polynomial stress distribution is as follows:

P x P P xt

P xt

P xt

P xto

( ) FHGIKJ FHGIKJ FHGIKJ FHGIKJ1 2

2

3

3

4

4

(D.3)

b. The equivalent membrane and bending stress distributions for the fourth order polynomialstress distribution are:

P P P P P Pm 0 1 2 3 42 3 4 5(D.4)

P P P P Pb 1 2 3 42 2920

615

(D.5)

D.2.2.4 Fourth Order Polynomial Stress Distribution With Net Section Bending Stress This distribution isused to represent a through-wall fourth order polynomial stress and a net section or global bendingstress applied to a circumferential crack in a cylindrical shell.

P x x y P P xt

P xt

P xt

P xt

Px

R tP

yR t

g g o

g

i

g

i

( , , ) FHGIKJ FHGIKJ FHGIKJ FHGIKJ

FHGIKJ FHGIKJ

1 2

2

3

3

4

4

5 6

(D.6)


D.2.2.5 Membrane and Through-Wall Bending Stress Distribution The membrane and bending stressdistribution is linear through the wall thickness and represents a common subset of the general stressdistribution (see paragraph D.2.2.2). Attributes of this stress distribution are discussed in AppendixC, paragraph C.2.2.5. The components of this stress distribution can be used directly in thereference stress solutions in this appendix.

D.2.3 Surface Correction Factor

D.2.3.1 A surface correction (also referred to as the Folias or bulging factor) is used to quantify the localincrease in the state of stress at the location of a crack in a shell type structure which occurs becauseof local bulging. The magnified state of stress is then used together with a reference stress solutionfor a plate with a similar crack geometry to determine the reference stress for the shell. Surfacecorrection factors are typically only applied to the membrane part of the reference stress because thisrepresents the dominant part of the solution.

D.2.3.2 The surface correction factors for through-wall cracks in cylindrical and spherical shells subject tomembrane stress loading are normally defined in terms of a single shell parameter, , given by thefollowing equation:

1818. c

R ti(D.7)

However, recent work indicates that the surface correction factors for cylindrical shells are also afunction of the shell radius-to-thickness ratio [D.14.9].

a. Cylindrical shell Longitudinal through-wall crack

1. Data fit from references [D.14.10] and [D.14.11] (recommended for use in allassessments):

Mt

FHG

IKJ

102 0 4411 0 00612410 0 02642 1533 10

2 4

2 6 4

0 5. . .

. . . ( )

.

(D.8)

2. Approximate expression from references [D.14.12] and [D.14.13]:

M fort 1 0 3797 0 001236 9 12 4 0 5. . .

. c h (D.9)

M fort 0 01936 33 9 12. . . (D.10)

3. Upper bound expression from reference [D.14.14]:

Mt 1 0 48452 0 5.

.c h (D.11)

4. General expression for membrane stress loading is given by the following equationwhere the coefficients Amm and Amb can be calculated using the equations in AppendixC, paragraph C.5.1. This expression is considered to be the most accurate and itincludes an R ti ratio dependency which can be significant.

M A A A At mm mb mm mb max ,b g b g (D.12)b. Cylindrical shell Circumferential through-wall crack


1. Data fit from reference [D.14.15] ] (recommended for use in all assessments):

Mt

FHG

IKJ

10078 010368 37894 1010 0 021979 15742 10

2 4 4

2 6 4

0 5. . . ( )

. . . ( )

.

(D.13)

2. General expression for membrane stress loading is given by the following equationwhere the coefficients Amm and Amb can be calculated using the equations in AppendixC, paragraph C.5.2.

M A A A At mm mb mm mb max ,b g b g (D.14)c. Spheres Circumferential through-wall crack

1. Data fit from references [D.14.10] and [D.14.11] (recommended for use in allassessments):

Mt

10005 0 49001 0 3240910 050144 0 011067

2

2

. . .. . .

(D.15)

2. Approximate expression [D.14.16]:

Mt 1 0 427 0 006662 3 0 5. .

. c h (D.16)

3. General expression for membrane stress loading is given by the following equationwhere the coefficients Amm and Amb can be calculated using the equations in AppendixC, paragraph C.6.1.

M A A A At mm mb mm mb max ,b g b g (D.17)D.2.3.3 The surface correction factors for surface cracks can be approximated using the results obtained for

a through-wall crack by using one of the following methods. In all of these methods, the equations forMt are provided in paragraph D.2.3.2.

a. Cylindrical or Spherical Shell The following is an empirical equation which does not produceconsistent results when the crack approaches a through-wall configuration, see reference[D.14.14]. The factor C in the equation is used to define a model for the cross sectional areaof the surface crack to be included in the analysis. A value of C 10. corresponds to arectangular model and a value of C 0 67. is used to model a parabolic shape. Experimentalresults indicate that a value of C 085. provides an optimum fit to experimental data [D.14. 7],[D.14.8]. The results from this equation are usually associated with a local limit load solution;the superscript L in the following equation designates a local limit load solution.

MC a

t M

C at

sL t

FHGIKJFHGIKJ

FHGIKJ

1 1

1(D.18)


b. Cylindrical or Spherical Shell This equation is based on a lower bound limit load solution andproduces a consistent result as the crack approaches a through-wall configuration, seereference [D.14.17].

1. In the following equation, the term Mt c( ) signifies that Mt is evaluated using theequations cited for a through-wall crack with the a shell parameter as opposed to the shell parameter (compare Equation (D.7) with Equation (D.20)). The results from thisequation are usually associated with a net section limit load solution; the superscript NSin the following equation designates a net section limit load solution.

.

Mat

at M

sNS

t a

FHG

IKJ

1

1 1( )

(D.19)

where,

ai

cR a

1818.(D.20)

2. In reference [D.14.17], the crack area is idealized as an equivalent rectangle with a areaequal to the elliptical crack area. In this appendix, this approximation is not used and thearea chosen to evaluate Mt is a rectangular area based on the component thicknessand the full length of the crack. If desired, the equivalent elliptical area approximationcan be introduced into the assessment by multiplying Equation (D.20) by 4 .

3. Equation (D.19) is written in terms of the component thickness and maximum depth ofthe flaw. If the flaw shape is characterized by a nonuniform thickness profile, Equation(D.19) can be written in terms of areas as follows:

MAA

AA M

sNS

o o t a

FHG

IKJ

1

1 1( )

(D.21)

c. The results from equations (D.18) and (D.19) are approximately the same for flaws up toa t 05. . Above this value, the use of Equation (D.18) to compute Ms will produce valueswhich significantly exceed those obtained using Equation (D.19). This will result inconservatism in the computation of the stress intensity ratio ( Kr ), if the stress intensity factoris a function of Ms , and the load ratio ( Lr ) in the FAD assessment for a given materialtoughness and yield stress. Experimental results indicate that Equation (D.19) producesconsistent results for a t 0 5. . Therefore, Equation (D.19) is recommended for use tocompute the stress intensity factor (numerator in Kr ) factor and reference stress (numeratorin Lr) unless additional conservatism is desired in the assessment. In summary, the followingvalues can be used to compute the surface correction factor:

M M assessment based on local ligament criterias sL

(D.22)

M M assessment based on net section collapse recommendeds sNS

b g (D.23)


D.2.4 Load Ratio and Reference Stress

D.2.4.1 The load ratio is the horizontal coordinate on the failure assessment diagram (see Section 9), and isdefined as

L PPr

l

ly

(D.24)

D.2.4.2 Alternatively, the load ratio can be written in terms of a reference stress:

Lrref

ys

(D.25)

with,

refl

lyys

PP

FHGIKJ (D.26)

D.2.4.3 This Appendix contains reference stress solutions for selected configurations. The solutions inparagraph D.2.4.2 can be converted into a yield load solution by rearranging equation (D.26). Thelimit load can be inferred by replacing the yield strength with an appropriate flow stress, see AppendixF.

D.2.5 Plastic Collapse In The Assessment Of Crack-Like Flaws

D.2.5.1 The position of an assessment point K Lr r,b g on the FAD represents a particular combination offlaw size, stresses and material properties. This point can be used to demonstrate whether the flawis acceptable and an associated in-service margin can be computed based on the location of thispoint. If the flaw is unacceptable, the location of the assessment point on the FAD can indicate thetype of failure which would be expected.

a. The failure assessment diagram can be divided into three zones as illustrated in Figure D.1. Ifthe assessment point lies in Zone 1, the predicted failure mode is predominantly fracturecontrolled and could be associated with brittle fracture. If the assessment point lies in Zone 3,the predicted failure mode is collapse controlled with extensive yielding resulting in largedeformations in the component. If the assessment point lies in a Zone 2 the predicted failuremode is elastic plastic fracture.

b. The significance of the Lr parameter in a FAD assessment can be described in terms ofcrack-tip plasticity. If fracture occurs under elastic plastic conditions, the Kr value defined bythe failure assessment line at the corresponding Lr value represents the elastic component ofthe crack driving force. The limiting value of Kr reduces from unity as Lr increases. Thus1 Krb g represents the enhancement of the crack driving force due to plasticity. Therefore,

the value of the Lr parameter represents a measure of the crack tip plasticity as long as theLr parameter is less than the maximum permitted or cut-of value (see paragraph D.2.5.2.b).

D.2.5.2 The value of Lr depends on the type of plastic collapse load solution utilized in the assessment.

a. Plastic collapse solutions can be defined in three ways:


1. Local Collapse Plastic collapse of the remaining ligament adjacent to the flaw beingassessed. The reference stress solutions shown for plates in paragraphs D.3 and D.4are based on a local collapse solutions. The reference stress solutions shown forcylinders and spheres which utilize the plate ligament equations (see paragraph D.3)with a surface correction factor, Ms , based on a local limit load (see paragraphs D.2.3.3and C.2.3.3 of Appendix C) are also considered to be local collapse solutions.

2. Net Section Collapse Plastic collapse of the structural section containing the flaw. Thereference stress solutions shown for cylinders and spheres which do not utilize the plateligament formulas of paragraph D.3 are considered to be net section collapse solutions.In addition, the reference stress solutions shown for cylinders and spheres which utilizethe plate ligament equations (see paragraph D.3) with a surface correction factor, Ms ,based on a global limit load (see paragraphs D.2.3.3) are also considered to be netsection collapse solutions. The reference stress solutions for bars and bolts inparagraphs D.11 are net section collapse solutions.

3. Gross Collapse Plastic collapse of the structure by unconstrained or gross strainingthroughout the structure. This occurs when a plastic collapse mechanism is formed inthe structure and may be unaffected by the presence of the crack.

b. It is acceptable to use the local plastic collapse solution to determine the reference stresswhen computing the value of Lr . However, this may be excessively conservative forredundant structures. If the structure or component has degrees of redundancy, plasticity atthe cracked ligament may be contained by the surrounding structure until conditions for grosscollapse are reached. In such cases, it may be possible to use more appropriate estimatesof Lr based on modified lower bound collapse solutions which are based on the response ofthe entire structure. For this approach to be adopted, it is essential to confirm by analysis thatthe plasticity at the cracked section is contained sufficiently by the remaining structure, so thatthe use of the standard assessment diagram gives conservative results. In ferritic steels, caremust also be exercised to ensure that local constraint conditions are not sufficient to inducebrittle fracture by a cleavage mechanism. Where global collapse can be shown to occur afterthe attainment of Lr maxb g the Lr cut offb g can be extended to the value relating to global collapseas described.

c. If the assessment point falls outside the acceptable region, then recategorization of the flawbeing evaluated can be undertaken and a reassessment made (see Section 9). In general, therecategorization procedures described in Section 9 will only be effective if the assessmentpoint falls within the elastic plastic fracture controlled zone or beyond Lr maxb g (in the collapsecontrolled zone).

D.2.5.3 The reference stress solutions in this appendix are based on the assessment of a single flaw.Multiple flaws which interact should be recategorized according to Section 9. However, multiple flawswhich do not interact according to Section 9 may still effect the plastic collapse conditions, andallowances should be made to the collapse solutions to accommodate these effects.

D.2.5.4 It is recommended that a gross collapse assessment be performed to ensure that the appliedstresses derived for local conditions do not cause failure of the structure in other regions.

a. In many cases a simple calculation can be performed to identify the highest applied stresscondition which will result in the attainment of the flow strength on a significant cross section.In certain structures, gross collapse may occur in regions away from the flaw being assessedbecause of thinned areas, or where design conditions cause yielding of the general structureprior to collapse of the local regions.

b. To facilitate understanding of the relative importance of local, net section and gross collapseloads, it is useful to calculate the minimum collapse load for regions away from the cracked


section, as well as that involving the cracked section and determining the Lr parameter forboth conditions. The minimum ratio of the gross collapse load for regions away from thecracked section to the local or net section collapse load at the cracked section represents amaximum value or cut off on the Lr -axis. The cut off limit may be less than one and in suchcases the assessment diagram is effectively restricted by this cut-off. The failure assessmentdiagram is generally limited at higher values of Lr to a cut-off at Lr maxb g which is based onmaterial properties rather than structural behavior. In displacement controlled applications, theassessment diagram may be extended beyond the Lr maxb g limit to the structural cut off limit.

D.3 Reference Stress Solutions For Plates

D.3.1 Plate Through-Wall Crack, Through-Wall Membrane And Bending Stress (RPTC)

D.3.1.1 The Reference Stress is [D.14.3]:

ref

b b mP P P

2 2 0 593 1c hb g

.

(D.27)

where,

cW

(D.28)

D.3.1.2 Notes:

a. See Figure C.1 for the component and crack geometry.

b. See paragraph D.2.2.3 for determination of Pm and Pb .

D.3.2 Plate Surface Crack, Infinite Length, Through-Wall Fourth Order Polynomial Stress Distribution(RPSCL1)

D.3.2.1 The Reference Stress is given by Equation (D.31) with the following definition of :

at

(D.29)

D.3.2.2 Notes:

a. See Figure C.2(b) for the component and crack geometry.


D.3.3 Plate Surface Crack, Infinite Length, Through-wall Arbitrary Stress Distribution (RPSCL2)


D.3.3.1 The Reference Stress in paragrapgh D.3.2 can be used.

D.3.3.2 Notes: see paragraph D.3.2.2.

D.3.4 Plate Surface Crack, Semi-Elliptical Shape, Through-wall Membrane And Bending Stress(RPSCE1)

D.3.4.1 The Reference Stress is [D.14.3], [D.14.18]:

With bending restraint:

ref

b b mgP gP P

b g b gb g2 2 2

0 5

2

9 1

3 1

.

(D.30)

With negligible bending restraint (e.g. pin-jointed):

ref

b m b m mP P P P P

3 3 9 1

3 1

2 2 20 5

2

b g b gb g

.

(D.31)

where

g ac

FHGIKJ1 20 2

0 753

.

(D.32)

at

tc

for W c t1

b g (D.33)

FHGIKJFHGIKJ

at

cW

for W c tb g (D.34)

D.3.4.2 Notes:

a. See Figure C.2(a) for the component and crack geometry.


c. The normal bending restraint solution can be obtained by setting g 10. [D.14.18].

d. If a c , compute g based on a c2 0 5 . .

D.3.5 Plate Surface Cracks, Semi-Elliptical Shape, Through-Wall Fourth Order Polynomial StressDistribution (RPSCE2)


D.3.5.1 The Reference Stress in paragraph D.3.4 can be used.

D.3.5.2 Notes: see paragraph D.3.4.

D.3.6 Plate Surface Crack, Semi-Elliptical Shape, Through-wall Arbitrary Stress Distribution (RPSCE3)


D.3.6.2 Notes: see paragraph D.3.4.

D.3.7 Plate Embedded Crack, Infinite Length, Through-Wall Fourth Order Polynomial Stress Distribution(RPECL)


ref

b m b m mP P P P Pdt

dt

RST

UVWLNM

OQP

LNM

OQP

3 3 9 1 4

3 1 4

2 2 20 5

2

b g b g

b g

.

(D.35)

where,

d d a 1 (D.36)

2at

(D.37)

D.3.7.2 Notes:

a. See Figure C.3(b) for the component and crack geometry.


D.3.8 Plate Embedded Crack, Elliptical Shape, Through-Wall Membrane and Bending Stress (RPECE1)

D.3.8.1 The Reference Stress is given by Equation (D.35) with following definitions of d and :

d d a 1 (D.38)

2

1

at

tc

for W c tb g (D.39)


FHGIKJFHGIKJ

2at

cW

for W c tb g (D.40)

D.3.8.2 Notes:



D.3.9 Plate Embedded Crack, Elliptical Shape, Through-Wall Fourth-Order Polynomial Stress Distribution(RPECE2)


D.3.9.2 Notes:



D.4 Reference Stress Solutions For Plates with Holes

D.4.1 Plate With Hole Through-Wall Single Edge Crack, Through-Wall Membrane And Bending Stress(RPHTC1)


att a tb g (D.41)

D.4.1.2 Notes:



D.4.2 Plate With Hole Through-Wall Double Edge Crack, Through-Wall Membrane And Bending Stress(RPHTC2)


22

att a tb g (D.42)

D.4.2.2 Notes:




D.4.3 Plate With Hole Surface Crack, Semi-Elliptical Shape, Through-Wall Membrane Stress (RPHSC1)

D.4.3.1 The Reference Stress is:

ref

m m mP P Pdt

dt

RST

UVWLNM

OQP

LNM

OQP

3 3 9 1 4

3 1 4

2 2 20 5

2

b g b g

b g

.

(D.43)

where,

d t c (D.44)

2

1

ct

ta

(D.45)

D.4.3.2 Notes:


b. See paragraph D.2.2.3 for determination of Pm .

D.4.4 Plate With Hole, Corner Crack, Semi-Elliptical Shape, Through-Wall Membrane and Bending Stress(RPHSC2)


22

act a tb g (D.46)

D.4.4.2 Notes:



D.5 Reference Stress Solutions For Cylinders

D.5.1 Cylinder Through-Wall Crack, Longitudinal Direction, Through-Wall Membrane and Bending Stress(RCTCL)



ref

b b t mP P M P

2 2

0 59

3

l qe j.

(D.47)

D.5.1.2 Notes:


b. See paragraph D.2.2.3 for determination of Pm and Pb . For internal pressure loading:

P PRtm

i (D.48)

P pRR R

tR

tR

tRb

o

o i i i i

FHGIKJ FHGIKJ

LNMM

OQPP

2

2 2

2 332

95

(D.49)

c. See paragraph D.2.3 to determine Mt for a through-wall crack in a cylinder.

D.5.2 Cylinder Through-Wall Crack, Circumferential Direction, Through-Wall Membrane and BendingStress (RCTCC1)


ref

b b mP P ZP

2 2

0 59

3

l qe j.

(D.50)

where,

ZR RR t

o i

o

2 2

2 2c h

b g b g (D.51)

tRo

(D.52)

FHGIKJarccos

sin2

(D.53)

c

Rm(D.54)

D.5.2.2 Notes:



b. See paragraph D.2.2.3 for determination of Pm and Pb . For internal pressure with a net sectionaxial force:

P pRR R

FR Rm

i

o i o i

2

2 2 2 2 c h (D.55)

Pb 0 0. (D.56)

D.5.3 Cylinder Through-Wall Crack, Circumferential Direction, Pressure With Net Section Axial Force andBending Moment (RCTCC2)


refys m m

ysM

R t pR

LNMM

OQPP2 2 22 3cos sin cosb g

(D.57)

where,

ys m

ys m m

R t F

R t pR2

2 2(D.58)

c

Rm(D.59)

D.5.3.2 Notes:


b. If the net-section bending moment is zero, the solution in paragraph D.5.2. must be used.

D.5.4 Cylinder Surface Crack, Longitudinal Direction Infinite Length, Internal Pressure (RCSCLL1)

D.5.4.1 The Reference Stress [D.14.1], [D.14.3]:

ref

b b s mP P M P

2 2

0 59

3

l q .(D.60)

where,

M at

s

10

10

.

.(D.61)

D.5.4.2 Notes:



b. See paragraph D.5.1.2.b for determination of Pm and Pb .

D.5.5 Cylinder Surface Crack, Longitudinal Direction Infinite Length, Through-Wall Fourth OrderPolynomial Stress Distribution (RCSCLL2)


D.5.5.2 Notes: see paragraph C.5.4.2.

D.5.6 Cylinder Surface Crack, Longitudinal Direction Infinite Length, Through-wall Arbitrary StressDistribution (RCSCLL3)


D.5.6.2 Notes: see paragraph C.5.4.2.

D.5.7 Cylinder Surface Crack, Circumferential Direction 360 Degrees, Pressure With Net Section AxialForce And Bending Moment (RCSCCL1)


refr

rrM N M

FHG

IKJ2 4

22 0 5.

(D.62)

For an inside surface crack

NP R R

R R ar

m o i

o i

2 2

2 2b g(D.63)

M P R RR R R ar bg

o i

o o i

LNMM

OQPP

316

4 4

4 3

b g (D.64)

For an outside surface crack

NP R R

R a Rr

m o i

o i

2 2

2 2b g(D.65)

M P R RR R a Rr bg

o i

o o i

LNMM

OQPP

316

4 4

3 4

b g (D.66)


D.5.7.2 Notes:


b. Pm and Pb are determined using the following equations:

P pRR R

FR Rm

i

o i o i

2

2 2 2 2c h c h (D.67)

P MRR Rbg

o

o i

0 25 4 4. c h (D.68)

D.5.8 Cylinder Surface Crack, Circumferential Direction 360 Degrees, Through-Wall Fourth OrderPolynomial Stress Distribution (RCSCCL2)


ref

b b mP P ZP

2 2

0 59

3

l qe j.

(D.69)

where,

Z x x

FHG

IKJ

LNM

OQP

1 2 22

1

(D.70)

tRo

(D.71)

x at

(D.72)

D.5.8.2 Notes:



D.5.9 Cylinder Surface Crack, Circumferential Direction 360 Degrees, Through-wall Arbitrary StressDistribution (RCSCCL3)


D.5.9.2 Notes: see paragragh D.5.8.2.


D.5.10 Cylinder Surface Crack, Longitudinal Direction Semi-Elliptical Shape, Internal Pressure(RCSCLE1)


ref

b b s mgP gP M P

b g b g2 2 0 593

.

(D.73)

where g is given by Equation (D.32) with the following definition of :

at

tc

1(D.74)

D.5.10.2 Notes:



c. See paragraph D.2.3 to determine Ms for a surface crack in a cylinder.

D.5.11 Cylinder Surface Crack, Longitudinal Direction Semi-Elliptical Shape, Through-Wall Fourth OrderPolynomial Stress Distribution (RCSCLE2)

D.5.11.1 The Reference Stress in paragraph D.5.10.1 can be used.

D.5.11.2 Notes: see paragrapgh C.5.10.2.

D.5.12 Cylinder Surface Crack, Longitudinal Direction Semi-Elliptical Shape, Through-wall ArbitraryStress Distribution (RCSCLE3)


D.5.12.2 Notes: see paragrapgh C.5.10.2.

D.5.13 Cylinder Surface Crack, Circumferential Direction Semi-Elliptical Shape, Internal Pressure andNet-Section Axial Force (RCSCCE1)


ref

b b mP P ZP

2 2

0 59

3

l qe j.

(D.75)

where,


P pRR R

FR Rm

i

o i o i

2

2 2 2 2 c h (D.76)

Pb 0 0. (D.77)

Z x x

FHG

IKJ

LNM

OQP

2 2 22

1

(D.78)

arccos sinAb g (D.79)

A xx x

LNMM

OQPP

1 2 2 12 1 2 1

2

b gb g b gb gb gm r (D.80)

tRo

(D.81)

x at

(D.82)

cR

for an internal cracki4

(D.83)

cR

for an external cracko4

(D.84)

D.5.13.2 Notes:


b. This solution can be used for any applied through-wall stress distribution if paragraph D.2.2.3is used to determine of Pm and Pb .

D.5.14 Cylinder Surface Crack, Circumferential Direction Semi-Elliptical Shape, Through-Wall FourthOrder Polynomial Stress Distribution With Net Section Bending Stress (RCSCCE2)


ref

m

M

R t at

FHG

IKJ2 2

2 sin sin(D.85)

where,


FHGIKJFHGIKJ

LNMM

OQPP2

1 at

P Pm bys

b g(D.86)

cR

for an internal cracki4

(D.87)

cR

for an external cracko4

(D.88)

If b g ,

ref

m ys

M

R t at

FHGIKJ2 2

2 sin(D.89)

where,

FHG

IKJ

1

2

at

P P

at

m b

ys

b g(D.90)

D.5.14.2 Notes:



c. The inclusion of the term Pb in Equation (C.90) will produce conservative results.

d. If the net section bending moment is zero, the solution in paragraph D.5.13 can be used withF 0 0. and Pb equal to the value determined in subparagraph b above.

D.5.15 Cylinder Surface Crack, Circumferential Direction Semi-Elliptical Shape, Through-wall ArbitraryStress Distribution (RCSCCE3)


D.5.15.2 Notes:




D.5.16 Cylinder Embedded Crack, Longitudinal Direction Infinite Length, Through-Wall Fourth OrderPolynomial Stress Distribution (RCECLL)


D.5.16.2 Notes:



D.5.17 Cylinder Embedded Crack, Circumferential Direction 360 Degrees, Through-Wall Fourth OrderPolynomial Stress Distribution (RCECCL)


D.5.17.2 Notes:



D.5.18 Cylinder Embedded Crack, Longitudinal Direction Elliptical Shape, Through-Wall Fourth OrderPolynomial Stress Distribution (RCECLE)

D.5.18.1 The Reference Stress is given by Equation (D.35) with the following definitions for d and :

d d a 1 (D.91)

2

1

at

tc

(D.92)

D.5.18.2 Notes:



D.5.19 Cylinder Embedded Crack, Circumferential Direction Elliptical Shape, Through-Wall Fourth OrderPolynomial Stress Distribution (RCECCE)


D.5.19.2 Notes:




D.6 Reference Stress Solutions For Spheres

D.6.1 Sphere Through-Wall Crack, Through-Wall Membrane and Bending Stress (RSTC)

D.6.1.1 The Reference Stress solution in paragraph D.5.1. can be used.

D.6.1.2 Notes:


b. See paragraph D.2.2.3 for determination of Pm and Pb . For internal pressure loading only:

P pRR Rm

i

o i

2

2 2 (D.93)

P pRR R

tR

tR

tRb

o

o i i i i

FHGIKJ FHGIKJ FHGIKJ

LNMM

OQPP

3

3 3

2 334

32

94

(D.94)

c. See paragraph D.2.3 to determine Mt for a through-wall crack in a sphere.

D.6.2 Sphere Surface Crack, Circumferential Direction 360 Degrees, Internal Pressure (RSSCCL1)


D.6.2.2 Notes:



c. See paragraph D.2.3 to determine Ms for a surface crack in a sphere.

D.6.3 Sphere Surface Crack, Circumferential Direction 360 Degrees, Through-Wall Fourth OrderPolynomial Stress Distribution (RSSCCL2)



D.6.4 Sphere Surface Crack, Circumferential Direction 360 Degrees, Through-wall Arbitrary FourthOrder Polynomial Stress Distribution (RSSCCL3)




D.6.5 Sphere Surface Crack, Circumferential Direction Semi-Elliptical Shape, Internal Pressure(RSSCCE1)

D.6.5.1 The Reference Stress in paragraph D.5.10. can be used.

D.6.5.2 Notes:



c. See paragraph D.2.3 to determine Ms for a surface crack in a sphere.

D.6.6 Sphere Surface Crack, Circumferential Direction Semi-Elliptical Shape, Through-Wall FourthOrder Polynomial Stress Distribution (RSSCCE2)



D.6.7 Sphere Surface Crack, Circumferential Direction Semi-Elliptical Shape, Through-wall ArbitraryStress Distribution (RSSCCE3)



D.6.8 Sphere Embedded Crack, Circumferential Direction 360 Degrees, Through-Wall Fourth OrderPolynomial Stress Distribution (RSECCL)


D.6.8.2 Notes:



D.6.9 Sphere Embedded Crack, Circumferential Direction Elliptical Shape, Through-Wall Fourth OrderPolynomial Stress Distribution (RSECCE)



D.6.9.2 Notes:



D.7 Reference Stress Solutions For Elbows And Pipe Bends

The reference stress solutions for cylinders can be used for elbows and pipe bends if the equivalentmembrane and bending stress at the location of the crack is determined considering the bendgeometry and applied loads. A discussion regarding the stress analysis for elbows is provided inAppendix C, paragraph C.7.

D.8 Reference Stress Solutions For Nozzles And Piping Tees

D.8.1 Nozzle Corner Crack, Radial Direction, Quarter-Circular Shape, Membrane Stress At The Corner(RNCC1)


ref m

n n

n n

Pt q r t

t q r t a

FHG

IKJ

2 52 5 0 25

2

2 2

.. .

l ql q (D.95)

where,

q r r t tn n n max ,2 l q (D.96)

r d tn n n

2(D.97)

D.8.1.2 Notes:

a. See Figure C.25 (Crack labeled G) and Figure C.26 for the component and crack geometry.

b. Pm is the primary membrane stress at the nozzle, the effects of the stress concentration areneglected in the calculation of the reference stress because this stress is localized.

D.8.2 Nozzle Corner Crack, Radial Direction, Quarter-Circular Shape, Cubic Polynomial StressDistribution (RNCC2)

D.8.2.1 The Reference Stress is computed using equations in paragraph D.8.2 with an equivalent membranestress.

D.8.2.2 Notes:

a. See Figure C.25 (Crack labeled G) and Figure C.26 for the component and crack geometry.


b. See paragraph D.2.2.3 for determination of Pm .

D.8.3 Surface Cracks At Nozzles General Solution

The reference stress solutions shown below can be used for nozzles if the equivalent membrane andbending stress at the location of the crack is determined considering the nozzle geometry and appliedloads. A discussion regarding the stress analysis for nozzles is provided in Appendix C, paragraphC.8.

Nozzle Neck or Branch (see Figure C.25)

Crack A Use KCTCC1, KCTCC2, KCSCCL3, KCSCCE3, KCECCL or KCECCE

Crack B Use KCTCL, KCSCLL3, KCSCLE3, KCECLL or KCECLE

Shell or Run Pipe (see Figure C.25)

Crack D & F Use KPTC, KPSCE3, KPECL, or KPECE2

Crack E & C Use KPTC, KPSCE3, KPECL, or KPECE2

Crack G Use the solutions in paragraph D.8

D.9 Reference Stress Solutions For Ring-Stiffened Cylinders

D.9.1 Ring-Stiffened Cylinder Internal Ring, Surface Crack At The Toe Of One Fillet Weld,Circumferential Direction 360 Degrees, Pressure Loading (RRCSCCL1)

D.9.1.1 The Reference Stress in paragraph D.5.8 can be used with an equivalent membrane and bendingstress.

D.9.1.2 Notes:


b. See paragraph A.8.3 of Appendix A for determination of the equivalent membrane stress, Pm,and bending stress, Pb based on the stress results at the inside and outside surface, or

Pms ID s OD

, ,

2(D.98)

Pbs ID s OD

, ,

2(D.99)

D.9.2 Ring-Stiffened Cylinder Internal Ring, Surface Crack At The Toe Of Both Fillet Welds,Circumferential Direction 360 Degrees, Pressure Loading (RRCSCCL2)




D.10 Reference Stress Solutions For Sleeve Reinforced Cylinders

The reference stress solutions shown below can be used for sleeve reinforced cylinders if the stressat the location of the crack is determined considering the actual component geometry and appliedloads. A discussion regarding the stress analysis is provided for sleeve reinforced cylinders inAppendix C, paragraph C.10.

Cracks At Sleeve Reinforced Cylinders (see Figure C.28)

Crack A Use KCTCC1, KCTCC2, KCSCCL3, KCSCCE3, KCECCL or KCECCE

Crack B Use KCTCL, KCSCLL3, KCSCLE3, KCECLL or KCECLE

D.11 Reference Stress Solutions for Round Bars and Bolts

D.11.1 Round Bar, Surface Crack 360 Degrees, Membrane and Bending Stress (RBSCL)

D.11.1.1 The Reference Stress is:

refr

rrM N M

FHG

IKJ2 4

22 0 5.

(D.100)

where,

N P RR ar

m o

o

2

2b g (D.101)

M P RR R ar bg

o

o o

LNMM

OQPP

316

4

3

b g (D.102)

D.11.1.2 Notes:


b. The primary membrane and global bending stresses are computed using the followingequations:

P FRm o

2 (D.103)


P MRbg o

43

(D.104)

D.11.2 Round Bar Surface Crack, Straight Front, Membrane and Bending Stress (RBSCS)


ref

m gbP P

212

2

316sin

(D.105)

where,

F

HGIKJarcsin

R aRo

o

(D.106)

FHGIKJ

FHGIKJ

FHGIKJ10002 3 9927 2 58491 2 2 8550 2

1 5 2 5 3

. . . .. .

aR

aR

aRo o o

(D.107)

D.11.2.2 Notes:

a. For the component and crack geometry see Figure C.30.

b. The primary membrane and global bending stresses can computed using the equations inparagrapgh D.11.2.b.

D.11.3 Round Bar, Surface Crack, Semi-Circular, Membrane and Bending Stress (RBSCC)


D.11.3.2 Notes:


b. The semi-elliptical flaw is evaluated as an equivalent to a straight front flaw.

D.11.4 Bolt, Surface Crack, Semi-Circular or Straight Front Shape, Membrane and Bending Stress (RBSC)

D.11.4.1 The Reference Stress in paragraph D.11.2 can be used by replacing Ro with Rth.

D.11.4.2 Notes:


b. The solution applies to a semi-circular or straight front surface crack.


D.12 Reference Stress Solutions For Cracks At Fillet Welds

D.12.1 Cracks at Fillet Welds Surface Crack At A Tee Joint, Semi-Elliptical Shape, Through-WallMembrane and Bending Stress (KFWSCE1)


D.12.1.2 Notes:



D.12.2 Cracks at Fillet Welds of Tee Junctions In Pressurized Components General Solution

The reference stress solutions shown below can be used for cracks at fillet welds in pressurecontaining components if the stress at the location of the crack is determined considering the actualcomponent geometry and applied loads. A discussion regarding the stress analysis is provided inAppendix C, paragraph C.12.

Cracks At Fillet Welds of Tee Junctions In Pressurized Components (see Figure C.32)

Flat Plate Tee Junctions Use RPTC, RPSCE3, RPECL, or RPECE2

Longitudinal Tee Junctions in Cylinders Use RCTCL, RCSCLL3, RCSCLE3, RCECLL orRCECLE

Circumferential Tee Junctions in Cylinders Use RCTCC1, RCTCC2, RCSCCL3, RCSCCE3,RCECCL or RCECCE

Circumferential Tee Junctions in Spheres Use RSTC, RSSCCL3, RSECCL or RSECCE

D.13 Reference Stress Solutions For Cracks In Clad Or Weld Overlayed Plates And Shells

The reference stress solutions in this appendix can be use to evaluate clad or weld overlayed plateand shell components. A discussion regarding the stress analysis for clad and weld overlayed plateand shell components is provided in Appendix C, paragraph C.13.

D.14 References

D.14.1 Miller, A.G., Review of Limit Loads of Structures Containing Defects, International Journal ofPressure Vessels & Piping, Vol. 32, 1988.

D.14.2 Zahoor, A., "Ductile Fracture Handbook", Electric Power Research Institute, Palo Alto, CA, 1989.


D.14.3 Willoughby, A.A. and Davey, T.G., Plastic Collapse in Part-Wall Flaws in Plates,FractureMechanics: Perspectives and Directions (Twentieth Symposium), ASTM STP 1020, R.P. Wei andR.P. Gangloff, Eds., American Society for Testing and Materials, Philadelphia, 1989, pp. 390-409.

D.14.4 Bamford, W.H., Landerman, E.I., and Diaz, E., Thermal Analysis of Cast Stainless Steel, and itsImpact on Piping Integrity, Circumferential Cracks in Pressure vessels and Piping Vol. II, ASMEPVP Vol. 95, G.M. Wilkowski, American Society of Mechanical Engineers, 1984, pp. 137-172.

D.14.5 Bergman, M., Bjorn, B., Dahlberg, L., Nilsson, F., and Sattari-Far, I., A Procedure For SafetyAssessment of Components with Cracks Handbook, SA/FoU-Report 91/01, The Swedish PlantInspectorate, Stockholm, Sweden, December, 1991.

D.14.6 BSI, Draft Revision to PD6493 Fracture Assessment, TWI, March, 1996.

D.14.7 Kiefner, J.F. and Vieth, P.H., Project PR 3-805, A Modified Criterion for Evaluating the RemainingStrength of Corroded Pipe, Battelle Report to the Pipeline Committee of the American GasAssociation, 1989.

D.14.8 Stephens, D.R., Krishnaswamy, P, Mohan, R., Osage, D.A., Sims, J.R., and Wilkowski, G., AReview of Analysis Methods and Acceptance Criteria for Local Thinned Areas in Piping and PipingComponents, 1997 Pressure Vessels and Piping Conference, Orlando, Florida, July, 1997.

D.14.9 Green, D. and Knowles, J., The Treatment of Residual Stress in Fracture Assessment of PressureVessels, ASME, Journal of Pressure Vessel Technology, Vol. 116, November 1994, pp. 345-352.

D.14.10 Folias, E.S., On the Effect of Initial Curvature on Cracked Sheets, International Journal of FractureMechanics, Vol. 5, No. 4, December, 1969, pp. 327-346.

D.14.11 Sih, G.C., Handbook of Stress Intensity Factors, Institute of Fracture and Solid Mechanics, LehighUniversity, Bethlehem, Pa.

D.14.12 Kramer, G.S., Wilkowski, G.M., and Maxey, W.A., Flaw Tolerance of Spiral Welded Pipe, BattelleNG-18 Report No. 154, January, 1987.

D.14.13 Kiefner, J.F. and Vieth, P.H., Project PR 3-805, A Modified Criterion for Evaluating the RemainingStrength of Corroded Pipe, Battelle Report to the Pipeline Committee of the American GasAssociation, 1989.

D.14.14 Eiber, R.J., Maxey, W.A., Duffy, A.R., and Atterbury, T.J., Investigation of the Initiation and Extent ofDuctile Pipe Rupture, Battelle Report Task 17, June, 1971.

D.14.15 Murakami, Y., Stress Intensity Factors Handbook, Pergamon Press, Oxford, 1987, pp. 1356-1358.

D.14.16 Tada, H., Paris, P.c. and Irwin, G.R, The Stress Analysis Of Cracks Handbook Second Edition,Paris Productions Inc., St. Louis, Missouri, 1985.

D.14.17 Chell,G.G, Application of the CEGB Failure Assessment Procedure, R6, to Surface Flaws, FractureMechanics: Twenty-First Symposium, ASTM STP 1074, J.P. Gudas, J.A. Joyce, and E.M. Hackett,Eds., American Society for Testing and Materials, Philadelphia, 1990, pp. 525-544.

D.14.18 Sattari-Far, I., Finite Element Analysis of Limit Loads For Surface Cracks in Plates, Int. J. Pres. Ves.& Piping, 57, 1994, pp. 237-243.

D.15 Tables and Figures


Figure D.1Failure Regions On The Failure Assessment Diagram

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Lr

K r

Zone 3Collapse Controlled

UnacceptableRegion

Lr(max)

Zone 1Fracture (Elastic)

Controlled

AcceptableRegion

Zone 2Fracture (Elastic-Plastic)And Collapse Controlled

Structured bookmarksTable of ContentsD.1 GeneralD.2 Stress AnalysisD.3 Reference Stress Solutions For PlatesD.4 Reference Stress Solutions For Plates With HolesD.5 Reference Stress Solutions For CylindersD.6 Reference Stress Solutions For SpheresD.7 Reference Stress Solutions For Elbows And Pipe BendsD.8 Reference Stress Solutions For Nozzles And Piping TeesD.9 Reference Stress Solutions For Ring-Stiffened CylindersD.10 Reference Stress Solutions For Sleeve Reinforced CylindersD.11 Reference Stress Solutions for Round Bars and BoltsD.12 Reference Stress Solutions For Cracks At Fillet WeldsD.13 Reference Stress Solutions For Cracks In Clad Or Weld Overlayed Plates And ShellsD.14 ReferencesD.15 Tables and FiguresFigure D.1

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yield stress

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