- Email: [email protected]
Modal and response spectrum analyses of an ITER divertor module
Fusion Engineering and Design 146 (2019) 1872–1876
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Modal and response spectrum analyses of an ITER divertor module a
a
Sang Yun Je , Jae Min Sim , Yoon-Suk Chang a b
b,⁎
T
Graduate School of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea Department of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Divertor module Finite element analysis Modal analysis Response spectrum analysis Tungsten block
The present study is to investigate resistance of an ITER divertor module against typical seismic loads. Two kinds of complex finite element (FE) models, which consists of cassette body, inner and outer vertical targets, dome and stabilizers, were developed; one is 1- or 2-layer simplified models without consideration of inner tubes. The other is a 3-layer detailed model with the inner tubes as well as cooling water. Modal analyses to predict dynamic characteristics were conducted in use of block-Lanczos algorithm for all the FE models. Subsequently, response spectrum analyses were performed by employing three modal combination techniques and taking into account different seismic magnitudes. Analysis results from the detailed model led to higher von-Mises stresses than those obtained from the simplified model. Moreover, calculated stress intensities at critical locations were compared with corresponding design stress intensities, according to ITER structural design criteria for in-vessel components, which satisfied sufficiently the structural assessment requirement.
1. Introduction In-vessel plasma facing components (IPFCs) such as first-wall, blanket and divertor modules should withstand harsh design conditions. Since the divertor modules among IPFCs undergo extreme loads, lots of analyses have been carried out. Particularly, mock-ups in most cases and a few single module itself were examined. While the majority of previous numerical analyses were carried out under thermal and electromagnetic loads due to their significance, severe dynamic loads may also threat structural integrity. The purpose of this study is to investigate resistance of an ITER divertor module against typical seismic loads. Two kinds of complex finite element (FE) models, which consists of cassette body, inner vertical target (IVT) and outer vertical target (OVT), dome and stabilizers, were developed. One is simplified models attaching 1- or 2-layer tungsten (W) armor blocks on the plasma contact surfaces without consideration of inner tubes. The other is a detailed model attaching actual 3-layer W-armor blocks with the inner tubes as well as cooling water. At first, modal analyses to predict dynamic characteristics such as frequencies and mode shapes were conducted by employing block-Lanczos algorithm for all the FE models. Subsequently, response spectrum analyses (RSA) were performed with three modal combination techniques by taking into account different seismic magnitudes based on American Society of Mechanical Engineers (ASME) boiler and pressure vessel (B&PV) code Sec. III Appendix N. Equivalent static analyses (ESA) were also conducted for
⁎
comparison. Finally, calculated stress intensities at critical locations were compared with corresponding design stress intensities according to ITER structural design criteria for in-vessel components (SDC-IC) [1] from integrity assessment point of view. 2. Development of FE model 2.1. Analysis models The divertor module considered in this study is classified into two major parts that have different intended functions. As shown in Fig. 1, one is plasma facing part like the dome, IVT and OVT, which called as high heat flux (HHF) components, to discharges heat load. The other part is cassette body and stabilizers to properly keep in place the HHF components. Especially, surfaces of the IVT and OVT arranged with the W-armor blocks to resist the high heat fluxes and mechanical loads etc. In this study, to investigate resistance of the divertor module under seismic loads, three-dimensional FE models were developed by using ANSYS based on ITER design document [2]. FE mesh was constructed by employing 10-node tetrahedral structural elements (SOLID187 in ANSYS element library). Number of nodes and elements used for the FE models were 410,000–820,000 and 240,000–440,000 approximately. Table 1 represents mass of the divertor module dependent on simplified and detailed modeling.
Corresponding author. E-mail address: [email protected] (Y.-S. Chang).
https://doi.org/10.1016/j.fusengdes.2019.03.054 Received 29 September 2018; Received in revised form 4 March 2019; Accepted 8 March 2019 Available online 13 April 2019 0920-3796/ © 2019 Elsevier B.V. All rights reserved.
Fusion Engineering and Design 146 (2019) 1872–1876
S.Y. Je, et al.
3. Numerical analyses 3.1. Modal analysis Modal analyses were conducted to predict dynamic characteristics such as frequencies and mode shapes due to difficulty in tests of real scaled divertor module. Block-Lanczos algorithm was also adopted, which is known as useful for finding solutions of a system with large sparse matrices. Number of modes used in the response spectrum analysis has to cover that the missing mass is less than 10% of the total system mass. Table 2 summarizes representative modal analysis results obtained from the 2- and 3-layered model. Fig. 4 compares the first mode shapes and corresponding frequencies determined by each model. While results of 1- and 2-layer models were similar, discrepancy was observed in the 3-layer model. Despite total masses were almost the same, the calculated frequencies were 73 Hz approximately from the simplified models and 6.91 Hz from the detailed model. Moreover, remarkable deformation occurred at the interface between Cu and CuCrZr tubes of the 3-layer model unlike that at the cassette body and dome of the 1- and 2-layer models. These differences were caused by modeling of the two tubes and coolant. Fig. 5 represents frequencies obtained from three modal analyses, which were extracted until 70% sum of effective mass ratios. Alike the mode shapes, resulting frequencies of the simplified models were almost the same but different with those of the detailed model. Hence, the 2- and 3-layer models were used for dynamic analyses.
Fig. 1. FE models of major parts at divertor module. Table 1 Mass of divertor module assigned in this study. Analysis model
1-layer 2-layer 3-layer
Mass (Ton) Cassette body
Dome
IVT
OVT
Total
9.09
1.45
1.70 1.61 1.57
1.74 1.70 1.61
13.98 13.85 13.72
2.2. Analysis conditions
3.2. Equivalent static analysis
Geometry and boundary conditions were set in accordance with relevant design documents. The cassette body is connected by vacuum vessel (VV) so that degree of freedoms (DOFs) at the green-colored joint locations were fully fixed as shown in Fig. 2. Since the dome, IVT and OVT are coupled by stabilizers, general metal friction coefficient of 0.3 was considered at contact regions [3]. Dimensions of the W-armor block were also indicated in the figure. Relating to the blocks attached by inner Cu alloy (CuCrZr) and Cu tubes, which arranged onto the structures made of type 316 L(N)-IG stainless steel (SS), three analysis models were generated as depicted in Fig. 3. In order to examine possibility of computational effort reduction, the 1- and 2-layer simplified models were postulated and analyzed. The 3-layer model reflecting real geometry of two tubes as well as materials was also used for practical analyses and their results were compared. All the properties of each material were referred to ITER handbook [4]. With regard to fluid flow inside the cooling tube influencing on frequencies, the added-mass assumption was adopted as convention.
There are two typical approaches for seismic analysis such as static one and dynamic ones. The static approach assumes that the structure responds in its fundamental mode. Since the inertia force is treated as static load, as equal to the product of structural mass and seismic acceleration, the calculation is easy but the accuracy is relativity low. Provided the first frequency exceeds 33 Hz, the equivalent static analysis is recognized as reasonable to estimate structural behavior against dynamic loads. On the other hand, seismic influences on ITER can be classified into two events such as the seismic level (SL)-1 for operation basis earthquake (OBE) and SL-2 for safety shutdown earthquake (SSE) [5]. Usually, SSE loads have been approximated three times greater than OBE loads. Seismic analyses were performed under four conditions of the aforementioned SL-1 and SL-2 for ITER, SL-K corresponding to 0.5 g for augmented new design and beyond design basis earthquake (BDBE) corresponding to 0.6 g considered provisionally in Korea. Equivalent static loads were adjusted by multiplying a factor of 1.5 to peak spectral accelerations of each case [6]. Relevant response spectra with 5% damping derived in accordance with ASME B&PV code Sec. III Appendix N, as depicted in Fig. 5, were used for the following dynamic analyses. From ESA, the critical location having the maximum stress was identified as the stabilizer in both 2- and 3-layer models. Table 3 summarizes maximum von-Mises stresses for each case, of which details will be discussed by coming up with those from the dynamic analyses in the later section. 3.3. Response spectrum analysis Dynamic analysis under the seismic loads has been carried out employing either response spectrum method or time history method; the latter evaluates structural behavior as a function of time by applying acceleration in usual. It provides accurate results but requires highly time consuming and complicate procedure. On the other hand, the former is relatively simple method so that approximately estimates structural behavior but provides results with reasonable accuracy. In the present study, dynamic analyses of the huge divertor module were conducted by using the response spectrum method for efficient computation and prevention of divergence issue. The vertical response
Fig. 2. Geometry and boundary conditions. 1873
Fusion Engineering and Design 146 (2019) 1872–1876
S.Y. Je, et al.
Fig. 3. Schematics of analysis models employed in this study. Table 2 Summary of modal analysis results. Mode number
Frequency(Hz)
Effective mass (Ton) x-direction
2 layered 1 72.7285 2 79.3086 3 81.9294 4 136.422 5 140.847 6 149.75 7 154.89 8 174.122 9 196.514 10 200.268 … 31 637.751 32 645.272 33 667.775 34 707.53 35 736.182 36 755.468 37 764.769 38 778.753 39 801.435 40 838.746 … 295 2616.74 296 2642.83 297 2652.86 298 2663.49 299 2689.55 300 2690.98 Sum (Ton) Effective mass / Total mass (%)
y-direction
z-direction
3 layered
2 layered
3 layered
2 layered
3 layered
2 layered
3 layered
6.914 6.945 6.945 6.946 7.267 7.268 7.279 7.280 7.343 7.344
3.79E-00 2.97E-06 1.00E-04 1.58E-03 4.14E-05 1.99E-00 1.54E-04 2.54E-05 6.85E-02 6.47E-05
1.31E-03 5.33E-05 5.65E-04 5.98E-04 1.83E-03 1.92E-03 5.61E-06 1.34E-05 1.01E-07 4.70E-10
1.41E+00 3.51E-06 6.41E-05 4.12E+00 1.49E-05 3.01E-01 4.70E-08 6.12E-06 3.52E+00 6.25E-03
6.44E-08 2.76E-09 3.07E-08 5.02E-08 8.21E-08 9.09E-08 2.60E-10 6.62E-10 1.41E-03 5.77E-06
2.25E-05 8.03E+00 1.33E-01 2.66E-05 9.69E-02 3.79E-06 5.47E-01 8.41E-01 5.30E-06 5.54E-03
1.26E-08 5.66E-10 6.13E-09 9.89E-09 8.74E-08 9.82E-08 2.67E-10 7.07E-10 3.44E-04 1.42E-06
33.169 33.170 76.293 81.283 98.857 92.897 98.903 98.949 101.595 101.613
3.40E-01 2.22E-03 4.51E-02 3.73E-04 2.95E-02 9.48E-06 1.62E-05 9.82E-02 1.43E+00 3.63E-05
7.42E-10 2.18E-09 1.44E-07 7.72E+00 1.05E-03 2.70E-08 7.71E-06 7.83E-05 5.06E-05 1.00E-06
4.04E-02 4.97E-04 1.34E-01 1.27E-03 2.41E-01 3.52E-06 1.36E-04 6.80E-04 1.18E-01 5.67E-07
3.63E-06 3.66E-06 3.76E-00 1.38E-06 2.22E-06 6.00E-10 2.69E-08 2.53E-07 5.25E-04 9.64E-07
5.60E-05 8.03E-03 1.99E-04 6.02E-03 1.41E-04 2.91E-02 1.89E-02 1.53E-05 9.73E-06 2.20E-01
4.22E-07 4.24E-07 1.83E-00 4.88E-06 2.84E-06 8.84E-10 3.91E-08 4.16E-07 1.41E-03 2.60E-06
2157.33 2173.07 2174.88 2184.99 2187.41 2187.67
6.51E-05 3.57E-03 1.56E-04 9.39E-04 6.41E-04 5.92E-04 13.50 97.33
2.66E-04 4.52E-04 1.49E-03 3.38E-03 1.71E-04 8.23E-04 13.01 94.14
7.59E-03 3.01E-04 1.66E-02 1.56E-04 1.32E-02 2.60E-03 13.75 99.13
2.39E-04 1.00E-05 8.82E-04 5.86E-04 3.33E-04 6.31E-07 12.78 92.52
1.02E-03 3.47E-04 9.90E-03 7.18E-05 2.79E-03 6.55E-04 13.25 99.42
4.15E-03 1.92E-04 1.10E-03 3.20E-04 6.98E-04 1.28E-06 12.69 91.84
Fig. 4. First mode shapes and frequencies of divertor module.
is expressed by
spectrum along z-direction as well as horizontal response spectrum along x- and y- directions in Fig. 6 were considered on fixed points as input with the modal analysis data at all significant modes. Three modal combination techniques such as square root of sum of the squares (SRSS), complete quadratic combination (CQC) and Rosenblueth (ROSE) were also employed to combine structural responses of the 2and 3-layer models. In the well-known SRSS technique, the maximum response (γ) is obtained by square root of sum of square of response in each mode and
n
γSRSS =
∑ γi2 i=1
(1)
It is fundamentally sound when modal frequencies are well separated. However, poor results are yielded if frequencies of major contributing modes are quite close together. As an alternative, the CQC can be used. In this technique, the maximum response from all of modes is 1874
Fusion Engineering and Design 146 (2019) 1872–1876
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Fig. 5. Interesting frequencies of modes. Table 3 Comparison of maximum von-Mises stresses. Analysis model
2-layer
Seismic load
SL-1 SL-2 SL-K BDBE SL-1 SL-2 SL-K BDBE
3-layer
ESA (MPa)
0.89 3.57 5.95 10.72 1.01 4.06 6.67 12.19
RSA (MPa) SRSS
CQC
ROSE
0.39 1.57 3.92 4.71 3.96 15.84 39.61 47.54
0.40 1.60 4.00 4.71 3.19 12.78 31.96 38.36
0.39 1.57 3.92 4.80 3.19 12.78 31.96 38.35
Fig. 7. The critical locations and von-Mises equivalent stress distributions of 3layer model under BDBE condition.
n
γROSE =
calculated as n
n
3.4. Structural assessment
∑ ∑ γi αij γj i=1 j=1
(3)
The response spectra applied at the location between cassette body and VV. In detail, the vertical response spectrum along z-direction as well as horizontal response spectra along x- and y-directions was considered. Each direction can be found from the coordinates in Fig. 2. Thereafter, similar trend of RSA results was observed to that of modal analyses results, i.e., the highest stress occurred at the interface between CuCrZr and Cu tubes of the 3-layer model unlike that at the stabilizer of the 2-layer model. If limited to only RSA results, mean difference of the maximum von-Mises stresses between these two models was 88.6%, which might be originated by the difference of interesting frequencies within 33 Hz as shown in Fig. 5. Fig. 7 represents critical locations and von-Mises equivalent stress distributions of the 3-layer model under BDBE condition as a typical case. From comparison of ESA and RSA results, differences of the maximum von-Mises stresses ranged from 50.8–126.3% and their mean value was 90.2%. In particular, ESA provided higher stresses than RSA in the 2-layer model but lower stresses than RSA in the 3-layer model. Furthermore, mean difference of the maximum von-Mises stresses among modal combination techniques was 19.3% in the 3-layer model while influence of them was ignorable in the 2-layer model as well as results by the CQC and ROSE techniques were comparable for both models.
Fig. 6. Response spectra used in dynamic analyses. (a) Equivalent static analysis. (b) Response spectrum analysis.
γCQC =
n
⎛ ⎞ ε γγ ⎜∑ ∑ ij i j ⎟ i j ⎝ ⎠
(2)
The SDC-IC and ITER Load Specification classifies seismic loads SL-1 as level A, SL-2 as level D monotonic loading (M-type) and categorizes stress components as general primary membrane stress intensity (Pm) and primary bending stress intensity (Pb). It also provides acceptance criteria that should be satisfied for each specific stress intensity and combination of them. As a part of the present study, structural
where γi and γj are maximum responses in the ith and jth modes, respectively, and αij is correlation coefficient. Another alternative is the ROSE technique for close mode combination having the following form.
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1) ESA without consideration of resonance effect identified stabilizer as the critical location and similar structural behaviors were observed between the simplified and detailed models. 2) Two kinds of modal analyses led to remarkably distinct dynamic characteristics. Mode shapes were different and frequency at the first mode obtained from detailed model was 10 times lower than those of simplified models. 3) Two kinds of RSA made also discrepancies. In particular, interface between CuCrZr and Cu tubes was determined as the critical location from the detailed model and its maximum von-Mises stresses were 88.6% higher in average than those of the simplified model. 4) RSA with the CQC employing 3-layer detailed model is recommended for the divertor module analyses. On the average, deviations of the maximum von-Mises stresses from the optimum one were approximately 90% in ESA and 19% in other modal combination techniques.
Table 4 Stress intensities calculated by RSA with CQC. Seismic loads
Pm (MPa)
Pb (MPa)
Safety margin
SL-1 SL-2 SL-K BDBE
2.38 9.53 23.85 28.49
0.08 0.35 0.46 0.89
107.31 32.02 12.80 10.94
assessment was carried out at the critical location. For this purpose, RSA results of 3-layer model with CQC technique were used and the following allowable condition for the cooling tube, made of Cu alloy, was considered.
Level A: Pm + Pb ≤ 3Sm
(4)
Level D : Pm + Pb ≤ 1.5min[2.4Sm, 0.7Su]
(5)
where Sm is the allowable design stress intensity and Su is the tensile strength. Stress intensities were compared with Sm of 88 MPa at 400 °C defined in the SDC-IC appendix A. Table 4 summarizes calculated membrane and bending stress intensities with safety margins converted from Eq. (4) and (5). As represented in the table, structural integrity was ascertained under the four seismic loading conditions. Even though fatigue assessment is also necessary, which will be introduced as another research taking into account further various cyclic loading in the near future.
Acknowledgments This research was supported by National R&D Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science & ICT (2017M1A7A1A01016222). References [1] ITER Organization, ITER Structural Design Criteria for In-Vessel Components, G 74 MA 8 01-05-28 (2012). [2] G. Mazzone, L. Petrizzi, M. Roccela, E. Mainardi, F. Lucca, A. Marin, G. Zanotelli, Final Report on the Revision of ITER Divertor Design, ENEA Frascati Research Center, 2003. [3] Z. Piec, J. Nowacki, Friction materials operating under high stress, cryogenic temperature, and in vacuum, J. Ach. Mater. Manuf. Eng. 18 (2006) 15–24. [4] ITER Organization, ITER Materials Properties Handbook, ITER Doc. NO. S 74 MA 2 (2007). [5] ITER Organization, Design Description Document, G 17 DDD 6 R0.2 (2004). [6] USNRC, Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis, NUREG.CR-6645 (1999).
4. Concluding remark In this study, resistance of a complex divertor module was investigated against typical seismic loads. Comparative equivalent static and dynamic analyses were conducted by using 1- or 2-layer simplified and 3-layer detailed FE models with 3 modal combination techniques. Thereby, the following conclusions were derived.
1876
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Modal and response spectrum analyses of an ITER divertor module a
a
Sang Yun Je , Jae Min Sim , Yoon-Suk Chang a b
b,⁎
T
Graduate School of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea Department of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Divertor module Finite element analysis Modal analysis Response spectrum analysis Tungsten block
The present study is to investigate resistance of an ITER divertor module against typical seismic loads. Two kinds of complex finite element (FE) models, which consists of cassette body, inner and outer vertical targets, dome and stabilizers, were developed; one is 1- or 2-layer simplified models without consideration of inner tubes. The other is a 3-layer detailed model with the inner tubes as well as cooling water. Modal analyses to predict dynamic characteristics were conducted in use of block-Lanczos algorithm for all the FE models. Subsequently, response spectrum analyses were performed by employing three modal combination techniques and taking into account different seismic magnitudes. Analysis results from the detailed model led to higher von-Mises stresses than those obtained from the simplified model. Moreover, calculated stress intensities at critical locations were compared with corresponding design stress intensities, according to ITER structural design criteria for in-vessel components, which satisfied sufficiently the structural assessment requirement.
1. Introduction In-vessel plasma facing components (IPFCs) such as first-wall, blanket and divertor modules should withstand harsh design conditions. Since the divertor modules among IPFCs undergo extreme loads, lots of analyses have been carried out. Particularly, mock-ups in most cases and a few single module itself were examined. While the majority of previous numerical analyses were carried out under thermal and electromagnetic loads due to their significance, severe dynamic loads may also threat structural integrity. The purpose of this study is to investigate resistance of an ITER divertor module against typical seismic loads. Two kinds of complex finite element (FE) models, which consists of cassette body, inner vertical target (IVT) and outer vertical target (OVT), dome and stabilizers, were developed. One is simplified models attaching 1- or 2-layer tungsten (W) armor blocks on the plasma contact surfaces without consideration of inner tubes. The other is a detailed model attaching actual 3-layer W-armor blocks with the inner tubes as well as cooling water. At first, modal analyses to predict dynamic characteristics such as frequencies and mode shapes were conducted by employing block-Lanczos algorithm for all the FE models. Subsequently, response spectrum analyses (RSA) were performed with three modal combination techniques by taking into account different seismic magnitudes based on American Society of Mechanical Engineers (ASME) boiler and pressure vessel (B&PV) code Sec. III Appendix N. Equivalent static analyses (ESA) were also conducted for
⁎
comparison. Finally, calculated stress intensities at critical locations were compared with corresponding design stress intensities according to ITER structural design criteria for in-vessel components (SDC-IC) [1] from integrity assessment point of view. 2. Development of FE model 2.1. Analysis models The divertor module considered in this study is classified into two major parts that have different intended functions. As shown in Fig. 1, one is plasma facing part like the dome, IVT and OVT, which called as high heat flux (HHF) components, to discharges heat load. The other part is cassette body and stabilizers to properly keep in place the HHF components. Especially, surfaces of the IVT and OVT arranged with the W-armor blocks to resist the high heat fluxes and mechanical loads etc. In this study, to investigate resistance of the divertor module under seismic loads, three-dimensional FE models were developed by using ANSYS based on ITER design document [2]. FE mesh was constructed by employing 10-node tetrahedral structural elements (SOLID187 in ANSYS element library). Number of nodes and elements used for the FE models were 410,000–820,000 and 240,000–440,000 approximately. Table 1 represents mass of the divertor module dependent on simplified and detailed modeling.
Corresponding author. E-mail address: [email protected] (Y.-S. Chang).
https://doi.org/10.1016/j.fusengdes.2019.03.054 Received 29 September 2018; Received in revised form 4 March 2019; Accepted 8 March 2019 Available online 13 April 2019 0920-3796/ © 2019 Elsevier B.V. All rights reserved.
Fusion Engineering and Design 146 (2019) 1872–1876
S.Y. Je, et al.
3. Numerical analyses 3.1. Modal analysis Modal analyses were conducted to predict dynamic characteristics such as frequencies and mode shapes due to difficulty in tests of real scaled divertor module. Block-Lanczos algorithm was also adopted, which is known as useful for finding solutions of a system with large sparse matrices. Number of modes used in the response spectrum analysis has to cover that the missing mass is less than 10% of the total system mass. Table 2 summarizes representative modal analysis results obtained from the 2- and 3-layered model. Fig. 4 compares the first mode shapes and corresponding frequencies determined by each model. While results of 1- and 2-layer models were similar, discrepancy was observed in the 3-layer model. Despite total masses were almost the same, the calculated frequencies were 73 Hz approximately from the simplified models and 6.91 Hz from the detailed model. Moreover, remarkable deformation occurred at the interface between Cu and CuCrZr tubes of the 3-layer model unlike that at the cassette body and dome of the 1- and 2-layer models. These differences were caused by modeling of the two tubes and coolant. Fig. 5 represents frequencies obtained from three modal analyses, which were extracted until 70% sum of effective mass ratios. Alike the mode shapes, resulting frequencies of the simplified models were almost the same but different with those of the detailed model. Hence, the 2- and 3-layer models were used for dynamic analyses.
Fig. 1. FE models of major parts at divertor module. Table 1 Mass of divertor module assigned in this study. Analysis model
1-layer 2-layer 3-layer
Mass (Ton) Cassette body
Dome
IVT
OVT
Total
9.09
1.45
1.70 1.61 1.57
1.74 1.70 1.61
13.98 13.85 13.72
2.2. Analysis conditions
3.2. Equivalent static analysis
Geometry and boundary conditions were set in accordance with relevant design documents. The cassette body is connected by vacuum vessel (VV) so that degree of freedoms (DOFs) at the green-colored joint locations were fully fixed as shown in Fig. 2. Since the dome, IVT and OVT are coupled by stabilizers, general metal friction coefficient of 0.3 was considered at contact regions [3]. Dimensions of the W-armor block were also indicated in the figure. Relating to the blocks attached by inner Cu alloy (CuCrZr) and Cu tubes, which arranged onto the structures made of type 316 L(N)-IG stainless steel (SS), three analysis models were generated as depicted in Fig. 3. In order to examine possibility of computational effort reduction, the 1- and 2-layer simplified models were postulated and analyzed. The 3-layer model reflecting real geometry of two tubes as well as materials was also used for practical analyses and their results were compared. All the properties of each material were referred to ITER handbook [4]. With regard to fluid flow inside the cooling tube influencing on frequencies, the added-mass assumption was adopted as convention.
There are two typical approaches for seismic analysis such as static one and dynamic ones. The static approach assumes that the structure responds in its fundamental mode. Since the inertia force is treated as static load, as equal to the product of structural mass and seismic acceleration, the calculation is easy but the accuracy is relativity low. Provided the first frequency exceeds 33 Hz, the equivalent static analysis is recognized as reasonable to estimate structural behavior against dynamic loads. On the other hand, seismic influences on ITER can be classified into two events such as the seismic level (SL)-1 for operation basis earthquake (OBE) and SL-2 for safety shutdown earthquake (SSE) [5]. Usually, SSE loads have been approximated three times greater than OBE loads. Seismic analyses were performed under four conditions of the aforementioned SL-1 and SL-2 for ITER, SL-K corresponding to 0.5 g for augmented new design and beyond design basis earthquake (BDBE) corresponding to 0.6 g considered provisionally in Korea. Equivalent static loads were adjusted by multiplying a factor of 1.5 to peak spectral accelerations of each case [6]. Relevant response spectra with 5% damping derived in accordance with ASME B&PV code Sec. III Appendix N, as depicted in Fig. 5, were used for the following dynamic analyses. From ESA, the critical location having the maximum stress was identified as the stabilizer in both 2- and 3-layer models. Table 3 summarizes maximum von-Mises stresses for each case, of which details will be discussed by coming up with those from the dynamic analyses in the later section. 3.3. Response spectrum analysis Dynamic analysis under the seismic loads has been carried out employing either response spectrum method or time history method; the latter evaluates structural behavior as a function of time by applying acceleration in usual. It provides accurate results but requires highly time consuming and complicate procedure. On the other hand, the former is relatively simple method so that approximately estimates structural behavior but provides results with reasonable accuracy. In the present study, dynamic analyses of the huge divertor module were conducted by using the response spectrum method for efficient computation and prevention of divergence issue. The vertical response
Fig. 2. Geometry and boundary conditions. 1873
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S.Y. Je, et al.
Fig. 3. Schematics of analysis models employed in this study. Table 2 Summary of modal analysis results. Mode number
Frequency(Hz)
Effective mass (Ton) x-direction
2 layered 1 72.7285 2 79.3086 3 81.9294 4 136.422 5 140.847 6 149.75 7 154.89 8 174.122 9 196.514 10 200.268 … 31 637.751 32 645.272 33 667.775 34 707.53 35 736.182 36 755.468 37 764.769 38 778.753 39 801.435 40 838.746 … 295 2616.74 296 2642.83 297 2652.86 298 2663.49 299 2689.55 300 2690.98 Sum (Ton) Effective mass / Total mass (%)
y-direction
z-direction
3 layered
2 layered
3 layered
2 layered
3 layered
2 layered
3 layered
6.914 6.945 6.945 6.946 7.267 7.268 7.279 7.280 7.343 7.344
3.79E-00 2.97E-06 1.00E-04 1.58E-03 4.14E-05 1.99E-00 1.54E-04 2.54E-05 6.85E-02 6.47E-05
1.31E-03 5.33E-05 5.65E-04 5.98E-04 1.83E-03 1.92E-03 5.61E-06 1.34E-05 1.01E-07 4.70E-10
1.41E+00 3.51E-06 6.41E-05 4.12E+00 1.49E-05 3.01E-01 4.70E-08 6.12E-06 3.52E+00 6.25E-03
6.44E-08 2.76E-09 3.07E-08 5.02E-08 8.21E-08 9.09E-08 2.60E-10 6.62E-10 1.41E-03 5.77E-06
2.25E-05 8.03E+00 1.33E-01 2.66E-05 9.69E-02 3.79E-06 5.47E-01 8.41E-01 5.30E-06 5.54E-03
1.26E-08 5.66E-10 6.13E-09 9.89E-09 8.74E-08 9.82E-08 2.67E-10 7.07E-10 3.44E-04 1.42E-06
33.169 33.170 76.293 81.283 98.857 92.897 98.903 98.949 101.595 101.613
3.40E-01 2.22E-03 4.51E-02 3.73E-04 2.95E-02 9.48E-06 1.62E-05 9.82E-02 1.43E+00 3.63E-05
7.42E-10 2.18E-09 1.44E-07 7.72E+00 1.05E-03 2.70E-08 7.71E-06 7.83E-05 5.06E-05 1.00E-06
4.04E-02 4.97E-04 1.34E-01 1.27E-03 2.41E-01 3.52E-06 1.36E-04 6.80E-04 1.18E-01 5.67E-07
3.63E-06 3.66E-06 3.76E-00 1.38E-06 2.22E-06 6.00E-10 2.69E-08 2.53E-07 5.25E-04 9.64E-07
5.60E-05 8.03E-03 1.99E-04 6.02E-03 1.41E-04 2.91E-02 1.89E-02 1.53E-05 9.73E-06 2.20E-01
4.22E-07 4.24E-07 1.83E-00 4.88E-06 2.84E-06 8.84E-10 3.91E-08 4.16E-07 1.41E-03 2.60E-06
2157.33 2173.07 2174.88 2184.99 2187.41 2187.67
6.51E-05 3.57E-03 1.56E-04 9.39E-04 6.41E-04 5.92E-04 13.50 97.33
2.66E-04 4.52E-04 1.49E-03 3.38E-03 1.71E-04 8.23E-04 13.01 94.14
7.59E-03 3.01E-04 1.66E-02 1.56E-04 1.32E-02 2.60E-03 13.75 99.13
2.39E-04 1.00E-05 8.82E-04 5.86E-04 3.33E-04 6.31E-07 12.78 92.52
1.02E-03 3.47E-04 9.90E-03 7.18E-05 2.79E-03 6.55E-04 13.25 99.42
4.15E-03 1.92E-04 1.10E-03 3.20E-04 6.98E-04 1.28E-06 12.69 91.84
Fig. 4. First mode shapes and frequencies of divertor module.
is expressed by
spectrum along z-direction as well as horizontal response spectrum along x- and y- directions in Fig. 6 were considered on fixed points as input with the modal analysis data at all significant modes. Three modal combination techniques such as square root of sum of the squares (SRSS), complete quadratic combination (CQC) and Rosenblueth (ROSE) were also employed to combine structural responses of the 2and 3-layer models. In the well-known SRSS technique, the maximum response (γ) is obtained by square root of sum of square of response in each mode and
n
γSRSS =
∑ γi2 i=1
(1)
It is fundamentally sound when modal frequencies are well separated. However, poor results are yielded if frequencies of major contributing modes are quite close together. As an alternative, the CQC can be used. In this technique, the maximum response from all of modes is 1874
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Fig. 5. Interesting frequencies of modes. Table 3 Comparison of maximum von-Mises stresses. Analysis model
2-layer
Seismic load
SL-1 SL-2 SL-K BDBE SL-1 SL-2 SL-K BDBE
3-layer
ESA (MPa)
0.89 3.57 5.95 10.72 1.01 4.06 6.67 12.19
RSA (MPa) SRSS
CQC
ROSE
0.39 1.57 3.92 4.71 3.96 15.84 39.61 47.54
0.40 1.60 4.00 4.71 3.19 12.78 31.96 38.36
0.39 1.57 3.92 4.80 3.19 12.78 31.96 38.35
Fig. 7. The critical locations and von-Mises equivalent stress distributions of 3layer model under BDBE condition.
n
γROSE =
calculated as n
n
3.4. Structural assessment
∑ ∑ γi αij γj i=1 j=1
(3)
The response spectra applied at the location between cassette body and VV. In detail, the vertical response spectrum along z-direction as well as horizontal response spectra along x- and y-directions was considered. Each direction can be found from the coordinates in Fig. 2. Thereafter, similar trend of RSA results was observed to that of modal analyses results, i.e., the highest stress occurred at the interface between CuCrZr and Cu tubes of the 3-layer model unlike that at the stabilizer of the 2-layer model. If limited to only RSA results, mean difference of the maximum von-Mises stresses between these two models was 88.6%, which might be originated by the difference of interesting frequencies within 33 Hz as shown in Fig. 5. Fig. 7 represents critical locations and von-Mises equivalent stress distributions of the 3-layer model under BDBE condition as a typical case. From comparison of ESA and RSA results, differences of the maximum von-Mises stresses ranged from 50.8–126.3% and their mean value was 90.2%. In particular, ESA provided higher stresses than RSA in the 2-layer model but lower stresses than RSA in the 3-layer model. Furthermore, mean difference of the maximum von-Mises stresses among modal combination techniques was 19.3% in the 3-layer model while influence of them was ignorable in the 2-layer model as well as results by the CQC and ROSE techniques were comparable for both models.
Fig. 6. Response spectra used in dynamic analyses. (a) Equivalent static analysis. (b) Response spectrum analysis.
γCQC =
n
⎛ ⎞ ε γγ ⎜∑ ∑ ij i j ⎟ i j ⎝ ⎠
(2)
The SDC-IC and ITER Load Specification classifies seismic loads SL-1 as level A, SL-2 as level D monotonic loading (M-type) and categorizes stress components as general primary membrane stress intensity (Pm) and primary bending stress intensity (Pb). It also provides acceptance criteria that should be satisfied for each specific stress intensity and combination of them. As a part of the present study, structural
where γi and γj are maximum responses in the ith and jth modes, respectively, and αij is correlation coefficient. Another alternative is the ROSE technique for close mode combination having the following form.
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1) ESA without consideration of resonance effect identified stabilizer as the critical location and similar structural behaviors were observed between the simplified and detailed models. 2) Two kinds of modal analyses led to remarkably distinct dynamic characteristics. Mode shapes were different and frequency at the first mode obtained from detailed model was 10 times lower than those of simplified models. 3) Two kinds of RSA made also discrepancies. In particular, interface between CuCrZr and Cu tubes was determined as the critical location from the detailed model and its maximum von-Mises stresses were 88.6% higher in average than those of the simplified model. 4) RSA with the CQC employing 3-layer detailed model is recommended for the divertor module analyses. On the average, deviations of the maximum von-Mises stresses from the optimum one were approximately 90% in ESA and 19% in other modal combination techniques.
Table 4 Stress intensities calculated by RSA with CQC. Seismic loads
Pm (MPa)
Pb (MPa)
Safety margin
SL-1 SL-2 SL-K BDBE
2.38 9.53 23.85 28.49
0.08 0.35 0.46 0.89
107.31 32.02 12.80 10.94
assessment was carried out at the critical location. For this purpose, RSA results of 3-layer model with CQC technique were used and the following allowable condition for the cooling tube, made of Cu alloy, was considered.
Level A: Pm + Pb ≤ 3Sm
(4)
Level D : Pm + Pb ≤ 1.5min[2.4Sm, 0.7Su]
(5)
where Sm is the allowable design stress intensity and Su is the tensile strength. Stress intensities were compared with Sm of 88 MPa at 400 °C defined in the SDC-IC appendix A. Table 4 summarizes calculated membrane and bending stress intensities with safety margins converted from Eq. (4) and (5). As represented in the table, structural integrity was ascertained under the four seismic loading conditions. Even though fatigue assessment is also necessary, which will be introduced as another research taking into account further various cyclic loading in the near future.
Acknowledgments This research was supported by National R&D Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science & ICT (2017M1A7A1A01016222). References [1] ITER Organization, ITER Structural Design Criteria for In-Vessel Components, G 74 MA 8 01-05-28 (2012). [2] G. Mazzone, L. Petrizzi, M. Roccela, E. Mainardi, F. Lucca, A. Marin, G. Zanotelli, Final Report on the Revision of ITER Divertor Design, ENEA Frascati Research Center, 2003. [3] Z. Piec, J. Nowacki, Friction materials operating under high stress, cryogenic temperature, and in vacuum, J. Ach. Mater. Manuf. Eng. 18 (2006) 15–24. [4] ITER Organization, ITER Materials Properties Handbook, ITER Doc. NO. S 74 MA 2 (2007). [5] ITER Organization, Design Description Document, G 17 DDD 6 R0.2 (2004). [6] USNRC, Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis, NUREG.CR-6645 (1999).
4. Concluding remark In this study, resistance of a complex divertor module was investigated against typical seismic loads. Comparative equivalent static and dynamic analyses were conducted by using 1- or 2-layer simplified and 3-layer detailed FE models with 3 modal combination techniques. Thereby, the following conclusions were derived.
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