COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Finite Element Method for Analyzing Stress Intensity Factor of a Surface
Crack in Tubular Joints
Y. B. Shao*, Z. F. Du, W. D. Hu
School of Civil Engineering, Yantai University, Yantai, 264005 China
Email: cybshao@ytu.edu.cn
Abstract Fatigue failure is very common for tubular joints used in offshore engineering because they are frequently
subjected to cyclic loading. In the fatigue failure process of these tubular structures, surface crack will initiate at the
weld toe and propagate along the chord/brace intersecting curve. The residual life of a tubular joint is dependent much
on the accurate estimation of the stress intensity factor of the surface crack. As the chord/brace intersecting curve is
very complicated, it is critically important to present a reasonable method to produce high-quality mesh in finite
element analysis. In this paper, a sub-zone method for generating the FE mesh of a surface crack in a tubular joint is
proposed. In this method, the surface crack can be located at any position along the weld toe and be of any length. To
avoid the element around the surface crack badly distorted, five element types are used to model the crack region.
Thereafter, two methods, namely the interaction J-integral and the displacement extrapolation methods, are used to
compute the stress intensity factors of a tubular joint along the crack front. The numerical models seem to be effective
for analyzing the stress intensity factor of a surface crack in a tubular joint.
Key words: tubular joints, surface crack, stress intensity factor, interaction J-integral method, displacement
extrapolation method
INTRODUCTION
Tubular members, especially hollow circular sections, are widely encountered in the construction of offshore
engineering structures. These tubular structures are formed by welding the brace members onto the circumference of
the chord member. Due to the non-uniformity of the stiffness around the chord/brace intersection, stress concentration
in this region exists, and thus fatigue failure is important in practice for these structures. As mentioned above, the
behaviour of the tubular joints is an essential factor that will affect the overall performance of the whole structure.
Therefore, it is crucial to be able to estimate the residual life of these damaged joints containing flaws or cracks. The
most commonly used method in predicting the remaining life of a tubular joint containing a surface crack is to employ
a fracture mechanics approach, based on accurate estimation of the stress intensity factors (SIFs) along the crack front.
Generally, the accuracy of the SIF values depends very much on three most important factors: (1) the geometrical
models used to describe the crack detail, (2) the grading of the finite element mesh used near the joint intersection and
around the crack front, and (3) the aspect ratio of the elements along the crack front. In the past, a considerable amount
of research effort had been directed to the research topic on developing a consistent model to analyze a cracked tubular
joint. Detailed geometric analysis for the intersection curve between a brace and a chord without welding details was
carried out by Cao et al. (1997) [1]. For the generation of finite element meshes for cracked joints, Cao et al. (1998) [2]
developed a procedure for transforming crack elements around a plane curve into crack elements for a doubly curved
semi-elliptical surface crack around the joint intersection while Bowness and Lee (1995) [3] developed another mesh
generation procedure to discretize the surface crack. Bowness and Lee (2000) [4] and Cao et al. (1998) [2] have used
two types of elements to model the surface crack. They used 3D prism singular elements to model the crack front and
3D hexahedral elements at the field far away from the crack front. The problem of using this method is that some
generated elements, especially the elements near the crack front, have a very high aspect ratio which in turn produces
numerical instability in the SIF values. In practice, using only two types of elements can not produce a good quality
mesh for any tubular joint containing a surface crack due to the complexity of the geometry and the weld.
⎯ 386 ⎯
Due to the previous reasons, a systematic modelling procedure for a general welded and cracked tubular joint is first
proposed in this study. A new definition of a surface crack in a tubular joint is used in the geometrical model. It can
cover a wide range of practical tubular joints, including T, Y, K and KK-joints. In the mesh generation, five types of
elements are used to model the surface crack and the other zones of the tubular joint. To avoid high aspect ratio of the
elements around the surface crack, a sub-zone mesh generation scheme is proposed in this study. In the sub-zone mesh
generation scheme, the entire joint structure is divided into several regions according to the computation requirements.
The mesh of the region containing the surface crack is generated separately because this region can be extracted from
the entire structure. Using this mesh generation method, the density and the aspect ratio of the elements can be
controlled easily.
Using an interaction J-integral method and a displacement extrapolation method, the stress intensity factor distribution
along the surface crack of a tubular joint is analyzed.
GEOMETRICAL AND NUMERICAL MODELLING OF A CRACKED JOINT
1. Geometrical modelling of a surface crack in a tubular joint When a crack initiates from the surface of the chord
of a welded tubular joint, it will propagate through the chord thickness in a special direction in which the energy
requirement is minimal. The crack front will propagate on 3-D curves which together form a surface which is called the
crack surface where the crack front lies on. As shown in Fig. 1, the crack surface is formed by joining a series of
straight lines WoD along the weld path. WoD is passing through the Z-axis and the thickness of the crack surface is
always equal to tc. Wo ( X Wo , YWo , Z Wo ) is the point on the weld profile and on the outer horizontal cylinder,
Wo D = tC = R1 − R2 , and the line Wo D will pass through Z-axis.
The coordinate of point D can be obtained and expressed as follow:
⎧ R2
⎫
⎪ R X Wo ⎪
⎧X D ⎫ ⎪ 1
⎪
⎪ ⎪ ⎪R
⎪
D = ⎨ YD ⎬ = ⎨ 2 YWo ⎬
⎪ Z ⎪ ⎪ R1
⎪
⎩ D ⎭ ⎪ ZW ⎪
o
⎪⎩
⎪⎭
(1)
Figure 1: Definition of a crack surface
After the crack surface is defined, it is necessary to define the crack front which can exist at any location on a surface
crack. From past experimental tests on the T-joints (Huang, 2002 [5]) as well as the K-joint (Shao, 2005 [6]), the crack
shape resembles a semi-ellipse. Therefore, this assumption is used to model the crack in any tubular joint in the present
study. In order to model the crack front conveniently in 3-D space, it is often more convenient to define it on a
normalized u ' - v' plane and then map it onto the crack surface as shown in Fig. 2. The u '-axis relates to the crack
length, lCr , while the v' relates to the crack depth, d. Apparently, it is easier to define the u '-axis by the polar angle α
though it has no direct relationship with the physical length of the crack. In this mapping approach, a crack with any
length and size can be modelled at any location. As illustrated in Fig. 3, the two crack tips will be located and defined
by the polar angles α Cr1 and α Cr2 in the u-v system. The coordinates (u ' , v' ) are defined as:
u' =
v' =
( α − α Cr1 − α Crange )
α Crange
, α Crange =
α Cr2 − α Cr1
(2)
2
d
tC
(3)
⎯ 387 ⎯
Figure 2: Mapping of a surface crack
Crack Tip 2
v
O
2αcrange
αcr2
αcr1
Crack Tip 1
u
Figure 3: Location of surface crack on a u-v coordinate system
where α is the polar angle corresponding to the point (u ' , v' ) , α Cr1 is the polar angle which defines the location of
crack tip 1, α Cr2 is the polar angle which defines the location of crack tip 2, d is the depth of the crack, t C is the
thickness of the chord member. Note that u ' ∈ [ −1 , 1 ] , v'∈ [0 , 1 ) and α Cr1 , α Cr2 , α ∈ (0o , 360o ] .
Outer Weld Path
(Weld Toe)
Y
Crack Tip 2
Z
O
lcr
lcr1
Note : lcr2 = lcr1 + lcr
Crack Tip 1
Y
Figure 4: Location of surface crack defined on global X-Y-Z coordinate system shown on Y-Z plane
In practice, the crack tip positions are frequently described by defining (or measuring) the arc lengths, lCr1 and lCr2 on
the global X-Y-Z coordinate system as shown in Fig. 4. The crack length, l Cr , will depend on the position of the crack
tips and is defined as lCr = lCr1 − lCr2 . Since the weld is defined by the polar angles α Cr1 and α Cr2 (Eq. 9), it is required
to compute the values of α Cr1 and α Cr2 from lCr1 and lCr2 respectively. In this study, the value of α Cr1 and α Cr2 are
computed from lCr1 and lCr2 by using a sample approximation procedure. For example, in order to compute α Cr1 from
⎯ 388 ⎯
lCr1 , starting from the v'-axis , a sequent of points will be generated by increasing their polar angles gradually in small
step equal to Δα . For each of this point, the corresponding arc length, l * , is computed until l* ≥ lCr1 . The estimated
value for α Cr1 is then defined as the one corresponding to the arc length which is closest to lCr1 . In practice, it is found
that a value of Δα = 0.1ο will be accurate enough for virtually all applications and the computational cost needed is
modest (Wong, 2001 [7]).
Suppose that the crack front curve is defined by the point Cr' in the u '−v' space as shown in Fig. 2. For any point
(u ' , v' ) on the curve, by using Eqs. (4, 5), the corresponding value of α can be obtained. Once α is known, the
coordinates of the point Wo ( X Wo , YWo , Z Wo ) could be computed. In order to define the location of the crack front, the
point Wo will be further modified. Assume the crack front is defined by the point Cr ( X Cr , YCr , Z Cr ) with depth equal
to d (Fig. 2), then 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:
⎧⎛
⎫
d ⎞
0
0⎪
⎪⎜⎜1 − ⎟⎟
R1 ⎠
⎪⎧ X ⎫
⎧ X Cr ⎫ ⎪⎝
W
⎛
d ⎞ ⎪⎪⎪ o ⎪
⎪
⎪ ⎪⎪
⎜⎜1 − ⎟⎟ 0⎬⎨ YWo ⎬
Cr = ⎨ YCr ⎬ = ⎨ 0
R1 ⎠ ⎪⎪
⎝
⎪Z ⎪ ⎪
Z ⎪
⎩ Cr ⎭ ⎪ 0
0
1⎪⎩ Wo ⎭
⎪
⎪
⎪⎩
⎪⎭
From Eq. (4), any point on the crack front can be determined in a 3-D space.
(4)
2. Finite element mesh generation of a tubular joint containing a surface crack In the present study, a sub-zone
technique is used in the mesh generation whereby the entire structure, as shown in Fig. 5, is divided into three main
zones, namely, refined zones (Zone CF1 and Zone CRBLOCK), coarse zones (Zone A, Zone ER, Zone EL,
EXTENCHL, EXTENCHR and Zone H) and transition zones (Zone B, Zone D, Zone G1 and Zone G2). In the zones
with refined mesh, three layers of elements are generated in the thickness direction so as to model the crack depths. In
the coarse mesh zones, only one layer of elements are generated in thickness direction
.
Zone-H
Zone-G2
Zone-CF1
Zone-G1
Zone-D
CRBLOCK
EXTENCHL
Zone-B
Zone-E
Zone-A
EXTENCHR
Figure 5: Sub-zone mesh generation
DCUBE-B
DCUBE-B
SFBLOCK-A
SFBLOCK-B
DCUBE-A
DCUBE-A
Figure 6: Mesh of zone CRBLOCK
⎯ 389 ⎯
Fig. 6 shows the mesh of CRBLOCK in a detailed view. CRBLOCK is extracted from Zone CF1 (refer to Fig. 7). It
must be noted here that, in Fig. 6, DCUBE-B is part of the sub-mesh in Zone CF1 and DCUBE-A is part of the
sub-mesh in Zone D. It also should be emphasized that the number and location of elements extracted from Zone CF1
will depend on the crack length and position. Once the crack length and the crack position are determined, the number
of elements to be extracted will be calculated automatically. This means a surface crack with any length at any fixed
position can be generated automatically.
After extracting CRBLOCK from Zone CF1, all the elements in Zone CF1 are hexahedral elements since there is no
crack in this zone. The mesh of CRBLOCK will be generated separately since it can be extracted from Zone CF1. The
modelling of the surface crack in detail is illustrated in Fig. 8. In the present study, five types of elements, as shown in
Table 1, are used to generate the mesh of any cracked tubular joint. The elements in the first ring are quarter-point
crack tip elements which consist of the crack front. The second ring consists of prism elements, which are used to
connect the crack elements and pyramid elements. SFBLOCK-A, which is to connect with tube elements, consists of
pyramid, prism and tetrahedral elements. It is a transition zone. However, as different types of elements are used in this
zone, when merging CRBLOCK with Zone CF1 and Zone D, the incompatibility of the surface becomes a big
problem. Because of this reason, tetrahedral and pyramid elements are used in DCUBE-A and DCUBE-B. These two
blocks are used to link the side faces of block SFBLOCK-A.
Figure 7: Zone CF1 after extracting the CRBLOCK
Tetrahedral Elements
Surface Crack Front
Prism Elements
SFBLOCK-A
Pyramid Elements
Second ring – (Prism
Elements)
First ring (QuarterPoint/Crack Elements)
Face to be
connected to
DCUBE-A
Figure 8: Detailed mesh along the surface crack
Table 1 Element types used in the mesh generation for cracked tubular joints
Element Types
No. of Nodes
1.
Hexahedral / Cubic Element – (H20)
20
2.
Prism/Wedge – (P15)
Quarter Point / Crack Element – (QP15)
(Collapsed Prism)
15
3.
15
Tetrahedron – (T10)
4.
5.
10
Pyramid – (PR20)
(Collapsed Hexahedral)
20
⎯ 390 ⎯
After the meshes of all the zones have been generated, they are then merged together to form the mesh of the entire
structure. Figs. 9a to 9c show the meshes of a T, a Y and a K-joint after merging respectively. As it can be seen clearly,
the advantage of generating the mesh zone by zone is that only the region near the intersection are required to be
modified when the surface crack details are added to the mesh.
a. a T-joint
b. a K-joint
c. a KK-joint
Figure 9: Meshes of tubular joints using sub-zone generation method
In order to study the convergence of the SIF values of the tubular joint, one uniform refinement is carried out by
doubling the mesh density (Lee et al., 2001 [8]). During the mesh generation procedure, the mesh generator will
always refer to a carefully predefined geometrical model of the joint to capture all the detail features such as welding
thickness and surface crack depth of the structures. Figs. 10a to 10c show the original and the doubled finite element
meshes of zone CF1 respectively. This mesh density upgrade scheme is used in this study to do the convergence study
of tubular joints in FE analysis.
a. Original mesh
b. Refined mesh
c. Mesh at the surface crack
Figure 10: Mesh refinement of zone CF1 containing the surface crack block
The mesh generator has been designed and developed to model a wide range of geometry for tubular joints with a
surface crack. In the current implementation of the program, the valid ranges of geometrical parameters which the
mesh generator can handle are listed below:
1) Intersecting angle between the brace and the chord: 30° ≤ θ ≤ 90°
2) Ratio of brace to chord radius: 0.05 < r3/R1 < 0.8
3) Ratio of brace’s thickness to brace’s radius: 0.03 < tb/r3 < 0.3
In the analysis of the stress intensity factor of a surface crack in a tubular joint, two most popularly used finite element
methods are: (1) J-integral method, and (2) displacement extrapolation method. J-integral method is insensitive to
mesh refinement, but it can not be used directly in mixed mode problems. To overcome this problem, Shih and Asaro
(1988) [9] presented an interaction J-integral method to solve the Mode I, II & III SIFs. Displacement extrapolation
method is based on Wetergaard’s equations which relate the displacements in the vicinity of the crack front to the
stress intensity factors. It should be noted here that the J-integral method lacks path independence in the region where
the crack meets the weld toe because the stress at the toe and the crack tip is singular.
3. Interaction J-integral method J-integral is a measure of the strain energy in the region of the crack tip. The
relationship between the J-integral and the SIFs is expressed as follow:
J=
1 T
K ⋅B⋅K
8π
(5)
where K = [ K I , K II , K III ]T and B is called the pre-logarithmic energy factor matrix. For homogeneous isotropic
materials, B is a diagonal matrix and the above equation can be simplified as:
J=
1
1
2
( K I2 + K II2 ) +
K III
2G
E
(6)
⎯ 391 ⎯
where E = E for plane stress and E = E ( 1 − ν ) for plane strain, axisymmetry, and 3D problems.
2
It is obvious from Eq. (6) that it is easy to obtain the value of J-integral from K I , K II and K III . However, it is not
feasible to obtain K I , K II and K III from J-integral directly. This means the J-integral method can not be easily used
to analyze a mixed mode problem directly. Due to this problem, Shih and Asaro (1988) [9] had proposed an interaction
integral method to calculate K from the J-integral.
By introducing an interaction J-integral, J int , the relationship between the SIFs and J int is obtained as follow:
K = 4 π B ⋅ J int
(7)
The detailed calculations of J int can be found in the paper published by Shih and Asaro (1988) [9], and this method
has been implemented in the ABAQUS (2001) [10] general finite element software.
4. Displacement extrapolation method Displacement extrapolation method uses the elastic solutions of the
displacements in the vicinity of the crack tip. The local radial, normal and tangential directions of a surface crack are
defined as shown in Fig. 11. The equations to calculate the SIFs of Mode I, Mode II and Mode III are given as
KI =
2π
G
vn
2 (1 − υ ) r
K II =
2π
G
ur
2 (1 − υ ) r
(8)
2π
wt
r
where ur, vn, wt denote the local radial, normal and tangential displacements of the nodes on the crack surface
respectively, G is the shear modulus and υ is the Poisson’s ratio, r is distance to crack front. The above equations are
applicable for plane strain condition. In this paper, plane strain condition is assumed to be suitable everywhere along
the crack front.
K III = G
n
r
Cracked
surface
t
Figure 11: Definition of the local coordinate at crack front
In evaluating SIFs by displacement extrapolation method, the near tip displacements are obtained from finite element
analysis, and thus the corresponding stress intensity factors can be calculated from Eq. (8).The stress intensity factors
at the crack tips can be obtained by extrapolating the SIFs at the crack vicinity along the crack front.
5. Numerical results of the SIFs of cracked tubular joints Using the above presented numerical method, the stress
intensity factor distributions of the surface crack in a tubular T-, a tubular K- and a tubular KK-joint are obtained. In
the FE analysis for a cracked T-joint under axial load (nominal stress caused by the axial load is 1.0 MPa), half
structure as shown in Fig. 12a is used due to the loading and the geometrical symmetry. The surface crack is located
symmetrically at the crown position. For the analyzed K-joint, a combined loading condition (axial load and in-plane
bending load) is used in the FE analysis. The surface crack is also symmetrically located the crown position because
the stress at the crown is maximum. The values of the combined loads are: the axial load AX=150 kN and the in-plane
bending load is IPB=13.5 kN. The tubular KK-joint in the FE analysis is subjected to balanced axial load. A tensile
stress of 1.0 MPa is applied at the ends of the two left braces, and a compression stress of 1.0 MPa is applied at the ends
of the two right braces. The crack shape information of the three joints is tabulated in Table 2.
⎯ 392 ⎯
Table 2 Crack shapes of the analyzed tubular joints
αCr1
Joint type
T-joint
K-joint
KK-joint
αCr2
0
0
-22.5
1420
1710
22.5
2180
279.50
d (mm)
12.7
12.75
10
AX
IPB
1.0 MPa
1.0 MPa
AX
0
80
90°
10
°
68.25
79.96
Dia116.35
45°
45
565.42
45°
45°
Dia136.35
20
Dia58.25
Dia68.25
90°
2100
1.0 MPa
1.0 MPa
Note:The parameters of four braces are all same.
a. T-joint
b. K-joint
c. KK-joint
Figure 12: Loading and boundary conditions for tubular joints in numerical analysis
The stress intensity factor distributions of the T-, K- and KK-joints are plotted in Figs. 13a to 13c against the crack
angle φ (refer to Figure 14). From Figs. 13a to 13c, it can be seen that for the analyzed tubular joints under axial or
combined axial and in-plane bending loads, SIFs of crack Mode I, KI, are much bigger than the values of KII and KIII.
This means that the relative opening displacement between the two crack faces is dominant. The detail of the opening
crack of the T-joint can be seen in Figure 15. The SIF distributions along the crack front of the T- and the K-joints are
symmetrical because the surface crack is symmetrical at the crown position. The SIF distribution of the KK-joint is not
absolutely symmetrical due to non-symmetry of the position of the surface crack.
60
KII
KI
KIII
KIII
45
1/2
SIF (MPa*m )
1/2
K / σn (mm )
45
30
15
0
-15
0.0
KII
Ke
1/2
KI
K (MPa*mm )
60
30
15
0
-15
0.2
0.4
0.6
0.8
1.0
-30
0.00
0.15
0.30
0.45
0.75
0.90
1.05
KI
KIII
φ (degree)
b. K-joint
c. KK-joint
Figure 13: SIF results of tubular joints
2c
ϕ
tc
Crack
Figure 14: Definition of crack angle
⎯ 393 ⎯
KII
Ke
0 20 40 60 80 100 120 140 160 180 200
φ/π
2φ /π
a. T-joint
0.60
14
12
10
8
6
4
2
0
-2
-4
Figure 15: The opening crack in the T-joint
CONCLUSIONS
This paper presents a numerical method of modelling a circular hollow section joint containing a surface crack at the
weld. In the modelling technique, five types of elements are used to produce the mesh of the surface crack. This mesh
generation scheme can avoid high aspect ratio of the elements around the crack front, and thus the accuracy of the finite
element SIF results can be guaranteed. Additionally, the mesh entire structure of a tubular joint is obtained by merging
the meshes of several sub-zones. Using this sub-zone mesh generation method, the mesh densities in different region
can be controlled easily. Finally, the stress intensity factor distributions of different tubular joints can be analyzed from
two commonly used FE methods using the proposed model in this study. From the FE results, it seems that the
presented numerical modelling method is reasonably reliable to be used in engineering analysis and design.
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