Article history:Received 2 April 2007Received in revised form 22 March 2008Accepted 26 March 2008Available online 31 March 2008

Application of the crack compliance method to the analysis of thin-walled rings with a

Thus for cracked dent of complex geometry the more urgent task is the development of simple engineering procedure ofSIF calculation rather than graphical presentation of FEA results or construction of simplifying formulas. As example of thelater approach we mention a well-known defect assessment manual [1], which provides the corresponding formula. Themanual [1] uses Hopkins solution [2] based on the strip yield model. This in fact is the simplest form [3] of a two-criteriaapproach and formed the basis for the rst version of the well-known R6 document proposed in 1976. Thus, Hopkins

0013-7944/$ - see front matter 2008 Published by Elsevier Ltd.

* Corresponding author.E-mail address: or@ipp.kiev.ua (I.V. Orynyak).

Engineering Fracture Mechanics 75 (2008) 40524065

Contents lists available at ScienceDirect

Engineering Fracture Mechanicsdoi:10.1016/j.engfracmech.2008.03.0081. Introduction

The modern means of in-line inspection of pipelines allows revealing a large number of cracks and dents. For each of themthe operator has to make a justied decision about their acceptability or the kinds and terms of repair. The stress analysis ofpipes, including the computation of stress intensity factors for cracks, is an important constituent of such a decision.

There are two problems when analyzing a pipe with a dent under a large internal pressure: the rerounding effect of thepressure and the poorly dened geometry of the dent being a local defect. Dents are characterized not only by the conven-tional dimensions depth, width, and length but also by such a subjective parameter as the smoothness of its contour(sharp, plain, etc.), which complicates the development of general methods for their analysis. In the case of a crack emanat-ing from the dent, the problem formulation becomes more difcult because of the geometrical features of the crack. Thismakes the standard FEM based analysis a rather inconvenient one.Keywords:CrackPipeDentStress intensity factorMethod of initial parametersCrack compliance methodradial crack has two features: a crack is considered as a concentrated angular complianceand the deformation of all other sections of the rings is calculated as for a curvilinear beam.The latter can be most conveniently found by the method of initial parameters where thevalues of generalized forces and displacements at the end of some zone are determined as alinear combination of their values at the beginning of the zone. The goal of the study is toderive and apply the method of initial parameters equations taking into account the inu-ence of circumferential stresses on the ring curvature. As far as the authors know, this is therst time that the stress intensity factor has been derived for an elastic thin-walled pipewith a radial crack in a geometrically nonlinear formulation. Here, an increase in pressureleads to a somewhat slowed increase in the stress intensity factor. In addition, a number ofproblems for dents are considered. The effect of the dent shape on the stressstrain state isanalyzed. An expression for the stress intensity factor for a complex defect, a crack emanat-ing from the dent apex, is presented.

2008 Published by Elsevier Ltd.Application of the crack compliance method to long axial cracksin pipes with allowance for geometrical nonlinearityand shape imperfections (dents)

I.V. Orynyak *, Ye.S. YakovlevaG.S. Pisarenko Institute for Problems of Strength, National Academy of Sciences of Ukraine, 2 Tymiryazevska str., Kyiv, Ukraine

a r t i c l e i n f o a b s t r a c t

journal homepage: www.elsevier .com/locate /engfracmech

GabrielResaltado

GabrielNota adhesivapara el capitulo 3

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4053Nomenclature

KI stress intensity factor (SIF)(r, u) polar coordinatesR0 the initial radius of curvature of the ideal ringR(u) polar radius of the middle surface of the ringw, u radial and circumferential displacementsE, m Youngs modulus and Poissons ratio~n;~l normal and tangent vectorsh rotation angle of the cross-sectionDh, Du jumps in rotation angle and displacements in the cracked sectionbi, ci dimensionless values of these jumps in the cracked section for i-loadinga depth of a cracka = a/t dimensionless depth of a crackYi(a) dimensionless SIF for i-loadingM bending moment per unit width of ringN, Q circumferential and transverse forces per unit width of ringNN nominal force from internal pressure (as for ideal ring)N1 additional force due to the shape imperfectionrN, rM intensities of stress from circumferential force and bending momentrq(x) distribution of the circumferential stresses in the cracked section due to all outer forces, shape imperfection,

residual stress in case of the crack absencesolution [2] uses both nominal stresses from internal pressure and the original expression for the stress intensity factor (SIF)based on formulas for a strip with a crack under membrane and bending stresses.

The second above mentioned problem, namely the rerounding effect of the pressure action, is a more complicated and aless evident one. This phenomena is exhibited, for example, in decreasing of coefcient of pipe bend exibility at loading bybending moment due to additional action of internal pressure. As it was said in resent paper [4] The pressure reductioneffect in smooth piping elbows is well known, but little understood. In fact the inner pressure increase the apparent stiffnessof the pipe wall and the primary goal of this paper is foremost to draw attention to it in case of the presence of crack.

With respect to thin-walled shells, this problem was thoroughly studied by Calladine [5]. He have shown that small ini-tial, or acquired due to the outer loading, deections from the ideal cylindrical shell form can be accounted for as some addi-tional distributed loading applied to the cylindrical shell. His method was called the equivalent load method. Practicalformulas were obtained only for innitely long axial imperfections [5] while a 3D analysis of actual defects using the ap-proach was performed in [6]. But Papkovich [7] was the rst to obtain a correct solution taking into account the action ofinternal pressure on the ring. Considering that the imperfection prole can be expressed as a Fourier series, he obtained ageometrically nonlinear analytical solution for displacements and stresses. Note that an identical solution is presented inthe well known API 579 Fitness-For-Service Standard [8]. For completeness of presentation, it will be given in this work too.

The proposed below method consider the crack as a concentrated compliance when the jumps in displacements and an-gles of rotation in the cracked section are linearly related to the values of force and moment in it. The proportionality factorsare calculated by integrating SIF over the crack length. This technique has found wide application in linear fracture mechan-ics, especially for beams, thin-walled shells, and plates. It was suggested for the analysis of elliptic cracks in plates under thename of line spring model by Rice and Levy [9] where the classical Kirchhoffs plate theory was used. Further developmentwas made, in particular, by Delale and Erdogan [10] where an analysis of cracked shells was performed based on Reissnersgoverning equations.

rqx dimensionless law of the above stress distributionq intensity of these stressesP internal pressurep PR30 12

E0t3dimensionless pressure

rP(x) the law of the circumferential stress distribution in the thick-walled pipe due to inner pressure (Lames formula)r1Px the sum of the above Lames stresses and uniform stresses from inner pressurep0 intensity of the above stressesru = PR0/t circumferential stresses in thin-walled pipex coefcient of the SIF reductionW dent depth at the point of the center of the dentDu angular length of the dentw1 angle jump at the point of the center of the dentw2 angle jump at the point which is the transition from the dent to the undistorted part of the pipeMN PR0t Wt nominal bending moment for a small dentNe equivalent axial force by Trescas condition

The technique of concentrated compliance is used together with the determination of displacements in the remaining,defect-free part of the body, which can be done by FEM calculation, as well as by analytical formulas. It can be most effec-tively applied to two-dimensional bodies for which it was thoroughly elaborated in works of Cheng and Finnie [11] and hascome to be known as crack compliance method. It received further development and wide application both in destructivemethods for measuring residual stresses [12] and in methods of SIF determination for so-called statically indeterminatebodies with cracks when kinematic boundary conditions are used [13,14].

The consideration of the concentrated compliance has practical signicance in the SIF computation for transverse cracksin spatial piping systems. The stress state induced by thermal expansion depends on the compliance of supports and ex-ibility of piping. An additional compliance of the section with a crack results in some unloading of the piping. Taking intoaccount this effect leads to a decrease in the SIF value and is favorable in leak-before-break analysis of NPP piping [15] orheat-exchange tubes of steam generator [16].

This paper analyzes a ring with a radial crack and makes use of the common features of Chengs and Finnies approach[17]:

A ring is considered as a curvilinear beam with a transverse crack. There are residual stresses in the cracked section, aswell as the stresses caused by some external loading.

4054 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 A crack is a concentrated compliance with two degrees of freedom (jumps in displacements, Du, and rotation angle, Dh,take place in the section with a crack). These jumps depend on all the stresses in the cracked section.

Due to the jumps in the cracked section, the continuity (or boundary conditions) of the ring-beam would be formally vio-lated if it were not for the additional bending momentM and longitudinal force N, which arise in the section with a crackand lead to a redistribution of displacement and forces in the beam.

In Cheng and Finnies work [11] the displacements of a curvilinear beam were obtained by aid of Castiglianos reciprocitytheorem. The novelty of this paper consists mainly in a more rigorous calculation of the beam. In our former work [18] theimprovements were related to the calculation of a thick-walled ring as a curvilinear beam. The present work mainly differs inthe adoption of rst order nonlinear governing equations for determining displacements and rotation of the curvilinear beamunder the action of internal pressure. There are a great number of methods for solution of such equations in structuralmechanics. The authors advocate the method of initial parameters (MIP) in conjunction with the sweep method [19]. Its es-sence for a plane beam consists in that six unknown values of generalized forces and displacement at the end of some zoneare determined as a linear combination of their values at the beginning of the zone. Thus the solution of equations for thebeam will be given here in a form convenient for subsequent application in the MIP.

2. Problem formulation

2.1. Geometry and governing equations

A pipe with a long surface axial crack can be treated as a ring with an radial crack. Thus consider a ring (pipe) with a con-stant wall thickness t, R(u) is the polar radius of the ring, u is the polar angle, and point O is the center of coordinates (Fig. 1).

Assume a radial crack of depth a. We relate the coordinate u = 0 with the section containing a crack. Introduce the localunit vectors:~n is the normal directed to the instantaneous center of curvature at the considered point u,~l is the tangent vec-tor directed clockwise. By two nearby sections, which are parallel to the vector ~n, we put into consideration an elementaryzone ds of a curvilinear beam (Fig. 2). The stressstrain state of this zone is characterized by six basic parameters. They are:radial displacement w (along ~n), circumferential displacement u (along~l), rotation angle of the cross-section, h directedclockwise; transverse force Q, longitudinal force N, and bending moment M directed as shown in Fig. 2.

OR()

x

Pt

a

Fig. 1. A curvilinear ring with a redial crack.

We

Taking

whereWr

Here Ewhere

Then

We sivalue

where

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4055NN N1: 8b

The asrangemax{Qthe re(8b) itEq. (5N NN N1; 8aN 0

additional force N1:mplify the governing equations using the above mentioned equivalent load method [5]. Then we present the soughtof the longitudinal force N as the sum of the nominal force caused by internal pressure N = PR = const. and someq

R1

RE0J

: 7a general expression for the curvature is written as follows:

1 1 R00

Maction of the bending moments, i.e., the problem of ring deformation is considered in a geometrically nonlinear formulation.is the Youngs modulus and the generalized modulus E0 is equal to E for a ring and to E/(1 m2) for an innite pipe,m is the Poissons ratio. When solving Eqs. (5), (6) we take into account the change in the ring curvature caused by thedhRudu

ME0J

; duRudu

wq 0; dw

Rdu uq h: 6R0 = oR/ou, and R00 = o2R/ou2.ite down six differential equations, which describe the deformation of the ring:

dQRudu

Nq P; dN

RuduQq 0; dM

Rudu Q ; 5ds R R du Rudu: 3Then, as a rst approximation, the expression for the curvature K is:

K 1q 1

R1 R

00

R

; 4into account condition (2), it is easy to show that the elementary length of the ring ds is approximately equal to:2 02

qSuppose that the shape of the ring differs insignicantly from a circular one, i.e.:

b2n; a2n 1: 2Ru R0 1n1

an cosnun1

bn sinnu : 1

X1 X1 !expand the radius-vector of the mean surface R(u) into a harmonic series:NN+dN

M+dMMw

Fig. 2. Directions of forces and displacement vectors.Q+dQ Q u

dssumption (8b) is the key one in the subsequent analysis. In spite that it is widely applied in analysis of the shells thes of its applicability are rarely discussed in the literature. Keeping in mind that Q and N1 are comparable and} sin hmax, where hmax is the maximal angle of the distortion of the form of the ring from the ideal circular form,quirement hmax 1 will justify the correctness of application of the assumption (8b). Taking into account conditioncan be shown that a change in the curvature as it is given by Eq. (7) is essential only in the rst one of the equilibrium). Thus, instead of Eqs. (5) and (6), we have:

dQR0 du

N1R0

PR20ME0J

PR0X1n1

n2 1an cosnuX1n1

n2 1bn sinnu !

;dN1R0 du

QR0 0; dM

R0 du Q ; 9

dhR0 du

ME0J

; dudu

w 0; dwdu

u hR0: 10

From Eq. (9) we get:

Q 00 v2Q PR0 X1n1

nn2 1an sinnuX1n1

nn2 1bn cosnu !

: 11a

Here

v2 1 PR30

E0J 1 p; 11b

where, for the sake of convenience, the dimensionless pressure p is introduced.

2.2. The jumps of displacements and rotation angles in the cracked section

Asscan be

where

wherefrom

The sitotal S

Note ting ju

2.3. T

4056 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065Fig. 3. Scheme of deformation of a beam near the cracked section.a

tq(x)As noted before, the stressstrain state of a curvilinear beam is characterized by six parameters. We denote them in thegeneral form as~Xu. The MIP is applicable when for each zone of the beam, which is free from outer loads (or loaded only bypressure), there exist six equations describing the relation between the values of these parameters at the beginning of thezone and those at any point of the zone. Thus

~Xu Au~X0; 17

u

xhat the sign of the h value presented in Fig. 3 is negative and formulas (12) and (13) give only a half of the correspond-mps.

he use of the method of initial parameters (MIP)KI KIN KIM KIq: 16gn means that compressive stresses are positive here. Then, taking into account the superposition principle, theIF can be written asformulas

KIq pa

pqYqa; KIN

pa

prNYNa and KIM

pa

prMYMa: 15Yi(a) is the dimensionless SIF, which can be obtained, for example, by the weight function method [2022] or takenhandbooks. The relation between the dimensionless SIF values and the SIF for the corresponding loading is given bycia a

0aYNaYiada and bia

a

0aYMaYiada; 14ci and bi are compliance coefcients for any type of loading designated by subscript i, which can be found asZ Zthose stresses. For example, the stresses from longitudinal force N, and bending momentM can be presented in similar formrN(x) = rN 1 and rM(x) = rM (1 2x/t) where the values rN = N/t and rM = 6M/t2 are, in fact, the intensities. Then, in accor-dance with the crack compliance method, the jumps of displacements and rotation angles (Fig. 3) depend from the intensityof stresses q as well as from stresses due to additional force and moment. They are found by the following formulas (see, forinstance, [18]):

Du t pE0cqq cMrM cNrN; 12

Dh 6pE0bqq bMrM bNrN; 13ume that circumferential stresses rq in the considered section with a crack, but under the assumption of its absence,presented in the form: rqx qrqx, where rqx is some basic law of the stress distribution and q is the intensity of

where

denom

Noble 1

The co

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4057and rotation angles (13) is the main idea and novelty of the present paper.

4. Computation of the SIF for a single crack in a ring

Consider a pipe under the action of internal pressure P only. Then, the distribution of circumferential stresses in everysection under the assumption of the crack absence is given by a well-known Lame formula: rP p0 r22=r2 1, wherethe intensity of these stresses p0 r21=r22 r21 P (here r1 is the internal and r2 the external radius). Let a crack be locatedin the section u = 0.

Consider a full set of initial parameters in the section u = Du, where Du is negligibly small. Due to symmetry, the longi-tudinal displacement, rotation angle, and transversal force Q in the section u = 0 are exactly equal to zero. But just beyondE0J 2 2

R30Q0E0J

1 u sinu2

cosu

R20M0E0J

u sinu; 20

w w0 cosu u0 sinu h0R0 sinu R30N0E0J

1 cosu u sinu2

R20M0E0J

1 cosu R30Q0E0J

sinu2

u cosu2

: 21

mbined usage of the MIP equations given in Table 1 as well as the expressions for the jump of the transverse force (19)u u0 cosuw0 sinu h0R01 cosu R30N0 u u cosu 3 sinu

te that for l = 1 (p 0), i.e., for geometrically linear formulation, the expressions for u and w in the rst column of Ta-are reduced to the known form:nal pressure.To obtain the solution sought for the SIF, we need to have a general solution of system (9)(11). In the given formulation,

geometrical nonlinearity has two features. The rst one consists in the inuence of the large value of the longitudinal forceNN = PR0 on the magnitude of the transverse force. Similarly to the jump of the rotation angles Dh in the cracked section, ajump of the transverse force DQ0 also takes place. It occurs because the force NN due to an abrupt change in the direction ofthe tangent vectors~t gives a projection on the normal:

DQ0 NN sinDh PR0Dh: 19The second feature of the present solution consists in explicit expressions for the matrix [A(u)] obtained within the linear-ized formulation of (9) and (10). Depending on the value of v2, three different cases are distinguished for which a generalsolution is presented in the form convenient for subsequent application in the MIP. It is given in Table 1.inator in expressions (18a) becomes equal to zero, so buckling of the ring is going to happen due to the action of exter-An n2 1ann2 v2 ; Bn

n2 1bnn2 v2 : 18b

Here and below, for the sake of uniformity we write N instead of N1.Some comments on the particular solution are presented. Firstly, it does not contain the coefcients at n = 1, as the latter

dene the displacement of a ring as a rigid body; and secondly, the solution is meaningful only for v2 < 4. For v2 = 4, theM PR20Xn2

An cosnu Bn sinnu; h PR30

E0J

Xn2

Ann

sinnu Bnn

cosnu

;

u PR40

E0J

Xn2

Annn2 1 sinnu

Bnnn2 1 cosnu

;

w PR40AnE0J

Xn2

An cosnun2 1

Bn sinnun2 1

;

18awhere the matrix [A(u)] is known and at the point u = 0 is unitary and ~Xu 0 ~X0u0;w0; h0;N1;0;Q0;M0, where the lowerindex 0 indicates that the parameters pertain to the point u = 0. Then we break up the beam into zones at whose bound-aries the concentrated forces (supports) are applied or a crack is placed. For these boundaries we write the conditions ofequality of all six parameters with allowance for the above jumps.

3. Complete solution for a ring in a geometrically nonlinear formulation

We seek the solution to system (9)(11) as the sum of a particular solution of an inhomogeneous system and a generalone for an homogeneous system. The particular solution is given by formulas [7,8,23]

Q PR0Xn2

nAn sinnu nBn cosnu; N PR0Xn2

An cosnu Bn sinnu;

the abthe Marbitrand bname

Table 1General solution to Eqs. (9) and (10) by the MIP

v2 > 0, l = v v2 = 0 v2 < 0; l v2

pN0 N0 M0R0 N0 N0

M0R0

N0 N0 M0R0Q Q0 cos lu N0l sin lu Q0 N0u Q0chlu N0l shluN N0 Q0l sinlu N0 1cos lul2

N0 Q0u N0 u

2

2 N0 Q0l shlu N0 chlu1l2

M M0 R0Q0 sin lul R0N0 1cos lul2

M0 R0Q0u N0R0 u2

2M0 R0Q0 shlul R0N0 chlu1l2

h h0 R0M0E0 J u

R20Q0E0 J

1cos lul2

R20N0E0 J ul2 sin lul3

h0 M0R0E0 J u

Q0R20

E0 Ju2

2 N0R20

E0 Ju3

6 h0 R0M0E0 J uR20Q0E0 J

chlu1l2

R20N0E0 J shlul3 ul2

u u0 cosuw0 sinu h0R01 cosu

R2M0 R3Q cos lu cosu u0 cosuw0 sinu h0R01 cosu

R2M0 R3Q u2 u0 cosuw0 sinu h0R01 cosu

R2M0 R3Q chlu

4058 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065where n0 pR20

EJ N0 and m0 pR0EJ M0. The expressions for calculation of constants ai are presented in Table 2. Note that the val-ues of a1 and a3 are related to the jump of the transverse force in the cracked section (19).

Eq. (22) allow us to nd the magnitudes ofM0 and N0. Note that RN0 = 0(M0). Then the axial stress rN = t/(6R) 0(rM) and itcan be neglected in the SIF calculation. Thus, in the ultimate expression for the SIF (16) we need to take into account thestresses from the internal pressure and bending momentM0 only. Eventually, similarly to our work [18], we get an analyticalexpression for a dimensionless SIF for a ring with a radial crack:

Yp Y0 1 YMY0fph

R09t kp fmm

! Y01 xa; p; 23

where kp 31 2a2 a43 4a3 4a2 4a2a3 a1 a4 a1a4 : 24Here,respecizes thstrip w

Forlinearin par

Table 2Express

ai

a1

a2

a3

a4h01 a1 n01 a2 m0 0 and 2h01 a3 n02 1 a4 m0 0; 22We used this procedure for the SIF computation for thick-walled cylinders with single or several cracks [18]. For a thin-walled pipe it is possible to use a simplifying assumption u0 = 0. Let us justify this. Since YM = 0(YN) (these functions are com-parable), from Eq. (14) follows that ci(a) = 0(bi(a)) and eventually, with allowance for Eqs. (12) and (13), we have u0 = t 0(h0).Thus, it is obvious that u0 h0R. Therefore, in the formulas for u(u) given in Table 1 it is possible to accept that u0 = 0. Theerror caused by this assumption can be evaluated by ratio t/(pR0) and for R0/t > 5 can be considered as quite satisfactory. Theabove conditions of symmetry at the point u = p yieldove section they are equal to the jumps of the corresponding values of Du (12), Dh (13), and PR0h0 (19), respectively. InIP, these values are considered as initial ones u0, h0 and Q0. The initial value of the radial displacement w can be chosenarily, for example, w0 = 0. Thus, in section u = Du, we have only two independent unknowns: the longitudinal force N0ending moment M0. They can be found by the MIP taking into account the condition of symmetry at the point u = p,ly, u(p) = h(p) = 0.0E0 J u sinu 0 0E0 J 1l2 l2 1l2 1l2

R30N0E0 J

ul2 sin lul31l2 sinu1l2 0E0 J u sinu 0 0E0 J cosu 1 2

R30N0E0 J

u3

6 u sinu 0E0 J u sinu 0 0E0 J l21l2

1l2 cosu1l2

R30N0E0 J

shlul31l2 ul2 sinu1l2

w w0 cosu u0 sinu h0R0 sinuR20M0E0 J 1 cosu

R30Q0E0 J

sin lul1l2 sinu1l2

R30N0E0 J

1l2 cos lul21l2 cosu1l2

w0 cosu u0 sinu h0R0 sinuR20M0E0 J 1 cosu

R30Q0E0 J u sinu

R30N0E0 J

u2

2 1 cosu

w0 cosu u0 sinu h0R0 sinuR20M0E0 J 1 cosu

R30Q0E0 J

shlul1l2 sinu1l2

R30N0E0 J

chlul21l2 1l2 cosu1l2 YM and Y0 are the dimensionless SIFs for bending and Lames stresses calculated for an innite strip with an edge crack,tively. Here, for the sake of convenience, the notion of the SIF reduction coefcient, x, is introduced, which character-e inuence of the completeness of the contour (ring geometry) as compared with the basic geometry, i.e., an inniteith an edge crack.mula (23) is the main result of the present study. At p 0 (then kp 0 1) it yields the SIF values for a ring in theformulation, and this result has already been obtained in [18]. Verication of the approach in general and formula (23)ticular for thick-walled cylinders was performed in [18], where the determination of Y0 due to circumferential Lames

ions for the values of constants ai depending on the dimensionless pressure p

v2 > 0, l = v v2 = 0 v2 < 0;l v2

pp 1cos lpl2

p22

p chlp1l2

1 1l2 sin lpl3p 1 p2

6 1 shlpl3p 1l2p 1cos lp2l

2

2l21l2p24 1 p chlp12l

2

2l21l22l2 2 sin lpl3p1l2 1 p

2

3 3 2shlpl3p1l2 2l2 1

stresses was made by the weight functions method [2022]. We present here the result of [18], where the additional actionof internal pressure on the surfaces of a radial crack emanating from the cylinder inner surface are taken into account,r1P p0 r22=r21 r22=r2. Table 3 lists the results of computation of the dimensionless SIF characterized by the valueY2 K1=P

pa

p 2r22r22r21

. For the sake of comparison, the results of computations by Zahoors formula [24] are also presented. Here

parameter b means the ratio of the inner radius to the wall thickness: b = (R0/t 0.5). Taking into account that Zahoors for-mulaa linea

A c

Table 3Dimensionless SIF values, Y2, for a radial crack from the inner surface of a pressurized cylinder with allowance for the pressure on the crack faces

a b = 5 b = 10 b = 20

Formula (23) Zahoor [24] Formula (23) Zahoor [24] Formula (23) Zahoor [24]

0.1 1.3787 1.3442 1.2924 1.2589 1.2444 1.22210.2 1.5215 1.4829 1.4512 1.4233 1.4101 1.42940.3 1.7458 1.7181 1.7055 1.7021 1.6788 1.78090.4 2.0648 2.0559 2.0836 2.1026 2.0889 2.28560.5 2.4945 2.5047 2.6313 2.6346 2.7107 2.95620.6 3.0563 3.0786 3.4263 3.3150 3.6822 3.8138

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4059was not explored before. Two arguments can be offered to prove the correctness of it. First, the wide application in the lit-erature of the crack compliance method, in general, and good correspondence attained with the known results for a limitcase k = 1, in particular. Second, the procedures for linear and nonlinear cases are the same, while the difference is onlyin the beam solutions given in Table 1. But all three beam solutions coincide at v? 0 and, in general, they correspond tothe homogeneous solution for a exible ring of Papkovich [7].

Now return to the analysis of formula (23). Obviously, the larger the denominator in the expression for x, the bigger is thedifference between the SIF for a crack in a ring and that in an innite strip. The reason is that the ring is stiffer than the latter.Therefore, the ratio 9t/R0 characterizes the stiffness of the ring. The inuence of the pressure p reveals itself through the mul-tiplier kp at R0/(9t) that can be interpreted as an increase in the apparent stiffness of the ring caused by internal pressure.Therefore, for analysis of the inuence of p on the SIF values it is important to have the dependence kp. It is shown in Fig. 4.Note that at p 1 the value of k is k = 9/(3 + p2). Thus, combined consideration of formula (23) with Fig. 4 allows us to statethat with increasing pressure the apparent stiffness increases and the dimensionless SIF decreases. As far as we know, theeffect of the decrease in the dimensionless SIF values with increasing internal pressure for a ring with a crack is obtained forthe rst time.

It is important to understand whether this reduction effect is of practical importance or it can be neglected. Look again atthe denominator in the expression for x (23). First of all note that the values of bM(a) for thin-walled cylinders with small andintermediate crack lengths (a 6 0.6) are proportional to a2 and are negligibly small as compared to the value of R0/9t. Thenthe SIF reduction coefcient x is approximately proportional to the multiplier F1a YMY0 bN (see formula (23)). The graph ofthis function is given in Fig. 5. The availability of the F1(a) value allows making a quick estimation of the reduction coefcientx, which is approximately equal to

x F1a 9tkp R0 : 25

1

0.8

involves a 5% error, we can state a good agreement between the results. This conrms the correctness of the results forr problem.omplete verication of formula (23) is difcult to do, as the inuence of geometrical nonlinearity in the SIF calculation0.6

0.4

0.2

00 1 1.9 2 10 20 30

P

Fig. 4. The dependence of the multiplier k on the dimensionless pressure p.

It is evident that an increase in p leads to a increase in x and decrease in SIF. Yet this effect can be revealed only when thereduction coefcient at zero p, i.e. xp 0, is not negligibly small as compared with 1. Otherwise, taking into account thedecrease in the small initial value of x at p 0 with increasing p has no practical sense. This takes place when the crack isshort and/or the R0/t ratio is very large. As an example consider a ring with R0/t = 18 and relative crack depth a = 0.5. Then,F1 0.2 and the coefcient of reduction x 0.1 at p 1. If p 2, then as it follows from Fig. 4, kp 0:5 and x increasesapproximately twofold. Then, the dimensionless SIF will decrease from 0.9 to 0.8, i.e., 0.9/0.8 = 1.125 times, which is anoticeable effect. But we need to understand whether in practice the magnitude of p can be large enough to be taken intoaccount.

The value of p cannot vary in wide limits and is restricted by the mechanical strength of pipes. Consider the inuence ofcircumferential stresses ru = PR/t on the coefcient x in some possible cases.The value of p can be presented as

p 12ru1 m2R2

Et2: 26

Assuming that E = 2 105 MPa and m = 0.3, we plot graphs of the inuence of the nominal stress ru on the coefcient of theSIF reduction (Fig. 6) for some specic values of the R /t ratio (20 and 40) and relative crack depth (0.4 and 0.6). As it follows

00.30.60.91.2

0 0.2 0.4 0.6 0.8

F 1

Fig. 5. The dependence of the multiplier F1a YMY0 bN on the relative crack depth a.

4060 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 4052406500.030.060.090.120.15

0 200 400 600PR/t

R 0 /t=20

R 0 /t=40Fig. 6.(a) a = 0 =0.4a0

from the graphs, a noticeable reduction of the dimensionless SIF for typical pipes at a practically possible level of circumfer-ential stresses is observed. Thus, for a pipe with a crack for a = 0.4, R0/t = 20, and ru = 200 MPa, an additional decrease in thecoefcient x due to the action of the internal pressure is equal to Dx = x(200) x(0) = 0.06. The inuence of the pressure fordeeper cracks is more noticeable, for example, for a = 0.6, Dx = x(200) x(0) = 0.15. The above values of x are large enoughto be accounted for in practical calculation of the SIF. =0.6

0

0.1

0.2

0.3

0.4

0 200 400 600PR/t

R 0 /t=20

R 0 /t=40

b

The inuence of nominal circumferential stresses due to internal pressure on the SIF reduction coefcient for two dimensionless depths of the crack:.4; (b) a = 0.6.

5. A thin-walled ring with combined defects

5.1. Symmetric dents under the action of internal pressure P

It can be seen from formula (18) that the dent depth predetermines mainly the maximum value of additional stressesfrom the bending moment. Furthermore, for a geometrically linear case (with no account for the straightening effect of inter-nal pressure), expression (18) for a bending moment fully corresponds to the expression for the dent geometry (1). In thiscase, the maximum bending stress from the additional bending moment is equal to: rM = PR/t 6W/t, where W is the max-imum dent depth. Thus, the total stresses with allowance for hoop stresses are equal to r = PR/t (1 + 6W/t). However, itshould be remembered that these results were obtained with simplications (2) and (4), which impose some inaccuracyfor relatively deep and sharp dents. The analysis by the FEM can be more universal [25] and free from restrictions. The resultof the FEM calculations demonstrates that sharp defects are usually more dangerous. Here, based on the equations of the MIPobtained, we develop a simple approach to consideration of the shape of a dent in the solution for the total stress in a pres-surized pipe.

To this end we consider three types of symmetric defects in a pipe with the initial curvature radius of R0 (Fig. 7). To sim-plify the application of the MIP equations to the given cases we assume that the deformed part of a pipe (dent geometry) canalso be described by a part of a circumference of some constant radius R1. Due to symmetry, we consider a half section only.For convenience, we designate three points for each geometry (Fig. 7) as: B is the center of the dent, A is the transition fromthe dent to the undistorted part of the pipe, and C is the center of the undistorted part of the pipe. Each shape of the dent ischaracterized by the following geometrical parameters: W is the dent depth at the point B; R1 is the dent radius (it can benegative, as for the geometry shown in Fig. 7b); Du is the angular length of the dent which is counted off from the initial

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4061center O of the circumference; w1 is the angle jump at the point B between the axis x (initial direction of the tangent tothe contour) and the tangent to the dent, and w2 is the angle jump at the point A between the tangent to the undistortedcircumference and the tangent to the dent.

Consider each dent separately. For the rst type of the dent (Fig. 7a) we have R1 = R0 and w1 = w2 = Du. The dependencebetween the depth of the dent W and its angular length Du is given by the following expression: W = 2R0(1 cos Du).The second type of the dent (Fig. 7b) is characterized by the following values: R1 = R0, w1 = 0, and w2 = 2Du, andW = 2R0(1 cos Du). Contrary to the rst and the second types, the third one (Fig. 7c) does not retain the original lengthof the deformed part of the circumference. Its specic feature is that the angle w2 is set to be equal to zero. Then from thiscondition, the value of R1 can be determined as: R1 R0 sinDu2 cosDuw cosw, w1 = Du + 2w, and tanw WR0R0 cosDuR0 sinDu .

Note that for the third type of geometry we consider only such dent depth W for which the value of the angle w1 < p/2.Now proceed to the calculation. Assume that a pipe is loaded with internal pressure p only. For the sake of comparability

of the results to be obtained, consider the problem in a geometrically linear formulation. The procedures of the solution forall types of the dents are identical and based on the MIP. Our travel starts from point B. We have here six initial parametersas six unknowns. Using the equations of the rst column of Table 1 at l? 1 and at R = R1, we nd the values of thoseparameters at the end of the zone, i.e. at the point A.

Similarly, we put here six initial conditions (unknowns) for the second (undeformed) zone. We write down six conditionsof equality of those parameters at the end of the rst zone and the beginning of the second one. For example, the condition ofthe force equality with allowance for the angular jump of the tangents to the contour at the point A has the form:

A

R0

R

21

B

O

A

R0

2

B

O

A

R0

R1

1WW

W

B

a b c

Fig. 7. Three types of the dent geometry: (a) rst type of dent; (b) second type of dent; (c) third type of dent.

N1f cosw2 Q1f sinw2 N2b; 27aQ1f cosw2 N1f sinw2 Q2b; 27b

where the upper indexes mean belonging to a certain zone and the lower ones mean the beginning (b) and the end (f) ofthe corresponding zone. Similar conditions are written for the displacements u and w. Due to symmetry, another six bound-ary conditions are written by xing the displacement components of a pipe as a rigid body. At the end of the second zone (forthe point C) they are:

w2f 0; u2f 0 and h2f 0: 28The boundary conditions at the point B are slightly more complicated. Here we have to take into account that, due to sym-metry, the projections of the forces on the axis y and the projections of the displacements on the axis x should be equal tozero. The boundary conditions are:

N1b sinw1 Q1b cosw1 0; h0 0 and u1b cosw1 w1b sinw1 0: 29Conditions (27)(29) allow us to solve the system of equations and dene all 12 unknowns which completely determine thestressstrain state in a pipe.

In this work we consider the stresses, which are induced by the bending moment, axial and transverse forces, at the pointB only. Denote the nominal bending moment by MN and nominal axial force by NN:

MN rMt2=6 PR0 Wtt 30

For the sake of convenience the results obtained for each dent are presented as the ratio of the real moment to the nominalone (30), i.e., m =M/MN, and as the ratio of the difference between the real and nominal axial forces to the same nominalforce at the point B, i.e., Dn = (NN N)/NN = (N + PR0)/PR0.

The dependence of the relative bending moment m from the ratio of the maximum dent depth W to the initial radius R0for each type of the dent is shown in Fig. 8a. Note that in contrast to the rst and second types of the dent, where the value ofW uniquely denes the dent length, the length of the dent of the third type, Du, is an independent variable. Thus, to compare

Type I

Fig. 8.relative

4062 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 4052406500.00 0.06 0.12 0.18 0.24 0.30

Inuencing of relative depth of dent in a central (most deep) point for three types of geometry of dent: (a) the relative bending moment; (b) thechange of the force Dn.W/R0.00

0.200.400.60 Type II

Type III0.80 nb0.60

0.70

0.00 0.06 0.12 0.18 0.24 0.30 W/R00.80the results for dents of different types, we take that the length of the dent of the third type is determined by its depth usingthe formulaW = 2R0(1 cosD u). As is evident from the graphs, the values ofm decrease with the growth of the relative dentdepthW/R0, while the maximum value is attained for the third type. With an increase in the relative dent depth the valuemdecreases insignicantly. It is of interest that in work [26] the estimation of m yielded m = 0.85. As can be seen from thegraphs, this estimate is valid for practically important cases of 0.05 6W/R0 6 0.1.

Fig. 8b shows the dependences of the relative change in the axial force Dn for dents of all three types. In work [26] it wassuggested that this relative change can be described by the following equation: Dn = 0.9W/R0. The graph shown in Fig. 8b for

0.90

1.00

m Type I

Type II

Type III

a

a dent of the second type corresponds best to the above equation. The change in the axial force Dn here has a maximum for adent of the third type. Remembering that the total axial force, N, is equal to N = PR0 (1 + Dn), it can be stated that N is min-imal for this dent type. Note, that the values of N for types I and III are given in the immediate vicinity to the point B.

To establishwhich of the three types of defects is themost dangerous, i.e., for strength analysis, we should additionally takeinto account the angular jump of the tangent vector at the point B (types I and III). This leads to the appearance of the trans-verse force Q in the immediate vicinity of the point B. Obviously, for strength analysis we need to have some combined char-acteristic that would take into account both forces N and Q. Using Trescas condition we determine the equivalent force Ne as

Ne PR01 Dn2 4Q=PR02

q: 31

We introduce the notion of the equivalent relative force, Dne, which is described by Dne = Ne/PR0. The dependence of theequivalent relative force Dne on the relative dent depth is shown in Fig. 9. Note that for the second type of the dentQ(u = 0) = 0 and Ne = N. As can be seen from Fig. 9, the third dent type gives the maximum values of Ne, which, e.g., forW/R0 = 0.15 is equal to 1.37PR0. The same is observed for the bending moment, which also has the maximum values forthe third dent type (Fig. 8a). Thus, the third dent type is the most dangerous one from the strength point of view.

Continue the analysis for a dent of the third type. As it was stated above, its shape is described by two independent vari-ables: the length, Du, and depth, W. To analyze the inuence of the dent length we x the depth ratio W/R0. Fig. 10 presentsan example of such computations at W/R0 = 0.1 for different values of Du. Note that for dents of type I and II the angularlength Du = 18.3 corresponds to the above value of the depth. Reduction of the length at constant depth increases the sharp-ness of the dent. Fig. 10 clearly demonstrates the inuence of the dent sharpness on the stress components. While the bend-ing moment grows insignicantly with increasing sharpness and equals to 0.9MN, the maximum value of the equivalent forcegrows with it and equals to 1.8PR0. It is easy to account for the latter magnitude. For a very shallow but sharp (w1? p/2) dentwe have limw1!p=2N ! 0 and limw1!p=2Q ! PR0. Thus Trescas condition (31) yields Dne? 2.

5.2. SIF computation for a crack emanating from a dent

Here we have a problem similar to that treated in Section 4. The only difference is that in the cracked section we initiallyhad not only an axial force but also a bendingmoment. Designate it byMD and introduce a bending stress rMD byMD = rMDt2/6.

Type I

Type II1.60ne

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 40630,75

1

1,25

5 10 15 20 25 30

Fig. 10. The inuence of the dent length at the xed depth W/R0 = 0.1 on the values of the bending moment and relative equivalent force at the point B forthe third type of dent.1,5W/R0

Type III

0.60

0.80

1.00

1.20

1.40

0.00 0.06 0.12 0.18 0.24 0.30

Fig. 9. The inuence of relative dent depth on the values of the relative equivalent force Ne/PR0 at the dent central point for three types of dents.

1,75

2m , ne

n em

[12] Sc[13] M

[15] Bain

[16] W

4064 I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 405240652003;125:8590.[17] Cheng W, Finnie I. Determination of stress intensity factor for partial penetration axial crack in thin-walled cylinders. J Engng Mater Technol

1986;108:3683.[18] Orynyak IV, Rozgonyuk VV, Yakovleva Ye. Generalization of the Cheng and Finnie method for calculating stress intensity factor for part-through

longitudinal cracks in a thick-walled ring. J Mach Manuf Reliab 2006;2:11624.[19] Orynyak IV, Radchenko SA. A mixed-approach analysis of deformations in pipe bends, Part 3. Calculation of bend axis displacements by the method of

initial parameters. Strength Mater 2004;5:46372.rtholom G, Keim E, Senski G, Steinbuch R, Wellein R. Determination of critical circumferential through-wall crack sizes regarding load reduction bycreasing exibility in piping systems. Int J Press Vessel Piping 1997;71:15564.ang X, Reinhardt W. On the assessment of through-wall circumferential cracks in steam generator tubes with tube supports. J Press Vessel Tech[14] Vainshtok VA. Engineering methods of computational fracture mechanics based on the application of weight functions. Strength Mater1988;3:30916.hindler HJ, Cheng W, Finnie I. Experimental determination of stress intensity factors due to residual stresses. Exp Mech 1997;37(3):2729.archand N, Parks DM, Pelloux RM. KI-solutions for single notch specimens under xed end displacements. Int J Fracture 1986;31:5365.To determine an additional bending moment induced due to the concentrated compliance in the cracked section, we use Eq.(22).

By repeating the procedures described in Section 4, considering that the dent depth is insignicant (thus N = PR0), andneglecting the geometrical nonlinearity, it is easy to get a simple formula for an additional bending moment induced by acrack:

M0 rMt2=6 PR0=tbN rMDbMR0=9t bM: 32

Thus, the nal formula for the SIF computation has the form:

KI PR0=tY0 rMD rMYMp ap : 33

This formula allows obtaining readily the SIF values for cracks emanating from the dents at low level of internal pressure. Fora more correct determination of the SIF it is necessary to implement the numerical procedure based on the MIP formulasgiven in Table 1.

6. Conclusion

The Cheng and Finnie crack compliance method and the method of initial parameters are used jointly for the SIF compu-tation in an elastic thin-walled ring with radial cracks. The peculiarities of realization of their joint usage dene the principalnovelty of the work. The main results obtained are:

1. For a circular ring (pipe) with initial imperfections the simplifying system of governing equations in a geometrically non-linear formulation is written and solved as the sum of particular solution and a general solution of homogeneous system.The former takes into account the imperfection of the form while the latter is presented in the form convenient to appli-cation in the method of initial parameters, i.e. it gives relation of displacements and forces in any point of part of ring withrespect to those at the beginning of the part.

2. A simplied analytical expression for the SIF in a pressurized pipe for R0/t > 5 with a radial crack in a geometrically non-linear formulation is obtained for the rst time. Thus, the dimensionless SIF does not remain a constant but slowlydecreases with increasing internal pressure. This effect is of practical signicance for thin walled pipes if the relative crackdepth is bigger than 0.4 and the value of circumferential stresses exceeds at least 50 MPa.

3. Dents of three different types which consist of two parts each with constant radius of curvature have been investigatedwith the account taken of the inuence of their depth on the bending moment and equivalent axial force in the deepestpoint. The sharper the dents the more dangerous it is from the strength point of view.

4. A simplied formula for the SIF computation for a crack emanating from the deepest point of the dent is proposed. Theapplication of it is restricted to the cases when the maximal angle of the distortion of the form of the ring from the idealcircular form is very small as compared with unity.

References

[1] Cosham A, Hopkins P. The pipeline defect assessment manual. In: Proceedings of IPC 2002 international pipeline conference, Calgary, Alberta, Canada;29 September3 October 2002.

[2] Hopkins P. The application of tness for purpose methods to defects detected in offshore transmission pipelines. In: Conference on welding and weldperformance in the process industry, London; 1992.

[3] Dowling AR, Towley CHA. The effect of defect on structural failures: a two-criterial approach. Int J Press Vessel Piping 1975;3(2):77107.[4] Lubis A, Boyle JT. The pressure reduction effect in smooth piping elbows revisited. Int J Press Vessel Piping 2004:11925.[5] Calladine CR. Structural consequences of small imperfections in elastic thin shells of revolution. Int J Solids Struct 1972;8:67997.[6] Croll JGA, Kaleli F, Kemp KO. A simplied approach to the analysis of geometrically imperfect cooling tower shells. Engng Struct 1979;1(1):928.[7] Papkovich PF, editor. Proceedings of structural mechanics of the ship, vol. 2. Leningrad: Sudpromgiz; 1962.[8] Fitness-For-Service. API Recommended Practice 579, 1st ed., American Petroleum Institute; January 2000.[9] Rice JR, Levy N. The part-through surface crack in a elastic plate. J Appl Mech 1972;39:18594.[10] Delale F, Erdogan F. Line spring model for surface crack in Reissner plate. Int J Engng Sci 1981;19:1331.[11] Cheng W, Finnie I. Measurement of residual hoop stresses in cylinders using the compliance method. J Engng Mater Technol 1986;108:8792.

[20] Orynyak IV. Constructing the weight function for at solids with cracks. Strength Mater 1990;8:11227.[21] Orynyak IV, Borodii MV. Use of an approximate fundamental solution for a half plane with an edge crack in a combined method of weight functions.

Mater Sci 1995;30:1058.[22] Orynyak IV, Borodii MV. The combined weight function method application for a hole emanated crack. Engng Fract Mech 1994;48(6):8914.[23] Ong LS. Allowable shape deviation in a pressurized cylinder. J Press Vessel Tech 1994;116(3):2747.[24] Zahoor A. Closed from expressions for fracture mechanics analysis of cracked pipes. J Press Vessel Tech 1985;107(2):2035.[25] Carroll B, Dinovitzer A, Lazor R. A defect-assessment model to prioritize dents for excavation and repair. J Pipeline Integrity 2002:2.[26] Shannon RW. Failure behavior of line pipe defects. Int J Press Vessel Piping 1974;2:24355.

I.V. Orynyak, Ye.S. Yakovleva / Engineering Fracture Mechanics 75 (2008) 40524065 4065

Application of the crack compliance method to long axial cracks in pipes with allowance for geometrical nonlinearity and shape imperfections (dents)IntroductionProblem formulationGeometry and governing equationsThe jumps of displacements and rotation angles in the cracked sectionThe use of the method of initial parameters (MIP)

Complete solution for a ring in a geometrically nonlinear formulationComputation of the SIF for a single crack in a ringA thin-walled ring with combined defectsSymmetric dents under the action of internal pressure PSIF computation for a crack emanating from a dent

ConclusionReferences