Chapter 6 Application of Developed Formulations
6.1 Applicability of developed formulations in LBB design 144
In the previous chapter, it was demonstrated that the restraint coefficient can predict the reduction ratio of applied moment or axial force at the cracked section by comparing with the FE analysis results and experimental data. In this section, the additional FE analyses were conducted to compare the COD and J integral that are the primary fracture mechanics parameters in LBB design (see Figure 1.2) to validate the applicability of the developed formulations from practical aspects.
6.1.1 Validation methods
To validate the applicability of the restraint coefficient to the LBB analysis, three cases of FE analyses were conducted. The detailed descriptions of each case are summarized in Figure 6.1. For the loading condition, only the distributed vertical load along the pipe was considered.
i) Case 1: Not considering pipe restraint effect
This case represents a current LBB evaluation procedure. The applied moment (Mapp) at the cracked section of an uncracked pipe can be simply calculated by solving beam equation for pipe subjected a distributed load. Mapp was then applied to the pipe end of a 3D FE model containing a circumferential throw-wall crack under free-ended
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boundary conditions to calculate COD or J-integral.
ii) Case 2: Considering pipe restraint effect – 3D FEA including restrained boundary conditions
In this case, 3D FEA models of the pipe containing a circumferential TWC under the fixed-ended boundary conditions were prepared.
Distributed vertical load which can represent various types of loading in nuclear piping systems was directly applied to the 3D FE models to calculated COD or J-integral.
iii) Case 3: Considering pipe restraint effect – Using the restraint coefficient
A 3D FEA model under free-ended boundary conditions same with that of case 1 was employed. Instead of the applied moment of an uncracked pipe (Mapp), the effective applied moment (Meff,app) was used, which is calculated by multiplying the linear elastic restraint coefficient by Mapp.
A commercial finite element analysis code, ABAQUS (Dassault Systémes, 2012) was used. A through-wall circumferential cracked pipe was simulated as a half-model using the 20-noded continuum element with reduced integration shown in Figure 6.2, 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 were ignored.
For the loading condition, the distributed vertical loads and internal pressures were considered. The distributed load was applied as a type of the gravity to the lower part of the continuum model. In the case of the internal pressure, a pressure and corresponding axial force were applied to the pipe inner surface and pipe ends, respectively. In addition, the half value of the pressure was applied to the crack face.
TP316 stainless steel 12 inch diameter pipe which is used in a typical primary side of the nuclear power plant was considered. The high temperature tensile property (327 ℃) of TP316 is represented in Figure 6.3. Three crack lengths (θ/π=0.125, 0.25, 0.5) were prepared, and a symmetric model with a crack in the center of the pipe (L1/Do:L2/Do=5:5, 10:10, 20:20) and an asymmetric model with a crack in the off-center of the pipe (L1/Do:L2/Do
=1:10, 1:20) were considered. Table 6.2 summarizes details on the analysis cases.
6.1.2 Validation results of COD and J-integral
Figure 6.4 and Figure 6.5 show the comparison results of COD and J-integral, in which the internal pressure was not included. X axis of each graph means the applied moment at the cracked section calculated from the uncracked
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elastic pipe (Mapp). The difference in the results between for case 1 and 2 is due to the effects of the restrained boundary conditions. Since the case 2 considered the restraint effect, the COD and J integral are lower than for case 1. The degree of the pipe restraint is increased with the increase of the crack length (θ/π) and the decrease of the restraint length (L1/Do or L2/Do). Analysis results in which the internal pressure was included are represented in Figure 6.6 and Figure 6.7. Values of COD and J-integral corresponded with those the case of without an internal pressure at the pressure equivalent moment.
General trends that were observed in both cases agreed well with each other.
It should be noted that in the case of the symmetric model with a crack length of 0.125(θ/π), the case 2 tends to overestimate than the case 1 when a large amount of moment is applied (See graph (a) of Figure 6.5 and graph (a) of Figure 6.7). If a crack is at the position where the anticipated moment is relatively low in the piping system, the effective applied moment can be increased because of the load redistribution due to the plastic deformation of the pipe. In virtually, this does not likely to occur since the region subjected to a low value of loading has the low probability of the crack initiation, and is not considered as a critical location of evaluations. Nevertheless, this should be carefully discussed under the extremely beyond design basis conditions.
Results predicted by using the linear elastic restraint coefficient (case 3) agreed well with case 2 while the case 3 overestimated COD and J-integral when the applied load was large enough for plastic deformations to occur. If the elastic-plastic restraint coefficient is used, results of case 3 could be close
to those of case 2 even in the plastic region.
Generally, it can be seen that the results predicted using the linear elastic restraint coefficient were closer to the realistic case (case 2) than the results calculated through the current LBB analysis method (case 1), slightly overestimating than the case 2. It was confirmed that the restraint coefficient could enhance the accuracy of the prediction of COD and J integral considering the pipe restraint effect without losing the conservatism for LBB evaluation.
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