Chapter 4 Development of Generalized Formulations on
5.1 Validation under static loading conditions
89
handbook (Zahoor, 1989) as follows;
,
, 4 4
4 m b 4 Press Eq m
Unrestrained LE
o i
M R
R V
E R R
(5.1)
, ,
, 4 4
4 m b 4 Press Eq eff m Restrained LE
o i
M R
R V
E R R
(5.2)
Vb is given in the EPRI ductile fracture handbook as a function of the ratio of the pipe mean radius to pipe thickness (Rm/t) and the crack length (θ/π). By dividing Eq. 5.1 by Eq. 5.2, the ratio of the COD of a restrained pipe to the COD of an unrestrained pipe (rCOD,LE) is obtained as Eq. 5.3:
, , ,
, ,1 , ,
, ,
Press Eq eff Restrained LE
COD LE Rest D M LE
Unrestrained LE Press Eq
r M C
M
(5.3) where the ratio is equal to the moment restraint coefficient, CRest,1D,M,LE. In case of the 1D pipe, the moment restraint coefficient was derived in Eq.4.8 as follow;
,1 , ,
, , ,
1 2
1
1 2 1 3
Rest D M LE
crack M LE
N N
C G EI
L L L
(5.4)
To obtain rCOD,LE, the linear elastic compliance of the crack (Gcrack,ψ,M,LE) is needed. Gcrack,ψ,M,LE can be determined by dividing rotation due to the crack (ψC,M,LE) by an applied moment (M).
, , , , ,
C M LE crack M LE
G M
(5.5)
The EPRI ductile fracture handbook (Zahoor, 1989) gives the linear
elastic rotation due to the crack under a bending moment as follows:
3
, , 2
C M LE '
m
B M
E R t
(5.6)
or
, , 3
, , , 2
'
C M LE crack M LE
m
G B
M E R t
(5.7) where E’ denotes E/(1-ν2) for plane strain and E for plan stress (ν is the Poisson’s ratio.). The dimensionless function B3 is given in the EPRI ductile fracture handbook as a function of the crack length (θ/π) and pipe mean radius to thickness ratio (Rm/t). By substituting Eq. 5.7 into Eq.5.4, we can evaluate the effect of PIB restraint on COD.
5.1.1.2 Comparison with linear elastic finite element analysis results
As described in the preceding section, a series of the linear elastic FEA to examine the PIB restraint effects on COD was conducted in the BINP program and in Miura’s research (Miura, 2001; Scott et al., 2005b). Miura has analyzed three crack length, and two types of restraint models; a symmetric model (L1=L2) and an asymmetric (L1≠L2). The proposed formula of rCOD,LE
was compared with these results.
Figure 5.1 shows the results of the developed formulation and FEA for the symmetric model. rCOD,LE predicted by using the moment restraint
91
of Rm/t and θ/π. The results of the asymmetric model for three restraint lengths and three crack lengths are represented in Figure 5.2. When the pipe length of one side of the crack is short (L1/D=1 or L2/D=1), the formula tends to slightly overestimate rCOD,LE versus FEA. Nevertheless, the comparison results show that the proposed evaluation method could well predict rCOD,LE
(the effect of PIB restraint) for the linear elastic model.
5.1.1.3 Calculation of the restrained COD for elastic-plastic model
The elastic-plastic behavior was also considered. It is difficult to express the restraint effect as a closed form formula because of the nonlinearity of the elastic-plastic model. In the same manner with the linear elastic model, the COD of the unrestrained pipe and restrained pipe can be derived by substituting MPress,Eq and MPress,Eq,eff into the elastic-plastic COD formula.
2
2 , 0 2 ,
,
0 n
m Press Eq m Press Eq
Unrestraind EP n
f R M R H M
EI M
(5.8)
2
2 , , 0 2 , ,
,
0 n
m Press Eq eff m Press Eq eff
Restraind EP n
f R M R H M
EI M
(5.9)
The dimensionless functions f2, H2 and M0 are given in the EPRI ductile fracture handbook (Zahoor, 1989). rCOD,EP can be obtained by dividing
Eq. 5.9 by Eq. 5.8 as follow;
2
2 , , 0 0 2 , ,
,
, 2
, 2 , 0 0 2 ,
n n
m Press Eq eff m Press Eq eff
Restraind EP
COD EP n n
Unrestraind EP m Press Eq m Press Eq
f R M M EI R H M
r f R M M EI R H M
(5.10)
In contrast with the linear elastic case, rCOD,EP depends on the magnitude of the applied moment due to the nonlinearity. Thus, the additional calculation procedure is required as follows;
First, the pressure equivalent moment MPress,Eq which means the bending moment that can induce exactly same rotational displacement with a given tension load arising from pressure (the bending moment when ψC,M,EP=ψC,T,EP) should be obtained. However, the EPRI ductile fracture handbook gives the formula of the rotation due to the crack of the elastic- plastic pipe caused by a bending moment (ψC,M,EP) but not by an axial tension load (ψC,T,EP). Thus, a new formula for an axial tension load and coefficient H4T were developed, and this will be discussed in more detail in the next subsection. Using two formulae, the equation for calculating the pressure equivalent moment (MPress,Eq) can be represented as Eq. 5.11.
4 , 0 4 ,
2 2
0
2 2
4 0 4
2 0
/ 1 0.5 / 2
/ 1 0.5 /
n
Press Eq B Press Eq
m
n
i in T i in
m
f M H M
R tE M
f R P H R P
R tE P
(5.11)
The left-hand side means the rotation due to the crack caused by a bending moment, and the right-hand side is caused by an internal pressure (Pin). The pressure equivalent moment MPress,Eq cannot be derived as a closed
93
form from Eq. 5.11; thus, it needs to be determined by an iterative calculation.
Second, the effective applied at the cracked section MPress,Eq,eff needs to be determined. The relation between MPress,Eq and MPress,Eq,eff is represented by using the moment restraint coefficient as follows;
, , ,1 , , ,
,1 , ,
, , , 1 2
, 2 1 3
Press Eq eff Rest D M EP Press Eq
Rest D M EP
crack M EP N N
M C M
where C L
L G EI L L
(5.12)
To obtain CRest,1D,M,EP, the elastic-plastic compliance of the crack (Gcrack,ψ,M,EP) is needed. In the same manner as linear elastic case, Gcrack,ψ,M,EP
can be determined by dividing rotation due to the crack (ψC,M,EP) by an applied moment (M).
, , , , ,
C M EP crack M EP
G M
(5.13)
The EPRI ductile fracture handbook (Zahoor, 1989) gives the elastic- plastic rotation due to the crack under a bending moment as follows:
0 4
4
, , 2 2
/ 1 0.5 / 0
n B
C M EP n
m
f H M
ER t M M
(5.14)
or
1
, , 4 0 4
, , , 2 2
/ 1 0.5 / 0
n
C b EP B
crack M EP n
m
H
f M
G M ER t M
(5.15)
where the dimensionless functions f4 and H4B are also given in the handbook.
It should be noted that M in Eq. 5.15 is the applied moment at the cracked
section MPress,Eq,eff, whereupon Eq. 5.15 can be written as:
1 , ,
0 4
4
, , , 2 2
/ 1 0.5 / 0
n
Press Eq eff B
crack M EP n
m
H M G f
ER t M
(5.16)
Thus, substituting Eq. 5.16 into Eq. 5.12, we obtain the relation between MPress,Eq and MPress,Eq,eff .
, ,
1 , , ,
0 4
4
1 2
2 2
0
2 1 3
/ 1 0.5 /
Press Eq eff
Press Eq n
Press Eq eff B
N N
n m
M
L M
H M
L f EI L L
ER t M
(5.17)
To obtain MPress,Eq,eff, Eq. 5.17 needs to be solved iteratively using MPress,Eq determined from Eq. 5.11. Finally, by substituting MPress,Eq
determined from Eq. 5.11 and MPress,Eq,eff determined from Eq. 5.17 into Eq.
5.10, we can evaluate rCOD,EP.
5.1.1.4 Comparison with elastic-plastic finite element analysis results
As mentioned in the subsection 5.1.1.3, to evaluate rCOD,EP, the rotation due to the crack caused by an axial tension load (ψC,T,EP) must be calculated. The new formula was developed referring from the rotation formula of a bending moment and the axial displacement formula for an axial tension load (P) from
95
the EPRI ductile fracture handbook(Zahoor, 1989) as follow;
0 4
4
, , 2
0
2
/ 1 0.5 /
n T
C T EP
m
H
f P P
ER t P
(5.18)
The dimensionless function H4T depends on Rm/t, θ/π, and the Ramberg-Osgood coefficient n. A series of the finite element analyses were conducted to tabulate the H4T for the case of Rm/t=10, using the finite element program ABAQUS (Dassault Systémes, 2012). Figure 5.3 shows a half- symmetry analysis model of a pipe with a circumferential through-wall crack.
The reduced integration 20-noded continuum elements were employed, and a focused mesh was applied at the crack tip. The multi-point constraint option in ABAQUS was utilized to make the displacement and rotation at the nodes on the pipe end plane equal to those of a reference node on the axis of the pipe. The effects of geometric nonlinearity and ovalization were ignored. For the material properties, Young’s modulus of 200 GPa and yield stress of 400 MPa were taken. The various crack lengths (θ/π) and Ramberg-Osgood coefficients (n) were considered, the dimensionless function (H4T) then was tabulated base on the FEA results.
The development of Eq. 5.18 aims to calculate the pressure equivalent moment which means the bending moment that can induce exactly same rotation with given tension load from pressure (the bending moment when ψC,M,EP= ψC,T,EP) in Eq. 5.11. The dimensionless function such as H4T and H4B
can be strongly controlled by the details of the modeling approach. In this regards, to maintain consistency, H4B was also newly tabulated using same FE
model with the case of H4T as shown in Table 5.1.
The PIB restraint effect on the elastic-plastic COD has been evaluated by J. Kim through FEA for several cases (Kim, 2008; Kim, 2004). By comparing with these FEA results, the developed evaluation procedure for the elastic-plastic model in the subsection 5.1.1.3 was validated. The various restraint lengths, crack sizes, and materials cases were considered. The results of the comparisons are shown in Figure 5.4. As the internal pressure (axial tension load) increases, rCOD,EP decreases because of the effect of the plastic behavior. These trends are observed in the results of the finite element analysis and the results predicted from the proposed evaluation procedure as well. Similar to the case of the linear elastic model, the formula tends to slightly overestimate rCOD,EP then the FEA results when the pipe length is short (L1/D=L2/D=1). Generally, the estimated values using Eqs. 5.10 to 5.18 coincide well with the FEA results. Thus, the proposed method can reliably be applied to the calculation of the restrained COD for the elastic-plastic model.
In this section, the PIB restraint effects on COD predicted using the developed formulations were validated with the various FEA results. For both linear elastic and elastic-plastic model, the restraint coefficient can be used to adjust the numerical expressions of COD for a free-ended pipe to take into account the restraint effects.
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5.1.2 Evaluation of effective applied moment for 3D pipe under static loading conditions
A crucial aspect of the developed formulation is that the amount of decrease in the applied load caused by the compliance change of the piping system due to the presence of a crack can be predicted using the restraint coefficient (see section 4.3.1). To demonstrate this aspect, the restraint coefficient estimated based on the procedure in section 4.4 were compared with the ratio of moment or force reduction predicted from the linear elastic finite element analyses. A series of FEA was conducted for the uncracked and cracked pipe separately, the ratio of the applied moment at the crack position between two cases was calculated.
The particular model shown in Figure 5.5 is analyzed for verification.
The piping system has two horizontal pipe segments of length 10Do, and a vertical pipe which are linked to two elbows of radius 1.5Do where Do is the pipe outer diameter. The reduced integration 20-noded continuum elements were employed, and a focused mesh was applied at the crack tip. The multi- point constraint option in ABAQUS was utilized to make the axial displacement and rotation at the nodes on the pipe end plane equal to those of a reference node on the axis of the pipe while the radial displacement is allowed. The effects of geometrical nonlinearity were ignored.
The nominal diameter and mean radius to thickness ratio are 12 inch and 5, respectively. Both pipe ends are fixed rigidly except the radial displacement, and the horizontal pipe segment connected the anchor 1 contains a circumferential crack at a distance of L1 from the anchor 1 while
nine values of L1 (crack position) were considered. For loading condition, not only three kinds of loading that were assumed for the development of the formulation, but also the thermal expansion load was considered in this analysis. Another information about analysis model and material properties are summarized in Table 5.2.
To determine Gcrack, the typical mesh of a circumferential through- wall cracked pipe using ABAQUS was used as shown in Figure 5.3. The rotations and axial displacements of the uncracked pipe and cracked pipe were obtained respectively, the difference was determined as the rotation or axial displacement of a pure crack (Zhang et al., 2010). Dividing the rotation or axial displacement due to the crack by the applied bending moment or axial force, Gcrack,ψ,Mz and Gcrack,ψ,Fx were obtained as 5.15E-12 rad/N∙mm and 4.03E-10 rad/N for crack length (θ/π) of 0.25, and 4.34E-11 rad/N∙mm and 4.15E-9 rad/N for θ/π=0.5, respectively.
To determine Gpipe, the simple FE model using beam element were employed as depicted in Figure 5.6. The straight pipe segments and elbows were created using pipe element and elbow element, respectively. The kinematic boundary conditions were applied to the connections of an elbow and straight pipe to account the effects of ovalization and radial expansion at the elbows while the warping is not allowed.
Figure 5.7 and Figure 5.8 show the applied loads and applied nominal stress at the cracked section of the uncracked and cracked pipes depending on the crack positions. Comparing with the results of the uncracked pipe,
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system compliance, and the degree of reduction in case of crack length (θ/π) of 0.5 is larger than 0.25.
The restraint effects on the bending moment and axial force are summarized in Figure 5.9. The value of 1 means that the behavior of cracked pipe is same with the uncracked pipe. It can be confirmed again that the crack and the pipe restraint affect the applied moment at cracked section regardless of types of loading. The results calculated from proposed evaluation methods agree well with the FEA results, while some differences are observed. This could be due to the simplified assumptions considered in the process of formulations development. Generally, it can be confirmed that using the restraint coefficient developed in this dissertation, the effective applied moment at cracked section can be calculated, and consequently, the accurate COD and allowable moment can be determined.