Chapter 4 Development of Generalized Formulations on
5.2 Validation under dynamic loading conditions
5.2.1 Benchmark dynamic analysis using cracked pipe model100
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5.2.1.1 Dynamic analysis methods for nonlinear piping system
This subsection describes the procedures of time history analysis using the finite element (FE) models of the uncracked and cracked piping systems subjected to a seismic loading. The time history analysis is one of the dynamic analysis methods, and the process of solving the equation of motion as a function of time (Chopra, 2007; Kim, 2014). This method can give more accurate results compared with the response spectrum analysis that is typically used in the seismic designs of structures. Key procedures and parameters should be considered in the time history analysis as follows;
i) Finite element (FE) model
The piping system can be simulated by using the beam element to calculate the applied moment at each point. The number of nodes should be determined to make the appropriate vibration shapes feed into the analysis.
For crack modeling, the “connector element” in ABAQUS 6.12 (Dassault Systémes, 2012) can be adopted at the position of a crack (Zhang et al., 2010). The connector element can join the position of two nodes and provide a rotational connection. To make the connector element behave like a crack, the relation between load and displacement due to the crack is applied as the behavior of the connector element (Zhang et al., 2010). An additional connector
element needs to be used in parallel to limit the crack closure.
ii) Input of applied loading
The seismic inputs of piping system are the excitations of the anchors or supports that are obtained from reactor building analyses using ground acceleration time histories. The requirements regarding time increment and frequency content for the design time histories to achieve reliable estimations are stated in Standard Review Plan (SRP) 3.7.1 (US NRC, 2007c).
If the supports have different motions, the response time history of each supports can be applied separately so that the effect of inertial load and relative anchor motion can be considered simultaneously.
To consider the uncertainties of the natural frequency of a structure, the analysis can be conducted three or five times for given time history while changing the time increments (ASME, 2010b).
iii) Modal analysis
Before implementing the time history analysis, the dynamic characteristics of structures must be evaluated using the modal analysis. The appropriate boundary conditions and preloading need to be reflected in the analysis model.
The information obtained from the results of the modal analysis
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such as the natural frequency, mode shape and effective mass are utilized in the dynamic analysis in a variety of ways. For example, by examining the mode shapes and mass participation factors considering the directions of applied excitations, the minimum and maximum modes that should be considered in the time history analysis can be determined. The appropriate time step for the time history analysis depends on the maximum frequency to be considered.
iv) Consideration of damping
The effects of damping due to the energy dissipation can be considered.
Regulatory guide 1.61 (US NRC, 2007a) gives the acceptable damping ratio to be used. In the case of the pipe subjected to a safety shutdown earthquake (SSE), 4% of the damping ratio is recommended.
One of the common damping models for dynamic analysis is Rayleigh damping model which describes damping ratio as follow:
2 2
n n
n
(5.19)
where ξn and ωn denote the damping ratio and the natural frequency of the n-th mode, respectively. α and β are the damping coefficients.
The coefficients α and β can be determined using the natural frequencies of two modes with specific damping ratio (Chopra, 2007).
These modes should be selected so that the dominant vibration mode
can be considered in the time history analysis.
v) Calculation of applied load
Using FE model and inputs data described above, one can implement the time history analysis. The maximum value of the applied moment at cracked section during loading can be used for calculating crack opening displacement and allowable moment of a crack.
5.2.1.2 Verification of nonlinear dynamic analysis methods
To validate the applicability of the time history analysis methods using the cracked pipe model described in the proceeding subsection, detailed analysis of simulated seismic pipe system experiment (Experiment 1-1) of the second international piping integrity research group (IPIRG-2) program was conducted (Hopper et al., 1996; Scott et al., 1996).
The main purpose of Experiment 1-1 was to investigate the behavior of piping system containing a circumferential surface crack under simulated seismic loading condition. Figure 5.10 shows the FE model of the piping system used in the experiment. The straight pipe was fabricated from ASTM A710, Grade A, Class 3 pipe (sch. 100) and interconnected elbows were of the type WPHY-65 (sch. 100 and 160). First of all, the straight pipe and elbow were simulated using PIPE31 which in one of the beam element of the finite
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restraint device (320kg) in the cracked section were represented as the point mass and distributed mass respectively. The detail information about pipe configurations are described in the reports (Hopper et al., 1996; Scott et al., 1996), and Table 5.3 summarizes the material properties applied to the analysis referred from ASME Code (ASME, 2010a).
For simulation of the crack, two connector elements were adopted that represent the crack opening and crack closure, respectively, as described in the preceding subsection. It was assumed that the crack could deform only in the rotational direction. The measured crack moment versus rotation curve was used as input for the behavior of the connector element in the crack opening direction (see Figure 5.11). The decrease of load-carrying capacity of the crack after reaching the maximum load was not considered.
Prior to the time history analysis, the modal analysis was conducted first in which only the elastic behaviors of the crack and pipe were considered.
As the preloading, the dead weight, internal pressure (15.5 MPa), and operating temperature (288 °C) were applied. Table 5.4 shows the comparisons of the natural frequencies between measured and predicted results. It can be seen that the analysis predicted the natural frequencies accurately except the second mode. However, this may not affect the pipe analysis significantly because the dominant mode shape of the 2nd mode is in the vertical direction while the simulated seismic loading was applied in E- W direction, and the piping system dominantly behaves in N-S and E-W direction during excitation due to the boundary conditions.
The measured value 4.5% was applied as the damping ratio, and the
minimum and maximum mode were determined as the 1st (4.60 Hz) and 9th (43.51 Hz) modes respectively to consider the behavior of the piping system both in N-S and E-W directions. The Rayleigh damping coefficient in Eq.
5.19 then were calculated as α=2.354257 and β=0.000298.
Based on the results of the modal analysis, the time history analysis was conducted. Before the dynamic analysis, the initial response (t=0 second) was calculated using static analysis considering the dead weight, internal pressure and thermal loading. Figure 5.12 shows the comparison of the applied moment at the cracked section between the measured and predicted results. It can be seen that overall waveform of vibration agrees well with the experiment results. In the experiment, the wall penetration of the crack occurred at 14.035 second. After that, the difference is observed since the applied moment is reduced because of the decrease in load-carrying capacity of crack that is not considered in the analysis. Nevertheless, the maximum applied moment was accurately predicted (600.01 kN∙m) compared with the measured value (597.66 kN∙m) within 0.5%. It can be concluded that the time history method using the elastic-plastic cracked pipe model provides an accurate estimation under the seismic loading conditions.
5.2.2 Validation of developed formulations using experimental measurements and dynamic analysis results
In this subsection, an example analysis was conducted to validate the developed formulations under the dynamic loading condition. The effective
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applied moment predicted using the restraint coefficient was compared with the experimental measurements of the experiment 1-1 of IPIRG-2 and dynamic analysis results of section 5.2.1. According to the evaluation procedure stated in section 4.4, the effective applied moment was calculated as follows;
i) Calculate Mapp from the uncracked pipe analysis: An additional dynamic analysis for an uncracked pipe was conducted as the same process of section 5.2.1.2. The calculated applied moment at the crack position is shown in Figure 5.18. The maximum value of 699.87 kN∙m was determined as Mapp.
ii) Define the compliance of the crack, Gcrack: The test specimen including the surface crack of IPIRG-2 piping system experienced the plastic behavior. Thus, the compliance of the crack was determined based on the elastic-plastic finite element analysis using the 3D solid model which is depicted in Figure 5.16. The rotation due to only a bending moment was considered because the effect of axial force was negligible in the experiment. The rotational compliance of crack was represented as a function of applied bending moment as shown in Figure 5.17.
iii) Define the compliances of the pipes: To calculate the compliance
matrices, the piping system was separated into region 1 (from anchor 1 to crack) and region 2 (from crack to anchor 2). Based on the linear elastic FE analysis using beam model in Figure 5.10, the compliance matrices were derived as follows;
,1
5.93E-04 6.56E-04 5.82E-06 -6.89E-10 -9.46E-10 1.71E-07 6.56E-04 1.30E-03 2.82E-06 -3.02E-10 -4.72E-10 3.33E-07 5.82E-06 2.82E-06 2.03E-03 -1.14E-07 -4.16E-07 9.77E-10 -6.89E-10 -3.02E-10 -1.14E-07 8.61E-11 1.33E-1
Gpipe
1 -1.04E-13 -9.46E-10 -4.72E-10 -4.16E-07 1.33E-11 1.04E-10 -1.63E-13 1.71E-07 3.33E-07 9.77E-10 -1.04E-13 -1.63E-13 1.00E-10
,2
3.00E-03 8.99E-04 6.96E-05 0.00E+0 3.51E-08 -4.54E-07 8.99E-04 3.68E-04 1.59E-05 4.63E-10 7.61E-09 -1.89E-07 6.96E-05 1.59E-05 6.15E-04 4.71E-08 2.75E-07 -7.61E-09 0.00E+0 4.63E-10 4.71E-08 5.20E-11 -5.32E-13 0.00E+
Gpipe
0 3.51E-08 7.61E-09 2.75E-07 -5.32E-13 1.48E-10 -3.84E-12 -4.54E-07 -1.89E-07 -7.61E-09 0.00E+0 -3.84E-12 1.04E-10
iv) Determine the restraint coefficient and calculate the effective applied moment: When the plastic behavior is considered, the compliance depends on the magnitude of applied load. Thus an iterative calculation should be performed to obtain the effective applied moment using Eq. 5.20.
, , ,
6,6
, , ,
1 z det
eff app Rest M eff app app
crack M eff app app
M C M M
G M M M
A
(5.20)
6,6
,1 ,2
minor of matrix A
pipe pipe crack
where M
A G G G
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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