Evaluation procedure to determine effective applied moment

Chapter 4 Development of Generalized Formulations on

4.4 Evaluation procedure to determine effective applied moment

4.4 Evaluation procedure to determine effective applied

are available. The deformation of the crack is allowed only one direction of rotation due to bending moment and axial force. The compliance of a crack is a rotation (ψ) due to the crack per unit moment (M) or axial force (F), and can be easily calculated using the analytic solutions or the finite element analysis as follows:

, ,

cracked pipe uncracked pipe crack M

cracked pipe uncracked pipe crack F

G M M

G F F



 



 



  

 

  

  



(4.41)

iii) Define the compliances of the pipein global coordinate (Gpipe,i,global):

The compliance of pipe should be determined for both side of the crack. To calculate Gpipe,1,global, exclude boundary conditions of region 2. Then impose the unit forces (Fx, Fy, Fz) and moments (Mx, My, Mz) on the crack position and calculate the deformation (x, y, z, θ, ϕ, ψ) to produce the compliance matrix as Eq. 4.42. For a case of region 2, repeat vice versa and determine Gpipe,2,global.

, ,

i x i y i z i x i y i z

x F x F x F x M x M x M

y F y F y F y M y M y M

z F z F z F z M z M z M

pipe i global

F F F M M M

G G G G G G

G G G G G G G

G G G G G G

     

     

     













 



(4.42)

iv) Transform the compliances of the pipe to the crack coordinate: The coordinate of the crack decided in step ii) does not always coincide with the global coordinate. Thus, the compliances of the pipe in global coordinate should be transformed to the crack coordinate using the usual tensor transformation rule as Eq. 4.43.

, , , ,

pipe i crack pipe i global

G T G T (4.43)

When the z’-axis (crack coordinate) make the angle α with respect to the z-axis (global coordinate), then the transformation matrix T is as follow;

1 0 0 0 0 0

0 cos sin 0 0 0

0 sin cos 0 0 0

0 0 0 1 0 0

0 0 0 0 cos sin

0 0 0 0 sin cos

 

 

  

 

  

 

  

 

 

(4.44)

v) Determine the restraint coefficient by substituting the compliance of the pipes and crack on the Eq. 4.45.

6,6

, , ,

6,1

, , ,

,1, ,2,

6,6 6,1

1 det

, minor of matrix A

z x

app

Rest M crack M crack F

app

Rest F crack M crack F

app pipe crack pipe crack crack

F M

C G G

M A

M M

C G G

F A

A G G G

M M

 

  

    

  

  

  

     

  



  



(4.45)

vi) Calculate the effective applied moment and force by multiplying the restraint coefficient on Mapp or Fapp using Eq. 4.46.

, ,

eff app Rest M app eff app Rest F app

M C M

F C F

 

  

 (4.46)

Although the restraint coefficient has been derived based on the elastic beam theory, the nonlinearity of crack behavior or pipe material can be reflected in the developed formulation. When it comes to determine the compliance, one can consider following combinations:

- Linear elastic pipe (Gpipe,LE) + Elastic-plastic crack (Gcrack,LE) - Linear elastic pipe (Gpipe,LE) + Elastic-plastic crack (Gcrack,EP) - Elastic-plastic pipe (Gpipe,EP) + Elastic-plastic crack (Gcrack,EP)

In order to compare the conservatism of above cases, an example analysis was conducted using the beam model subjected the distributed as shown in Figure 4.5. 4 cases of the pipe length were analyzed (2L/Do=5, 10, 15, 20) while L1/Do was fixed to 1.

As a result, Figure 4.10 shows the comparisons the effective applied moment at cracked section (Meff,app) depending upon the Mapp calculated from the elastic analysis of the uncracked pipe. The difference between graph (a) and (b) indicates that if only the plastic behavior of a crack is considered, the applied moment can be reduced more than the case of the elastic model.

However, if the plastic deformation of the pipe occurs, the cracked section subjected more significant moment than the elastic pipe + plastic crack case

due to the load redistribution.

When the elastic-plastic behavior is considered, the compliance depends on the amount of applied load. Therefore, above steps from i) to vi) should be repeated to calculate the effective applied moment and axial force.

For convenience, following suggestion regarding the consideration of nonlinearity can be available: Under the design basis conditions, an assumption that crack and pipe do not experience the plastic behavior can lead conservative results while the case that allows the non-linear behavior of crack only is more accurate. If the applied load is large enough to cause the plastic deformation of the pipe, the elastic-plastic compliances of crack and pipe must be considered.

Figure 4.1 The concept of effective applied moment at the cracked section M_eff,app

Crack M_app

Crack

Reaction M, F Initial Applied Moment

Load redistribution

Effective Applied Moment

Figure 4.2 Beam model of fixed-ended pipe with a circumferential crack subjected to a pressure induced bending for development of the moment

restraint coefficient

M Δψ

G_crack,ψ,M

Crack : M_{Press, Eq.}

Region-1 Region-2

M_React,Rest x

L₁ L₂

2L=L₁+L₂

F_React,Rest

Figure 4.3 Schematic descriptions of the compliance approach 1. Separate the cracked section from a piping system

2. Define the compliances of pipe (G_pipe) and crack (G_crack)

3. Derive the restraint coefficient in terms of compliances

Crack

y z

yF y yM z

F y M z

y G F y G M

G_ F G_ M

 

 

   

 

     

 



L₁ L₂

M Δψ

GCrack

Crack

G M



  

Mz F_y ψ y

Crack

L₁ L₂

1. Separation of the pipe and crack

2. Definition of the compliance

L₁ L₂

Crack

=

L₁ L₂

Pure Crack

+

M Δψ

G_crack,ψ,M

, , crack M

G _ M

Figure 4.4 Beam model and free body diagram of fixed-ended pipe with a circumferential crack subjected to a pressure induced bending for development of the moment restraint coefficient based on the compliance

approach M_app

L₁ L₂

2L=L₁+L₂

x z

M_C

M_Z M_Z

F_y F_y M_app=M_C+M_Z

Figure 4.5 Beam model and free body diagram of fixed-ended pipe with a circumferential crack subjected to a distributed load for development of the

moment restraint coefficient based on the compliance approach

L₁

w

L₂

2L=L₁+L₂

L₁

w

L₂

2L=L₁+L₂

M_C,CPipe M_C,UcPipe

M_B M_A

F_A F_B

M_B M_A

F_A F_B

Figure 4.6 Beam model and free body diagram of fixed-ended pipe with a circumferential crack subjected to a relative displacement of the supports for

development of the moment restraint coefficient based on the compliance approach

F_A

F_B

d M_A

M_B

L₁ L₂

2L=L₁+L₂

F_A

F_B

d M_A

M_B

L₁ L₂

2L=L₁+L₂

M_C,CPipe M_C,UcPipe

Figure 4.7 Beam model and free body diagram of 2D piping system containing a circumferential crack for development of the restraint

coefficient based on the compliance approach M_Z

F_Y F_X

F_Y

M_Z F_X

M_app=M_C+M_Z F_app = F_C+ F_X Region-1 Region-2 y

x z

M_app, F_app

M_C,F_C

Figure 4.8 Beam model and free body diagram of the generalized 3D piping system containing a circumferential crack for development of the restraint

coefficient based on the compliance approach M_app, F_app

M_C,F_C

M_app=M_C+M_Z F_app = F_C+ F_X

Region-1 Region-2

x z

Fx,Mx

F_Y,M_Y

Fz,Mz

F_Y,M_Y

Fz,Mz

Fx,Mx

Figure 4.9 The procedure for calculation of the effective applied moment and force

i) Calculate applied moment and axial force at crack position, and determine orientation of crack to make the crack subjected to maximum M

ii) Define the compliances of crack (G

_crack

) for M-rotation & F-rotation behavior

G_crack,ψMz=

z y'

α

z' Global

coordinate Crack

coordinate

0 0 0 0 0 0

0 0 0 0

x z

crack

F M

G_ G_

 

 

  

 

 

y x z

Crack position

G_crack,ψFx=

M, Ψ_uncrack/2

M, Ψ_crack/2 F, Ψ_uncrack/2

F, Ψ_crack/2

Figure 4.9 The procedure for calculation of the effective applied moment and force (continued) iii) Define the compliances of pipe (Gpipe,i,global) for 6 D.O.Fs in global coordinate

<Region 1>

Exclude boundary conditions of region 2

Apply a unit load on crack position and calculate the deformation Determine Gpipe,1,global

<Region 2>

Exclude boundary conditions of region 1

Apply a unit load on crack position and calculate the deformation Determine Gpipe,2,global

, ,

ix i y iz i x i y i z

i x i y i z i x i y i z

ix i y i z i x i y i z

i x i y i z i x i y i z

x F x F x F x M x M x M

y F y F y F y M y M y M

z F z F z F z M z M z M

pipe i global

F F F M M M

G G G G G G

G G G G G G G

G G G G G G

     

     

     













 

 Region-1

x z

FZ,MZ

FY,MY

F_X,M_X

Region-2

F_Z,M_Z FY,MY

F_X,M_X

Figure 4.9 The procedure for calculation of the effective applied moment and force (continued) iv) Transform the compliances of pipe (Gpipe,i,global) to crack coordinate (Gpipe,i,crack)

v) Determine the restraint coefficient (C_Rest)

vi) Calculate the effective applied moment and axial force (M_eff,app, F_eff,app)

,1, ,2,

6,6 minor of matrix A

pipe crack pipe crack crack

A G G G

  



, , , ,

pipe i crack pipe i global

G  T  G  T

Transformation matrix T 

z y' α

z' Global

coordinate Crack

coordinate

1 0 0 0 0 0

0 cos sin 0 0 0

0 sin cos 0 0 0

0 0 0 1 0 0

0 0 0 0 cos sin

0 0 0 0 sin cos

 

 

  

 

  

 

  

 

 

6,6

, , ,

6,1

, , ,

1 det

z x

app

Rest M crack M crack F

app

Rest F crack M crack F

app

F M

C G G

M A

M M

C G G

F A

 

  

    

  

  

  

     

  



, ,

eff app Rest M app eff app Rest F app

M C M

F C F

 

   



(a) Elastic pipe + Elastic crack

(b) Elastic Pipe + Plastic Crack

Figure 4.10 Effect of nonlinear behavior on the applied moment at the

0 100 200 300 400 500

0 50 100 150 200 250

Limit of design moment Limit moment

of crack

(L1+L

2)/D

5 10 15 20

Applied moment at crack position of cracked pipe [kN-m]

Applied moment at crack position of uncracked pipe [kN-m]

Elastic Pipe + Elastic Crack

0 100 200 300 400 500

0 50 100 150 200 250

Elastic Pipe + Plastic Crack (L₁+L₂)/D_o

5 10 15 20

Applied moment at crack position of cracked pipe [kN-m]

Applied moment at crack position of uncracked pipe [kN-m]

Limit moment of crack

Limit of design moment

Figure 4.10 Effect of nonlinear behavior on the applied moment at the cracked section (continued)

0 100 200 300 400 500

0 50 100 150 200 250

Plastic Pipe + Plastic Crack (L1+L

2)/D

5 10 15 20

Applied moment at crack position of cracked pipe [kN-m]

Applied moment at crack position of uncracked pipe [kN-m]

Limit moment of crack

Limit of design moment

Chapter 5 Validation of Developed Formulations

To verify the generalized analytical formulation and concept of the effective applied moment derived in this dissertation, benchmark studies are carried out against both FEA-based numerical analysis as well as well-documented large-scale experimental data. In the benchmark against numerical analysis, both static and dynamic loading conditions are included by a series of finite element analysis. In dynamic loading conditions, a very detailed time history analysis was included.

First, in the same context as in the development process of the restraint coefficient, the effects of pressure induced bending restraint on COD were predicted using FEA from earlier studies and the results from generalized formulations were compared. Then the static analyses were performed to evaluate the amount of restraint considering the anticipated loads at the cracked section under the normal operating conditions. In addition, the crack stability analysis assumes the faulted loading condition in which the seismic load is considered. Finally, the dynamic analysis results using cracked pipe model were compared with experimental data to demonstrate that restraint coefficient is valid under the dynamic loading conditions with excellent agreements.

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