Analysis of stress intensity factor (SIF) for cracked tubular K-joints subjected to balanced axial load

Engineering Failure Analysis 13 (2006) 44–64 www.elsevier.com/locate/engfailanal

Analysis of stress intensity factor (SIF) for cracked tubular K-joints subjected to balanced axial load Yong-Bo Shao School of Civil Engineering, Yantai University, Yantai City 264005, PR China Received 29 April 2004; accepted 7 December 2004 Available online 31 March 2005

Abstract Numerical modelling of cracked tubular K-joint is presented in this study. A new mesh generator for a tubular K-joint with a surface crack was developed and finite element analysis of stress intensity factor (SIF) for cracked K-joint was conducted by using the proposed numerical modelling. Numerical results of SIF were then verified from experimental tests on a full-scale K-joint specimen. The verifications show that numerical results agree reasonably well with experimental results. Based on the accuracy of numerical analysis, 5120 numerical models of K-joints subjected to balanced axial load were analyzed. SIF results of these numerical models were used to build up a database. An interpolation method was introduced to calculate SIF of each model in a valid range by using the analyzed SIF results in the database. The accuracy of the SIF obtained from interpolation method was verified from 972 numerical models which were also analyzed from finite element analysis. The comparisons between SIFs from interpolation method and numerical results show that the introduced interpolation method can provide accurate SIF values for cracked tubular K-joints and thus it is both reliable and efficient to be used in design in practice.
2005 Elsevier Ltd. All rights reserved. Keywords: Numerical modelling; Tubular K-joints; Mesh generator; Stress intensity factor (SIF); Balanced axial load

1. Introduction Tubular K-joints have been widely encountered in offshore structures and they are always subjected to cyclic loads caused by seawater waves, and hence fatigue failure has become a very common phenomenon. For this reason, many researchers are interested to study the fatigue, fracture and crack growth behavior of these K-joints so as to predict the remaining service life. The prediction of the damaged tubular K-joints depends very much on the accuracy of the stress intensity factors (SIFs). Although many researchers have E-mail address: [email protected]. 1350-6307/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2004.12.031

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

45

carried out extensive work on SIFs for the past decades, reliable and accurate formulas of SIFs are still very scarce for K-joints. Generally, both experimental tests and finite element analysis can be used to analyze SIFs of cracked tubular K-joints. However, it is known that fatigue tests are very time-consuming and crack initiation and propagation are extremely difficult to be monitored accurately. Fatigue tests are also limited from the complicated geometry of K-joints and instruments which can provide actual fatigue loadings. Up to today, no reports have been found in fatigue tests on full-scale tubular K-joints. Hence, in this study, fatigue test on a full-scale tubular K-joint specimen was conducted and the fatigue performance was investigated. Crack initiation and propagation were monitored step-by-step in fatigue tests by using alternate current potential drop (ACPD) technique. SIFs of the specimen were then obtained from Paris equation and they can be used as benchmark to verify the accuracy of numerical analysis of SIF. Compared to experimental tests, numerical analysis can estimate SIF of cracked tubular K-joints very quickly. However, the accuracy of the SIF values in numerical analysis depends very much on three most important factors: the geometrical models used to describe the weld size and the crack details, the grading of the finite element mesh used near the joint intersection and around the crack front, and the aspect ratio of the elements along the crack front. In this paper, a correct definition of the crack surface is used in the geometrical model and a systematic modelling procedure for a general welded and cracked tubular K-joint was proposed. In the mesh generation, five types of elements, i.e., hexahedral, prism, quarter-point collapsed prism, tetrahedron and pyramid elements, as shown in Table 1, are used to model the surface crack and the other zones of the tubular joint. In the mesh generation process for K-joint, the entire structure was divided into different distinct zone. For each zone, the mesh is generated separately using different densities based on mesh quality requirements. After the mesh of all the zones have been completed, they are merged to form the entire mesh of the structure. Using this method, the field with high stress gradient is refined with high quality mesh to ensure sufficient number of elements in this zone. At the same time, the fields far away from the crack and

Table 1 Element types used in mesh generation of cracked tubular K-joint Element types

No. of nodes

1.

Hexahedral/cubic element – (H20)

20

2.

Prism/wedge – (P15)

15

3.

Quarter point/crack element – (QP15) (collapsed prism)

15

4.

Tetrahedron – (T10)

10

5.

Pyramid – (PR20) (collapsed hexahedral)

20

46

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

the weld, which have little effects on the stress intensity factors, have relatively coarse mesh. The fields of different mesh densities are joined together by some other fields which are called the transition zones. Hence, the density and aspect ratio of the elements can be controlled easily. Finite element analysis of SIF using the proposed numerical modelling has been committed. Numerical results were evaluated from experimental test results and the comparisons agree very well. Thereafter, a database was built up from finite element analysis of 5120 numerical models of K-joints subjected to balanced axial load, which is the most common loading case in practice. A new interpolation method was introduced and this method can calculate SIF of any cracked tubular K-joint under balanced axial load in a valid range by using the analyzed numerical models in the database. The accuracy of SIF estimated from this method was then verified from other 972 numerical models and the verification shows that this new interpolation method is efficient and accurate and it is reliable to be used for design purpose.

2. Numerical analysis of SIF 2.1. Geometrical modelling of surface crack In numerical analysis of cracked tubular K-joint, geometric and numerical modelling of the weld and the surface crack are critical to the accuracy of SIF results. Generally, the intersecting curve between the brace and the chord is a complicated 3D curve. The curve along the weld toe, which is formed after the brace and the chord are jointed by welding, is extremely complicated. Cao et al. [1] developed the formula of the intersecting curve and thus it is practicable to obtain the weld toe curve after the weld size has been defined by some specifications. In this study, AWS specifications [2] which define the weld size are adopted. As the weld quality and weld size have significant effects on finite element analysis for tubular structures, the definition of the weld size in numerical analysis must satisfy with AWS specifications [2]. Lie et al. [3] proposed a detailed method to construct accurate and consistent geometric models for the weld and the weld size defined in this method can satisfy with AWS specifications [2] quite well. The method was adopted in present study to model the weld of tubular K-joint. Besides the geometric modelling of the weld, numerical modelling of surface crack is more critical in numerical analysis for SIF. In numerical modelling, semi-elliptical crack is generally accepted by most researchers. It is also adopted in this study. When a crack initiates from the surface of the chord of a K-joint, it will propagate through the chord thickness in a special direction in which the energy requirement is minimal. The crack will propagate on a 3D curve which is called the crack surface where the crack front lies on. As shown in Fig. 1, the crack surface is forming by joining a series of straight lines WoD along the weld path. WoD is passing through the Z-axis and the thickness of the cracked surface is always equal to tc. Wo(XWo, YWo, ZWo) is the point on the weld profile and on the outer horizontal cylinder, and point D will be located according to the assumptions that Point D is on the inner horizontal cylinder, ! ! j W o D j ¼ tc ¼ R1
R2 , and the line W o D will pass through Z-axis. The first assumption means that: X 2D þ Y 2D ¼ R22 :

ð1Þ

The second assumption implies that: ðR1
R2 Þ2 ¼ ðX W o
X D Þ2 þ ðY W o
Y D Þ2 þ ðZ W o
Z D Þ2 :

ð2Þ

From the third assumption, the following equation can be obtained: XWo XD ¼ : YWo YD

ð3Þ

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

47

Fig. 1. Definition of crack surface.

Thus the determination of the coordinate of point D can be obtained and expressed as follows: 9 8 R2 9 8 > = > = < R1 X W o > D ¼ Y D ¼ R2 Y W o > : > > > R1 ; ; : ZD ZW o

ð4Þ

After the crack surface is defined, it is necessary to define the crack front which can exist on any location of the crack surface. As mentioned previously, semi-ellipse is used to model the crack shape in this study. In order to model the crack front conveniently in 3D space, it is often more convenient to define it on a normalized u 0
v 0 plane and then map it onto the crack surface as shown in Fig. 2. The u 0 -axis relates to the half crack length, lcr, while v 0 relates to the crack depth, d. In the mapping approach, a crack with any length and size can be modelled at any location. Suppose that the crack front curve is defined by the point Cr 0 in the u 0
v 0 space with depth equal to d as shown in Fig. 2. By using a similar approach for the computation of point D, it can be shown that the coordinates of the point Cr are given by:

Fig. 2. Mapping a surface crack from 2D to 3D.

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

8
 9 > 8 d 1
> X > > > R 1 < Cr = < Cr ¼ Y Cr ¼ 0 > > ; > > : > Z Cr : 0

0
 1
Rd1 0

9 9 8 0> > > => = Y W 0 >> o > > ; : > ; ZW o 1

ð5Þ

From Eq. (5), any point on the crack front can be determined in a 3D space. Fig. 3 illustrates a sample model of half K-joint with a surface crack along the weld.

2.2. Numerical modelling The crack surface in tubular structures is a complicated double-curved surface and it must be defined reasonably in numerical modelling. The double-curved crack surface makes it very difficult to generate high-quality mesh around the crack front. Hence, different types of elements are used to avoid high aspect ratio of the elements in this region. To generate the mesh of tubular K-joints with a surface crack which can be located at any position with any length and depth, five types of elements listed in Table 1, are used in the present mesh generation procedure. Quarter-point crack tip elements are used along the crack front to simulate the displacements singularity. For these elements, the mid-side nodes are moved to the quarter point for the edge connected to the crack front. Prism elements are employed to model the transition zone between the region near the crack surface and the far field region. Tetrahedral elements are used to link the quarter-point crack tip elements and other types of elements which enclose the crack front. Pyramid elements are used to connect the prism elements with tetrahedral elements around the crack front. In the fields far away from the crack, hexahedral elements are used to model the remaining part of the members. The locations of these different types of elements are illustrated in Fig. 4. It can be seen in Fig. 4, the entire structure is divided into several distinct zones and in each zone the mesh can be generated separately. Therefore, different mesh densities for different zones can be obtained and thus it is easy to control the mesh quality according to different requirements. After the mesh of all the zones has been completed, all the mesh is then merged together to form the mesh of the entire structure. Fig. 5 shows the mesh of a K-joint after merging the mesh of all the zones.

Fig. 3. Crack surface along the weld for a half K-joint sample.

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

Fig. 4. Mesh generation for cracked K-joint.

49

50

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

Fig. 5. Mesh of K-joint after merging.

2.3. Evaluation of SIF by J-integral in finite element analysis In numerical analysis of SIF, J-integral is adopted in this study. J-integral is a measure of the strain energy in the region of the crack tip. Shih and Asaro [4] had proposed the relationship between the J-integral and the SIFs as: J¼

1 T K  B  K; 8p

ð6Þ

where K = [KI, KII, KIII]T is vector of stress intensity factors for mode I, mode II and mode III. B is called the pre-logarithmic energy factor matrix. For homogeneous isotropic materials, B is a 3 · 3 diagonal matrix and the above equation can be simplified as: J¼

1 2 1 2 K ; ðK I þ K 2II Þ þ 2G III E

ð7Þ

where E ¼ E for plane stress and E ¼ E=ð1
t2 Þ for plane strain, axisymmetry, and 3D problems. It is obvious from Eq. (7) that it is easy to obtain the value of J-integral from KI, KII and KIII. However, it is not feasible to obtain KI, KII and KIII from J-integral directly. This means the J-integral method can not be easily used to analyze mixed mode problem directly. Based on this problem, Shih and Asaro [4] had proposed an interaction integral method to calculate K from the J-integral. In the interaction integral method, generally, the J-integral for a given problem can be written as:  1 
1
1 K I B
1 J¼ ð8Þ 11 K I þ 2K I B12 K II þ 2K I B13 K III þ ðterms not involving K I Þ : 8p Here, J-integral is defined as an auxiliary pure Mode I crack-tip field with stress intensity factorI as follows: J Iaux ¼

1 k I  B
1 11  k I : 8p

ð9Þ

Superimposing the auxiliary field onto the actual field yields: J Itot ¼

1
1
1 ½ðK I þ k I ÞB
1 11 ðK I þ k I Þ þ 2ðK I þ k I ÞB12 K II þ 2ðK I þ k I ÞB13 K III 8p þ ðterms not involving K I or k I Þ:

ð10Þ

Since the terms not involving KI or kI in JtotI and J are equal, the interaction integral can be defined as: J Iint ¼ J Itot
J
J Iaux ¼

 k I 
1
1 B11 K I þ B
1 12 K II þ B13 K III : 4p

ð11Þ

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If the calculations are repeated for Mode II and Mode III, a linear system of equations can be obtained as:

2

J Iint

3

2


1
1 k I B
1 11 K I þ k I B12 K II þ k I B13 K III

3

1 6 6 II 7 7
1
1 4 J int 5 ¼ 4 k II B
1 21 K I þ k II B22 K II þ k II B23 K III 5: 4p
1
1 J III k III B
1 int 31 K I þ k III B32 K II þ k III B33 K III

ð12Þ

If the values of ka(a = I, II, III) are assigned unit values, the solution of the above equations leads to: ð13Þ

K ¼ 4pB  J int :

Therefore, once Jint is obtained, K can be easily calculated from Eq. (13). The detailed calculations of Jint can be found in the paper published by Shih and Asaro [4], and this method has been implemented in the ABAQUS [5] general finite element software. 3. Fatigue test on K-joint 3.1. K-joint specimen The basic dimensions of the K-joint specimen are given in Fig. 6. Structural steel pipes to API 5L Grade B specifications [6] were used for the specimens. A specially designed test rig as shown in Fig. 7 was used to test the tubular K-joint joints. The rig is capable of applying static loading on a joint specimen to determine the HSS distributions in the joint, as well as cyclic loading to study the fatigue performance and fracture behaviour of the joint. It is illustrated in Fig. 8 that three actuators can be operated individually or concurrently to enact a multi-axis loading condition.

Fig. 6. Configuration of the tubular K-joint specimen.

52

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

Fig. 7. General view of the test rig.

Fig. 8. Close view of the three actuators.

3.2. Stress distribution Fatigue crack on K-joint generally initiates from the location of peak hot spot stress (HSS). Therefore, static tests were first carried out to investigate the stress distribution around the intersection of Kjoint specimen subjected to combined axial load (AX) and in-plane bending load (IPB), and determine the peak HSS position on the joints before fatigue testing. Numerical analysis of corresponding uncracked K-joint was also conducted and the stress distributions along the weld toe then can be obtained both from experimental test and finite element analysis. In static test, the applied load is AX = 180 kN and IPB = 18 kN. The loading case and boundary conditions of the specimen are illustrated in Fig. 9. The stress distributions along the weld toe are plotted in Fig. 10 and some critical positions in tubular K-joint are illustrated in Fig. 11. Apparently, numerical results agree well with experimental results and the peak HSS locates at crown and almost symmetrical to the crown. From the stress distributions, it can be inferred that surface crack will initiate at crown and propagate towards to saddle in a nearly symmetrical way.

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

53

AX = 180 kN

IPB = 18 kN

Fig. 9. Loading case and boundary conditions for K-joint specimen. 600

Chord Brace

Stress (MPa)

400 200 0 -200 -400 -600 0

50

100

150

200

250

300

350

Angle from Heel (Degree) Fig. 10. Stress distribution along weld.

3.3. Fatigue test In fatigue test, alternating current potential drop (ACPD) technique was used to monitor crack developments in fatigue test of the K-joint specimen. ACPD is a well-established technique [7] and it can capture accurate information of crack growth in fatigue test. This technique was adopted in this study to monitor crack initiation and propagation during the fatigue test on the K-joint specimen.

54

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

Fig. 11. Definition of some critical positions.

For the fatigue test, the specimen was tested in air under a sinusoidal constant amplitude loading until failure using the test rig. A stress ratio of R = 0 and a frequency of 0.2 Hz was used throughout the test. The cyclic loading patterns applied to the specimen are shown in Fig. 12. During the test, all the actuators were preset under load control condition. Notice that in fatigue test combined AX and IPB load was adopted because IPB can provide higher peak HSS and thus make the crack initiate and propagate to fail in a relatively shorter period. Number of cycles of cyclic load was recorded during the test and simultaneously, crack depth and length were monitored from ACPD measurements. Crack developments were then able to be plotted in Fig. 13. In Fig. 13, crack length was measured from crown and it can be seen that the position of the deepest point was almost kept constant at the crown although in later stage the deepest point has a little offset. This small offset may be due to the actual weld size along weld toe on the chord is not absolutely symmetrical, thus the stress distributions shown in Fig. 10 are not absolutely symmetrical. When crack depth reached to 23 mm from ACPD readings, the test was stopped. Fig. 14 shows the crack along the weld toe after fatigue test. It is almost symmetrical to the crown. After the fatigue test finished, crack surface was split into two parts. The crack surface was then able to be seen and it is illustrated in Fig. 15. Fig. 15(a) shows that the surface crack is reasonably acceptable to be assumed as semi-ellipse. Apparently, it is clear to see from Fig. 15(b) that crack surface defined in Fig. 1 is reasonable compared to the observed crack surface. 200

Axial Load IPB Load

180 160

Load (kN)

140 120 100 80 60 40 20 0 0.0 -20

0.8

1.6

2.4

3.2

4.0

Time (Second) Fig. 12. Cyclic load applied in fatigue test.

4.8

Crack Depth (mm)

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64 24 20 16 12 8 4 0 -120 -100 -80 -60 -40 -20

0

20

40

60

80 100 120

Crack Length (mm) Fig. 13. Crack developments from ACPD readings.

Fig. 14. Surface crack at the crown.

Fig. 15. Close view of crack surface.

55

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

3.4. Evaluation of SIF from test The experimental stress intensity factors of the deepest point can be obtained from the well known Paris
equation: m

da=dN ¼ CðDKÞ ;

ð14Þ

where da/dN is the crack growth rate which are already obtained from the experimental test, DK is the range of the stress intensity factor, C and m are material constants. These values of C and m supplied by the steel manufacturer, are 1.45 · 10
11 (m/cycle) (MPa m1/2)
2.75 and 2.75, respectively. From the ACPD readings, da/dN against a/T is plotted in Fig. 16. The experimental stress intensity factors at the deepest point are deduced from Eq. (14), and they are plotted in Fig. 17 together with the numerical values of KI, KII, KIII and Ke. KI, KII and KIII are stress intensity factors for three crack Modes,

1E-3

da/dN (mm/Cycle)

1E-4

1E-5

1E-6

1E-7 0.0 0.1 0.2 0.3

0.4 0.5

0.6

0.7 0.8

0.9

1.0

a/T Fig. 16. Crack growth rate of the deepest point.

45.0 37.5

1/2

SIF (MPa*m )

30.0

Experimental Results KI KII KIII Ke

22.5 15.0 7.5 0.0 -7.5 -15.0 -22.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a/T Fig. 17. Comparisons of SIF results from experiment and numerical analysis.

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

57

respectively. Ke, which is the crack driving force parameter for a mixed mode fracture problem, is an equivalent stress intensity factor. This parameter was proposed by Chong Rhee [8] and it is expressed as: ð15Þ K e ¼ ½K 2I þ K 2II þ K 2III =ð1
tÞ1=2 : Obviously, numerical analysis provided reasonably accurate estimations of SIF for the cracked K-joint in the process of crack propagation. It can also be seen that the two values of KI and Ke are very close to each other. This means that crack Mode I is dominant in the mixed crack mode whilst KII and KIII are much smaller compared with KI. The comparisons of SIF from different methods show that the proposed numerical modelling in this study can provide accurate and reliable estimation of SIF for cracked tubular K-joint. 4. Interpolation method used to estimate SIF of K-joints subjected to balanced axial load 4.1. Parametric studies Based on the accurate and reliable modelling introduced previously, 5120 numerical models of cracked tubular K-joint subjected to axial load were analyzed in this study. Balanced axial load is a dominant loading case for tubular K-joint used in engineering structures although some other loading cases also exist. Under this loading case, the peak HSS always locates at crown symmetrically. Therefore, in all the analyzed models, the surface crack is located symmetrically to the crown. Balanced axial load, as illustrated in Fig. 18, is defined as follows: P 1 sin h1 ¼ P 2 sin h2 : ð16Þ In numerical analysis of SIF, some geometric parameters, which are illustrated in Fig. 19, are generally used. These geometric parameters are also adopted in this study. The valid range of these parameters in the 5120 numerical models is listed as follows: P2

P1

θ1

θ2

Fig. 18. Definition of balanced axial load. d or d2

t or t1

d or d1

Brace 2 Brace 1

t or t2 g Saddle

θ2

θ1

Heel

Crown

D e T

Chord L

α =2L/D

β = d/D

τ = t/T

ζ = g/D

γ = D/2T

Fig. 19. Geometrical parameters of tubular K-joint.

58

         

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

t1 = t2 = t. d1 = d2 = d. c = D/2T 2 [12, 30]. b = d/D 2 [0.3, 0.6]. s = t/T 2 [0.25, 1.0]. c/a 2 [5, 8]. a/T 2 [0.1, 0.7]. h1 2 [30, 60]. h2 2 [30, 60]. e = 0.

where a is crack depth and c is half of crack length. The details of the geometric parameters of these 5120 numerical models are tabulated in Table 3. All the analyzed numerical models were used to build up a database which is shown in Fig. 20.

Fig. 20. Database of SIF for K-joints under balanced axial load.

Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

59

4.2. Interpolation method In this study, Lagrange interpolation method was introduced to analyze SIF of K-joints subjected to balanced axial load from the database which stores the SIF results of 5120 numerical models. It is well known that for 1D problem, consider a set of N distinct points {x1, x2,   , xN} in x-axis and N corresponding function values {f1, f2,   , fN}. The idea of Lagrange
s method is to solve N simple interpolation problems obtained by taking the distinct abscissas x1, x2,   , xN as already presented, but taking the N primitive sets of f values. f1; 0;    ; 0g; f0; 1;    ; 0g;    ; f0; 0;    ; 1g in turn. In the case that N distinct abscissas x1,x2,   , xN are given, using the N data sets, N cardinal functions L1,L2,   , LN are constructed with the following properties:  1; i ¼ j; ð17Þ Li ðxj Þ ¼ 0; i 6¼ j: After Li(xj) was defined, a polynomial which passes through the N distinct points (xi, fi) (i = 1, 2,   , N) can be obtained as follows: N X pðxÞ ¼ f1 L1 ðxÞ þ f2 L2 ðxÞ þ    þ fN LN ðxÞ ¼ fi Li ðxÞ: ð18Þ i¼1

Obviously, from Eq. (18) it is found p(xi) = fi, i = 1, 2,   , N. In Eq. (18), Li(x) is called interpolation function and it is determined from the following equation: , N N ðx
x1 Þ    ðx
xi
1 Þðx
xiþ1 Þ    ðx
xN Þ Li ðxÞ ¼ ¼ P ðx
xj Þ P ðxi
xj Þ: ðxi
x1 Þ    ðx
xi
1 Þðx
xiþ1 Þ    ðx
xN Þ j ¼ 1 j¼1 ð19Þ j 6¼ i j 6¼ i Note that the factor x
xi is missing from the numerator and xi
xi is missing from the denominator. After Li(x) has been determined, p(x) can be obtained from Eq. (18). Then for any point xa (x1 6 xa 6 xN), the estimation of function value p(xa) can be obtained from Eq. (18). Generally, when N is big enough, p(xa) is as close as possible to the real function value f(xa). N-dimensional problem can be implemented by simply getting the product of N One-Dimensional interpolation functions. Take Two-Dimensional problem as an example. Assume the cardinal coordinate system is {a1, a2}. A set of nodes are arranged in a net with a column of r + 1 and a row of p + 1. It is shown in Fig. 21. Now it is necessary to construct the interpolation function Ni of the node on Ith row and Jth column. It is known that for One-Dimensional problem: . rþ1 ða1
a10 Þða1
a11 Þ    ða1
a1I
1 Þða1
a1Iþ1 Þ    ða1
a1r Þ ðrÞ 1 1 ¼ P ða
a Þ ða1I
a1j Þ; LI ða1 Þ ¼ 1 j ðaI
a10 Þða1I
a11 Þ    ða1I
a1I
1 Þða1I
a1Iþ1 Þ    ða1I
a1r Þ j ¼ 0 ð20Þ j 6¼ I

ðpÞ

LJ ða2 Þ ¼

pþ1 ða2
a20 Þða2
a21 Þ    ða2
a2J
1 Þða2
a2J þ1 Þ    ða2
a2p Þ ¼ P ða2
a2j Þ=ða2J
a2j Þ: 2 2 2 2 2 2 2 2 2 ðaJ
a0 ÞðaJ
a1 Þ    ðaJ
aJ
1 ÞðaJ
aJ þ1 Þ    ðaJ
a2p Þ j ¼ 0 j 6¼ J

ð21Þ

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

1 (0, p) (I, J)

(r, p)

(0, 0)

(r, 0)

1 Fig. 21. Lagrange interpolation scheme.

ðrÞ

Then the interpolation function Ni to be constructed can be expressed as a product of LI ða1 Þ and

ðpÞ LJ ða2 Þ,

ðrÞ

ðpÞ

N i ¼ N IJ ¼ LI ða1 Þ  LJ ða2 Þ:

ð22Þ

It can be checked that Ni is equal to 1 at node i and equal to zero at other nodes. Form Eq. (22), interpolation function Ni in N-Dimensional problem can be deduced. For N-Dimensional problem, the cardinal coordinate system can be expressed as {a1, a2,   , aN}. For each dimension, Mi distinct nodes are used to get the interpolation function. Then the interpolation function at node i (determined by a set number {k1, k2,   , kN}) can be obtained by the same idea as expressed in Eq. (22): ðM
1Þ

N i ¼ Lk1 1

ðM
1Þ

ða1 Þ  Lk2 2

ðM
1Þ

ða2 Þ     LkN N

ðaN Þ:

ð23Þ

Note that for above N-Dimensional problem, there are altogether M1 Æ M2    MN interpolation functions. Each function is obtained by multiplying N terms. Once Ni is determined, any function value f ða1I ; a2I ;    ; aNI Þ can be obtained by the following equation: f ¼

M X

ð24Þ

N j fj ;

j

where M ¼ M 1 þ M 2 þ    þ M N ¼ lue at the jth node.

PN

i¼1 M i

is the total number of interpolating nodes. fj is the function va-

4.3. Implementation of interpolation method in analysis of SIF In analyzed numerical models, SIF is determined by five geometrical parameters (c, s, h1, h2 and b) and two crack shape parameters (a/T and c/a) together with nominal stress rnom. For tubular K-joints subjected to balanced axial load, nominal stress can be obtained from following equation: rnom ¼

4P 1 2

2

p½d
ðd
2tÞ 

;

where P1 is illustrated in Fig. 18. d is diameter of brace and t is thickness of brace.

ð25Þ

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As nominal stress can be determined from Eq. (25), SIF can be normalized by dividing SIF over nominal stress. The normalized SIF, expressed as K/rnom, is determined by geometrical parameters and crack shape parameters. Therefore, interpolation method of 7D dimensional problem can be used to analyze SIF of each cracked K-joint model by using Eq. (24). In Eq. (24), fj (j = 1, M) is the SIF of each model in database. N is equal to 7. As it can be seen from Table 2, for parameters c, s, a/T, c/a and b, four interpolating nodes were used. Five interpolating nodes were used for parameters h1 and h2. Then interpolation function expressed in Eq. (23) can be determined. Using this interpolation method, SIF of any cracked tubular Kjoints under balanced axial load can be estimated. 4.4. Evaluation of SIF from interpolation method To evaluate the accuracy of SIF obtained from interpolation method, comparisons of SIF results between interpolation method and numerical analysis were conducted. Fig. 22 shows the comparisons of SIF values obtained from two methods. In Fig. 22, SIF is normalized by dividing SIF with nominal stress. Apparently, the agreements are reasonably well. From Fig. 22 the relationship between SIF and some parameters can be recognized. Fig. 22(a) shows that SIF will decrease with c increasing although the variation is not fast. Fig. 22(b) shows SIF is decreasing with b increasing. As b is the ratio of diameter of brace over diameter of chord, a big value of b means the diameter of brace is bigger if diameter of chord is kept constant. Therefore, it shows that SIF is smaller when diameter of brace is bigger while diameter of chord is kept constant for K-joint. Relationship between SIF and s is illustrated in Fig. 22(c). Apparently, bigger s means thickness of brace is bigger if thickness of chord is kept constant and the stress concentration along the weld on the chord is smaller. Therefore, it is reasonable that SIF is smaller in this case. The relationship between SIF and crack shape can be shown in Fig. 22(d) and Fig. 22(e). Generally, SIF is bigger when the crack is deeper and longer. Effects of intersecting angle h1 on SIF can be seen in Fig. 22(f) and SIF is increasing with larger h1. Moreover, 972 numerical models which are tabulated in details in Table 3 were analyzed to benchmark the interpolation method. Numerical results and estimated SIF values from interpolation method were plotted together in Fig. 23. It is clear that the tendency is quite good. The errors between numerical results and estimated SIF values were plotted in Fig. 24. In Fig. 24, the relative error is defined as follows: Relative error ¼

K INT
K N  100%; KN

ð26Þ

where KINT and KN are normalized stress intensity factors obtained from interpolation method and numerical analysis, respectively. In Fig. 24, For most models, the errors lie in a range of ±15%. For safety, an upper bound and a lower band can be determined from Fig. 24 that ±20% can be taken as the bound values. Although when relative

Table 2 Details of numerical models in database Model Case

c

Case Case Case Case Case

12, 12, 12, 12, 12,

1 2 3 4 5

Total No.

s 18, 18, 18, 18, 18,

24, 24, 24, 24, 24,

30 30 30 30 30

4 * 4 * 4 * 4 * 4 * 5 = 5120

0.25, 0.25, 0.25, 0.25, 0.25,

0.5, 0.5, 0.5, 0.5, 0.5,

0.75, 0.75, 0.75, 0.75, 0.75,

1.0 1.0 1.0 1.0 1.0

h1

h2

c/a

30 40 45 50 60

60 50 45 40 30

5, 5, 5, 5, 5,

6, 6, 6, 6, 6,

a/T 7, 7, 7, 7, 7,

8 8 8 8 8

0.1, 0.1, 0.1, 0.1, 0.1,

b 0.3, 0.3, 0.3, 0.3, 0.3,

0.5, 0.5, 0.5, 0.5, 0.5,

0.7 0.7 0.7 0.7 0.7

0.3, 0.3, 0.3, 0.3, 0.3,

0.4, 0.4, 0.4, 0.4, 0.4,

0.5, 0.5, 0.5, 0.5, 0.5,

0.6 0.6 0.6 0.6 0.6

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64 9.0

3.5

SIF from Lagrange Interpolation SIF from Numerical Analysis

7.5

K/σnom (mm )

2.5

1/2

1/2

K/σnom (mm )

3.0

SIF from Lagrange Interpolation SIF from Numerical Analysis

2.0 1.5

6.0 4.5 3.0

1.0

1.5

0.5

0.0 0.0

0.0 9

12

15

18

21

24

27

30

33

γ

(a)

θ 1=300, θ2=600, c/a=5.5, a/T=0.15, β=0.35, τ =0.28

0.2

0.3

0.4

0.5

0.6

0.7

0.8

a/T θ 1=500, θ2=400, c/a=5.5, τ =0.6, γ =20, β =0.55

(d)

6

SIF from Lagrange Interpolation SIF from Numerical Analysis

7.2

0.1

SIF from Lagrange Interpolation SIF from Numerical Analysis

5 1/2

K/σnom (mm )

1/2

K/σnom (mm )

6.0 4.8 3.6 2.4

4 3 2 1

1.2 0.0 0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0 5.0

0.65

5.5

6.0

6.5

β (b)

θ 1=40 , θ2=50 , c/a=6.5, a/T=0.4, γ =20, τ =0.4 0

0

8.0

8.5

SIF from Lagrange Interpolation SIF from Numerical Analysis

7.5 1/2

K/σnom (mm )

1/2

K/σnom (mm )

9.0

SIF from Lagrange Interpolation SIF from Numerical Analysis

10 .0 7.5 5.0

6.0 4.5 3.0 1.5

2.5 0.0 0.2

0.0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

24

1.0

τ (c)

7.5

θ1=600, θ2=300, a/T=0.15, τ =0.6, γ =20, β =0.55

(e)

15 .0 12 .5

7.0

c/a

θ1=450, θ2=450, c/a=7.5, a/T=0.6, γ =16, β =0.45

30

36

42

48

54

60

θ1 (f)

c/ a=6.5, a/T=0.4, τ =0.6,γ =20, β =0.55

Fig. 22. Comparisons of SIF obtained from numerical analysis and interpolation method.

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

63

Table 3 Details of numerical models used to verify interpolation method Model case

c

Case Case Case Case Case

15, 15, 15, 15, 15,

1 2 3 4 5

Total No.

s 21, 21, 21, 21, 21,

27 27 27 27 27

0.375, 0.375, 0.375, 0.375, 0.375,

0.625, 0.625, 0.625, 0.625, 0.625,

0.875 0.875 0.875 0.875 0.875

h1

h2

c/a

30 40 45 50 60

60 50 45 40 30

5.5, 5.5, 5.5, 5.5, 5.5,

a/T 6.5, 6.5, 6.5, 6.5, 6.5,

7.5 7.5 7.5 7.5 7.5

0.2, 0.2, 0.2, 0.2, 0.2,

b 0.4, 0.4, 0.4, 0.4, 0.4,

0.6 0.6 0.6 0.6 0.6

0.35, 0.35, 0.35, 0.35, 0.35,

0.45, 0.45, 0.45, 0.45, 0.45,

0.55 0.55 0.55 0.55 0.55

3 * 3 * 3 * 3 * 3 * 4 = 972

18

Numerical Results SIF Estimated from Interpolation Method

16

1/2

K/σnom (mm )

14 12 10 8 6 4 2 0 0

100 200 300 400 500 600 700 800 900 1000

Model No. Fig. 23. Evaluation of SIF from interpolation method.

error is negative it means SIF obtained from interpolation underestimates the values compared to numerical results, numerical results of SIFs are generally conservative because in numerical analysis, SIFs are usually higher than experimental results as weld size obtained from AWS specifications [2] is smaller. Based on this conservativeness,
20% is an acceptable error. 100 80

Relative Error (%)

60 40 20 0 -20 -40 -60 -80 -100 0

100 200 300 400 500 600 700 800 900 1000

Case No. Fig. 24. Error estimations.

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Y.-B. Shao / Engineering Failure Analysis 13 (2006) 44–64

5. Conclusions This paper presented geometrical modelling of surface crack in tubular K-joints. A new mesh generator was developed and the mesh generated from this mesh generator can provide different mesh densities and avoid high element aspect ratio. Numerical analysis of stress intensity factor (SIF) for cracked tubular K-joints was able to be conducted by using the new numerical modelling scheme. To evaluate numerical analysis, a full-scale K-joint was tests under cyclic load. ACPD technique was adopted to monitor crack initiation and propagation. Crack growth rate was then obtained from experimental data. By using Paris
equation, SIFs at the deepest point for different crack shapes can be calculated. Numerical results agree quite well with experimental results. This proves that the proposed numerical modelling of cracked tubular K-joints is accurate and reliable. Based on the accuracy of numerical modelling, 5120 numerical models of K-joints subjected to balanced axial load were analyzed and the results were used to build up a database. Then an interpolation method was introduced to analyze SIF of any model in a valid range. The accuracy of SIF obtained from this method was verified by comparing the interpolating results with numerical results of 972 numerical models. Analysis of errors shows that interpolation method is efficient and accurate to estimate SIF of cracked K-joints under balanced axial load.

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