Chapter 4 Development of Generalized Formulations on
4.3 Development of generalized formulation
4.3.2 Consideration of the complex piping configurations
restraint coefficient, CRest,1D,M.
As stated earlier, the earlier studies to express the restraining effects on COD analytically have only focused on the pressure induced bending. As Eqs. 4.22 and 4.27 imply, regardless of the types of loadings, the constraint of pipe has an influence on reducing the applied moment at the cracked section. This is also the strong evidence to broaden the practical aspects of the restraint coefficient. By multiplying the restraint coefficient on the uncracked pipe analysis results, one can calculate the effective applied moment at the cracked section of a cracked pipe, as follows;
, ,1 , ,
C CPipe Rest D M C UcPipe
M C M
. (4.28)
This means that, therefore, the restraint coefficient can be used not only to normalize the pipe restraining effect as derived in Eq. 4.17, but also to predict the amount of decrease in the driving force caused by the compliance change of the piping system due to the presence of a crack under any loading conditions.
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restricted to the particular geometries were replaced to by the compliance of pipe. In this subsection, the compliance approach will be extended to a two- dimensional piping system.
Figure 4.7 shows an arbitrary 2D pipe containing a circumferential crack. In this case, the piping system may include the elbows bent at various angles or pipe joints. Unlike the case of 1D pipe, the moment or force applied to a complex piping system can cause a large axial force on the cracked section. Therefore, it was assumed that both the bending moment and axial force at the cracked section are affected by the change of compliance due to the restraint effect. The rotational compliances of the pure crack then can be represented as follows;
, ,
, ,
cracked pipe uncracked pipe crack M
cracked pipe uncracked pipe crack F
G M M
G F F
.
(4.29)
Suppose that the applied moment and axial force at the postulated position of a crack calculated from the uncracked pipe analysis are Mapp and Fapp, respectively. A free body diagram of cracked pipe subjected the Mapp and Fapp is illustrated based on the compliance approach in Figure 4.7. The pure crack was separated from the pipe system, and the effective applied moment and axial force are expressed by MC and FC, respectively. The axial and vertical forces (Fx, Fy), and bending moment (Mz) at the cut-off section of pipe then can be described as shown in Figure 4.7. Based on this free body diagram, vertical displacement (y), axial displacement (x) and rotation (ψ) at the cut- off point of pipe are expressed as a function of compliances of pipe and
loadings as Eq. 4.30.
1 1 1
2 2 2
1 1 1
2 2 2
1 1 1
2 2 2
1 2
1 2
1 2
x y z
x y z
x y z
x y z
x y z
x y z
x F x x F y x M z
x F x x F y x M z
y F x y F y y M z
y F x y F y y M z
F x F y M z
F x F y M z
x G F G F G M
x G F G F G M
y G F G F G M
y G F G F G M
G F G F G M
G F G F G M
(4.30)
In accordance with the continuity of deformation at cracked section, the boundary conditions are given by the expressions, as follows;
1 2
1 2
1 2 , , , ,
0 0
C crack M C crack F C
x x y y
G M G F
(4.31)
The equilibrium of force and moment are given as below;
app C z
app C x
M M M
F F F
. (4.32)
By substituting the boundary conditions, Eq. 4.30 can be reduced as following relation;
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2 , ,
,1 ,2
0
x x y y z z
x x y y z z
x x y y z z
x F x F x F x F x M x M x
y F y F y F y F y M y M y
F F F F M M crack M z
x x
pipe pipe crack y y
z z
G G G G G G F
G G G G G G F
G G G G G G G M
F F
G G G F A F
M M
, , , ,
0
crack M app crack F app
G M G F
(4.33)
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From the Eq. 4.33, Mz and Fx can be obtained by using Cramer’s rule (Brunetti, 2014):
3,3
, , , ,
3,1
, , , ,
det det
z crack M app crack F app
x crack M app crack F app
M G M G F M
A
F G M G F M
A
(4.34)
where Mi,j is the minor of the matrix A that is the determinant of the smaller matrix formed by eliminating the i-th row and the j-th column from the matrix A. Solving Eq. 4.34, the reduction ratio of the applied bending moment and axial force due to the presence of a crack can be derived as follows;
,
3,3
, , , ,
3,3
, , , ,
,
, , , ,
1
1 det
1 det
1
1
app z
C z
Rest M
app app app
crack M app crack F app
app
app
crack M crack F
app
app x
C x
Rest F
app app app
crack M app crack F ap
M M
M M
C M M M
G M G F M
A M
F M
G G
M A
F F
F F
C F F F
G M G F
3,13,1
, , , ,
det
1 det
p
app
app
crack M crack F
app
M A F
M M
G G
F A
,1 ,2
pipe pipe crack
where AG G G
(4.35)
The restraint coefficient for 2D complex pipe is also expressed as a function of the compliances of pipe and crack, not the parameters that are
limited to the specific pipe configurations. In addition, it should be noted that the amount of decrease in the bending moment and axial force depends on the ratio of Mapp and Fapp.
4.3.2.2 Generalized formulation considering three-dimensional piping system
Finally, the generalized formulation can be derived that can be adopted regardless of the configurations of the pipe and boundary conditions by implementing the compliance approach to a piping system in the space.
Figure 4.8 represents a generalized three-dimensional piping system containing a circumferential crack. In this case, three directions of forces (Fx, Fy, Fz) and moments (Mx, My, Mz) can produce the deformations in six degrees of freedom (x, y, z, θ, ϕ, ψ) as depicted in Figure 4.8. By the similar process employed for the derivation with the 2D pipe, it was assumed that only the bending moment and axial force can produce the rotational displacement of a pure crack so that Eq. 4.29 is available in this case.
Based on the free body diagram in Figure 4.8, the relations between the responses and loadings at the cut-off section of each region of the pipe can be given by a matrix form, as follows:
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1 1
ix iy iz i x i y i z
i x i y iz i x i y i z
ix iy iz i x i y i z
ix iy iz i x i y i z
ix iy iz i x i y i z
ix i y
x F x F x F x M x M x M
i
y F y F y F y M y M y M
i
z F z F z F z M z M z M
i i
i F F F M M M
i F F F M M M
i
F F
G G G G G G
x
G G G G G G
y
G G G G G G
z
G G G G G G
G G G G G G
G G G
1 1 ,
iz i x i y i z
x x
y y
z i z
pipe i
x x
y y
z z
F M M M
F F
F F
F F
M G M
M M
M M
G G G
(4.36)
where subscript i represents the region index (1 or 2) of the pipe, and Gpipe,i is the pipe compliance matrix of region i. By the similar process with the previous section, the assumption that the deformations at the crack position are continuous except the direction of z-rotation gives boundary conditions, as follows;
1 2
1 2
1 2
1 2
1 2
1 2 , , , ,
0 0 0 0 0
c crack M C crack F C
x x y y z z
G M G F
(4.37)
Next, applying these boundary conditions to 4.36 gives:
,1 ,2
, , , ,
0 0 0 0 0
x y z
pipe pipe crack
x y
crack M C crack F C
z
F F G G G F
M M
G M G F
M
(4.38)
Using Cramer’s rule(Brunetti, 2014), one can show that Mzand Fx can be obtained, as follows:
6,6
, , , ,
6,1
, , , ,
,1 ,2
det det
z crack M app crack F app
x crack M app crack F app
pipe pipe crack
M G M G F M
A
F G M G F M
A where A G G G
(4.39)
where Mi,j is the minor of the matrix A that is the determinant of the smaller matrix formed by eliminating the i-th row and the j-th column from the matrix A. The generalized form of the restraint coefficient is then derived as follows;
6,6
, , ,
6,1
, , ,
1 det
1 det
z x
z x
app
Rest M crack M crack F
app
app
Rest F crack M crack F
app
F M
C G G
M A
M M
C G G
F A
.
(4.40)
The efforts in Section 4.3 for the development of the generalized formulation of the restraint coefficient have resulted in a number of important findings that may be worthwhile to be summarized before moving into other subjects. The major findings are summarized as follows;
i) The pipe restraint can reduce the applied moment (crack driving force) at the cracked section.
ii) The presence of a crack in the piping system may results of decrease in the applied moment and axial force at the cracked section.
iii) The load reduction due to the pipe restraint and the presence of a crack
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can be treated as the same phenomenon.
iv) Equation 4.40 is the generalized formulation of the restraint coefficient that means the load reduction ratio at the cracked section. This can be utilized to predict the fracture mechanics parameters of a circumferential crack irrespective of the types of applied loadings and the configuration of the piping system.