Evaluation of effective applied moment for 3d pipe under

Chapter 4 Development of Generalized Formulations on

5.2 Validation under dynamic loading conditions

5.2.3 Evaluation of effective applied moment for 3d pipe under

109

given above, CRest,M and Meff,app were determined as 0.8511 and 595.69 kN∙m, respectively.

It was found that the prediction based on the developed procedure (595.69 kN∙m) has good agreement with both the experimental data (597.66 kN∙m) and dynamic analysis (600.01 kN∙m). Therefore, the concept of effective applied moment can be utilized for the pipe integrity assessment under transient operating conditions.

5.2.3 Evaluation of effective applied moment for 3d pipe under

from 16 inch nominal diameter scheduled 100 TP 304 stainless steel and has a circumferential through-wall crack. Three crack lengths (θ/π=0.250, 0.375, 0.500) were considered, and Figure 5.19 describes the relation between the moment and rotation due to the crack obtained from the finite element analysis using the model created with the continuum element as shown in Figure 5.3. In the same manner with the previous section, this was applied to the behavior of the connector element.

For the input seismic motion, the results of a reactor building analysis from an earlier study (Kim, 2014) were applied. The reactor building analysis was conducted using the measured data of El Centro earthquake (Vibrationdata) to make the response time histories of the anchors and supports. The geometries of containment building were referred from the 1000 MWe Korean Generation II pressurized water reactor (Optimized Power Reacter-100, OPR-1000). The simplified FE model was prepared using shell element without the consideration of the internal walls and other internal structures as illustrated in Figure 5.20. The building was fabricated from post- tensioned concrete, and 5% of the damping ratio was considered based on the regulatory guide 1.61 (US NRC, 2007a).

The piping systems are actually located in a region from the base up to a height of 27.5 m. Among this region, the bottom (0 m) and top (27.50 m) positions were selected to extract the response time histories. To make the piping system be subjected to large loading, it was assumed that the excitation of 0 m is applied to the anchors and supports of the pipe model (see Figure

111

5.20) and that of 27.5 m is applied to the actuator location.

5.2.3.2 Evaluation methods

To verify that the developed solution is available under the dynamic loading conditions from a practical perspective, three cases of analysis were prepared and summarized in Figure 5.21. In this evaluation, only the seismic loading was considered, and other normal operating loads were not included.

i) The current practice of LBB using response spectrum analysis and seismic anchor motion (SAM) analysis:

This represents the current procedure of the LBB analysis. The applied moments due to the inertial load (MRS) and seismic anchor motion (MSAM) are calculated from the response spectrum analysis and the static analysis using the relative displacement of anchors, respectively.

The combined applied moment (MCombined) can be determined as Eqs.

5.21 and 5.22 (US NRC, 2007b).

 Mi Combined  Mi RS  Mi SAM (5.21)

 M Combined  M1 ²Combined M2 ²Combined M3 ²Combined (5.22) where subscript i is the i-th component of moment (i=1, 2, 3). For response spectrum analysis, the response spectrum obtained from building analysis in Figure 5.22 was applied. Since it was assumed that the actuator is connected to different elevation with other supports,

the enveloped response spectrum was considered. In addition, to take into account of the uncertainty of natural frequency of the structure, the peaks were broadened (US NRC, 1973).

In the case of SAM analysis, the inputs data are the displacement time histories of each anchor. Figure 5.23 and Figure 5.24 show the displacement of two locations and relative displacement respectively (Kim, 2014). The maximum applied moment calculated from SAM analysis is MSAM.

ii) Time history analysis using linear elastic and elastic-plastic model:

The time history analysis tends to produce more realistic estimations compared with the above case. In the time history analysis, the effect of inertial load and relative anchor motion can be considered simultaneously if the different time histories are applied to the anchors.

The analysis should be performed repeatedly while changing the time interval of the same time history of input motion to reduce the effects of natural frequency uncertainty (ASME, 2010b). The maximum applied moment during the application of seismic load is then determined as MTH,UcPipe for an uncrakced pipe and MTH,CPipe for a cracked pipe. Both the linear elastic and elastic-plastic behaviors of

113

crack and pipe material are considered.

iii) Using the restraint coefficient

The developed restraint coefficient is the ratio of the effective applied moment at the cracked section considering the boundary conditions to the applied moment at the cracked section calculated from the uncracked pipe analysis. Therefore, CRest,M can be used as a correction factor to consider the presence of a crack to the uncracked pipe analysis results by using following equations;

 

, ,

eff app RS SAM Rest M

M  M  M C

(5.23)

 

, , ,

eff app TH UcPipe Rest M

M  M C

(5.24)

5.2.3.4 Evaluation results

Figure 5.25 describes the comparisons of the applied moment at the cracked section as a function of the crack length for each analysis case. Regarding the method of dynamic analysis, it is clear that the response spectrum analysis with SAM analysis overestimates the response than the time history analysis.

If the presence of a crack is considered in the analysis, the applied moment at the cracked section decreases regardless of analysis methods and this is coincident with the discussions of the earlier studies (Kim, 2014; Scott

et al., 2002). The degree of the reduction increases with the crack length (θ/π).

For example, when the crack length θ/π is 0.25, the applied moment declines 5 % from the case of uncracked pipe. In case of 0.375 and 0.5 of θ/π, the reduction ratio is about 10 % and 25 %, respectively. Additionally, the elastic- plastic behaviors also reduce the applied moment. The reduction ratios are 35%

and 60% for the linear elastic and elastic-plastic time history, respectively.

When it comes to the practical perspective regarding the restraint coefficient, two aspects can be inferred from the evaluation results. First, the implementing of the restraint coefficient into the current practice LBB analysis gives more accurate results of cracked pipe behavior without losing the conservatism due to the overestimation of the response spectrum analysis.

Furthermore, this can be used to secure the margin of the existing pipe analysis results.

Second, the restraint coefficient can enhance the efficiency of the time history analysis. The corrected applied moments by using the restraint coefficient (CRest,M∙MTH,UcPipe) agree with the results predicted from the time history analysis using the cracked pipe (MTH,CPipe) for all cases (see Figure 5.25). It can be indicated that the time history analysis of the piping system for various crack length can be replaced with a single uncracked pipe system analysis using the restraint coefficient for various crack lengths. It may help to improve the efficiency of the probabilistic fracture mechanics analysis or seismic fragility analysis that requires a significant number of time- consuming calculations.

115

Table 5.1 Dimensionless function H4B and H4T in the formula of rotation due to crack for circumferential through-wall cracked pipe determined from

FEA

θ/π

n

2 3 5 7

R/t=10

H4B

0.125 0.063 0.087 0.126 0.148 0.250 0.357 0.407 0.443 0.445 0.500 1.094 0.854 0.585 0.433

H4T

0.125 0.053 0.073 0.100 0.110 0.250 0.306 0.333 0.302 0.249 0.500 0.817 0.623 0.437 0.331

Table 5.2 Loading conditions, material property, and pipe geometries considered for verification of the developed formulation under the static loading conditions

Loading condition Elastic modulus [psi] Dn [inch] Rm/t

Crack Length [θ/π]

(L1/DO, L2/DO)

Internal pressure (2320.6 psi) Dead weight

Relative displacement (x axis: 5 inch, y axis: -5 inch)

Thermal load (563 °F)

2.73E+7 12 5 0.25, 0.5

(1,9), (2,8) (3,7), (4,6), (5,5), (6,4),

(7,3), (9,1)

117

Table 5.3 The material properties applied to the validation analysis of the experiment 1-1 IPIRG-2 program (ASME, 2010a)

(288 °C)

Elastic Modulus

Thermal Expansion

Density

Poisson’s ratio

Instantaneous Mean

[MPa] [m/m/C] [m/m/C] [ton/mm³]

A710 (straight pipe) 185469.0 14.76 13.14 7.76E-09 0.3

WPHY 65 (elbow) 185469.0 14.76 13.14 7.76E-09 0.3

TP304 (crack) 176505.8 19.08 17.64 8.03E-09 0.31

Table 5.4 Comparisons of the natural frequencies of IPIRG-2 piping system between measured data and FE analysis results

Natural frequency [Hz]

1st 2nd 3rd 4th

Experiment 4.5 8.5 14.2 19.2

Analysis 4.60 12.99 15.08 18.45

119

(a) Dependence of rCOD,LE on the crack length (Rm/t=5)

(b) Dependence of rCOD,LE on the crack length (Rm/t=10) Figure 5.1 Comparisons of rCOD,LE predicted using the developed formulations and linear elastic FEA – symmetric model (Miura, 2001)

0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

/= 0.125 : FEA - Miura

/= 0.125 : Eng. Formula

/= 0.250 : FEA - Miura

/= 0.250 : Eng. Formula

/= 0.500 : FEA - Miura

/= 0.500 : Eng. Formula

Symmetric Restraint R_m/t=5

Restraint COD ratio, r COD,LE

Normalized Restraint Length, L/D

0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Restraint COD ratio, rCOD,LE

Normalized Restraint Length, L/D

/= 0.125 : FEA - Miura

/= 0.125 : Eng. Formula

/= 0.250 : FEA - Miura

/= 0.250 : Eng. Formula

/= 0.500 : FEA - Miura

/= 0.500 : Eng. Formula

Symmetric Restraint R_m/t=10

(a) Dependence of rCOD,LE on the crack length (L2/D=10)

(b) Dependence of rCOD,LE on the crack length (L2/D=5) Figure 5.2 Comparisons of rCOD,LE predicted using the developed

0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Non-symmetric Restraint R_m/t=10

L₂/D=10

/= 0.125 : FEA - Miura

/= 0.125 : Eng. Formula

/= 0.250 : FEA - Miura

/= 0.250 : Eng. Formula

/= 0.500 : FEA - Miura

/= 0.500 : Eng. Formula

Restraint COD ratio, r COD,LE

Normalized Restraint Length, L

1/D

0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Non-symmetric Restraint R_m/t=10

L₂/D=5

/= 0.125 : FEA - Miura

/= 0.125 : Eng. Formula

/= 0.250 : FEA - Miura

/= 0.250 : Eng. Formula

/= 0.500 : FEA - Miura

/= 0.500 : Eng. Formula

Restraint COD ratio, rCOD,LE

Normalized Restraint Length, L₁/D

121

(c) Dependence of rCOD,LE on the crack length (L2/D=1) Figure 5.2 Comparisons of rCOD,LE predicted using the developed formulations and linear elastic FEA – asymmetric model (Miura, 2001)

(Continued)

0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.4 /= 0.125 : FEA - Miura

/= 0.125 : Eng. Formula

/= 0.250 : FEA - Miura

/= 0.250 : Eng. Formula

/= 0.500 : FEA - Miura

/= 0.500 : Eng. Formula

Restraint COD ratio, r COD,LE

Normalized Restraint Length, L

1/D Non-symmetric Restraint

Rm/t=10 L2/D=1

Figure 5.3 3D FE model of a circumferential through-wall cracked pipe used for tabulations of new dimensionless functions (H4T, H4B)

123

(a) Dependence of rCOD,EP on the crack length (L1/D= L2/D =1)

(b) Dependence of rCOD,EP on the crack length (L1/D= L2/D =10) Figure 5.4 Comparisons of rCOD,EP predicted using the developed formulations and elastic-plastic FEA – symmetric model (Kim, 2008)

0 50 100 150 200 250

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Restraint Length : L₁/D=L₂/D=1 Material : Ref. (Kim, 2008) R_m/t=10

Restraint COD ratio, rCOD,EP

Nominal Tensile Stress, _t [MPa]

/=0.125 : FEA - Kim

/=0.125 : Eng. Calc.

/=0.250 : FEA - Kim

/=0.250 : Eng. Calc.

/=0.500 : FEA - Kim

/=0.500 : Eng. Calc.

0 50 100 150 200 250

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Restraint COD ratio, rCOD,EP

Nominal Tensile Stress, 

t [MPa]

/=0.125 : FEA - Kim

/=0.125 : Eng. Calc.

/=0.250 : FEA - Kim

/=0.250 : Eng. Calc.

/=0.500 : FEA - Kim

/=0.500 : Eng. Calc.

Restraint Length : L₁/D=L₂/D=10 Material : Ref. (Kim, 2008) R_m/t=10

(c) Dependence of rCOD,EP on material property (L1/D= L2/D =10) - Mat. Ref. : Reference material property for 304SS (Kim, 2008) - Mat. 3 : CF8M(288°C) material property (Kim, 2008)

- Mat. 7 : A106(288°C) material property (Kim, 2008)

Figure 5.4 Comparisons of rCOD,EP predicted using the developed formulations and elastic-plastic FEA – symmetric model

(Kim, 2008) (Continued)

0 50 100 150 200 250

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Restraint Length : L

1/D=L

2/D=10

/=0.25 Rm/t=10

Restraint COD ratio, r COD,EP

Nominal Tensile Stress, _t [MPa]

Mat. Ref. : FEA - Kim Mat. Ref. : Eng. Calc.

Mat. 3 : FEA - Kim Mat. 3 : Eng. Calc.

Mat. 7 : FEA - Kim Mat. 7 : Eng. Calc.

125

Figure 5.5 3D FE model of 3D piping system containing a circumferential through-wall crack used for verification of the developed formulation

Figure 5.6 FE model using beam element of 3D piping system to calculate the pipe compliance for verification of the developed formulation

L/D_O=10 L/D_O=10

L₁

L₂

R/D_O=1.5

Anchor 1

Anchor 2

z x y

Anchor 2

Crack

L/D_O=10 L/D_O=10

L₁

L₂

R/D_O=1.5

Anchor 1

z x y

Figure 5.7 Comparisons of applied moment and axial force at the cracked section calculated from finite element analysis

Figure 5.8 Comparisons of applied nominal stress at the cracked section due

0 2 4 6 8 10

-4.0x10² -2.0x10² 0.0 2.0x10² 4.0x10² 6.0x10² 8.0x10²

Applied moment at cracked section [kN-m]

Crack position from Anchor 1 (L₁/D_o) Applied moment

Uncracked pipe

Cracked pipe (=0.25) Cracked pipe (=0.50)

8.0x10² 1.0x10³ 1.2x10³ 1.4x10³ Applied axial force

Uncracked pipe Cracked pipe (=0.25) Cracked pipe (=0.50)

Applied axial force at cracked section [kN]

0 2 4 6 8 10

-4.0x10² -2.0x10² 0.0 2.0x10² 4.0x10² 6.0x10² 8.0x10²

Nominal stress at cracked section [MPa]

Crack position from Anchor 1 (L₁/D_o)

bending

axial force

Uncracked pipe

Cracked pipe (=0.25) Cracked pipe (=0.50)

127

(a) Bending Moment

(b) Axial force

Figure 5.9 Comparisons of the restraint coefficient and the ratio of load reduction calculated from finite element analysis

0 2 4 6 8 10

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Restraint effect on aplied moment M eff,app/M app

Crack position from Anchor 1 (L₁/D_o) Formula FEA

=0.25 =0.50

0 2 4 6 8 10

0.6 0.7 0.8 0.9 1.0 1.1 1.2

Restraint effect on aplied axial force, F eff,app/F app

Crack position from Anchor 1 (L

1/D

o) Formula FEA

=0.25 =0.50

Figure 5.10 FE model used for analysis of simulated seismic pipe system analysis of IPIRG-2 program

129

Figure 5.11 Applied moment and rotation due to the crack of experimental results and input data used for connector element behavior

(a) From 0 to 15 second

(b) Full time history

Figure 5.12 Comparisons of applied moment time history at the cracked section between experiment and analysis result

0 5 10 15

-1000 -800 -600 -400 -200 0 200 400 600 800

Applied Moment at Cracked Section [kN-m]

time [sec]

Experiment Analysis

0 5 10 15 20 25

-1000 -800 -600 -400 -200 0 200 400 600 800

Applied Moment at Cracked Section [kN-m]

time [sec]

Experiment Analysis

131

Figure 5.13 Comparisons of reaction load time history at Node 6 between experiment and analysis results

0 5 10 15 20 25

-1000 -750 -500 -250 0 250 500 750 1000

Node 6 Reaction load [kN]

time [sec]

Experiment Analysis

(a) N-S

(b) E-W

Figure 5.14 Comparisons of displacement load time history at Elbow 3 between experiment and analysis results

0 5 10 15 20 25

-100 -75 -50 -25 0 25 50 75 100

Displacement at Elbow3 (N-S) [mm]

time [sec]

Experiment Analysis

0 5 10 15 20 25

-150 -100 -50 0 50 100 150

Displacement at Elbow3 (E-W) [mm]

time [sec]

Experiment Analysis

133

(a) N-S

(b) E-W

Figure 5.15 Comparisons of displacement load time history at Node 21 between experiment and analysis result

0 5 10 15 20 25

-30 -15 0 15 30

Displacement at Node 21 (N-S) [mm]

time [sec]

Experiment Analysis

0 5 10 15 20 25

-150 -100 -50 0 50 100 150

Displacement at Node 21 (E-W) [mm]

time [sec]

Experiment Analysis

Figure 5.16 3D FE model of pipe containing a surface crack to calculate the compliance of a crack

Figure 5.17 Elastic-plastic compliance of the surface crack (Equivalent crack length of (θ/π) = 0.383)

0.0 2.0x10⁸ 4.0x10⁸ 6.0x10⁸ 8.0x10⁸

0.00E+000 1.00E-011 2.00E-011 3.00E-011

G Crack,PE [rad/N-m]

Applied Moment [N-m]

Equivalent Crack length () 0.383

135

Figure 5.18 Applied moment at cracked section calculated from uncracked pipe analysis for experiment 1-1 of IPIRG-2 program

0 5 10 15

-1000 -800 -600 -400 -200 0 200 400 600 800

Applied Moment at Cracked Section [kN-m]

time [sec]

Uncracked pipe

M

_app

= 699.887 kN-m

Figure 5.19 Applied moment and rotation due to the crack applied as the behavior of connector element

0.00 0.02 0.04 0.06 0.08 0.10

0 200 400 600 800

Appli ed Mom ent [kN- m ]

Rotation due to the crack [rad]

Crack length () 0.250 0.375 0.500

137

Figure 5.20 Geometries of containment building of OPR-1000 type plant and FE model (Kim, 2014)

Pipe region

Anchor &

support Actuator

Figure 5.21 Schematic diagram of procedures and methods to calculate the effective applied moment to validation of the developed formulation

Displ. Time history Floor response

spectrum

Containment Analysis Ground acc.

Acc. Time history

Current practice

(w/o crack, LE)

Response Spectrum Analysis

Using time history analysis

(w/ crack, LE&EP)

M

_RS

Seismic Anchor Motion Analysis

M

_TH

M

_SAM

│M

_RS

│+│M

_SAM

│

Using restraint coefficient

(w/ crack, LE)

( ^│M

_RS

^│+│M

_SAM

^│ ) ^∙C

_Rest

139

Figure 5.22 Acceleration response spectrum obtained from containment building analysis (Kim, 2014)

10^-1 10⁰ 10¹ 10²

1 2

Acce lera ti on [g]

Frequency [Hz]

Anchors and supports Actuator

Enveloped spectrum

Figure 5.23 Displacement time histories of two selected locations obtained from containment building analysis (Kim, 2014)

Figure 5.24 Relative displacement time histories between two selected locations obtained from containment building analysis (Kim, 2014)

0 10 20 30 40 50

-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120

Displacement [mm]

Time [s]

Anchors and Supports Actuator

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

-5 -4 -3 -2 -1 0 1 2 3 4 5

Relative Displacement [mm]

Time [s]

141

Figure 5.25 Comparisons of the reduction ratios of the applied moment at the cracked section predicted using the time history analysis, restraint

coefficient compared with the current practice of LBB

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Using the restraint coeff. [C

Rest(M

RS+M

SAM)]

Time history analysis [M

TH] - Linear elastic model Time history analysis [M

TH] - Elastic plastic model Using the restraint coeff. [C_RestM_TH] -Linear elastic

Ratio of applied moment to current practice

Crack length []

0 200 400 600 800

Applied moment at cracked section [kN-m]

143